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From: heredia@enel.ucalgary.ca (Edwin Heredia)
Newsgroups: comp.dsp
Subject: 180 deg phase shift
Message-ID: <1991May9.160115.3494@cpsc.ucalgary.ca>
Date: 9 May 91 16:01:15 GMT
Reply-To: heredia@enel.ucalgary.ca (Edwin Heredia)
Organization: EE Department, U. of Calgary, Calgary, Alberta, Canada
Lines: 20
Nntp-Posting-Host: enelc.enel.ucalgary.ca

For those who can't believe that inverting the signal produces 
a 180 deg. phase shift and for those who think that this trick 
works only with "easy-to-work" signals as pure tones:

Given *ANY* signal  x(n)  with Fourier Transform  X(w)

Consider the transformation          y(n) = (-1) * x(n)

in the Fourier domain this gives     Y(w) = (-1) * X(w)

which is                         Y(w) = exp(j180) * X(w)

result: 180 deg. phase shift!

where *ANY* means really *any*, that is ECG's, speech, etc., and 
(-1)*x(n)  means really inverting the signal.

---------
If a 180 deg shift produced about 15 articles posted, how many 
would be produced by a 360 deg shift? Answer: 2*15 = 30