mirror of https://github.com/saitohirga/WSJT-X.git
3290 lines
72 KiB
Plaintext
3290 lines
72 KiB
Plaintext
#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\fancyhead[L]{Franke \& Taylor: {\it Open Source Soft-Decision Decoder \ldots}}
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\end_header
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\begin_body
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\begin_layout Title
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Open Source Soft-Decision Decoder for the JT65 (63,12) Reed-Solomon Code
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\end_layout
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\begin_layout SubTitle
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\emph on
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Under-the-hood description of the JT65 decoding procedure, including a wholly
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new algorithm for its powerful error-correcting code.
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\end_layout
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\begin_layout Author
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Steven J.
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Franke, K9AN and Joseph H.
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Taylor, K1JT
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\end_layout
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\begin_layout Section
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:Introduction-and-Motivation"
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\end_inset
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Background and Motivation
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\end_layout
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\begin_layout Standard
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\begin_inset ERT
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status open
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\begin_layout Plain Layout
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\backslash
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RaggedRight
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\end_layout
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\end_inset
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The JT65 protocol has revolutionized amateur-radio weak-signal communication
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by enabling operators with small or compromise antennas and relatively
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low-power transmitters to communicate over propagation paths not usable
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with traditional technologies.
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The protocol was developed in 2003 for Earth-Moon-Earth (EME, or
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\begin_inset Quotes eld
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\end_inset
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moonbounce
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\begin_inset Quotes erd
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\end_inset
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) communication
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\begin_inset CommandInset citation
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LatexCommand cite
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key "jt65_protocol"
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\end_inset
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, where the scattered return signals are always weak.
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It was soon found that JT65 also enables worldwide communication on the
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HF bands with low power, modest antennas, and efficient spectral usage.
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Thousands of amateurs now use JT65 on a regular basis, making contacts
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on all bands from 160 meters through microwaves.
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\end_layout
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\begin_layout Standard
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JT65 uses timed transmitting and receiving sequences one minute long.
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Messages are short and structured so as to streamline minimal exchanges
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between two amateur operators over potentially difficult radio paths.
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Most messages contain two callsigns and a grid locator, signal report,
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acknowledgment, or sign-off; one of the tokens CQ, QRZ, or DE may be substitute
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d for the first callsign.
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Alternatively, a message may contain up to 13 Latin characters of arbitrary
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text.
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All messages are efficiently compressed into exactly 72 bits of digital
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information.
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It should be obvious that the JT65 protocol is intended for the basic purpose
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of completing legitimate, documented two-way contacts, but not for extended
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conversations.
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Full details of the message structure and encoding procedure were presented
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in an earlier publication
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\begin_inset CommandInset citation
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LatexCommand cite
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key "jt65_protocol"
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\end_inset
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.
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For a concise description of the overall process of transmitting and receiving
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a JT65 message, see the accompanying sidebar
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\series bold
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JT65 Message Processing
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\series default
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.
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\end_layout
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\begin_layout Standard
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A major reason for the success and popularity of JT65 is its use of a strong
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error-correction code.
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Before transmission, each 72-bit message is divided into 12 six-bit
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\emph on
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symbols
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\begin_inset CommandInset nomenclature
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LatexCommand nomenclature
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symbol "{\\bf Symbol: }"
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description "The information carried in one signalling interval, usually an integral number of bits. JT65 uses 6-bit symbols."
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\end_inset
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\emph default
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and augmented with 51 additional symbols of error-correcting information.
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These 51
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\emph on
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parity symbols
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\emph default
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are computed according to information-theory rules that maximize the probabilit
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y of correctly decoding the message, even if many symbols are received incorrect
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ly.
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The JT65 code is properly described as a short block-length, low-rate Reed-Solo
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mon code based on a 64-symbol
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\emph on
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alphabet.
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\emph default
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\begin_inset CommandInset nomenclature
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LatexCommand nomenclature
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symbol "{\\bf Alphabet: }"
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description "A sequence of possible symbol values used for signaling. JT65 uses a 64-character alphabet, values in the range 0 to 63."
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\end_inset
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Characters in this alphabet are mapped onto 64 different frequencies for
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transmission.
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\end_layout
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\begin_layout Standard
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Reed Solomon codes are widely used to ensure reliability in data transmission
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and storage.
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In hardware implementations, decoding is generally accomplished with a
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procedure such as the Berlekamp-Massey (BM) algorithm, based on
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\emph on
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hard decisions
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\emph default
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\begin_inset CommandInset nomenclature
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LatexCommand nomenclature
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symbol "{\\bf Hard decision: }"
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description "Received symbols are assigned definite values by the demodulator."
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\end_inset
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for each of the symbol values received.
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\emph on
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Soft decisions
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\begin_inset CommandInset nomenclature
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LatexCommand nomenclature
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symbol "{\\bf Soft decision: }"
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description "Received symbols are assigned tentative values (most probable, second most probable, etc.) and quality indicators."
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\end_inset
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\emph default
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are potentially more powerful, however.
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For each received JT65 symbol we can estimate not only the value most likely
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to be correct, but also the second, third, etc., most likely.
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Most importantly, we can also estimate the probability that each of those
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possible values is the correct one.
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Decoders that make use of such information are called
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\emph on
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soft-decision decoders.
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\end_layout
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\begin_layout Standard
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Until now, nearly all programs implementing JT65 have used the patented
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Kötter-Vardy (KV) algebraic soft-decision decoder
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\begin_inset CommandInset citation
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LatexCommand cite
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key "kv2001"
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\end_inset
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, licensed to and implemented by K1JT as a closed-source executable for
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use only in amateur radio applications.
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Since 2001 the KV decoder has been considered the best available soft-decision
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decoder for Reed Solomon codes.
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\end_layout
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\begin_layout Standard
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We describe here a new open-source alternative called the Franke-Taylor
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(FT, or K9AN-K1JT) soft-decision decoding algorithm.
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It is conceptually simple, built on top of the BM hard-decision decoder,
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and in this application it performs even better than the KV decoder.
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The FT algorithm is implemented in the popular programs
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\emph on
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WSJT
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\emph default
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,
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\emph on
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MAP65
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\emph default
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, and
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\emph on
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WSJT-X
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\emph default
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, widely used for amateur weak-signal communication using JT65 and other
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specialized digital protocols.
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These programs are open-source, freely available
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\begin_inset CommandInset citation
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LatexCommand cite
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key "wsjt"
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\end_inset
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, and licensed under the GNU General Public License.
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\end_layout
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\begin_layout Standard
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The JT65 protocol specifies transmissions that start one second into a UTC
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minute and last for 46.8 seconds.
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Receiving software therefore has as much as ten seconds to decode a message
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before the start of the next minute, when the operator will send a reply.
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With today's personal computers, this relatively long time encourages experimen
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tation with decoders of high computational complexity.
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With time to spare, the FT algorithm lowers the decoding threshold on a
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typical fading channel by many dB over the hard-decision BM decoder, and
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by a meaningful amount over the KV decoder.
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In addition to its excellent performance, the new algorithm has other desirable
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properties, not least of which is its conceptual simplicity.
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Decoding performance and computational complexity scale in a convenient
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way, providing steadily increasing soft-decision decoding gain as a tunable
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parameter is increased over more than five orders of magnitude.
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Appreciable gain is available from our decoder even on very simple (and
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relatively slow) computers.
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On the other hand, because the algorithm benefits from a large number of
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independent decoding trials, further performance gains should be achievable
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through parallelization on high-performance computers.
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\end_layout
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\begin_layout Standard
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The remainder of this paper is organized as follows.
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Section 2 presents a brief overview of the nature of Reed Solomon codes
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and their error-correcting capabilities.
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Section 3 provides statistical motivation for the FT algorithm, and Section
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4 describes the algorithm in full detail.
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Material in these two sections is important because it documents our approach
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and underlines its fundamental technical contributions.
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These sections are heavier in formal mathematics than common in
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\emph on
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QEX
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\emph default
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; for this reason, some readers may choose to skip or skim them and proceed
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more quickly to the results.
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Most readers will benefit by reviewing the original paper on the JT65 protocol
|
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|
\begin_inset CommandInset citation
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|
LatexCommand cite
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|
key "jt65_protocol"
|
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|
|
\end_inset
|
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|
.
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A procedure for
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\emph on
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hinted decoding
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\emph default
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--- determining which one, if any, of a list of likely messages matches
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the one that was received --- is outlined in Section 5.
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Finally, in Section 6 we present performance measurements of the FT and
|
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hinted decoding algorithms and make explicit comparisons to the BM and
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KV decoders familiar to users of older versions of
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\emph on
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WSJT
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\emph default
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,
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\emph on
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MAP65
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\emph default
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and
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\emph on
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WSJT-X
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\emph default
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.
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Section 7 summarizes some on-the-air experiences with the new decoder.
