I ran across a problem in a DSP book that got me to thinking (the best kind= of problem). The author suggests that we can illustrate aliasing by sampl= ing an audio signal at a high frequency and then replacing samples with zer= oes to get a lower effective sampling frequency. For example, if we sample= at fs1 =3D 32 kHz and then keep every M =3D 4th sample (replacing the othe= r samples with zeroes) then the resulting signal is equivalent to sampling = at fs2 =3D 8 kHz (fs1/M). I agree that aliasing is present (if the original signal has content above = 4 kHz), but if you play the new signal back at 32 kHz you will also hear "r= eplica" distortion. To get a better sense of just aliasing due to sampling= at 8 kHz you should low pass filter the new signal with a cutoff frequency= of 4 kHz (fs2/2 or fs1/(2M)). The process appears to be the same as down-sampling (without an anti-aliasi= ng filter, since the goal is to demonstrate aliasing) followed by up-sampli= ng at the same rate. I think you still need to follow the up-sampler with = a low-pass (replica elimination) filter to generate a signal that when play= ed back at 32 kHz sounds the same as sampling (and replaying) the original = analog signal at 8 kHz (without using an anti-aliasing filter). Note: I am not arguing that the author is wrong. I am just looking for ver= ification that my thinking is correct (or not). The problem is presented ve= ry early in the book (and it is a very good book) before filtering is even = discussed. He says that this zero-replacement method will illustrate disto= rtion due to aliasing. He does not say that it is exactly equivalent to th= e signal that would be obtained by sampling at the lower rate without an an= ti-aliasing filter. It just got me to thinking about a better way (perhaps= ) to demonstrate only aliasing. Tony

# Simulation of Aliasing

Started by ●July 6, 2020

Reply by ●July 6, 20202020-07-06

On Monday, July 6, 2020 at 8:04:23 AM UTC-7, richard...@gmail.com wrote:> I ran across a problem in a DSP book that got me to thinking > (the best kind of problem). > The author suggests that we can illustrate aliasing by sampling an > audio signal at a high frequency and then replacing samples with zeroes > to get a lower effective sampling frequency. For example, if we > sample at fs1 = 32 kHz and then keep every M = 4th sample > (replacing the other samples with zeroes) then the resulting signal > is equivalent to sampling at fs2 = 8 kHz (fs1/M).> I agree that aliasing is present (if the original signal has content > above 4 kHz), but if you play the new signal back at 32 kHz you will > also hear "replica" distortion. To get a better sense of just aliasing > due to sampling at 8 kHz you should low pass filter the new signal > with a cutoff frequency of 4 kHz (fs2/2 or fs1/(2M)).Without getting too much detail, it does seem like two different things. I didn't know the name "replica distortion" before. Another choice is to generate four copies of every fourth sample. I think you should try them all and see what they sound like.

Reply by ●July 6, 20202020-07-06

On Monday, July 6, 2020 at 12:30:41 PM UTC-5, ga...@u.washington.edu wrote:> On Monday, July 6, 2020 at 8:04:23 AM UTC-7, richard...@gmail.com wrote: > > I ran across a problem in a DSP book that got me to thinking > > (the best kind of problem).=20 > > The author suggests that we can illustrate aliasing by sampling an > > audio signal at a high frequency and then replacing samples with zeroes > > to get a lower effective sampling frequency. For example, if we > > sample at fs1 =3D 32 kHz and then keep every M =3D 4th sample > > (replacing the other samples with zeroes) then the resulting signal > > is equivalent to sampling at fs2 =3D 8 kHz (fs1/M). >=20 > > I agree that aliasing is present (if the original signal has content > > above 4 kHz), but if you play the new signal back at 32 kHz you will > > also hear "replica" distortion. To get a better sense of just aliasing > > due to sampling at 8 kHz you should low pass filter the new signal > > with a cutoff frequency of 4 kHz (fs2/2 or fs1/(2M)). >=20 > Without getting too much detail, it does seem like two different things. >=20 > I didn't know the name "replica distortion" before. >=20 > Another choice is to generate four copies of every fourth sample. >=20 > I think you should try them all and see what they sound like.Oh, I have tried it. With a 400 Hz tone and original sampling at 32 kHz an= d then reducing the sample rate to 2 kHz you can hear the 1600 Hz replica. = There are spectral lines at (n*2000 +- 400). If you reduce the sample rate= to 500 Hz you can hear an alias at 100 Hz. The spectral lines are at (n*50= 0 +- 400). I am playing around with a replica elimination filter now so tha= t I only hear the distortion due to aliasing. The goal is to produce a sign= al that would sound the same as an undersampled signal.

