🚀 go-pugleaf

truman@zabriskie.berkeley.edu (Tom Truman) writes:


>With all of the commotion caused by a simple question regarding 180 deg.
>phase shift, I can't resist posting this one:

>" Suppose that a certain telephone channel can be characterized by
>a frequency response H(f) = exp(j * P(f)), where P(-f) = -P(f).

>Because P(f) may be a complicated function of frequency, such a
>channel may have serious distortions.  It has been suggested (rather
>impractically) that the signal on the recieving end be recorded on
>tape, flown by fast jet back to the transmitting end, played backwards
>through the original channel a second time, and recorded again on
>tape.

>The signal on this second tape, if played backwards, should (it is claimed)
>be the original signal, independent of P(f).

>Would this work ?"

OK, I'll bite.  The system looks like:


------>  H(f)  ------> (-t) ------>  H(f)  ------> (-t) ------>

  X(f)           A(f)        B(f)           C(f)         Y(f)

where (-t) indicates time reversal.  That is, b(t) = a(-t).  From
Fourier transform properties, B(f) = A(-f).  Tracing through the
system: 

A(f) = X(f) H(f)
B(f) = X(-f) H(-f)
C(f) = X(-f) H(-f) H(f)
Y(f) = X(f) H(f) H(-f)
     = X(f) e^{jP(f)} e^{jP(-f)}
     = X(f) e^{jP(f)} e^{-jP(f)}
     = X(f)

So it does work.  In fact, this time reversal trick gets used a lot to
get linear phase in systems using IIR filters.  Discrete-time systems,
that is, with finite-length signals, so time-reversal is easy.

An important factor is that the telephone channel introduces only
phase distortion, not frequency distortion.  That is, |H(f)| = 1.
If |H(f)| \neq 1, then the input-output relationship of the above
system is:

Y(f) = X(f) |H(f)|^2			(assuming h(t) is real)

This system has no phase distortion, but the frequency distortion is
channel-dependent.

Well, given what's happened recently in this group, I've tried to be
careful here, but if there's a flaw, fire away!
-- 
Steve Eddins	
eddins@brazil.eecs.uic.edu 	(312) 996-5771 		FAX: (312) 413-0024
University of Illinois at Chicago, EECS Dept., M/C 154, 1120 SEO Bldg,
Box 4348, Chicago, IL  60680