Re: Question about Linear systems

On 6 Okt, 15:45, Anja <anja.e...@xxxxxxxxxxxxxx> wrote:
Hello everyone,

I am trying to introduce myself to DSP through the free book at
(www.dspguide.com) on the Internet.

I was studying the chapter on linear systems and it mentions that
multiplying 2 signals together is not a linear operation. It gives the
example that when you multiply 2 sinusoids of different frequencies,
the result is clearly not a sinusoid.

The linear system model is useful because it leaves you
with a choise: Either express the input signal as one
"complicated" signal, and apply the system function
on this one signal. Or you can express the input as a
sum of "simple" signals, where you apply the system function
on each of these, and synthesize the total output signal
as the sum of "simple" outputs.

Now, for *linear* systems it odes not matter which of the
approaches you use< the results of both methods are equal.
Which is the reason why the Fourier transform is so useful.

For *nonlinear* systems the two approaches will yield
different results.

Assume

y[n] = x[n]^2

If

x[n] = sin[w_1*n] + sin[w_2*n]

Then

y[n] = sin[w_1*n]^2 + sin[w_2*n]^2 + sin[w_1*n]sin[w_2*n]

This is different from applying the system function
separately to the components sin[w_1*n] and sin[w_2*n]
of x[n]:

y'[n] = sin[w_1*n]^2 + sin[w_2*n]^2

where the cross term sin[w_1*n]sin[w_2*n] lacks.

The system is nonlinear because y[n] =/= y'[n].

Rune

.

Relevant Pages
Loading