Hi all,

Can anybody provide some deep thoughts on why the cascaded system response 
should be multiplication of each individual response in frequency domain?

Must both systems to be linear, or causal, or time-invariant, or linear + 
causal + timeinvariant in order for the multicative relationship to hold?

Any other conditions on the systems? 

lucy wrote:
> Hi all,
> 
> Can anybody provide some deep thoughts on why the cascaded system response 
> should be multiplication of each individual response in frequency domain?
> 
> Must both systems to be linear, or causal, or time-invariant, or linear + 
> causal + timeinvariant in order for the multicative relationship to hold?
> 
> Any other conditions on the systems? 
> 
> 

Multiplication of frequency response and convolution of impulse response
are the same under the convolution theorem. Linear Time Invariant 
systems with some regularity are just convolutions. Cascading is just
convolution of impulse responses.

All of this is the same mathematics as you were told about when you
asked about composing linear systems except you now call it cascading
in keeping with the signal processing subject matter.

If you stay with Linear Time Invariant systems then nice things happen.
Physical systems tend to be time invariant and they are usually quite
linear until they get near to their breaking points.

Images analyzed in their spatial domains are not spatially invariant so
Fourier analysis is not justified from the usual first principles but is
still often a useful tool because it is a unary transformation.

lucy wrote:

> Hi all,
> 
> Can anybody provide some deep thoughts on why the cascaded system response 
> should be multiplication of each individual response in frequency domain?


Think of the ogre who lives under the bridge as a filter. When you cross
the bridge, he demands half of your pennies*, but none of your other
money. You start the day with 128 pennies (which you spend only to cross
the bridge) and make three round trips across the bridge. How many
pennies do you have left at day's end? What arithmetic operation did you
use to calculate that?

> Must both systems to be linear, or causal, or time-invariant, or linear + 
> causal + timeinvariant in order for the multicative relationship to hold?

Causal is not necessary. Linear and time invariant guarantees that the
relationship holds. Their lack doesn't guarantee that it does not.

> Any other conditions on the systems? 

Not that I see.

Jerry
________________________________________
* Your penny cache goes 6 dB down every time you cross the bridge.
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

"Gordon Sande" <g...@worldnet.att.net> wrote in message
news:J2rhd.42591$df2.36567@edtnps89...
>
>
> lucy wrote:
> > Hi all,
> >
> > Can anybody provide some deep thoughts on why the cascaded system response
> > should be multiplication of each individual response in frequency domain?
> >
> > Must both systems to be linear, or causal, or time-invariant, or linear +
> > causal + timeinvariant in order for the multicative relationship to hold?
>
> If you stay with Linear Time Invariant systems then nice things happen.
> Physical systems tend to be time invariant and they are usually quite
> linear until they get near to their breaking points.

Even physical systems which are time-variant can often be treated/approximated
as time-invariant for analysis.  The approximation is often very good over short
periods.  For example, a simple analog gain stage has time-variant effects due
to heating and aging of components.  But these changes are so small and slow
that they are usually totally ignored with no problems.

"Gordon Sande" <g...@worldnet.att.net> wrote in message 
news:J2rhd.42591$df2.36567@edtnps89...
>
>
> lucy wrote:
>> Hi all,
>>
>> Can anybody provide some deep thoughts on why the cascaded system 
>> response should be multiplication of each individual response in 
>> frequency domain?
>>
>> Must both systems to be linear, or causal, or time-invariant, or linear + 
>> causal + timeinvariant in order for the multicative relationship to hold?
>>
>> Any other conditions on the systems?
>
> Multiplication of frequency response and convolution of impulse response
> are the same under the convolution theorem. Linear Time Invariant systems 
> with some regularity are just convolutions. Cascading is just
> convolution of impulse responses.
>
> All of this is the same mathematics as you were told about when you
> asked about composing linear systems except you now call it cascading
> in keeping with the signal processing subject matter.
>
> If you stay with Linear Time Invariant systems then nice things happen.
> Physical systems tend to be time invariant and they are usually quite
> linear until they get near to their breaking points.
>
> Images analyzed in their spatial domains are not spatially invariant so
> Fourier analysis is not justified from the usual first principles but is
> still often a useful tool because it is a unary transformation.
>

Why "Images analyzed in their spatial domains are not spatially invariant"?

So Fourier analysis only applies to LTI systems? Why?