🚀 go-pugleaf

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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