🚀 go-pugleaf

I find it interesting that this thread has dragged on as long as
it has.  I assume that this is due to a tendency to respond
without taking time to think about the problem.

To phase shift a signal means to multiply its Fourier transform
by exp(j A) where A is the desired phase shift in radians.
The inverse Fourier transform of the signal exp(j A) X(w) is
exp(j A) x(t).  (We're just multiplying by a scalar here. There is
no time shift.  If we had multiplied by exp(j A w) instead of
exp(j A) the corresponding time signal would be x(t + A).)
If A is equal to pi (180 degrees) then exp(j A) = -1 and so we
are just inverting the signal.

Notice that there is no requirement for the signal to be periodic.
It just has to be Fourier transformable.

I suspect the problem is that many people are thinking in terms
of Fourier series and phase shifting each component of the series.
We still get the same result though. If x(t) is periodic
we can represent it in a Fourier series as
      x(t) = Sum a(k) exp(j k w0 t)
where the Sum is over all k and w0 is the fundamental frequency.
Now shift each component by A radians.  Call this new signal y(t).
      y(t) = Sum a(k) exp[j (k w0 t + A)]
(Notice again that if A is equal to pi, then y(t) = -x(t).)
What is the time shift of each component?  Rewrite as
      y(t) = Sum a(k) exp[j k w0 (t + A/(k w0))].
The time shift of each component depends upon the frequency of the
component,  so the time shift of each component waveform is not
the same.  (This is obvious if you sketch two harmonically related
sinusoids and then sketch their 180 degree phase shifted version.)

This has all been said by others.  I just that I would throw in my
$.02 because the argument seemed to be shifting to the wrong answer.

Regards,
Tony Richardson

amr@egr.duke.edu