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# Discussion Groups | Comp.DSP | Sampling Complex Signal

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# Sampling Complex Signal - Walter Wego - 2012-08-05 09:38:00

```Hi,

I know you have to sample above Nyquist or you risk aliases folding into
the passband.

But it if your signal is complex, it seems to me that in a given interval
of time youâve collected twice the information you would have for a real
signal, or that youâve effectively doubled your sample rate.

I got that same message when looking through old posts on DSP Related.

I thought that implied that for a basebanded complex signal (for example)
the highest frequency you could correctly reconstruct would be twice that
for a real signal.

But Iâm not finding that to be the case in practice, as the Matlab
snippet below shows â the complex signal folds into the passband just as
the real signal does.

While I still have some hair left to tear out, can someone help me
understand where Iâm going astray, and why the part 2 of code below gives
the âwrongâ result?

Thanks, itâs much appreciated.

Walt

clear all

Fs = 2048;
t  = (0:1/Fs:4-1/Fs);
F1 = 1024+256;
LO = 32;

nyquistfreq = Fs/2
aliasfreq   = Fs-F1

% ----- Part 1 -----

% Create a real, undersampled signal that will fold into passband
x = cos(2*pi*F1*t);

% Calculate the spectrum
NFFT = 1024;
z    = fft(x, NFFT);
Pzz  = abs(z);

% Plot the spectrum
Dec = 1;
f   = ((Fs/2) / Dec) * linspace(0, 1, NFFT/2+1);
figure(100);
plot(f, 2*Pzz(1:1+NFFT/2)/NFFT);
axis([f(1), f(end), 0, 1.01*max(2*Pzz(1:1+NFFT/2)/NFFT)]);
title('Spectrum of Undersampled Real Signal');

% ----- Part 2 -----

% Translate spectrum of original signal (becomes complex in time)
xs = x .* exp(-i*2*pi*LO*t);

% Calculate the spectrum
NFFT = 1024;
z = fft(xs, NFFT);
Pzz = abs(z);

% Plot the spectrum
Dec = 1;
f = ((Fs/2) / Dec) * linspace(0, 1, NFFT/2+1) + LO;
figure(200);
plot(f, 2*Pzz(1:1+NFFT/2)/NFFT);
axis([f(1), f(end), 0, 1.01*max(2*Pzz(1:1+NFFT/2)/NFFT)]);
title('Spectrum of Complex Signal');

```
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# Re: Sampling Complex Signal - kaz - 2012-08-05 12:00:00

```>Hi,
>
>I know you have to sample above Nyquist or you risk aliases folding into
>the passband.
>
>But it if your signal is complex, it seems to me that in a given interval
>of time youâve collected twice the information you would have for a
real
>signal, or that youâve effectively doubled your sample rate.
>
>I got that same message when looking through old posts on DSP Related.
>
>I thought that implied that for a basebanded complex signal (for example)
>the highest frequency you could correctly reconstruct would be twice that
>for a real signal.
>
>But Iâm not finding that to be the case in practice, as the Matlab
>snippet below shows â the complex signal folds into the passband just
as
>the real signal does.
>
>While I still have some hair left to tear out, can someone help me
>understand where Iâm going astray, and why the part 2 of code below
gives
>the âwrongâ result?
>
>Thanks, itâs much appreciated.
>
>Walt
>
>clear all
>
>Fs = 2048;
>t  = (0:1/Fs:4-1/Fs);
>F1 = 1024+256;
>LO = 32;
>
>nyquistfreq = Fs/2
>aliasfreq   = Fs-F1
>
>% ----- Part 1 -----
>
>% Create a real, undersampled signal that will fold into passband
>x = cos(2*pi*F1*t);
>
>% Calculate the spectrum
>NFFT = 1024;
>z    = fft(x, NFFT);
>Pzz  = abs(z);
>
>% Plot the spectrum
>Dec = 1;
>f   = ((Fs/2) / Dec) * linspace(0, 1, NFFT/2+1);
>figure(100);
>plot(f, 2*Pzz(1:1+NFFT/2)/NFFT);
>axis([f(1), f(end), 0, 1.01*max(2*Pzz(1:1+NFFT/2)/NFFT)]);
>title('Spectrum of Undersampled Real Signal');
>
>% ----- Part 2 -----
>
>% Translate spectrum of original signal (becomes complex in time)
>xs = x .* exp(-i*2*pi*LO*t);
>
>% Calculate the spectrum
>NFFT = 1024;
>z = fft(xs, NFFT);
>Pzz = abs(z);
>
>% Plot the spectrum
>Dec = 1;
>f = ((Fs/2) / Dec) * linspace(0, 1, NFFT/2+1) + LO;
>figure(200);
>plot(f, 2*Pzz(1:1+NFFT/2)/NFFT);
>axis([f(1), f(end), 0, 1.01*max(2*Pzz(1:1+NFFT/2)/NFFT)]);
>title('Spectrum of Complex Signal');
>
>
>

But sampling only applies to real signal (call it I or Q, Re or Im). It
remains that the ADC must sample correctly each channel separately.

A complex signal cannot exist in real design, instead we have two channels
and process them as if they are members of a complex signal.

>
```
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# Re: Sampling Complex Signal - kaz - 2012-08-05 12:03:00

```Remember also a complex approach does not change frequency value of signal
but may eliminate a dc mirror copy