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Refer to the sidebar
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\series bold
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|
Glossary of Specialized Terms
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\series default
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|
for brief definitions of some potentially unfamiliar language.
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|
\end_layout
|
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\begin_layout Section
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\begin_inset CommandInset label
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LatexCommand label
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name "sec:JT65-messages-and"
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|
\end_inset
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|
JT65 Messages and Reed Solomon Codes
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\end_layout
|
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\begin_layout Standard
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The JT65 message frame consists of a short, compressed 72-bit message encoded
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for transmission with a Reed-Solomon code.
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Reed-Solomon codes are
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\emph on
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block codes
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\emph default
|
|
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|
\begin_inset CommandInset nomenclature
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|
LatexCommand nomenclature
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|
symbol "{\\bf Block code: }"
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|
description "An error-correcting code that treats data in blocks of fixed size."
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\end_inset
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characterized by
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\begin_inset Formula $n$
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\end_inset
|
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|
, the length of their
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\emph on
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codewords
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\emph default
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;
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\begin_inset CommandInset nomenclature
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|
LatexCommand nomenclature
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|
symbol "{\\bf Codeword:}"
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description "For the JT65 code, a vector of 63 symbol values each in the range 0 to 63."
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\end_inset
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\begin_inset Formula $k$
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\end_inset
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|
, the number of message symbols conveyed by the codeword; and the transmission
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alphabet, or number of possible values for each symbol in a codeword.
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The codeword length and the number of message symbols are specified with
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the notation
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\begin_inset Formula $(n,k)$
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\end_inset
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|
.
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JT65 uses a (63,12) Reed-Solomon code with an alphabet of 64 possible values
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for each symbol.
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Each of the 12 message symbols represents
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\begin_inset Formula $\log_{2}64=6$
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\end_inset
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|
message bits.
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|
The source-encoded
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|
\begin_inset CommandInset nomenclature
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|
LatexCommand nomenclature
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|
symbol "{\\bf Source encoding: }"
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|
description "Compression of a message to use a minimum number or bits. JT65 source-encodes all messages to 72 bits."
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|
\end_inset
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|
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|
message conveyed by a 63-symbol JT65 frame thus consists of 72 information
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|
bits.
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|
The JT65 code is
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|
\emph on
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|
systematic
|
|
\emph default
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|
, which means that the 12 message symbols are embedded in the codeword without
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modification and another 51 parity symbols derived from the message symbols
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|
are added to form a codeword of 63 symbols.
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|
|
\end_layout
|
|
|
|
\begin_layout Standard
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|
In coding theory the concept of
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\emph on
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|
Hamming distance
|
|
\emph default
|
|
|
|
\begin_inset CommandInset nomenclature
|
|
LatexCommand nomenclature
|
|
symbol "{\\bf Hamming distance: }"
|
|
description "The Hamming distance between two codewords, or between a received word and a codeword, is equal to the number of symbol positions in which they differ."
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|
\end_inset
|
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|
is used as a measure of disagreement between different codewords, or between
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|
a received word
|
|
\begin_inset CommandInset nomenclature
|
|
LatexCommand nomenclature
|
|
symbol "{\\bf Received word: }"
|
|
description "A vector of symbol values, possibly accompanied by soft information on individual reliabilities."
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|
\end_inset
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|
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|
and a codeword.
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|
Hamming distance is the number of code symbols that differ in two words
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|
being compared.
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|
Reed-Solomon codes have guaranteed minimum Hamming distance
|
|
\begin_inset Formula $d$
|
|
\end_inset
|
|
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|
, where
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
d=n-k+1.\label{eq:minimum_distance}
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|
\end{equation}
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|
\end_inset
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|
With
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|
\begin_inset Formula $n=63$
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|
\end_inset
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|
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|
and
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|
\begin_inset Formula $k=12$
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|
\end_inset
|
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|
the minimum Hamming distance of the JT65 code is
|
|
\begin_inset Formula $d=52$
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|
\end_inset
|
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|
.
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With 72 information bits in each message, JT65 can transmit any one of
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|
\begin_inset Formula $2^{72}\approx4.7\times10^{21}$
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|
\end_inset
|
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|
possible messages.
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|
The codeword for any message differs from every other codeword in at least
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|
52 of the 63 symbol positions.
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|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
A received word containing some
|
|
\emph on
|
|
errors
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|
\emph default
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|
(incorrect symbols) can be decoded into the correct codeword using a determinis
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|
tic,
|
|
\begin_inset CommandInset nomenclature
|
|
LatexCommand nomenclature
|
|
symbol "{\\bf Deterministic algorithm: }"
|
|
description "A series of computational steps that for the same input always produces the same output."
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|
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|
\end_inset
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|
algebraic algorithm provided that no more than
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|
\begin_inset Formula $t$
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\end_inset
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|
symbols were received incorrectly, where
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|
\begin_inset Formula
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|
\begin{equation}
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|
t=\left\lfloor \frac{n-k}{2}\right\rfloor .\label{eq:t}
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|
\end{equation}
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|
\end_inset
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For the JT65 code
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\begin_inset Formula $t=25$
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\end_inset
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|
, so it is always possible to decode a received word having 25 or fewer
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symbol errors.
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|
Any one of several well-known algebraic algorithms, such as the BM algorithm,
|
|
can carry out this hard-decision decoding.
|
|
Two steps are necessarily involved in this process.
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|
We must (1) determine which symbols were received incorrectly, and (2)
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|
find the correct value of the incorrect symbols.
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|
If we somehow know that certain symbols are incorrect, that information
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|
can be used to reduce the work involved in step 1 and allow step 2 to correct
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|
more than
|
|
\begin_inset Formula $t$
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|
\end_inset
|
|
|
|
errors.
|
|
In the unlikely event that the location of every error is known, and if
|
|
no correct symbols are accidentally labeled as errors, the BM algorithm
|
|
can correct up to
|
|
\begin_inset Formula $d-1=n-k$
|
|
\end_inset
|
|
|
|
errors.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm creates lists of symbols suspected of being incorrect and
|
|
sends them to the BM decoder.
|
|
Symbols flagged in this way are called
|
|
\emph on
|
|
erasures
|
|
\emph default
|
|
|
|
\begin_inset CommandInset nomenclature
|
|
LatexCommand nomenclature
|
|
symbol "{\\bf Erasure: }"
|
|
description "A received symbol may be ``erased'' when confidence in its value is so low that it is unlikely to provide useful information. "
|
|
|
|
\end_inset
|
|
|
|
.
|
|
With perfect erasure information up to
|
|
\begin_inset Formula $n-k=51$
|
|
\end_inset
|
|
|
|
incorrect symbols can be corrected for the JT65 code.
|
|
Imperfect erasure information means that some erased symbols may be correct,
|
|
and some other symbols in error.
|
|
If
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
symbols are erased and the remaining
|
|
\begin_inset Formula $n-s$
|
|
\end_inset
|
|
|
|
symbols contain
|
|
\begin_inset Formula $e$
|
|
\end_inset
|
|
|
|
errors, the BM algorithm can find the correct codeword as long as
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
s+2e\le d-1.\label{eq:erasures_and_errors}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
If
|
|
\begin_inset Formula $s=0$
|
|
\end_inset
|
|
|
|
, the decoder is said to be an
|
|
\emph on
|
|
errors-only
|
|
\emph default
|
|
decoder.
|
|
If
|
|
\begin_inset Formula $0<s\le d-1$
|
|
\end_inset
|
|
|
|
, the decoder is called an
|
|
\emph on
|
|
errors-and-erasures
|
|
\emph default
|
|
decoder.
|
|
The possibility of doing errors-and-erasures decoding lies at the heart
|
|
of the FT algorithm.
|
|
On that foundation we have built a capability for using soft information
|
|
on the reliability of individual symbol values, thereby producing a soft-decisi
|
|
on decoder.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:Statistical Framework"
|
|
|
|
\end_inset
|
|
|
|
Statistical Framework
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm uses the estimated quality of received symbols to generate
|
|
lists of symbols considered likely to be in error, thus enabling decoding
|
|
of received words with more than 25 errors.
|
|
Algorithms of this type are generally called
|
|
\emph on
|
|
reliability-based
|
|
\emph default
|
|
or
|
|
\emph on
|
|
probabilistic
|
|
\emph default
|
|
decoding methods
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "lc2004"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Such algorithms involve some amount of educating guessing about which received
|
|
symbols are in error or, alternatively, about which received symbols are
|
|
correct.
|
|
The guesses are informed by quality metrics associated with the received
|
|
symbols.
|
|
To illustrate why it is absolutely essential to use such soft-symbol informatio
|
|
n in these algorithms it helps to consider what would happen if we tried
|
|
to use completely random guesses, ignoring any available soft-symbol informatio
|
|
n.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
As a specific example, consider a received JT65 word with 23 correct symbols
|
|
and 40 errors.
|
|
We do not know which symbols are in error.
