From: syd@en.ecn.purdue.edu (Dennis P Hilgenberg)
Subject: Re: Why filter D/A output?
Organization: Purdue University Engineering Computer Network
Date: Sat, 21 Nov 92 15:10:41 GMT

Wait a minute here.  Someone seems to have strayed from the topic
at hand.

philg@martigny.ai.mit.edu wrote:

	Driving a nonlinear, i.e. distorting, system, i.e. amplifier/speaker,
	with a single frequency produces an output made up only of that
	frequency and its harmonics.  This can be shown easily by looking at a
	Taylor series expansion of a nonlinear function and seeing what
	happens to cosine squared and cosine cubed terms (use trig
	identities).

To which I responded:

	This is [mathematically] not true in general.  
	Example: f(x) = arccos (x).  The output of such a system 
	when x = cos(t) is simply t, which is not periodic, and
	hence does not have a Fourier series expansion.

	One could be more obnoxious by letting f(x) = cos (1.5 arccos(x)).
	
	The Taylor series must converge at every point in order for this
	argument to work.  (Note that arccos (x) does not meet this 
	requirement.)

	A sufficient condition for this is that the nonlinearity be continuous.
	(In the real world, this is probably a valid assumption.)

MATGBB@LURE.LATROBE.EDU.AU (BYRNES,Graham) countered with:

	I guess we can at least assume that the transfer function is bounded
	over any compact interval ? :-) One might even hope that it would be 
	continuous....so we can use Taylor's theorem to get an arbitrarily
	accurate approximation over the domain (ie possible input values).
	Cheers, GB
	(Yeah, I know, what a wank)

Now, out of the blue, fineberg@nas.nasa.gov writes:

	Actually you're both wrong.  The reason why we need filtering is 
	because real functions are sampled  over finite intervals, i.e., we 
	can't really produce a delta function.  This causes distortion, 
	however, it is all outside of the frequency range of the original 
	function (remember we filtered the original function to prevent 
	aliasing).  Try sampling and recovering a signal, then display it 
	on a scope or spectrum analyzer, it is pretty obvious.
	(Dennis, you should have done this or will do this in EE440 lab).

Although I shouldn't have to defend my qualifications, I will clarify
a few things.  (1) I am a PhD student in the field of digital signal 
processing, and think I understand sampling theory quite well.  (2) 
I am in fact a T.A. for the EE 440 lab that you mention (for those 
outside the scope of Purdue University, this is the senior-level 
"Transmission of Information" lab).  (3) The paragraph in the original 
post to which I was responding (I've included it for your convenience) 
had nothing to do with sampling.
 
Also, you have made several factual errors.  First, the issues of sampling
over a finite interval and of non-ideal sampling pulses are not the same.
Sampling over a finite interval (as opposed to over all time) will cause
the "ideal" (Fourier transform) spectrum to be _convolved_ with a (sin x)/x
function, while sampling with nonideal pulses (say rectangles) will 
cause the Fourier transform to be _multiplied_ by a (sin x)/x term.
This distortion takes place independent of additional distortion caused
by the anti-aliasing filter(s), and in fact should be compensated for
by well-designed filters.  I don't know what you mean when you say
"we filtered the original function to prevent aliasing."

In either case, the distortion is _not_ confined to the terms outside
the frequency range of interest.  I suggest that you conduct the 
experiment that you recommended with a flat-spectrum signal, e.g., 
a wideband FM modulation of a triangle wave, to see these effects 
clearly demonstrated.


-- 
Dennis Hilgenberg
syd@ecn.purdue.edu
LaRouche in '96
Newsgroups: rec.audio