```
______________________________

# Re: Sampling Complex Signal - Robert Adams - 2012-08-05 12:07:00

```I haven't run your code but I think the conceptual problem is that not all complex signals
are analytic signals. An analytic signal has no negative frequency components. When you
multiplied by exp(I ....) you just made a small circular shift in the frequency domain but you
still have negative frequencies. Instead you should construct your complex signal using x +
i*hilbert(x)

Bob
```
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# Re: Sampling Complex Signal - Rick Lyons - 2012-08-05 12:41:00

```On Sun, 05 Aug 2012 08:38:11 -0500, "Walter Wego" <61340@dsprelated>
wrote:

>Hi,
>
>I know you have to sample above Nyquist or you risk aliases folding into
>the passband.
>
>But it if your signal is complex, it seems to me that in a given interval
>of time youâve collected twice the information you would have for a real
>signal, or that youâve effectively doubled your sample rate.
>
>I got that same message when looking through old posts on DSP Related.
>
>I thought that implied that for a basebanded complex signal (for example)
>the highest frequency you could correctly reconstruct would be twice that
>for a real signal.
>
>But Iâm not finding that to be the case in practice, as the Matlab
>snippet below shows â the complex signal folds into the passband just as
>the real signal does.
>
>While I still have some hair left to tear out, can someone help me
>understand where Iâm going astray, and why the part 2 of code below gives
>the âwrongâ? result?
>
>Thanks, itâs much appreciated.
>
>Walt

Hello Walt,
If I may be so bold, in your spectrum analysis
modeling I suggest that you always plot the
full freq spectra range from 0 -to- Fs
(or -Fs/2 -to- approx. +Fs/2).

Your real signal has a non-zero spectral component at
768 Hz, and a non-zero spectral component at -768 Hz.

Your complex signal will have the same spectrum
as the real signal, only the spec components of
the complex signal will be shifted in the negative
freq direction by 32 Hz.  Those shifted components
will reside at -800 and +736 Hz.

Your 2nd plot has an incorrect freq-axis vector.
Suggest you plot using:

Freq_Axis = (-NFFT/2:(NFFT/2)-1)*(Fs/NFFT);
plot(Freq_Axis, fftshift(Pzz), '-bs')
hold on
plot(Freq_Axis, fftshift(Pzz_2), '-ro')
hold off
title('Blue = real sig,   Red = complex sig.')
xlabel('Hz'), grid on, zoom on

Good Luck,
[-Rick-]
PS.  I've been watching "Breaking Bad" on Netflix.
Walt, would your last name happen to be "White"?