|
|
Suppose that the decoder randomly selects
|
|
\begin_inset Formula $s=40$
|
|
\end_inset
|
|
|
|
symbols for erasure, leaving 23 unerased symbols.
|
|
According to Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:erasures_and_errors"
|
|
|
|
\end_inset
|
|
|
|
), the BM decoder can successfully decode this word as long as
|
|
\begin_inset Formula $e$
|
|
\end_inset
|
|
|
|
, the number of errors present in the 23 unerased symbols, is 5 or less.
|
|
The number of errors captured in the set of 40 erased symbols must therefore
|
|
be at least 35.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The probability of selecting some particular number of incorrect symbols
|
|
in a randomly selected subset of received symbols is governed by the hypergeome
|
|
tric probability distribution.
|
|
Let us define
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
as the number of symbols from which erasures will be selected,
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
as the number of incorrect symbols in the set of
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
symbols, and
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
as the number of errors in the symbols actually erased.
|
|
In an ensemble of many received words
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
will be random variables, but for this example we will assume that
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
is known and that only
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
is random.
|
|
The conditional probability mass function for
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
with stated values of
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
may be written as
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P(x=\epsilon|N,X,s)=\frac{\binom{X}{\epsilon}\binom{N-X}{s-\epsilon}}{\binom{N}{s}}\label{eq:hypergeometric_pdf}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $\binom{n}{k}=\frac{n!}{k!(n-k)!}$
|
|
\end_inset
|
|
|
|
is the binomial coefficient.
|
|
The binomial coefficient can be calculated using the function
|
|
\family typewriter
|
|
nchoosek(n,k)
|
|
\family default
|
|
in the numerical computing language
|
|
\emph on
|
|
GNU Octave
|
|
\emph default
|
|
, or with one of many free online calculators.
|
|
The hypergeometric probability mass function defined in Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:hypergeometric_pdf"
|
|
|
|
\end_inset
|
|
|
|
) is available in
|
|
\emph on
|
|
GNU Octave
|
|
\emph default
|
|
as function
|
|
\family typewriter
|
|
hygepdf(x,N,X,s)
|
|
\family default
|
|
.
|
|
The cumulative probability that at least
|
|
\begin_inset Formula $\epsilon$
|
|
\end_inset
|
|
|
|
errors are captured in a subset of
|
|
\begin_inset Formula $s$
|
|
\end_inset
|
|
|
|
erased symbols selected from a group of
|
|
\begin_inset Formula $N$
|
|
\end_inset
|
|
|
|
symbols containing
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
errors is
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P(x\ge\epsilon|N,X,s)=\sum_{j=\epsilon}^{s}P(x=j|N,X,s).\label{eq:cumulative_prob}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Example 1:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Suppose a received word contains
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
incorrect symbols.
|
|
In an attempt to decode using an errors-and-erasures decoder,
|
|
\begin_inset Formula $s=40$
|
|
\end_inset
|
|
|
|
symbols are randomly selected for erasure from the full set of
|
|
\begin_inset Formula $N=n=63$
|
|
\end_inset
|
|
|
|
symbols.
|
|
The probability that
|
|
\begin_inset Formula $x=35$
|
|
\end_inset
|
|
|
|
of the erased symbols are actually incorrect is then
|
|
\begin_inset Formula
|
|
\[
|
|
P(x=35)=\frac{\binom{40}{35}\binom{63-40}{40-35}}{\binom{63}{40}}\simeq2.4\times10^{-7}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
Similarly, the probability that
|
|
\begin_inset Formula $x=36$
|
|
\end_inset
|
|
|
|
of the erased symbols are incorrect is
|
|
\begin_inset Formula
|
|
\[
|
|
P(x=36)\simeq8.6\times10^{-9}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
Since the probability of erasing 36 errors is so much smaller than that
|
|
for erasing 35 errors, we may safely conclude that the probability of randomly
|
|
choosing an erasure vector that can decode the received word is approximately
|
|
|
|
\begin_inset Formula $P(x=35)\simeq2.4\times10^{-7}$
|
|
\end_inset
|
|
|
|
.
|
|
The odds of producing a valid codeword on the first try are very poor,
|
|
about 1 in 4 million.
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Example 2:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
How might we best choose the number of symbols to erase, in order to maximize
|
|
the probability of successful decoding? By exhaustive search over all possible
|
|
values up to
|
|
\begin_inset Formula $s=51$
|
|
\end_inset
|
|
|
|
, it turns out that for
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
the best strategy is to erase
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
symbols.
|
|
According to equation
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:erasures_and_errors"
|
|
|
|
\end_inset
|
|
|
|
, with
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $d=52$
|
|
\end_inset
|
|
|
|
then
|
|
\begin_inset Formula $e$
|
|
\end_inset
|
|
|
|
must be 3 or less.
|
|
Decoding will be assured if the set of erased symbols contains at least
|
|
|
|
\begin_inset Formula $40-3=37$
|
|
\end_inset
|
|
|
|
errors.
|
|
With
|
|
\begin_inset Formula $N=63$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
, the probability of successful decode in a single try is
|
|
\begin_inset Formula
|
|
\[
|
|
P(x\ge37)\simeq1.9\times10^{-6}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
This probability is about 8 times higher than the probability of success
|
|
when only 40 symbols were erased.
|
|
Nevertheless, the odds of successfully decoding on the first try are still
|
|
only about 1 in 500,000.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Paragraph
|
|
Example 3:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Examples 1 and 2 show that a random strategy for selecting symbols to erase
|
|
is unlikely to be successful unless we are prepared to wait a long time
|
|
for an answer.
|
|
So let's modify the strategy to tip the odds in our favor.
|
|
Let the received word contain
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
incorrect symbols, as before, but suppose we know that 10 received symbols
|
|
are significantly more reliable than the other 53.
|
|
We might therefore protect the 10 most reliable symbols and select erasures
|
|
from the smaller set of
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
less reliable ones.
|
|
If
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
symbols are chosen randomly for erasure in this way, it is still necessary
|
|
for the erased symbols to include at least 37 errors, as in Example 2.
|
|
However, the probabilities are now much more favorable: with
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=45$
|
|
\end_inset
|
|
|
|
, Eq.
|
|
(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:cumulative_prob"
|
|
|
|
\end_inset
|
|
|
|
) yields
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
linebreak
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $\mbox{\ensuremath{P(x\geq37)=0.016.}}$
|
|
\end_inset
|
|
|
|
Even better odds are obtained by choosing
|
|
\begin_inset Formula $s=47$
|
|
\end_inset
|
|
|
|
, which requires
|
|
\begin_inset Formula $x\ge38$
|
|
\end_inset
|
|
|
|
.
|
|
With
|
|
\begin_inset Formula $N=53$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $X=40$
|
|
\end_inset
|
|
|
|
, and
|
|
\begin_inset Formula $s=47$
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula $P(x\ge38)=0.027$
|
|
\end_inset
|
|
|
|
.
|
|
The odds for producing a codeword on the first try are now about 1 in 38.
|
|
A few hundred independently randomized tries would be enough to all-but-guarant
|
|
ee production of a valid codeword by the BM decoder.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:The-decoding-algorithm"
|
|
|
|
\end_inset
|
|
|
|
The Franke-Taylor Decoding Algorithm
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Example 3 shows how statistical information about symbol quality should
|
|
make it possible to decode received frames having a large number of errors.
|
|
In practice the number of errors in the received word is unknown, so our
|
|
algorithm simply assigns a high erasure probability to low-quality symbols
|
|
and relatively low probability to high-quality symbols.
|
|
As illustrated by Example 3, a good choice of erasure probabilities can
|
|
increase the chance of producing a codeword by many orders of magnitude.
|
|
Once erasure probabilities have been assigned to each of the 63 received
|
|
symbols, the FT algorithm uses a random number generator to decide whether
|
|
or not to erase each symbol, according to its assigned erasure probability.
|
|
The list of erased symbols is then submitted to the BM decoder, which produces
|
|
either a codeword or a flag indicating failure to decode.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The process of selecting the list of symbols to erase and calling the BM
|
|
decoder comprises one cycle of the FT algorithm.
|
|
The next cycle proceeds with a new selection of erased symbols.
|
|
At this stage we must treat any codeword obtained by errors-and-erasures
|
|
decoding as no more than a
|
|
\emph on
|
|
candidate
|
|
\emph default
|
|
.
|
|
Our next task is to find a metric that can reliably select one of many
|
|
proffered candidates as the codeword that was actually transmitted.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm uses quality indices made available by a noncoherent 64-FSK
|
|
demodulator (see the sidebar
|
|
\series bold
|
|
JT65 Message Processing
|
|
\series default
|
|
).
