heredia@enel.ucalgary.ca (Edwin Heredia) (05/13/91)
About 4 days ago, I posted an article showing that for any signal
x(n) with Fourier Transform X(w), the operation y(n) = -x(n) shifts
the phase of x(n) 180 deg.
Undoubtly, the only signals for which the demonstration is not valid
are those that do not admit a Fourier Transform (in which case we
probably cannot talk about magnitude or phase).
Two interesting but erroneous comments appeared after I posted the
original article.
comment #1: "...surely you mean x(n) periodic ..."
As karsh@sgi.com pointed out, this is not true because the Fourier
transform is defined for periodic as well as aperiodic signals. In
fact, most of the textbooks introduce the Fourier transform for
aperiodic signals and then show how the concept can be used for
periodic signals.
Again karsh@sgi.com pointed out that any absolutaly summable
sequence admits a Fourier transform (example: time-limited sequences).
This is o.k., but is a sufficient condition, not necessary,
which means that some non-absolutely-summable signals can have
a Fourier transform (example: periodic sequences).
comment #2: "...x(n) cannot be a DC signal ..."
This is not true either. A discretized "DC signal" is a train of
discrete impulses at ...n=-1, n=0, n=1,.... Even though this is not
an absolutely-summable sequence it admits a Fourier transform which is
Magnitude: a real-valued (not discrete) impulse
train at w=0, w= (+-) 2*pi, etc.
Phase: 0 or 180 (depending if the signal is positive or negative)
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