solution required urgently of following problem

[mabdulahad]

Let h(t)=1/2b exp(-|t|/b) be the impulse response a filter
Assume you have N such filters in series, i.e. one after the other, and that b=1/sqrt(N) : what is the impulse response
of the series in the limit for N that goes to infinitiy?

 

[Dalibor]

Do you suppose that this filter is causal (i.e. h(t)=0 for t<0) or not?
D.

 

[zorx]

Some facts:
1. Filter transfer function H(f) (in frequency domain) and filter impulse response h(t) (in time domain) are Fourier transform pair, i.e. F{h(t)} = H(f)
2. If you have the serie of N different filters with corresponding transfer functions H1(f), H2(f), ..., Hn(f), then overall transfer function is H(f)=H1(f) x H2(f) x ... x Hn(f) (the sign "x" stands for multiplication)
3. If you have the serie of N different filters with corresponding impulse responses h1(t), h2(t), ..., hn(t), then overall transfer function is h(t)=h1(t) * h2(t) * ... * hn(t) (the sign "*" stands for convolution)

Q: What is easier to calculate: multiple product or multiple convolution?
A (IMHO): Multiple product

This is what you should do:
1. Find the Fourier transform of impulse response F{h(t)} = H(f)
2. Calculate overall transfer function of serie of N filters Hn(f) = H(f) x H(f) x ... x H(f) = [N(f)]^N
3. Calculate impulse response of overall filter by taking inverse Fourier transform F^-1{Hn(f)} = hn(t)
4. Let the N->inf in hn(t)

Regards (and let us know the results)

 

[Dalibor]

One note should be mentioned: multiplication in the frequency domain is equivalent to convolution, not correlation, in the frequency domain.
I am afraid that Mamdulahad is interested in the analytical, not numeric form of the result. Any way, he should clearly state if his formula for h(t), which contains the absolute value of the time, defines the impulse response also for time<0. If yes, then the filter is non-causal and thus non-realizable, and the problem which should be solved is only a mathematical exercise.
He should also precise if the formula for the impulse response is really
h(t)=1/2b exp(-|t|/b)
or
h(t)=(1/2)*b*exp(-|t|/b)

D.

 

[zorx]

Hi Dalibor,

Thank you for the correction. I made a mistake, it should be convolution, not correlation. (P.S. you made a small typo - "multiplication in the frequency domain is equivalent to convolution, not correlation, in the frequency domain." - it should be "time domain")
Regarding the solution to the problem, I was working with
h(t)=1/(2b) * exp(-|t|/b)

F{h(t)} = H(w) = 1 / [1+b^2 * w^2]

The N cascaded filter structure gives
Hn(w) = [H(w)]^N = 1 / [1+b^2 * w^2]^N

and I guess in this step could N-> inf, and find the limit to the Hn(w) function, and then perform the inverse Fourier transform to calculate hn(t)

 

[Dalibor]

Hi Zorx,
you are right:-D.
Your result
Hn(w) = [H(w)]^N = 1 / [1+b^2 * w^2]^N
is computed on the assumption that the impulse response is even function of time, i.e. the response to the Dirac impulse precedes this impulse. In other words, the corresponding linear system is not causal.
Based on this assumption, and taking into account the Mabdulahad statement that b=1/sqrt(N), we have
H=lim Hn(w) for N going to infinity:
H=exp(-w^2).
Then according to the dictionary of Fourier transform, we have the resulting impulse response
h(t)=1/(2*sqrt(pi))*exp(-t^2/4)
Hope that I did not make a mistake.
Any way, the result again corresponds to non-causal system.