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
Where is the bug in this prove ?
Maybe the frequency analysis should be done for uspsampled sequences ?
Reply by Steve Pope●March 5, 20182018-03-05
>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,
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
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
Two sequences are added by pairwise addition of values with the
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.