|
|
The demodulator computes binned power spectra for each signaling interval;
|
|
the result is a two-dimensional array
|
|
\begin_inset Formula $S(i,j)$
|
|
\end_inset
|
|
|
|
, where the frequency index
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
assumes values 0 to 63 and the symbol index
|
|
\begin_inset Formula $j$
|
|
\end_inset
|
|
|
|
has values 1 to 63.
|
|
The most likely value for each symbol is taken as the frequency bin with
|
|
largest signal-plus-noise power over all values of
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
.
|
|
The fractions of total power in the two bins containing the largest and
|
|
second-largest powers, denoted respectively by
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $p_{2}$
|
|
\end_inset
|
|
|
|
, are computed for each symbol and passed from demodulator to decoder as
|
|
soft-symbol information.
|
|
The FT decoder derives two metrics from
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $p_{2}$
|
|
\end_inset
|
|
|
|
, namely
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank (the rank
|
|
\begin_inset Formula $\{1,2,\ldots,63\}$
|
|
\end_inset
|
|
|
|
of the symbol's fractional power
|
|
\begin_inset Formula $p_{1,\, j}$
|
|
\end_inset
|
|
|
|
in a sorted list of
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
values) and the ratio
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
.
|
|
High ranking symbols have larger signal-to-noise ratio than those with
|
|
lower rank.
|
|
When
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
is close to 1, the most likely symbol value is only slightly more reliable
|
|
than the second most likely one.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
We use 3-bit quantization of the metrics
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank and
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
to index the entries in an
|
|
\begin_inset Formula $8\times8$
|
|
\end_inset
|
|
|
|
table of symbol error probabilities.
|
|
The probabilities were derived empirically from a large data set of received
|
|
words that were successfully decoded.
|
|
The table provides an estimate of the
|
|
\emph on
|
|
a priori
|
|
\emph default
|
|
probability of symbol error based on the metrics
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
-rank and
|
|
\begin_inset Formula $p_{2}/p_{1}$
|
|
\end_inset
|
|
|
|
.
|
|
This table is a key element of the algorithm, as it determines which symbols
|
|
are effectively protected from erasure.
|
|
The
|
|
\emph on
|
|
a priori
|
|
\emph default
|
|
symbol error probabilities are close to 1 for low-quality symbols and close
|
|
to 0 for high-quality symbols.
|
|
Recall from Examples 2 and 3 that candidate codewords are produced with
|
|
higher probability when the number of erased symbols is larger than the
|
|
number of incorrect symbols.
|
|
Correspondingly, the FT algorithm works best when the probability of erasing
|
|
a symbol is somewhat larger than the probability that the symbol is incorrect.
|
|
For the JT65 code we found empirically that good decoding performance is
|
|
obtained when the symbol erasure probability is about 1.3 times the symbol
|
|
error probability.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm tries successively to decode the received word using independen
|
|
t educated guesses to select symbols for erasure.
|
|
For each iteration a stochastic erasure vector is generated based on the
|
|
symbol erasure probabilities.
|
|
The erasure vector is sent to the BM decoder along with the full set of
|
|
63 hard-decision symbol values.
|
|
When the BM decoder finds a candidate codeword it is assigned a quality
|
|
metric
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
, the
|
|
\emph on
|
|
soft distance
|
|
\begin_inset CommandInset nomenclature
|
|
LatexCommand nomenclature
|
|
symbol "{\\bf Soft distance: }"
|
|
description "The soft distance between a received word and a codeword is a measure of how greatly they differ, taking into account available soft information on symbol values."
|
|
|
|
\end_inset
|
|
|
|
|
|
\emph default
|
|
between the received word and the codeword:
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
d_{s}=\sum_{j=1}^{n}\alpha_{j}\,(1+p_{1,\, j}).\label{eq:soft_distance}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
Here
|
|
\begin_inset Formula $\alpha_{j}=0$
|
|
\end_inset
|
|
|
|
if received symbol
|
|
\begin_inset Formula $j$
|
|
\end_inset
|
|
|
|
is the same as the corresponding symbol in the codeword,
|
|
\begin_inset Formula $\alpha_{j}=1$
|
|
\end_inset
|
|
|
|
if the received symbol and codeword symbol are different, and
|
|
\begin_inset Formula $p_{1,\, j}$
|
|
\end_inset
|
|
|
|
is the fractional power associated with received symbol
|
|
\begin_inset Formula $j$
|
|
\end_inset
|
|
|
|
.
|
|
Think of the soft distance as made up of two terms: the first is the Hamming
|
|
distance between the received word and the codeword, and the second ensures
|
|
that if two candidate codewords have the same Hamming distance from the
|
|
received word, a smaller soft distance will be assigned to the one where
|
|
differences occur in symbols of lower estimated reliability.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In practice we find that
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
can reliably identify the correct codeword if the signal-to-noise ratio
|
|
for individual symbols is greater than about 4 in linear power units.
|
|
We also find that significantly weaker signals can be decoded by using
|
|
soft-symbol information beyond that contained in
|
|
\begin_inset Formula $p_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $p_{2}$
|
|
\end_inset
|
|
|
|
.
|
|
To this end we define an additional metric
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
, the average signal-plus-noise power in all received symbols according
|
|
to a candidate codeword's symbol values:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
u=\frac{1}{n}\sum_{j=1}^{n}S(c_{j},\, j).\label{eq:u-metric}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
Here the
|
|
\begin_inset Formula $c_{j}$
|
|
\end_inset
|
|
|
|
's are the symbol values for the candidate codeword being tested.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The correct JT65 codeword produces a value for
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
equal to the average of
|
|
\begin_inset Formula $n=63$
|
|
\end_inset
|
|
|
|
bins containing both signal and noise power.
|
|
Incorrect codewords have at most
|
|
\begin_inset Formula $k-1=11$
|
|
\end_inset
|
|
|
|
such bins and at least
|
|
\begin_inset Formula $n-k+1=52$
|
|
\end_inset
|
|
|
|
bins containing noise only.
|
|
Thus, if the spectral array
|
|
\begin_inset Formula $S(i,\, j)$
|
|
\end_inset
|
|
|
|
has been normalized so that the average value of the noise-only bins is
|
|
unity,
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
for the correct codeword has expectation value (average over many random
|
|
realizations) given by
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\bar{u}_{c}=1+y,\label{eq:u1-exp}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $y$
|
|
\end_inset
|
|
|
|
is the signal-to-noise ratio in linear power units.
|
|
If we assume Gaussian statistics and a large number of trials, the standard
|
|
deviation of measured values of
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
is
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\sigma_{c}=\left(\frac{1+2y}{n}\right)^{1/2}.\label{eq:sigma1}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
In contrast, the expected value and standard deviation of the
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
-metric for an incorrect codeword (randomly selected from a population of
|
|
all
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
worst case
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
codewords,
|
|
\emph on
|
|
i.e.
|
|
\emph default
|
|
, those with
|
|
\begin_inset Formula $k-1$
|
|
\end_inset
|
|
|
|
symbols identical to corresponding ones in the correct word) are given
|
|
by
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\bar{u}_{i}=1+\left(\frac{k-1}{n}\right)y,\label{eq:u2-exp}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\sigma_{i}=\frac{1}{n}\left[n+2y(k-1)\right]^{1/2},\label{eq:sigma2}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where the subscript
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
is an abbreviation for
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
incorrect
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
If
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
is evaluated for a large number of candidate codewords, one of which is
|
|
correct, we should expect the largest value
|
|
\begin_inset Formula $u_{1}$
|
|
\end_inset
|
|
|
|
to be drawn from a population with statistics described by
|
|
\begin_inset Formula $\bar{u}_{c}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\sigma_{c}.$
|
|
\end_inset
|
|
|
|
If no tested codeword is correct,
|
|
\begin_inset Formula $u_{1}$
|
|
\end_inset
|
|
|
|
is likely to come from the
|
|
\begin_inset Formula $(\bar{u}_{i},\,\sigma_{i})$
|
|
\end_inset
|
|
|
|
population and to be several standard deviations above the mean.
|
|
In either case the second-largest value,
|
|
\begin_inset Formula $u_{2},$
|
|
\end_inset
|
|
|
|
will likely come from the
|
|
\begin_inset Formula $(\bar{u}_{i},\,\sigma_{i})$
|
|
\end_inset
|
|
|
|
population, again several standard deviations above the mean.
|
|
If the signal-to-noise ratio
|
|
\begin_inset Formula $y$
|
|
\end_inset
|
|
|
|
is too small for decoding to be possible or the correct codeword is never
|
|
presented as a candidate, the ratio
|
|
\begin_inset Formula $r=u_{2}/u_{1}$
|
|
\end_inset
|
|
|
|
will likely be close to 1.
|
|
On the other hand, correctly identified codewords will produce
|
|
\begin_inset Formula $u_{1}$
|
|
\end_inset
|
|
|
|
significantly larger than
|
|
\begin_inset Formula $u_{2}$
|
|
\end_inset
|
|
|
|
and thus smaller values of
|
|
\begin_inset Formula $r$
|
|
\end_inset
|
|
|
|
.
