mirror of
https://github.com/saitohirga/WSJT-X.git
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More refinements to section 7 and figs.
git-svn-id: svn+ssh://svn.code.sf.net/p/wsjt/wsjt/branches/wsjtx@6317 ab8295b8-cf94-4d9e-aec4-7959e3be5d79
This commit is contained in:
Steven Franke
parent
f838755ae6
commit
b12290871d
@ -2,9 +2,9 @@
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# run: gnuplot fig_bodide.gnuplot
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# then: pdflatex fig_bodide.tex
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#
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set term epslatex standalone size 6in,4in
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set term epslatex standalone size 6in,2*6/3in
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set output "fig_bodide.tex"
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set xlabel "$E_s/N_0$ (dB)"
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set xlabel "$E_b/N_0$ (dB)"
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set ylabel "Word Error Rate"
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set style func linespoints
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set key off
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@ -13,8 +13,9 @@ set mxtics 2
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set mytics 10
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set grid
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set logscale y
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plot [3:9] [1e-4:1] "bmdata.dat" using ($1+29.66):(1-$2) with linespoints pt 4 title 'BM', \
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"bmtheory25.dat" using 1:3 with linespoints pt 5 title 'theory25', \
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"bmtheory40.dat" using 1:3 with linespoints pt 5 title 'theory40', \
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"bmtheory43.dat" using 1:3 with linespoints pt 5 title 'theory43', \
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"ftdata-100000.dat" using ($1+29.66):(1-$2) every ::1 with linespoints pt 4 title 'FT'
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plot [3:9] [1e-4:1] "bmdata.dat" using ($1+29.1):(1-$2) with linespoints lt 1 pt 4 title 'BM', \
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"bmtheory25.dat" using ($1-0.6):3 with linespoints lt 2 pt 5 title 'theory25', \
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"ftdata-100000.dat" using ($1+29.1):(1-$2) every ::1 with linespoints lt 1 pt 4 title 'FT', \
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"kvasd-15.dat" using ($1+29.1):(1-$2) every ::1 with linespoints lt 1 pt 4 title 'KV', \
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"bodide.lab" with labels
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@ -2,15 +2,14 @@
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# run: gnuplot fig_ntrials_vs_nhard.gnuplot
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# then: pdflatex fig_ntrials_vs_nhard.tex
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#
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set term epslatex standalone size 12cm,8cm
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set term epslatex standalone size 6in,2*6/3in
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set output "fig_ntrials_vs_nhard.tex"
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set xlabel "Errors in received word ($X$)"
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set ylabel "Number of trials"
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set title "AWGN, $\\frac{E_s}{N_o}=5.7$ dB"
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set title "AWGN, $\\frac{E_b}{N_o}=5.1$ dB"
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set tics in
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set mxtics 5
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set mytics 10
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#set grid
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set logscale y
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plot "stats-100000-24db-3.dat" using 1:4 pt 12 notitle
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#plot "stats-100000-24db-3.dat" using 1:4 pt 13 notitle
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Binary file not shown.
@ -2,7 +2,7 @@
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# run: gnuplot fig_wer2.gnuplot
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# then: pdflatex fig_wer2.tex
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#
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set term epslatex standalone size 5in,5*2/3in
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set term epslatex standalone size 6in,6*2/3in
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set output "fig_wer2.tex"
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set xlabel "SNR in 2500 Hz Bandwidth (dB)"
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set ylabel "Percent Copy"
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@ -17,8 +17,6 @@ plot [-27:-22] [0:110] \
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"ftdata-10000.dat" using 1:(100*$2) with linespoints lt 1 pt 7 title 'FT-10K', \
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"ftdata-1000.dat" using 1:(100*$2) with linespoints lt 1 pt 7 title 'FT-1K', \
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"ftdata-100.dat" using 1:(100*$2) with linespoints lt 1 pt 7 title 'FT-100', \
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"kvasd-8.dat" using 1:(100*$2) with linespoints lt 2 pt 8 title 'KV-8', \
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"kvasd-12.dat" using 1:(100*$2) with linespoints lt 2 pt 8 title 'KV-12', \
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"kvasd-15.dat" using 1:(100*$2) with linespoints lt 2 pt 8 title 'KV-15', \
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"bmdata.dat" using 1:(100*$2) with linespoints pt 11 title 'BM', \
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"wer2.lab" with labels
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@ -2,7 +2,7 @@
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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\textclass paper
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\use_default_options true
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\maintain_unincluded_children false
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\language english
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@ -84,6 +84,15 @@ Steven J.
