What is the Sampling Period Range for Recoverable Signals?

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Discussion Overview

The discussion revolves around determining the appropriate range of sampling periods for a signal generated by the convolution of two band-limited signals. The focus is on ensuring that the original signal can be accurately recovered from its sampled version, with considerations of the Nyquist rate and the implications of convolution in the frequency domain.

Discussion Character

  • Homework-related, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant describes the problem involving the convolution of two band-limited signals and seeks guidance on how to approach the sampling period for recovery.
  • Another participant suggests considering the convolution theorem as a relevant concept for the problem.
  • A participant questions how to implement the given CTF transforms in the context of the problem.
  • There is a query about what sets the lower limit on the sampling rate necessary for recovering the original signal.
  • A participant mentions the Nyquist rate as a critical factor, stating it should be equal to 2fmax to avoid aliasing.
  • One participant expresses confusion regarding the hints provided and seeks further clarification.
  • Another participant prompts consideration of the frequency representation Y(jω) and its zeros to deduce fmax, suggesting a deeper exploration of the frequency domain.

Areas of Agreement / Disagreement

Participants generally agree on the relevance of the Nyquist rate in the context of signal recovery, but there is no consensus on the specific implementation details or the exact range of sampling periods required.

Contextual Notes

Participants have not fully resolved the mathematical steps necessary to determine the sampling period range, and there are dependencies on the definitions of the signals involved.

Who May Find This Useful

This discussion may be useful for students and practitioners interested in signal processing, particularly those dealing with sampling theory and convolution in the context of band-limited signals.

satchmo05
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Homework Statement



The signal y(t) is generated by convolving a band limited signal x1(t) with another band limited signal x2(t) that is y(t)=x1(t)*x2(t) where:

--> X1(jω)=0 for|ω| > 1000Π
--> X2(jω)=0 for|ω| >2000Π

Impulse train sampling is performed on y(t) to obtain:
--> yp(t)= [summation from n = (−∞,∞)] y(nT)δ(t− nT)

Specify the range of values for sampling period T which ensures that y(t) is recoverable from yp(t).

Homework Equations


All of the equations that I would are most likely showing.

The Attempt at a Solution


My thoughts were to plug in (nT) for every t in both x1(t) and x2(t) and then take the Fourier transform of that, cut of the edges where the transforms are equal to zero and then that is where I go blank...

I imagine that is the right implementation to start the problem with, but please correct me if I am wrong. Thank you in advance to all who may be able to help - it is much appreciated!
 
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Hint: convolution theorem.
 
I can see where convolution comes into play, but how can I implement the CTF transforms that are given?
 
What sets the lower limit on the sampling rate if you want to be able to recover the original signal?
 
The Nyquist rate, to sample at the perfect rate (without aliasing/oversampling to occur) - it would be = 2fmax
 
I am pretty confused at what you're trying to hint at here. I appreciate the help, but my mind is still blank.
 
It's kind of hard to say anything without giving away the answer. Think about Y(jω). Where is it zero? Can you deduce fmax from that information?
 

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