|
|
We therefore apply a ratio threshold test, say
|
|
\begin_inset Formula $r<R_{1}$
|
|
\end_inset
|
|
|
|
, to identify codewords with high probability of being correct.
|
|
As described in Section
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "sec:Theory,-Simulation,-and"
|
|
|
|
\end_inset
|
|
|
|
, we use simulations to set an empirical acceptance threshold
|
|
\begin_inset Formula $R_{1}$
|
|
\end_inset
|
|
|
|
that maximizes the probability of correct decodes while ensuring a low
|
|
rate of false positives.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
As with all decoding algorithms that generate a list of possible codewords,
|
|
a stopping criterion is necessary.
|
|
FT accepts a codeword unconditionally if the Hamming distance
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
and soft distance
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
obey specified criteria
|
|
\begin_inset Formula $X<X_{0}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $d_{s}<D_{0}$
|
|
\end_inset
|
|
|
|
.
|
|
Secondary acceptance criteria
|
|
\begin_inset Formula $d_{s}<D_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $r<R_{1}$
|
|
\end_inset
|
|
|
|
are used to validate additional codewords that fail the first test.
|
|
A timeout is used to limit execution time if no acceptable codeword is
|
|
found in a reasonable number of trials,
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
.
|
|
Today's personal computers are fast enough that
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
can be set as large as
|
|
\begin_inset Formula $10^{5},$
|
|
\end_inset
|
|
|
|
or even higher.
|
|
Pseudo-code for the FT algorithm is presented in an accompanying box,
|
|
\series bold
|
|
Algorithm 1
|
|
\series default
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float algorithm
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Pseudo-code for the FT algorithm.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
For each received symbol, define the erasure probability as 1.3 times the
|
|
|
|
\emph on
|
|
a priori
|
|
\emph default
|
|
symbol-error probability determined from soft-symbol information
|
|
\begin_inset Formula $\{p_{1}\textrm{-rank},\, p_{2}/p_{1}\}$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Make independent stochastic
|
|
\begin_inset CommandInset nomenclature
|
|
LatexCommand nomenclature
|
|
symbol "{\\bf Stochastic algorithm: }"
|
|
description "An algorithm involving chance or probability in determining the series of computational steps to be taken."
|
|
|
|
\end_inset
|
|
|
|
decisions about whether to erase each symbol by using the symbol's erasure
|
|
probability, allowing a maximum of 51 erasures.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Attempt errors-and-erasures decoding using the BM algorithm and the set
|
|
of erasures determined in step 2.
|
|
If the BM decoder produces a candidate codeword, go to step 5.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If BM decoding was not successful, go to step 2.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Calculate the hard-decision Hamming distance
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
between the candidate codeword and the received symbols, along with the
|
|
corresponding soft distance
|
|
\begin_inset Formula $d_{s}$
|
|
\end_inset
|
|
|
|
and the quality metric
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
is the largest one encountered so far, preserve any previous value of
|
|
\begin_inset Formula $u_{1}$
|
|
\end_inset
|
|
|
|
by setting
|
|
\begin_inset Formula $u_{2}=u_{1}.$
|
|
\end_inset
|
|
|
|
Then set
|
|
\begin_inset Formula $u_{1}=u,$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $d_{1}=d_{s},$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $X_{1}=X,$
|
|
\end_inset
|
|
|
|
and save the codeword.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $X_{1}<X_{0}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $d_{1}<D_{0}$
|
|
\end_inset
|
|
|
|
, go to step 11.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If the number of trials is less than the timeout limit
|
|
\begin_inset Formula $T,$
|
|
\end_inset
|
|
|
|
go to 2.
|
|
|
|
\begin_inset Formula $ $
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $d_{1}<D_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $r=u_{2}/u_{1}<R_{1},$
|
|
\end_inset
|
|
|
|
go to step 11.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Otherwise, declare decoding failure and exit.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
An acceptable codeword has been found.
|
|
Declare a successful decode and return the saved codeword.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Inspiration for the FT decoding algorithm came from a number of sources.
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "lc2004"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
textsuperscript{,}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "lhmg2010"
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
textsuperscript{,}
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "lk2008"
|
|
|
|
\end_inset
|
|
|
|
After developing this algorithm, we became aware that our approach is conceptua
|
|
lly similar to a stochastic, erasures-only list decoding algorithm described
|
|
in another reference
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "ls2009"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
That algorithm is applied to higher-rate Reed-Solomon codes on a symmetric
|
|
channel using binary phase-shift keying (BPSK).
|
|
Our 64-ary input channel with 64-FSK modulation required us to develop
|
|
unique methods for assigning erasure probabilities and for defining acceptance
|
|
criteria to select the best codeword from the list of tested candidates.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:Hinted-Decoding"
|
|
|
|
\end_inset
|
|
|
|
Hinted Decoding
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The FT algorithm is completely general.
|
|
With equal sensitivity it can recover any one of the
|
|
\begin_inset Formula $2^{72}\approx4.7\times10^{21}$
|
|
\end_inset
|
|
|
|
different messages that can be transmitted with the JT65 protocol.
|
|
In some circumstances it's easy to imagine a
|
|
\emph on
|
|
much
|
|
\emph default
|
|
smaller list of messages (say, a few thousand messages or less) that would
|
|
be among the most likely ones to be received.
|
|
One such favorable situation exists when making short ham-radio contacts
|
|
that exchange minimal information including callsigns, signal reports,
|
|
perhaps Maidenhead locators, and acknowledgments.
|
|
On the EME path or a VHF or UHF band with limited geographical coverage,
|
|
the most common received messages frequently originate from callsigns that
|
|
have been decoded before.
|
|
Saving a list of previously decoded callsigns and associated locators makes
|
|
it easy to generate a list of hypothetical messages and their corresponding
|
|
codewords at very little computational expense.
|
|
The resulting candidate codewords can be tested in almost the same way
|
|
as those generated by the probabilistic method described in Section
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "sec:The-decoding-algorithm"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
We call this approach
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
hinted decoding;
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
it is sometimes referred to as the
|
|
\emph on
|
|
Deep Search
|
|
\emph default
|
|
algorithm.
|
|
In certain limited situations it can provide enhanced sensitivity for the
|
|
principal task of any decoder, namely to determine precisely what message
|
|
was sent.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
For hinted decoding we again invoke a ratio threshold test, but in this
|
|
case we use it to answer a more limited question.
|
|
Over the full list of messages considered likely, we want to know whether
|
|
a suitable metric can distinguish with confidence between the one correct
|
|
codeword and all others in the generated list --- or, alternatively, to
|
|
determine that the correct codeword is
|
|
\emph on
|
|
not
|
|
\emph default
|
|
contained in the list.
|
|
We again find that the most effective metric involves a comparison of
|
|
\begin_inset Formula $u_{1}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $u_{2},$
|
|
\end_inset
|
|
|
|
the largest and second-largest values of total signal-plus-noise power
|
|
among all the tested codewords.
|
|
The criterion for comparison is chosen empirically to maximize the number
|
|
of correct decodes while ensuring that false decodes are rare.
|
|
Because tested candidate codewords are drawn from a list typically no longer
|
|
than a few thousand entries, rather than
|
|
\begin_inset Formula $2^{72},$
|
|
\end_inset
|
|
|
|
the limit can can be more relaxed than that used in the FT algorithm.
|
|
Thus, for the limited subset of messages suggested by previous experience
|
|
to be likely, hinted decodes can be obtained at lower signal levels than
|
|
required for the full universe of
|
|
\begin_inset Formula $2^{72}$
|
|
\end_inset
|
|
|
|
possible messages.
|
|
Pseudo-code for the hinted-decoding algorithm is presented as
|
|
\series bold
|
|
Algorithm 2
|
|
\series default
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float algorithm
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
Pseudo-code for hinted decoding
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Generate a list of
|
|
\begin_inset Formula $L$
|
|
\end_inset
|
|
|
|
codewords considered likely to be received.
|
|
Set a pointer to the start of this list.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Fetch the next candidate codeword and calculate its metric
|
|
\begin_inset Formula $u.$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $u$
|
|
\end_inset
|
|
|
|
is the largest metric encountered so far, preserve any previous value of
|
|
|
|
\begin_inset Formula $u_{1}$
|
|
\end_inset
|
|
|
|
by setting
|
|
\begin_inset Formula $u_{2}=u_{1}.$
|
|
\end_inset
|
|
|
|
Then set
|
|
\begin_inset Formula $u_{1}=u$
|
|
\end_inset
|
|
|
|
and save the codeword.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If the number of tested codewords is less than
|
|
\begin_inset Formula $L,$
|
|
\end_inset
|
|
|
|
go to step 2.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
If
|
|
\begin_inset Formula $r=u_{2}/u_{1}<R_{2},$
|
|
\end_inset
|
|
|
|
go to step 7.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Otherwise, declare decoding failure and exit.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
An acceptable codeword has been found.
|
|
Declare a successful result and return the codeword and the value
|
|
\begin_inset Formula $q=100\,(u_{1}-bu_{2})$
|
|
\end_inset
|
|
|
|
as a confidence indicator.
|
|
(By default we use the value
|
|
\begin_inset Formula $b=1.12$
|
|
\end_inset
|
|
|
|
for submode JT65A.)