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Taylor, K1JT
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\end_layout
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\begin_layout Standard
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\begin_inset CommandInset toc
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LatexCommand tableofcontents
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\end_inset
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\end_layout
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\begin_layout Abstract
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The JT65 protocol has revolutionized amateur-radio weak-signal communication
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by enabling amateur radio operators with small antennas and relatively
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@ -333,7 +342,16 @@ Statistical Framework
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The FT algorithm uses the estimated quality of received symbols to generate
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lists of symbols considered likely to be in error, thus enabling decoding
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of received words with more than 25 errors.
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As a specific example, consider a received JT65 word with 23 correct symbols
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\end_layout
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\begin_layout Standard
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(SF: provide brief overview of literature survey and discuss the inspiration
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for the FT approach).
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\end_layout
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\begin_layout Standard
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As a specific example, consider a received JT65 word with 23 correct symbols
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and 40 errors.
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We do not know which symbols are in error.
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Suppose that the decoder randomly selects
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@ -738,7 +756,7 @@ The FT algorithm uses quality indices made available by a noncoherent 64-FSK
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\begin_inset Formula $p_{1}$
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\end_inset
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and
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and
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\begin_inset Formula $p_{2}$
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\end_inset
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@ -1129,7 +1147,7 @@ Hinted Decoding
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\end_layout
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\begin_layout Standard
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Write this...
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To be written...
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\end_layout
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\begin_layout Section
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@ -1143,7 +1161,7 @@ Implementation in WSJT-X
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\end_layout
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\begin_layout Standard
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Write this...
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To be written...
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\end_layout
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\begin_layout Section
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@ -1153,75 +1171,84 @@ name "sec:Theory,-Simulation,-and"
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\end_inset
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Decoder Performance Evaluation: Simulations and Real Data
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Decoder Performance Evaluation
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\end_layout
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\begin_layout Subsection
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Simulated results on the AWGN channel
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\end_layout
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\begin_layout Standard
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The fraction of time that
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\begin_inset Formula $X$
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Comparisons of decoding performance are usually presented in the professional
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literature as plots of word error-rate versus
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\begin_inset Formula $E_{b}/N_{0}$
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\end_inset
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, the number of symbols received incorrectly, is expected to be less than
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some number
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\begin_inset Formula $D$
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, the ratio of the energy collected per information bit to the one-sided
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noise power spectral density,
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\begin_inset Formula $N_{0}$
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\end_inset
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depends on signal-to-noise ratio.
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For the case of additive white Gaussian noise (AWGN) and noncoherent 64-FSK
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demodulation this probability is easy to calculate.
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Representative examples for
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\begin_inset Formula $D=25,$
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.
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In amateur radio circles performance is usually plotted as the probability
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of successfully decoding a received word vs signal-to-noise ratio in a
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2.5 kHz reference bandwidth,
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\begin_inset Formula $\mathrm{SNR}{}_{2.5\,\mathrm{kHz}}$
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\end_inset
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\strikeout off
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\uuline off
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\uwave off
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\noun off
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\color none
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\begin_inset Formula $D=40$
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.