```
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# Re: Sampling Complex Signal - Eric Jacobsen - 2012-08-05 12:43:00

```On Sun, 05 Aug 2012 08:38:11 -0500, "Walter Wego" <61340@dsprelated>
wrote:

>Hi,
>
>I know you have to sample above Nyquist or you risk aliases folding into
>the passband.
>
>But it if your signal is complex, it seems to me that in a given interval
>of time youâve collected twice the information you would have for a real
>signal, or that youâve effectively doubled your sample rate.
>
>I got that same message when looking through old posts on DSP Related.
>
>I thought that implied that for a basebanded complex signal (for example)
>the highest frequency you could correctly reconstruct would be twice that
>for a real signal.
>
>But Iâm not finding that to be the case in practice, as the Matlab
>snippet below shows â the complex signal folds into the passband just as
>the real signal does.
>
>While I still have some hair left to tear out, can someone help me
>understand where Iâm going astray, and why the part 2 of code below gives
>the âwrongâ result?
>
>Thanks, itâs much appreciated.
>
>Walt
>
>clear all
>
>Fs = 2048;
>t  = (0:1/Fs:4-1/Fs);
>F1 = 1024+256;
>LO = 32;
>
>nyquistfreq = Fs/2
>aliasfreq   = Fs-F1
>
>% ----- Part 1 -----
>
>% Create a real, undersampled signal that will fold into passband
>x = cos(2*pi*F1*t);
>
>% Calculate the spectrum
>NFFT = 1024;
>z    = fft(x, NFFT);
>Pzz  = abs(z);
>
>% Plot the spectrum
>Dec = 1;
>f   = ((Fs/2) / Dec) * linspace(0, 1, NFFT/2+1);
>figure(100);
>plot(f, 2*Pzz(1:1+NFFT/2)/NFFT);
>axis([f(1), f(end), 0, 1.01*max(2*Pzz(1:1+NFFT/2)/NFFT)]);
>title('Spectrum of Undersampled Real Signal');
>
>% ----- Part 2 -----
>
>% Translate spectrum of original signal (becomes complex in time)
>xs = x .* exp(-i*2*pi*LO*t);
>
>% Calculate the spectrum
>NFFT = 1024;
>z = fft(xs, NFFT);
>Pzz = abs(z);
>
>% Plot the spectrum
>Dec = 1;
>f = ((Fs/2) / Dec) * linspace(0, 1, NFFT/2+1) + LO;
>figure(200);
>plot(f, 2*Pzz(1:1+NFFT/2)/NFFT);
>axis([f(1), f(end), 0, 1.01*max(2*Pzz(1:1+NFFT/2)/NFFT)]);
>title('Spectrum of Complex Signal');

For real-valued signals sampling at Fs provides low-pass support for
signals from zero to Fs/2.     For complex-valued signals sampling at
Fs provides baseband support from -Fs/2 to +Fs/2, or double the
bandwidth of the real-valued case.

So when you put real-valued signal energy at higher than Fs/2, it will
"alias" in either case.

Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com
```
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# Re: Sampling Complex Signal - Rick Lyons - 2012-08-05 12:45:00

```On Sun, 05 Aug 2012 09:41:45 -0700, Rick Lyons
<R...@_BOGUS_ieee.org> wrote:

>
>Your 2nd plot has an incorrect freq-axis vector.
>Suggest you plot using:
>
>  Freq_Axis = (-NFFT/2:(NFFT/2)-1)*(Fs/NFFT);
>  plot(Freq_Axis, fftshift(Pzz), '-bs')
>  hold on
>  plot(Freq_Axis, fftshift(Pzz_2), '-ro')
>  hold off
>  title('Blue = real sig,   Red = complex sig.')
>  xlabel('Hz'), grid on, zoom on
>

Oops.
I forgot to mention; the above "Pzz_2" is the
complex signal's power spectrum.  Sorry 'bout that.

[-Rick-]
```
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# Re: Sampling Complex Signal - Rick Lyons - 2012-08-05 12:57:00

```On Sun, 05 Aug 2012 11:03:13 -0500, "kaz" <37480@dsprelated> wrote:

>Remember also a complex approach does not change frequency value of signal
>but may eliminate a dc mirror copy
>

Whoa, wait!  Multiplying a real-valued signal
by a negative-frequency complex exponential
yields a complex-valued time signal whose
spectrum is different from the real-valued
signal's spectrum.

[-Rick-]
```
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# Re: Sampling Complex Signal - kaz - 2012-08-05 13:14:00

```>On Sun, 05 Aug 2012 11:03:13 -0500, "kaz" <37480@dsprelated> wrote:
>
>>Remember also a complex approach does not change frequency value of
signal
>>but may eliminate a dc mirror copy
>>
>
>Whoa, wait!  Multiplying a real-valued signal
>by a negative-frequency complex exponential
>yields a complex-valued time signal whose
>spectrum is different from the real-valued
>signal's spectrum.
>
>[-Rick-]
>

Of course you can shift signal anywhere to positive or negative frequency
but I am talking about a pair of a given signal as Re/Im. Each of Re or Im
will have mirror image around dc but the pair as a complex signal may have
one copy only depending on their balance.

```
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# Re: Sampling Complex Signal - Tim Wescott - 2012-08-05 14:55:00

```On Sun, 05 Aug 2012 08:38:11 -0500, Walter Wego wrote:

> Hi,
>
> I know you have to sample above Nyquist or you risk aliases folding into
> the passband.
>
> But it if your signal is complex, it seems to me that in a given
> interval of time youâve collected twice the information you would have
> for a real signal, or that youâve effectively doubled your sample rate.
>
> I got that same message when looking through old posts on DSP Related.
>
> I thought that implied that for a basebanded complex signal (for
> example) the highest frequency you could correctly reconstruct would be
> twice that for a real signal.
>
> But Iâm not finding that to be the case in practice, as the Matlab
> snippet below shows â the complex signal folds into the passband just as
> the real signal does.
>
> While I still have some hair left to tear out, can someone help me
> understand where Iâm going astray, and why the part 2 of code below
> gives the âwrongâ result?
>
> Thanks, itâs much appreciated.

Walter:

It looks like Rick already gave you a good answer to the question you
were asking, so I'll let that stand (besides, I'm too lazy to look at
someone else's code -- I have to spend at least 30 seconds per line
telling myself "no, this is not necessarily crap, it's just that I didn't
write it").

So instead I'll comment on your assertion that you must sample above
Nyquist or risk aliasing.  This (and related assertions of varying
degrees of erroneousness) have been made on this group before, often
enough that I have an answer all ready for all of them:

http://www.wescottdesign.com/articles/Sampling/sampling.pdf

I'm not sure if the answer for your case is in there.  It is: (a), yes,
sampling a complex signal gets you (essentially) two samples per
interval, and hence could Nyquist with half as many samples as you'd
think, (b), the Nyquist criterion says you need 2 * B total _independent_
haven't filtered out the lower sideband) then they won't help.

And (c), if you read the paper you'll find that the Nyquist criterion, as
you state it, is unreachable, because there are no humanly-accessible
signals of finite bandwidth.  To really use Nyquist's work correctly you
have to understand how aliasing mucks things up, why it's impossible to
fully meet the criterion, and (like the story of the mathematician, the
engineer, and the pretty girl) what you need to do to make your system
work close enough.

--
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com
```
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