Digital Signal Processing - Digital Filter Question

[Candle111]

Hello, I have been newly introduced to DSP, and I'm finding some of the concepts quite tricky. I've got a problem I'm stuck with, and I'm hoping someone might be able to enlighten me on how to go about solving it. The question is as follows:

A pole zero filter design gave the following difference equation:

y = x + b.y(n - 1)1

Describe the output when using an input signal given as:

x= 0 for n <0
1 for n >= 0

Use appropriate analysis in your description.


Thank you! I appreciate any assistance.

 

[rahul.achates]

This is a feedback filter , output sample is depend on previous output value , multiple with coefficient b ...

X is to make sure system sampling from time 0 ... below time zero is not a realistic situation.

Rahul

 

[zorro]

... below time zero is not a realistic situation.

I don't agree; n=0 can be an arbitrary time reference. It is as realistic as to consider the time before 0:00 of today.

A pole zero filter design gave the following difference equation:

y = x + b.y(n - 1)

Describe the output when using an input signal given as:

x= 0 for n <0
1 for n >= 0

(I removed the "1" character at the end of the difference equation considering it's a typo.)

It should be specified an initial condition in order that the problem b completely specified.
Let's consider that, for example y(-2)=0, and let's see what happens for n>=-1

For any n:
y = x + b.y(n - 1)

Knowing that y(-2)=0 and that x=0 for n<0, now we replace n for the different values starting at n=-1:

y(-1) = x(-1) + b.y(-1 - 1) = x(-1) + b.y(-2) = 0 + b.0 = 0
y(0) = x(0) + b.y(-1) = 1 + b.0 = 1
y(1) = x(1) + b.y(0) = 1 + b.1 = 1 + b
y(2) = x(2) + b.y(1) = 1 + b.(1+b) = 1 + b + b^2
... an so on ...

The next step is to realize that methods of signal processing allow to get the same result for this problem (an for others, much more complicated) in a simple and clever way.
Regards

Z