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The relationship between
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\begin_inset Formula $E_{b}/N_{o}$
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\end_inset
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\family default
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\series default
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\shape default
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\size default
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\emph default
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\bar default
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\strikeout default
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\uuline default
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\uwave default
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\noun default
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\color inherit
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, and
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\begin_inset Formula $D=43$
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and
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\begin_inset Formula $\mathrm{SNR}{}_{2.5\,\mathrm{kHz}}$
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\end_inset
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are plotted in Figure
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is described in Appendix
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "sec:Appendix:SNR"
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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Results of simulations using the BM, FT, and KV decoding algorithms on the
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JT65 (63,12) code are presented in terms of word error-rate vs
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\begin_inset Formula $E_{b}/N_{o}$
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\end_inset
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in Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:bodide"
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\end_inset
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for a range of SNRs as filled squares with connecting lines.
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The rightmost such curve shows that on the AWGN channel the hard-decision
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BM decoder should succeed about 90% of the time at
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\begin_inset Formula $E_{s}/N_{0}=7.5$
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.
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For these tests we generated at least 1000 signals at each signal-to-noise
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ratio, assuming the additive white gaussian noise (AWGN) channel, and processed
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the data using each algorithm.
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For word error-rates less than 0.1 it was necessary to process 10,000 or
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even 100,000 simulated signals in order to capture enough errors to make
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the estimates of word-error-rate statistically meaningful.
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As a test of the fidelity of our numerical simulations, Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:bodide"
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\end_inset
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dB, 99% of the time at 8 dB, and 99.98% at 8.5 dB.
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For comparison, the righmost curve with open squares shows that simulated
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results agree with theory to within less than 0.2 dB.
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also shows theoretical results (filled squares) for comparison with the
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BM results.
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The simulated BM results agree with theory to within about 0.1 dB.
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As expected, the soft-decision algorithms FT and KV are about 2 dB better
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than the hard-decision BM algorithm.
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In addition, FT has a slight edge (about 0.2 dB) over KV.
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\end_layout
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\begin_layout Standard
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@ -1248,56 +1275,33 @@ name "fig:bodide"
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\end_inset
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Word error rates as a function of
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\begin_inset Formula $E_{s}/N_{0},$
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\begin_inset Formula $E_{b}/N_{0},$
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\end_inset
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the signal-to-noise ratio in bandwidth equal to the symbol rate.
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Filled squares illustrate theoretical values for
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\begin_inset Formula $D=25,$
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\end_inset
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\strikeout off
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\uuline off
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\uwave off
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\noun off
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\color none
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\begin_inset Formula $D=40$
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\end_inset
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\family default
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\series default
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\shape default
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\size default
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\emph default
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\bar default
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\strikeout default
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\uuline default
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\uwave default
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\noun default
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\color inherit
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, and
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\begin_inset Formula $D=43$
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\end_inset
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.
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Open squares illustrate measured results for the BM and FT (
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the signal-to-noise ratio per bit.
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The single curve marked with filled squares shows a theoretical prediction
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for the BM decoder.
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Open squares illustrate simulation results for an AWGN channel with the
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BM, FT (
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\begin_inset Formula $T=10^{5}$
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\end_inset
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) decoders in program
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) and KV (
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\begin_inset Formula $\lambda=15$
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\end_inset
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) decoders used in program
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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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The KV results are for decoding complexity coefficient
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\begin_inset Formula $\lambda=15$
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\end_inset
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, the most aggressive setting that has historically been used in earlier
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versions of the WSJT programs.
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\end_layout
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\end_inset
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@ -1311,38 +1315,170 @@ WSJT-X
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\end_layout
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\begin_layout Standard
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Received JT65 words with more than 25 incorrect symbols can be decoded if
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sufficient information on individual symbol reliabilities is available.
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Using values of
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Because of the importance of error-free transmission in commercial applications,
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plots like that in Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:bodide"
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\end_inset
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often extend downward to much smaller error rates, say
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\begin_inset Formula $10^{-6}$
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\end_inset
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or less.
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The circumstances for minimal amateur-radio QSOs are very different, however.
