Question about Linear systems

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.

My question is how is that the test for the linearity of the system?
Why should the multiplication of 2 sinusoids in a linear system
produce another sinusoid?

I hope I am expressing my query clearly enough... Would really
appreciate if someone can help me clarify this doubt...

Thanks,

Anja

On Oct 6, 9:45 am, Anja <anja.e...@googlemail.com> 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.
>
> My question is how is that the test for the linearity of the system?
> Why should the multiplication of 2 sinusoids in a linear system
> produce another sinusoid?
>
> I hope I am expressing my query clearly enough... Would really
> appreciate if someone can help me clarify this doubt...
>
> Thanks,
>
> Anja

A definitiion and test for a linear system can be found here:

http://en.wikipedia.org/wiki/Linear_system

A system that multiplies two inputs fails the test.

John

> A definitiion and test for a linear system can be found here:
>
> http://en.wikipedia.org/wiki/Linear_system
>
> A system that multiplies two inputs fails the test.

I looked at this page but am still unable to understand how and where
it fails the test. Mathematics is not my strong point (though I am
working on it every day!). Is it possible that you can explain a bit
as to what constraints are violated?

Thanks,
Anja

On Sat, 06 Oct 2007 13:45:47 -0000, Anja <anja.ende@googlemail.com>
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.
>
>My question is how is that the test for the linearity of the system?
>Why should the multiplication of 2 sinusoids in a linear system
>produce another sinusoid?

Hi Anja,
   to help answer your 2nd question, have a  
look in some maths reference book and you'll see 
a "trig identity" that says:

   sin(A)*sin(B) = cos(A-B)/2 -cos(A+B)/2   [1]

If we think of A = 2*pi*f1*t, and B = 2*pi*f2*t, 
then Eq. [1] tells us that the product of two sinusoids 
is equal to the sum of two "other" sinusoids.
The frequencies of those two "other" sinusoids are:
f1-f2, and f1+f2.

Actually, this universal truth in Eq. [1] is 
why AM radio works.

Good Luck,
[-Rick-]

Anja wrote:
>>A definitiion and test for a linear system can be found here:
>>
>>http://en.wikipedia.org/wiki/Linear_system
>>
>>A system that multiplies two inputs fails the test.
> 
> 
> I looked at this page but am still unable to understand how and where
> it fails the test. Mathematics is not my strong point (though I am
> working on it every day!). Is it possible that you can explain a bit
> as to what constraints are violated?
> 
> Thanks,
> Anja
> 

The key sentence of the article is "The behavior of the resulting system 
subjected to a complex input can be described as a sum of responses to 
simpler inputs."

"complex" in this case does refer _specifically_ to input(s) having 
"real" and "imaginary" components.

A wordy paraphrase might be "The behavior of the resulting system 
subjected to a set of inputs can be described as a sum of responses to 
each individual member of the set of inputs."

On 6 Okt, 15:45, Anja <anja.e...@googlemail.com> 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

So, if I understand correctly, this multiplier system fails linearity
test because it is not additive.

So, if the input is the first sinusoid: sin(x1) and let the output for
that be y1.

And if the output is the second sinusoid sin(x2), let the output be
y2.

When both of them are presented as inputs, the output must be y1 + y2
for the system to be called linear... am I correct in understanding it
this way?

Thanks for all your help guys. I really appreciate it.

Anja

On Oct 6, 4:11 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 6 Okt, 15:45, Anja <anja.e...@googlemail.com> 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

Ahhhhhh... that makes sense. Thanks!

Anja 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.
>
> My question is how is that the test for the linearity of the system?


Multiplication with a fixed signal is a linear operation. A
"system" (*) H is linear iff

H(a x1 + b x2) = a H(x1) + b H(x2)

for all a numbers a and b and all signals x1 and x2 (just showing the
above condition to be true for a specific pair of signals will not
do). Note that the system H takes one signal as an argument: H = H(x).
Now what if H multplies its argument by sin(w t)?. You get

H(a x1(t) + b x2(t))
= sin(w t)(a x1(t)+b x2(t))
= a sin(w t) x1(t) + b sin(w t) x2(t)
= a H(x1(t)) + b H(x2(t)),

so clearly H is a linear system. Systems can be generalized to more
than one input or output. In this case you still write

y = H(x),

but both x(t) and y(t) are vector functions, ie.

x(t) = ( x1(t), x2(t) , ..., x_n(t) )

and

y(t) = ( y1(t), y2(t), ...., y_m(t) ).

The linearity condition for such systems is exactly the same as above,
except that x1(t) and x2(t) as well as y(t) = H( x(t) ) are vectors.
So the simple case of the system H given by

y(t) = H(x1(t), x2(t))
= x1(t) x2(t),

(n=2, m=1), is not linear. Can you prove it?

Regards,
Andor

BTW, forget that single sinusoid in - single sinusoid out linearity
test. This is a stronger condition than linearity. For this to be
true, you additionally need time-invariance of the system.


(*) What is a "system"? Colloquially, it is an operation that takes an
input (some signal),  and produces an output (another signal or a
number - numbers are also signals).

>
> Multiplication with a fixed signal is a linear operation. A
> "system" (*) H is linear iff
>
> H(a x1 + b x2) = a H(x1) + b H(x2)
>
> for all a numbers a and b and all signals x1 and x2 (just showing the
> above condition to be true for a specific pair of signals will not
> do). Note that the system H takes one signal as an argument: H = H(x).
> Now what if H multplies its argument by sin(w t)?. You get
>
> H(a x1(t) + b x2(t))
> = sin(w t)(a x1(t)+b x2(t))
> = a sin(w t) x1(t) + b sin(w t) x2(t)
> = a H(x1(t)) + b H(x2(t)),
>
> so clearly H is a linear system. Systems can be generalized to more
> than one input or output. In this case you still write
>
> y = H(x),
>
> but both x(t) and y(t) are vector functions, ie.
>
> x(t) = ( x1(t), x2(t) , ..., x_n(t) )
>
> and
>
> y(t) = ( y1(t), y2(t), ...., y_m(t) ).
>
> The linearity condition for such systems is exactly the same as above,
> except that x1(t) and x2(t) as well as y(t) = H( x(t) ) are vectors.
> So the simple case of the system H given by
>
> y(t) = H(x1(t), x2(t))
> = x1(t) x2(t),

Shouldn't this be y1(t)y2(t) ???
>
> (n=2, m=1), is not linear. Can you prove it?

What are n and m denoting in this case???

Thanks for your reply. Helps a lot.

Anja

>
> Regards,
> Andor
>
> BTW, forget that single sinusoid in - single sinusoid out linearity
> test. This is a stronger condition than linearity. For this to be
> true, you additionally need time-invariance of the system.
>
> (*) What is a "system"? Colloquially, it is an operation that takes an
> input (some signal),  and produces an output (another signal or a
> number - numbers are also signals).