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:Theory,-Simulation,-and"
|
|
|
|
\end_inset
|
|
|
|
Decoder Performance Evaluation
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Comparisons of decoding performance are usually presented in the professional
|
|
literature as plots of word error rate versus
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
, the ratio of the energy collected per information bit to the one-sided
|
|
noise power spectral density.
|
|
For weak-signal amateur radio work, performance is more usefully presented
|
|
as the probability of successfully decoding a received word plotted against
|
|
|
|
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
|
|
\end_inset
|
|
|
|
, the signal-to-noise ratio in a 2500 Hz reference bandwidth.
|
|
The relationship between
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
|
|
\end_inset
|
|
|
|
is described in Appendix
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "sec:Appendix:SNR"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
Examples of both types of plot are included in the following discussion,
|
|
where we describe simulations carried out to compare performance of the
|
|
FT algorithm and hinted decoding with other algorithms and with theoretical
|
|
expectations.
|
|
We have also used simulations to establish suitable default values for
|
|
the acceptance parameters
|
|
\begin_inset Formula $X_{0},$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $D_{0},$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $D_{1},$
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $R_{1},$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $R_{2}.$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Simulated results on the AWGN channel
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Results of simulations using the BM, KV, and FT decoding algorithms on the
|
|
JT65 code are presented in terms of word error rate versus
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
in Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For these tests we generated at least 1000 signals at each signal-to-noise
|
|
ratio, assuming the additive white gaussian noise (AWGN) channel, and we
|
|
processed the data using each algorithm.
|
|
For word error rates less than 0.1 it was necessary to process 10,000 or
|
|
even 100,000 simulated signals in order to capture enough errors to make
|
|
the measurements statistically meaningful.
|
|
As a test of the fidelity of our numerical simulations, Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
also shows results calculated from theoretical probability distributions
|
|
for comparison with the BM results.
|
|
The simulated BM results agree with theory to within about 0.1 dB.
|
|
The differences are caused by small errors in the estimates of time and
|
|
frequency offset of the received signal in the simulated data.
|
|
Such
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
sync losses
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
are not accounted for in the idealized theoretical results.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
As expected, on the AWGN channel the soft-decision algorithms FT and KV
|
|
are about 2 dB better than the hard-decision BM algorithm.
|
|
In addition, FT has an edge over KV that increases from about 0.2 dB at
|
|
higher SNRs to nearly 0.5 dB at low SNR.
|
|
With timeout parameter
|
|
\begin_inset Formula $T=10^{5}$
|
|
\end_inset
|
|
|
|
execution time for FT is longer than that for the KV algorithm, but still
|
|
small enough to be fully practical on today's home computers.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_bodide.pdf
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
Word error rates as a function of
|
|
\begin_inset Formula $E_{b}/N_{0},$
|
|
\end_inset
|
|
|
|
the signal-to-noise ratio per information bit.
|
|
The curve labeled
|
|
\emph on
|
|
Theory
|
|
\emph default
|
|
shows a theoretical prediction for the hard-decision BM decoder.
|
|
Remaining curves represent simulation results on an AWGN channel for the
|
|
BM, KV, and FT decoders.
|
|
The KV algorithm was executed with complexity coefficient
|
|
\begin_inset Formula $\lambda=15$
|
|
\end_inset
|
|
|
|
, the most aggressive setting historically used in the
|
|
\emph on
|
|
WSJT
|
|
\emph default
|
|
programs.
|
|
The FT algorithm used timeout setting
|
|
\begin_inset Formula $T=10^{5}.$
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Error-free transmission is important in commercial applications, so plots
|
|
like Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
are often extended downward to error rates of
|
|
\begin_inset Formula $10^{-6}$
|
|
\end_inset
|
|
|
|
or even less.
|
|
The circumstances for minimal amateur-radio contacts are very different,
|
|
however.
|
|
Decoding failure rates of order 0.1 or higher may be perfectly acceptable:
|
|
they simply require repeat transmissions.
|
|
In such circumstances the essential information is more usefully presented
|
|
in a plot showing the percentage of transmissions copied correctly as a
|
|
function of signal-to-noise ratio.
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:WER2"
|
|
|
|
\end_inset
|
|
|
|
shows the FT and KV results from Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
in this format, along with additional FT results for
|
|
\begin_inset Formula $T=10^{4},\:10^{3},\:10^{2}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $10$
|
|
\end_inset
|
|
|
|
.
|
|
It's easy to see that the FT decoder produces more decodes than KV when
|
|
|
|
\begin_inset Formula $T\gtrsim3000$
|
|
\end_inset
|
|
|
|
.
|
|
As already noted in connection with Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:bodide"
|
|
|
|
\end_inset
|
|
|
|
, FT with
|
|
\begin_inset Formula $T=10^{5}$
|
|
\end_inset
|
|
|
|
has approximately
|
|
\begin_inset Formula $0.5$
|
|
\end_inset
|
|
|
|
dB gain over KV at low SNR.
|
|
It also provides very significant gains over the hard-decision BM decoder,
|
|
even when limited to very small
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_wer2.pdf
|
|
lyxscale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:WER2"
|
|
|
|
\end_inset
|
|
|
|
Percent of JT65 messages copied as a function of
|
|
\begin_inset Formula $\mathrm{SNR}{}_{2500},$
|
|
\end_inset
|
|
|
|
assuming additive white gaussian noise and no fading.
|
|
Numbers adjacent to curves specify values of timeout parameter
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
for the FT decoder.
|
|
Open circles and dotted line show results for the KV decoder with complexity
|
|
coefficient
|
|
\begin_inset Formula $\lambda=15$
|
|
\end_inset
|
|
|
|
.
|
|
Results for the BM algorithm are plotted with crosses and dashed line.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Parameter
|
|
\begin_inset Formula $T$
|
|
\end_inset
|
|
|
|
in the FT algorithm is the maximum number of symbol-erasure trials allowed
|
|
for a particular attempt at decoding a received word.
|
|
Most successful decodes take only a small fraction of the maximum allowed
|
|
number of trials.
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:N_vs_X"
|
|
|
|
\end_inset
|
|
|
|
shows the number of stochastic erasure trials required to find the correct
|
|
codeword plotted as a function of
|
|
\begin_inset Formula $X,$
|
|
\end_inset
|
|
|
|
the number of hard-decision errors in the received word.
|
|
This test run used 1000 simulated transmissions at
|
|
\begin_inset Formula $\mathrm{SNR_{2500}}=-24$
|
|
\end_inset
|
|
|
|
dB, just slightly above the decoding threshold, with timeout parameter
|
|
|
|
\begin_inset Formula $T=10^{5}.$
|
|
\end_inset
|
|
|
|
No points are shown for
|
|
\begin_inset Formula $X\le25$
|
|
\end_inset
|
|
|
|
because all such words are successfully decoded by a single run of the
|
|
errors-only BM algorithm.
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:N_vs_X"
|
|
|
|
\end_inset
|
|
|
|
shows that the FT algorithm decodes received words with as many as
|
|
\begin_inset Formula $X=43$
|
|
\end_inset
|
|
|
|
symbol errors.
|
|
It also shows that the average number of trials increases with the number
|
|
of errors in the received word.
|
|
The variability of decoding time also increases dramatically with the number
|
|
of errors in the received word.
|
|
These results provide insight into the mean and variance of execution time
|
|
for the FT algorithm, since execution time is roughly proportional to the
|
|
number of required erasure trials.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_ntrials_vs_nhard.pdf
|
|
lyxscale 120
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:N_vs_X"
|
|
|
|
\end_inset
|
|
|
|
Number of trials needed to decode a received word
|
|
\emph on
|
|
vs.
|
|
|
|
\emph default
|
|
Hamming distance
|
|
\begin_inset Formula $X$
|
|
\end_inset
|
|
|
|
between received word and decoded codeword.
|
|
We used 1000 simulated transmissions on an AWGN channel with no fading.
|
|
The signal-to-noise ratio was
|
|
\begin_inset Formula $\mathrm{SNR}{}_{2500}=-24$
|
|
\end_inset
|
|
|
|
dB, or
|
|
\begin_inset Formula $E_{b}/N_{o}=5.1$
|
|
\end_inset
|
|
|
|
dB.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Subsection
|
|
Simulated results for Rayleigh fading and hinted decoding
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
presents the results of simulations for signal-to-noise ratios ranging
|
|
from
|
|
\begin_inset Formula $-18$
|
|
\end_inset
|
|
|
|
to
|
|
\begin_inset Formula $-30$
|
|
\end_inset
|
|
|
|
dB, again using 1000 simulated signals for each plotted point.