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Error rates of order 0.1 or higher may be acceptable.
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In this case the essential information is better presented in a plot showing
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the percentage of transmissions copied correctly as a function of signal-to-noi
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se ratio.
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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:WER2"
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\end_inset
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shows the FT results for
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\begin_inset Formula $T=10^{5}$
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\end_inset
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and the KV results that were shown in Figure
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "fig:bodide"
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\end_inset
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in this format along with additional FT results for
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\begin_inset Formula $T=10^{4},10^{3},$
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\end_inset
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and
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\begin_inset Formula $10^{2}$
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\end_inset
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.
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The KV results are plotted with open triangles.
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It is apparent that the FT algorithm produces more decodes than KV when
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\begin_inset Formula $T=10^{4}$
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\end_inset
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or larger.
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\end_layout
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\begin_layout Standard
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\begin_inset Float figure
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wide false
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sideways false
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status open
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\begin_layout Plain Layout
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\align center
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\begin_inset Graphics
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filename fig_wer2.pdf
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lyxscale 120
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Caption Standard
|
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:WER2"
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\end_inset
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Percent of JT65 messages copied as a function of SNR in 2.5 kHz bandwidth.
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Solid lines with filled round circles are results from the FT decoder with
|
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\begin_inset Formula $T=10^{5},10^{4},10^{3},$
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\end_inset
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and
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\begin_inset Formula $10^{2}$
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\end_inset
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, respectively, from left to right.
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The dashed line with open triangles is the KV decoder with complexity coefficie
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nt
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\begin_inset Formula $\lambda=15$
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\end_inset
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|
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.
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Results from the BM algorithm are also shown with filled triangles.
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\end_layout
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||||
|
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\end_inset
|
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|
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|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
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The timeout parameter
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||||
\begin_inset Formula $T$
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\end_inset
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||||
|
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that are practical with today's personal computers and the soft-symbol
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information described above, we find that the FT algorithm nearly always
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||||
produces correct decodes up to
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\begin_inset Formula $X=40$
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\end_inset
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||||
|
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, and some additional decodes are found in the range 41 to 43.
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As an example, Figure
|
||||
employed in the FT algorithm is the maximum number of symbol-erasure trials
|
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allowed for a particular attempt at decoding a received word.
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Most successful decodes take only a small fraction of the maximum allowed
|
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number of trials.
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Figure
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\begin_inset CommandInset ref
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LatexCommand ref
|
||||
reference "fig:N_vs_X"
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\end_inset
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|
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plots the number of stochastic erasure trials required to find the correct
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codeword versus the number of hard-decision errors for a run with 1000
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simulated transmissions at
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\begin_inset Formula $SNR=-24$
|
||||
shows the number of stochastic erasure trials required to find the correct
|
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codeword versus the number of hard-decision errors in the received word
|
||||
for a run with 1000 simulated transmissions at
|
||||
\begin_inset Formula $\mathrm{SNR}=-24$
|
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\end_inset
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|
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dB, just slightly above the decoding threshold.
|
||||
Note that both mean and variance of the required number of trials increase
|
||||
steeply with the number of errors in the received word.
|
||||
Execution time of the FT algorithm is roughly proportional to the number
|
||||
of required trials.
|
||||
The timeout parameter was
|
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\begin_inset Formula $T=10^{5}$
|
||||
\end_inset
|
||||
|
||||
for this run.