Reply by ●July 6, 20202020-07-06

On Monday, July 6, 2020 at 1:03:21 PM UTC-5, richard...@gmail.com wrote:> On Monday, July 6, 2020 at 12:30:41 PM UTC-5, ga...@u.washington.edu wrote: > > On Monday, July 6, 2020 at 8:04:23 AM UTC-7, richard...@gmail.com wrote: > > > I ran across a problem in a DSP book that got me to thinking > > > (the best kind of problem). > > > The author suggests that we can illustrate aliasing by sampling an > > > audio signal at a high frequency and then replacing samples with zeroes > > > to get a lower effective sampling frequency. For example, if we > > > sample at fs1 = 32 kHz and then keep every M = 4th sample > > > (replacing the other samples with zeroes) then the resulting signal > > > is equivalent to sampling at fs2 = 8 kHz (fs1/M). > > > > > I agree that aliasing is present (if the original signal has content > > > above 4 kHz), but if you play the new signal back at 32 kHz you will > > > also hear "replica" distortion. To get a better sense of just aliasing > > > due to sampling at 8 kHz you should low pass filter the new signal > > > with a cutoff frequency of 4 kHz (fs2/2 or fs1/(2M)). > > > > Without getting too much detail, it does seem like two different things. > > > > I didn't know the name "replica distortion" before. > > > > Another choice is to generate four copies of every fourth sample. > > > > I think you should try them all and see what they sound like. > > Oh, I have tried it. With a 400 Hz tone and original sampling at 32 kHz > and then reducing the sample rate to 2 kHz you can hear the 1600 Hz > replica. There are spectral lines at (n*2000 +- 400). If you reduce the > sample rate to 500 Hz you can hear an alias at 100 Hz. The spectral lines > are at (n*500 +- 400). I am playing around with a replica elimination > filter now so that I only hear the distortion due to aliasing. The goal is > to produce a signal that would sound the same as an undersampled signal.The Octave/MATLAB code for this is below: %%%%%%%%%%%%%%%%%%%%%%%%%%%% % This procedure is equivalent to down-sampling (without a LPF anti-aliasing % filter) followed by up-sampling (without a LPF anti-replica) filter. % % To truly hear only aliasing a LPF anti-replica filter should be used. %%%%%%%%%%%%%%%%%%%%%%%%%%%% f0 = 400; fs = 32000; % sample rate t = 2; % seconds xf = @(f0, fs, n) 0.75*sin(2*pi*f0*n/fs); n = (0:t*fs-1); %%%%%%%%%%%%%%%%%%%%%%%%%%%% x = xf(f0, fs, n); X = fft(x); clf; subplot(3,2,1) plot((0:length(X)-1)*fs/length(X), 20*log10(abs(X))) xlim([0 5000]) drawnow(); player = audioplayer(x, fs); playblocking(player) %%%%%%%%%%%%%%%%%%%%%%%%%%%% fsn = 2000; r = fs/fsn; % 16 xn = zeros(size(x)); xn(1:r:end) = x(1:r:end); X = fft(xn); subplot(3,2,3) plot((0:length(X)-1)*fs/length(X), 20*log10(abs(X))) xlim([0 5000]) drawnow(); player = audioplayer(xn, fs); playblocking(player) %%%%%%%%%%%%%%%%%%%%%%%%%%%% fsn = 500; r = fs/fsn; % 64 xn = zeros(size(x)); xn(1:r:end) = x(1:r:end); X = fft(xn); subplot(3,2,5) plot((0:length(X)-1)*fs/length(X), 20*log10(abs(X))) xlim([0 5000]) drawnow(); player = audioplayer(xn, fs); playblocking(player)

Reply by ●July 6, 20202020-07-06

On Monday, July 6, 2020 at 12:30:41 PM UTC-5, ga...@u.washington.edu wrote:> On Monday, July 6, 2020 at 8:04:23 AM UTC-7, richard...@gmail.com wrote: > > I ran across a problem in a DSP book that got me to thinking > > (the best kind of problem).=20 > > The author suggests that we can illustrate aliasing by sampling an > > audio signal at a high frequency and then replacing samples with zeroes > > to get a lower effective sampling frequency. For example, if we > > sample at fs1 =3D 32 kHz and then keep every M =3D 4th sample > > (replacing the other samples with zeroes) then the resulting signal > > is equivalent to sampling at fs2 =3D 8 kHz (fs1/M). >=20 > > I agree that aliasing is present (if the original signal has content > > above 4 kHz), but if you play the new signal back at 32 kHz you will > > also hear "replica" distortion. To get a better sense of just aliasing > > due to sampling at 8 kHz you should low pass filter the new signal > > with a cutoff frequency of 4 kHz (fs2/2 or fs1/(2M)). >=20 > Without getting too much detail, it does seem like two different things. >=20 > I didn't know the name "replica distortion" before. >=20 > Another choice is to generate four copies of every fourth sample. >=20 > I think you should try them all and see what they sound like.Generating four copies of every fourth sample would be equivalent to convol= ving the zeroed signal with a 4-sample rect pulse. in the frequency domain= this would be equivalent to multiplying by a sinc whose first null is at t= he location of the first replica. I don't see this is being as effective i= n achieving my goal as a LP replica elimination filter. (The goal is to pr= oduce a signal which has the same spectrum as undersampling the original an= alog signal.)