The RealCepstrum(A) command computes the real logarithm of the magnitude of the Fourier transform of a sequence.

A must be a one-dimensional Array and must contain real numbers only.

$\mathrm{with}\left(\mathrm{SignalProcessing}\right)\:$

$\mathrm{Fs}≔100\:$

$\mathrm{tStart}≔0\:$

$\mathrm{tEnd}≔15\:$

$\mathrm{tEcho}≔4.5\:$

$N≔\left(\mathrm{tEnd}-\mathrm{tStart}\right)\mathrm{Fs}+1\:$

$t≔\mathrm{Vector}\left(\left[\mathrm{seq}\left(i\,i=\mathrm{tStart}..\mathrm{tEnd}\,\frac{1.0}{\mathrm{Fs}}\right)\right]\right)\:$

$\mathrm{cleanSignal}≔\mathrm{`~`}\left[\mathrm{`/`}\right]\left(\mathrm{`~`}\left[\sin \right]\left(2\mathrm{\pi }\cdot 1.3t\right)\,\mathrm{`\; \$`}\,\mathrm{`~`}\left[\mathrm{`+`}\right]\left(t\,\mathrm{`\; \$`}\,10\right)\right)+\mathrm{`~`}\left[\mathrm{`/`}\right]\left(\mathrm{`~`}\left[\cos \right]\left(2\mathrm{\pi }\cdot 2.4t\right)\,\mathrm{`\; \$`}\,\mathrm{`~`}\left[\mathrm{`+`}\right]\left(t\,\mathrm{`\; \$`}\,10\right)\right)\:$

$\mathrm{cleanSignalPlot}≔\mathrm{plot}\left(t\,\mathrm{cleanSignal}\,\mathrm{legend}=''Clean\; Signal''\right)$

$\mathrm{echoSignal}≔\mathrm{Vector}\left(\left[\mathrm{`\$`}\left(0\,\mathrm{floor}\left(\mathrm{Fs}\mathrm{tEcho}+1\right)\right)\,\mathrm{cleanSignal}\left[1..N-\mathrm{floor}\left(\mathrm{Fs}\mathrm{tEcho}+1\right)\right]\right]\right)\:$

$\mathrm{dirtySignal}≔\mathrm{cleanSignal}+0.3\mathrm{echoSignal}\:$

$\mathrm{dirtySignalPlot}≔\mathrm{plot}\left(t\,\mathrm{dirtySignal}\,\mathrm{color}=\mathrm{blue}\,\mathrm{legend}=''Signal\; with\; Echo''\right)$

$r≔\mathrm{RealCepstrum}\left(\mathrm{dirtySignal}\right)\:$

$\mathrm{plot}\left(t\,r\,\mathrm{view}=\left[\mathrm{tStart}..\mathrm{tEnd}\,-1.5..1\right]\right)$

$\mathrm{threshold}≔\mathrm{map}\left(x\mapsto \mathrm{piecewise}\left(x<0.2\&comma;0\&comma;1\right)\&comma;r\right)\&colon;$

$\mathrm{ind}≔\mathrm{ArrayTools}:-\mathrm{SearchArray}\left(\mathrm{threshold}\&comma;5\right)\&colon;$

$t\left[452\right]$

$4.510000000$

$\mathrm{filteredSignal}≔\mathrm{SignalProcessing}:-\mathrm{Filter}\left(\mathrm{dirtySignal}\&comma;\mathrm{Array}\left(\left[1\&comma;\mathrm{`\$`}\left(0\&comma;450\right)\&comma;0.35\right]\right)\&comma;\mathrm{Array}\left(\left[1\right]\right)\right)\&colon;$

$\mathrm{filteredSignalPlot}≔\mathrm{plot}\left(t\&comma;\mathrm{filteredSignal}\&comma;\mathrm{color}=\mathrm{black}\&comma;\mathrm{legend}=''Filtered\; Signal''\right)$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{dirtySignalPlot}\&comma;\mathrm{filteredSignalPlot}\right)$

$\mathrm{plots}:-\mathrm{display}\left(\mathrm{cleanSignalPlot}\&comma;\mathrm{filteredSignalPlot}\right)$

The SignalProcessing[RealCepstrum] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.