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No points are shown for
|
||||
\begin_inset Formula $X\le25$
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||||
\end_inset
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||||
|
||||
because all such words were successfully decoded by the BM algorithm.
|
||||
Figure
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "fig:N_vs_X"
|
||||
|
||||
\end_inset
|
||||
|
||||
shows that the FT algorithm decoded received words with as many as
|
||||
\begin_inset Formula $X=43$
|
||||
\end_inset
|
||||
|
||||
symbol errors.
|
||||
The results also show that, on average, the number of trials increases
|
||||
with the number of errors in the received word.
|
||||
The variability of the decoding time also increases dramatically with the
|
||||
number of errors in the received word.
|
||||
These results also provide insight into the mean and variance of the execution
|
||||
time for the FT algorithm, as execution time will be roughly proportional
|
||||
to the number of required trials.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
@ -1356,7 +1492,6 @@ status open
|
||||
\begin_inset Graphics
|
||||
filename fig_ntrials_vs_nhard.pdf
|
||||
lyxscale 120
|
||||
scale 120
|
||||
|
||||
\end_inset
|
||||
|
||||
@ -1377,7 +1512,7 @@ Number of trials needed to decode a received word versus Hamming distance
|
||||
between the received word and the decoded codeword, for 1000 simulated
|
||||
frames on an AWGN channel with no fading.
|
||||
The SNR in 2500 Hz bandwidth is -24 dB (
|
||||
\begin_inset Formula $E_{s}/N_{o}=5.7$
|
||||
\begin_inset Formula $E_{b}/N_{o}=5.1$
|
||||
\end_inset
|
||||
|
||||
dB).
|
||||
@ -1395,101 +1530,7 @@ Number of trials needed to decode a received word versus Hamming distance
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Comparison with Berlekamp-Massey and Koetter-Vardy
|
||||
\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 signal-to-noise ratio per information bit.
|
||||
Results of simulations using the Berlekamp-Massey, Koetter-Vardy, and Franke-Ta
|
||||
ylor decoding algorithms on the (63,12) code are presented in this way in
|
||||
Figure
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "fig:WER"
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
For these tests we generated 1000 signals at each signal-to-noise ratio,
|
||||
assuming the additive white gaussian noise (AWGN) channel, and processed
|
||||
the data using each algorithm.
|
||||
As expected, the soft-decision algorithms FT and KV are about 2 dB better
|
||||
than the hard-decision BM algorithm.
|
||||
FT has a slight edge (about 0.2 dB) over KV with the default settings for
|
||||
each algorithm, as implemented in our JT65 decoders.
|
||||
\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_wer.pdf
|
||||
lyxscale 120
|
||||
scale 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:WER"
|
||||
|
||||
\end_inset
|
||||
|
||||
Word error rate (WER) as a function of
|
||||
\begin_inset Formula $E_{b}/N_{0}$
|
||||
\end_inset
|
||||
|
||||
for non-fading signals in AWGN.
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
Because of the importance of error-free transmission in commercial applications,
|
||||
plots like that in Figure
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand ref
|
||||
reference "fig:WER"
|
||||
|
||||
\end_inset
|
||||
|
||||
often extend downward to much smaller error rates, say
|
||||
\begin_inset Formula $10^{-6}$
|
||||
\end_inset
|
||||
|
||||
or less, .
|
||||
The circumstances for minimal amateur-radio QSOs are very different, however.
|
||||
Error rates of order 0.1, or ever higher, may be acceptable.
|
||||
In this case the essential information is better presented in a plot showing
|
||||
the percentage of transmissions copied correctly as a function of signal-to-noi
|
||||
se ratio.