|
|
We include three curves for each decoding algorithm: one for the AWGN channel
|
|
and no fading, and two more for simulated Doppler spreads of 0.2 and 1.0
|
|
Hz.
|
|
These simulated Doppler spreads are comparable to those encountered on
|
|
HF ionospheric paths and also for EME at VHF and the lower UHF bands.
|
|
For comparison we note that the JT65 symbol rate is about
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
linebreak
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
2.7 Hz.
|
|
It is interesting to note that while Rayleigh fading severely degrades
|
|
the success rate of the BM decoder, the penalties are much smaller with
|
|
both FT and Deep Search (DS) decoding.
|
|
Simulated Doppler spreads of 0.2 Hz actually increased the FT decoding rate
|
|
slightly at SNRs close to the decoding threshold, presumably because with
|
|
the low-rate JT65 code, signal peaks provide the information needed for
|
|
good copy.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_psuccess.pdf
|
|
lyxscale 90
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:Psuccess"
|
|
|
|
\end_inset
|
|
|
|
Percentage of JT65 messages successfully decoded as a function of
|
|
\begin_inset Formula $\mathrm{SNR}{}_{2500}$
|
|
\end_inset
|
|
|
|
.
|
|
Results are shown for the hard-decision Berlekamp-Massey (BM) and soft-decision
|
|
Franke-Taylor (FT) decoding algorithms.
|
|
Curves labeled DS correspond to the hinted-decode (Deep Search) algorithm
|
|
with a codeword list of length
|
|
\begin_inset Formula $L\,$
|
|
\end_inset
|
|
|
|
= 5850.
|
|
Numbers adjacent to the curves are simulated Doppler spreads in Hz.
|
|
In the current version of
|
|
\emph on
|
|
WSJT-X
|
|
\emph default
|
|
the performance of the DS algorithm is limited by synchronization failures
|
|
when
|
|
\begin_inset Formula $SNR\lesssim-28\,\textrm{dB}$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
On-the-air Experience
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The JT65 protocol has proven remarkably versatile.
|
|
Today the mode is used by thousands of amateurs around the world, communicating
|
|
over terrestrial paths on the MF and HF bands and over terrestrial as well
|
|
as EME paths from 50 MHz through 10 GHz.
|
|
Three
|
|
\emph on
|
|
submodes
|
|
\emph default
|
|
are in use, together accommodating a wide range of Doppler spreads and
|
|
potential instrumental instabilities.
|
|
All three submodes transmit the 63 data symbols interspersed with 63 synchroniz
|
|
ation symbols at keying rate 11025/4096 = 2.69 baud.
|
|
Submode JT65A uses tone spacing equal to the symbol rate; its total occupied
|
|
bandwidth is
|
|
\begin_inset Formula $66\times2.69=177.6$
|
|
\end_inset
|
|
|
|
Hz.
|
|
Submodes B and C have tone spacings and occupied bandwidths 2 and 4 times
|
|
larger.
|
|
In practice JT65A is generally used at 50 MHz and below, JT65B on 144 through
|
|
432 MHz, and JT65C at 1296 MHz and above.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:JT65B_EME"
|
|
|
|
\end_inset
|
|
|
|
shows portions of the main window and spectrogram displays from program
|
|
|
|
\emph on
|
|
WSJT-X,
|
|
\emph default
|
|
illustrating replies to a CQ from K1JT on 144.118 MHz using submode JT65B
|
|
on the EME path.
|
|
Speckled vertical lines on the waterfall at 1494 and 1515 Hz are the synchroniz
|
|
ing tones of signals from DL7UAE and SP6GWB.
|
|
Other visible speckles (barely above the noise) up to about 1870 Hz are
|
|
some of the data tones from these two stations.
|
|
Two lines of decoded text show that the estimated average signal strengths
|
|
were
|
|
\begin_inset Formula $\mathrm{SNR}{}_{2500}=-23$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $-24$
|
|
\end_inset
|
|
|
|
dB, respectively, just one or two dB above decoding threshold for the FT
|
|
decoder.
|
|
Note that the two signals overlap throughout more than 90% of their occupied
|
|
bandwidths, yet both are decoded cleanly and without errors.
|
|
Such behavior is typical of the JT65 protocol.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename JT65B_EME.png
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:JT65B_EME"
|
|
|
|
\end_inset
|
|
|
|
Examples of JT65B EME signals recorded at K1JT.
|
|
Numbers above the spectrogram are audio frequencies in Hz, and the spectrogram'
|
|
s vertical span is one minute of time.
|
|
The horizontal green bar on the frequency axis indicates the bandwidth
|
|
occupied by the second decoded signal, a reply from SP6GWB.
|
|
See text for additional details.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
As another example, Figure
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "fig:spectrogram"
|
|
|
|
\end_inset
|
|
|
|
shows activity in submode JT65A during a single minute on the 20 m amateur
|
|
band.
|
|
At this time the band was crowded with overlapping signals.
|
|
With care you can count at least 19 distinct synchronizing tones (the speckled
|
|
vertical lines in the figure), and can see as many as four signals overlapping
|
|
in some places.
|
|
After signal processing to demodulate the signals and produce soft-symbol
|
|
data for the FT decoder, program
|
|
\emph on
|
|
WSJT-X
|
|
\emph default
|
|
extracts and decodes 21 error-free messages from this recorded data segment.
|
|
This result is achieved with a relatively small timeout parameter,
|
|
\begin_inset Formula $T=1000.$
|
|
\end_inset
|
|
|
|
For these results the decoder uses two successive sweeps over the spectrum.
|
|
The strongest signals (12 in this example) are sequentially decoded and
|
|
subtracted from the raw data after the first pass.
|
|
Another 9 signals are decoded in the second pass.
|
|
For comparison, the hard-decision BM decoder decodes only 12 messages from
|
|
this recording (9 in the first pass and 3 more in a second).
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Float figure
|
|
wide false
|
|
sideways false
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
\align center
|
|
\begin_inset Graphics
|
|
filename fig_waterfall.png
|
|
scale 60
|
|
BoundingBox 0bp 0bp 1124bp 200bp
|
|
clip
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset Caption Standard
|
|
|
|
\begin_layout Plain Layout
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "fig:spectrogram"
|
|
|
|
\end_inset
|
|
|
|
Spectrogram from
|
|
\emph on
|
|
WSJT-X
|
|
\emph default
|
|
showing one minute of data collected under crowded band conditions on the
|
|
20 m band.
|
|
Numbers on the scale are frequencies (in Hz) above 14.076 MHz.
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
Our implementation of the FT decoder, written in a combination of FORTRAN
|
|
and C, is freely available as open-source code
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "wsjt_sourceforge"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
For the Berlekamp-Massey part of the algorithm we use routines written
|
|
by Phil Karn, KA9Q
|
|
\begin_inset CommandInset citation
|
|
LatexCommand cite
|
|
key "karn"
|
|
|
|
\end_inset
|
|
|
|
, modified slightly so that the Reed-Solomon syndromes are computed only
|
|
once in our most time-consuming loop (steps 2 through 8,
|
|
\series bold
|
|
Algorithm 1
|
|
\series default
|
|
).
|
|
The FT algorithm has become an integral part of programs
|
|
\emph on
|
|
WSJT,
|
|
\emph default
|
|
|
|
\emph on
|
|
MAP65,
|
|
\emph default
|
|
and
|
|
\emph on
|
|
WSJT-X
|
|
\emph default
|
|
.
|
|
Improvement in sensitivity over the Kötter-Vardy decoder is small, only
|
|
a few tenths of a dB, but especially on the EME path such small advantages
|
|
are sometimes very important.
|
|
Perhaps even more essential, programs in the
|
|
\emph on
|
|
WSJT
|
|
\emph default
|
|
family are now entirely open source.
|
|
We no longer need to use the patented KV algorithm or the specially licensed
|
|
executable program
|
|
\family typewriter
|
|
kvasd[.exe]
|
|
\family default
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
Acknowledgments
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
We thank G3WDG, G4WJS, KD9DSW, PY2SDR, SM5BSZ, VK7MO, and W3SZ for helpful
|
|
comments on an earlier version of this paper.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newpage
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "1"
|
|
key "jt65_protocol"
|
|
|
|
\end_inset
|
|
|
|
“The JT65 Communications Protocol”, J.
|
|
Taylor, K1JT,
|
|
\emph on
|
|
QEX
|
|
\emph default
|
|
, September-October 2005, pp.
|
|
3-12.
|
|
Available also at http://physics.princeton.edu/pulsar/K1JT/JT65.pdf.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "2"
|
|
key "kv2001"
|
|
|
|
\end_inset
|
|
|
|
“Algebraic soft-decision decoding of Reed-Solomon codes,” R.
