Multiple ADCs and aliasing

W dniu 03.03.2018 o 19:51, Richard (Rick) Lyons pisze:
(...)

>> could you clarify why x(t), which spectrum is X(f) in Eq.(19), MUST BE a
>>    periodic signal ?
> 
> I'll do my best. Let's say I have a ten-second duration analog audio recording of a pure sine wave. First I compute the spectrum, Y1(f), of the first five seconds of the audio signal. Then I compute the spectrum, Y2(f), of five seconds of the sine wave audio starting at the beginning of the 2nd second out to the end of the 6th second.  For this periodic sine wave input signal, Y1(f) and Y2(f) will be related to each other as shown in Eqs. (18) and (19) in the cited Razavi. That is, Y2(f) will be equal to Y1(f) multiplied by a phase angle that is a linear function of frequency. Also, in this periodic signal case, |Y1(f)| = |Y2(f)|.

both examples are inadequate.
If the Fs is correct sampling frequency (Fs > 2 x BW of signal), then we 
are talking about situation where we have the first recording sampled 
from the beginning, at Fs/2, and second, starting from the moment 
delayed fom beginning by Ts, and also at Fs/2 rate.

Anyway, |Y1(f)| is not equal to |Y2(f)| generally. That is also true in 
case of infinite periodic signal like, on example, sinusoid of Fs/4 
frequency. The amplitude of this two sequences (+x1, -x1 for first, and 
+x2, -x2 for second) will be generally different, but equal for moments 
where sinusoid phase is pi/4+k*pi only.

As for mentioned paper, equation (19) has an error in exponent of e. The 
correct exponent related to Tck/2 delay (Fck = Fs/2) is:

-j*2*pi*f*delta(t))= -j*2*pi*f*Tck/2 = -j*pi*f*Tck .

As f = k*Fck (exponent values for other f are multiplied by 0) we get:
-j*2*pi*k*Fck*Tck/2 = -j*pi*k,
what gives the sequence: +1, -1, +1, -1 ... .
Even spectral copies are added, odd - subtracted, so we should get 
correct spectrum, as in Fs case, but because Y1(f) and Y2(f) are 
different, all the more for non-periodic signals ... so, the Figure 6 is 
not correct.

Where is the bug in this prove ?

Maybe the frequency analysis should be done for uspsampled sequences ?

Best regards,

Roman Rumian

>W dniu 03.03.2018 o 23:43, Richard (Rick) Lyons pisze:

> > Hi Roman. I forgot to ask you: Do you agree that the
> > addition of the two sequences,

> >      Seq# 1 = x(0), x(2), x(4), x(6)
> >
> >      Seq# 2 = x(1), x(3), x(5), x(7)
> >
> > produces the four-sample sequence:
> >
> >       x(0)+x(1), x(2)+x(3), x(4)+x(5), x(6)+x(7)

>Hi Richard,
>
>yes, of course.

This is of course how two sequences are added.  The question
however is how two signals are added.

Rick, you were the first to inject the word sequence into this
discussion, which had until then been discussing signals, when
you wrote:

     The paper you cited is interesting. On page
     1810 of the paper the author wrote: "Since
     multiplexing of discrete-time signals is equivalent
     to addition,... ." That's not true, of course. We
     don't simply add the two discrete sequences' samples,
     we interleave the two sequences' samples.  Perhaps the
     "addition" the author is referring to should be called
     "time-aligned addition" or "time-interleaved addition".

Sequences and signals are different objects; therefore, the
addition operator is (potentially) defined differently for the two.

It's like an object-oriented programming language - the " + " operator
is overloaded.

Two sequences are added by pairwise addition of values with the
same index.

Whereas two signals are added by pairwise addition of values which 
are at the same point along the time axis.

As you note the latter could be called "time aligned addtion", but
I propose this is the default notion of addition of signals.  
Certainly real-life signals, wherein a summing node adds two
signals that exist in the same local time frame, are added in 
a time-aligned fasion.

Steve