|
||||
|
||||
Simulated results for hinted decoding and Rayleigh fading
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
@ -1529,7 +1570,6 @@ status open
|
||||
\begin_inset Graphics
|
||||
filename fig_psuccess.pdf
|
||||
lyxscale 90
|
||||
scale 90
|
||||
|
||||
\end_inset
|
||||
|
||||
@ -1665,7 +1705,13 @@ Berlekamp-Massey decoder written by Phil Karn, http://www.ka9q.net/code/fec/
|
||||
|
||||
\begin_layout Section
|
||||
\start_of_appendix
|
||||
Signal to Noise Ratios
|
||||
\begin_inset CommandInset label
|
||||
LatexCommand label
|
||||
name "sec:Appendix:SNR"
|
||||
|
||||
\end_inset
|
||||
|
||||
Appendix: Signal to Noise Ratios
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
@ -1676,7 +1722,7 @@ The signal to noise ratio in a bandwidth,
|
||||
, that is at least as large as the bandwidth occupied by the signal is:
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
SNR_{B}=\frac{P_{s}}{N_{o}B}\label{eq:SNR}
|
||||
\mathrm{SNR}_{B}=\frac{P_{s}}{N_{o}B}\label{eq:SNR}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
@ -1685,18 +1731,18 @@ where
|
||||
\begin_inset Formula $P_{s}$
|
||||
\end_inset
|
||||
|
||||
is the signal power,
|
||||
is the signal power (W),
|
||||
\begin_inset Formula $N_{o}$
|
||||
\end_inset
|
||||
|
||||
is one-sided noise power spectral density, and
|
||||
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 $SNR_{2.5\,\mathrm{kHz}}$
|
||||
\begin_inset Formula $\mathrm{SNR}_{2.5\,\mathrm{kHz}}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
@ -1717,32 +1763,20 @@ In the professional literature, decoder performance is characterized in
|
||||
\begin_inset Formula $N_{o}$
|
||||
\end_inset
|
||||
|
||||
, or in terms of
|
||||
\begin_inset Formula $E_{s}/N_{o}$
|
||||
\end_inset
|
||||
|
||||
, the ratio of the energy collected per received symbol,
|
||||
\begin_inset Formula $E_{s}$
|
||||
\end_inset
|
||||
|
||||
, and
|
||||
\begin_inset Formula $N_{o}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Denote the duration of a channel symbol by
|
||||
\begin_inset Formula $\tau_{s}$
|
||||
\end_inset
|
||||
|
||||
(for JT65,
|
||||
\begin_inset Formula $\tau_{s}=0.375\,\mathrm{s}$
|
||||
\begin_inset Formula $\tau_{s}=0.3715\,\mathrm{s}$
|
||||
\end_inset
|
||||
|
||||
).
|
||||
Signal power is related to the energy per symbol by
|
||||
\begin_inset Formula
|
||||
\begin{equation}
|
||||
P_{s}=E_{s}/\tau_{s},\label{eq:signal_power}
|
||||
P_{s}=E_{s}/\tau_{s}.\label{eq:signal_power}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
@ -1780,36 +1814,28 @@ reference "eq:Eb_Es"
|
||||
\end_inset
|
||||
|
||||
),
|
||||
\begin_inset Formula $SNR_{2.5\,\mathrm{kHz}}$
|
||||
\begin_inset Formula $\mathrm{SNR}_{2.5\,\mathrm{kHz}}$
|
||||
\end_inset
|
||||
|
||||
can be written in terms of
|
||||
\begin_inset Formula $E_{b}/N_{o}$
|
||||
\end_inset
|
||||
|
||||
or
|
||||
\begin_inset Formula $E_{s}/N_{o}$
|
||||
\end_inset
|
||||
|
||||
:
|
||||
\begin_inset Formula
|
||||
\[
|
||||
SNR_{2.5\,\mathrm{kHz}}=1.08\times10^{-3}\frac{E_{s}}{N_{o}}=1.23\times10^{-3}\frac{E_{b}}{N_{o}}.
|
||||
\mathrm{SNR}_{2.5\,\mathrm{kHz}}=1.23\times10^{-3}\frac{E_{b}}{N_{o}}.
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
If all quantities are expressed in dB, then:
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\[
|
||||
SNR_{2.5\,\mathrm{kHz}}=(E_{b}/N_{o})_{\mathrm{dB}}-29.66\,\mathrm{dB}=(E_{s}/N_{o})_{\mathrm{dB}}-29.10\,\mathrm{dB}.
|
||||
SNR_{2.5\,\mathrm{kHz}}=(E_{b}/N_{o})_{\mathrm{dB}}-29.1\,\mathrm{dB}.
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
Loading…
Reference in New Issue
Block a user