|
|
Kötter and A.
|
|
Vardy,
|
|
\emph on
|
|
IEEE Transactions on Information Theory
|
|
\emph default
|
|
, Vol.
|
|
49, pp.
|
|
2809-2825, 2003.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "3"
|
|
key "wsjt"
|
|
|
|
\end_inset
|
|
|
|
|
|
\emph on
|
|
WSJT Home Page
|
|
\emph default
|
|
: http://www.physics.princeton.edu/pulsar/K1JT/.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "4"
|
|
key "lc2004"
|
|
|
|
\end_inset
|
|
|
|
|
|
\emph on
|
|
Error Control Coding, 2nd Edition
|
|
\emph default
|
|
, Shu Lin and Daniel J.
|
|
Costello, Pearson-Prentice Hall, 2004.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "5"
|
|
key "lhmg2010"
|
|
|
|
\end_inset
|
|
|
|
``Stochastic Chase Decoding of Reed-Solomon Codes'', Camille Leroux, Saied
|
|
Hemati, Shie Mannor, Warren J.
|
|
Gross,
|
|
\emph on
|
|
IEEE Communications Letters
|
|
\emph default
|
|
, Vol.
|
|
14, No.
|
|
9, pp.
|
|
863-865, 2010.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "6"
|
|
key "lk2008"
|
|
|
|
\end_inset
|
|
|
|
``Soft-Decision Decoding of Reed-Solomon Codes Using Successive Error-and-Erasur
|
|
e Decoding,'' Soo-Woong Lee and B.
|
|
V.
|
|
K.
|
|
Vijaya Kumar,
|
|
\emph on
|
|
IEEE
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
GLOBECOM
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
Proceedings,
|
|
\emph default
|
|
2008
|
|
\emph on
|
|
.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "7"
|
|
key "ls2009"
|
|
|
|
\end_inset
|
|
|
|
``Stochastic Erasure-Only List Decoding Algorithms for Reed-Solomon Codes,
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
Chang-Ming Lee and Yu T.
|
|
Su,
|
|
\emph on
|
|
IEEE Signal Processing Letters,
|
|
\emph default
|
|
Vol.
|
|
16, pp.
|
|
691-694, 2009.
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "8"
|
|
key "wsjt_sourceforge"
|
|
|
|
\end_inset
|
|
|
|
Source code for all programs in the
|
|
\emph on
|
|
WSJT
|
|
\emph default
|
|
project is stored in a Subversion repository at Sourceforge: https://sourceforg
|
|
e.net/projects/wsjt/
|
|
\end_layout
|
|
|
|
\begin_layout Bibliography
|
|
\begin_inset CommandInset bibitem
|
|
LatexCommand bibitem
|
|
label "9"
|
|
key "karn"
|
|
|
|
\end_inset
|
|
|
|
Errors-and erasures decoder for the Berlekamp-Massey algorithm written by
|
|
Phil Karn, KA9Q: http://www.ka9q.net/code/fec/
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newpage
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Section
|
|
\start_of_appendix
|
|
\begin_inset CommandInset label
|
|
LatexCommand label
|
|
name "sec:Appendix:SNR"
|
|
|
|
\end_inset
|
|
|
|
Appendix: Signal to Noise Ratios
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
The signal to noise ratio in a bandwidth,
|
|
\begin_inset Formula $B$
|
|
\end_inset
|
|
|
|
, that is at least as large as the bandwidth occupied by the signal is:
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\mathrm{SNR}_{B}=\frac{P_{s}}{N_{0}B}\label{eq:SNR}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
where
|
|
\begin_inset Formula $P_{s}$
|
|
\end_inset
|
|
|
|
is the average signal power (W),
|
|
\begin_inset Formula $N_{0}$
|
|
\end_inset
|
|
|
|
is one-sided noise power spectral density (W/Hz), and
|
|
\begin_inset Formula $B$
|
|
\end_inset
|
|
|
|
is the bandwidth in Hz.
|
|
In amateur radio applications, digital modes are often compared based on
|
|
the SNR defined in a 2.5 kHz reference bandwidth,
|
|
\begin_inset Formula $\mathrm{SNR}_{2500}$
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In the professional literature, decoder performance is characterized in
|
|
terms of
|
|
\begin_inset Formula $E_{b}/N_{0}$
|
|
\end_inset
|
|
|
|
, the ratio of the energy collected per information bit,
|
|
\begin_inset Formula $E_{b}$
|
|
\end_inset
|
|
|
|
, to the one-sided noise power spectral density,
|
|
\begin_inset Formula $N_{0}$
|
|
\end_inset
|
|
|
|
.
|
|
Denote the duration of a channel symbol by
|
|
\begin_inset Formula $\tau_{s}$
|
|
\end_inset
|
|
|
|
(for JT65,
|
|
\begin_inset Formula $\tau_{s}=0.3715\,\mathrm{s}$
|
|
\end_inset
|
|
|
|
).
|
|
JT65 signals have constant envelope, so the average signal power is related
|
|
to the energy per symbol,
|
|
\begin_inset Formula $E_{s}$
|
|
\end_inset
|
|
|
|
, by
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
P_{s}=E_{s}/\tau_{s}.\label{eq:signal_power}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
The total energy in a received JT65 message consisting of
|
|
\begin_inset Formula $n=63$
|
|
\end_inset
|
|
|
|
channel symbols is
|
|
\begin_inset Formula $63E_{s}$
|
|
\end_inset
|
|
|
|
.
|
|
The energy collected for each of the 72 bits of information conveyed by
|
|
the message is then
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
E_{b}=\frac{63E_{s}}{72}=0.875E_{s.}\label{eq:Eb_Es}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
Using equations (
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:SNR"
|
|
|
|
\end_inset
|
|
|
|
)-(
|
|
\begin_inset CommandInset ref
|
|
LatexCommand ref
|
|
reference "eq:Eb_Es"
|
|
|
|
\end_inset
|
|
|
|
),
|
|
\begin_inset Formula $\mathrm{SNR}_{2500}$
|
|
\end_inset
|
|
|
|
can be written in terms of
|
|
\begin_inset Formula $E_{b}/N_{o}$
|
|
\end_inset
|
|
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|
:
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\mathrm{SNR}_{2500}=1.23\times10^{-3}\frac{E_{b}}{N_{0}}.\label{eq:SNR2500}
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|
\end{equation}
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|
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|
\end_inset
|
|
|
|
If all quantities are expressed in dB, then:
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\mathrm{SNR}_{2500}=(E_{b}/N_{0})_{\mathrm{dB}}-29.1\,\mathrm{dB}=(E_{s}/N_{0})_{\mathrm{dB}}-29.7\,\mathrm{dB}.\label{eq:SNR_all_types}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newpage
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Box Doublebox
|
|
position "t"
|
|
hor_pos "c"
|
|
has_inner_box 1
|
|
inner_pos "t"
|
|
use_parbox 0
|
|
use_makebox 0
|
|
width "100col%"
|
|
special "none"
|
|
height "1in"
|
|
height_special "totalheight"
|
|
status open
|
|
|
|
\begin_layout Paragraph
|
|
|
|
\size large
|
|
Sidebar: JT65 Message Processing
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
User A enters or selects message consistent with formatting rules of JT65.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Transmitting software at A: compress message into 12 six-bit symbols, then
|
|
add 51 six-bit parity symbols.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Intersperse 63 synchronizing symbols among the 63 information-carrying symbols.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Start transmission 1 s into a UTC minute.
|
|
Transmit each symbol value at a distinct frequency.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Signal propagates from A to B, arriving much weaker and corrupted by noise,
|
|
fading, and Doppler spread.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Receiving software at B: remove impulsive noise; detect synchronizing signal,
|
|
measure its frequency and time offset.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Shift spectrum to put sync tone at zero frequency, correcting for any measured
|
|
drift.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Compute binned power spectra
|
|
\begin_inset Formula $S(i,j)$
|
|
\end_inset
|
|
|
|
for all information symbols.
|
|
(Index
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
runs over 64 possible symbol values, index
|
|
\begin_inset Formula $j$
|
|
\end_inset
|
|
|
|
over 63 symbol numbers.)
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Remove any possible spurs (signal appearing at same
|
|
\begin_inset Formula $i$
|
|
\end_inset
|
|
|
|
for all
|
|
\begin_inset Formula $j$
|
|
\end_inset
|
|
|
|
).
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Apply Algorithm 1, the FT algorithm.
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Optional: if FT decoding was unsuccessful apply Algorithm 2, hinted decoding.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Display decoded message for User B.
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\begin_inset ERT
|
|
status open
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
|
\backslash
|
|
newpage
|
|
\end_layout
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
\begin_inset CommandInset nomencl_print
|
|
LatexCommand printnomenclature
|
|
set_width "auto"
|
|
|
|
\end_inset
|
|
|
|
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|