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A continuous-time signal x(t) is such that its CTFT X(f) is zero when when |f|>1,400 Hz. You would like to low-pass-filter the signal x(t) with a cut off frequency of 800Hz and a gain of 7. Let's call this desired filtered signal y(t). | A continuous-time signal x(t) is such that its CTFT X(f) is zero when when |f|>1,400 Hz. You would like to low-pass-filter the signal x(t) with a cut off frequency of 800Hz and a gain of 7. Let's call this desired filtered signal y(t). | ||
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a) Assume that you are only given a sampling of x(t), specifically a sampling obtained by taking 6000 samples per second (samples equally spaced in time). Can one process this sampling in such a way that a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how. If you answered no, explain why not. | a) Assume that you are only given a sampling of x(t), specifically a sampling obtained by taking 6000 samples per second (samples equally spaced in time). Can one process this sampling in such a way that a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how. If you answered no, explain why not. | ||
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[[2015_Fall_ECE_438_Boutin|Back to ECE438, Fall 2014, Prof. Boutin]] | [[2015_Fall_ECE_438_Boutin|Back to ECE438, Fall 2014, Prof. Boutin]] | ||
Revision as of 16:31, 1 October 2015
Homework 4 Solution, ECE438, Fall 2015, Prof. Boutin
Question
A continuous-time signal x(t) is such that its CTFT X(f) is zero when when |f|>1,400 Hz. You would like to low-pass-filter the signal x(t) with a cut off frequency of 800Hz and a gain of 7. Let's call this desired filtered signal y(t).
a) Assume that you are only given a sampling of x(t), specifically a sampling obtained by taking 6000 samples per second (samples equally spaced in time). Can one process this sampling in such a way that a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how. If you answered no, explain why not.
- Answer: No. Since the sampling frequency is 6000 Hz, there will be no aliasing when we sample the signal which has the maximum frequency of 1400 Hz (fs > 2*fm). However, when we apply LPF with 800 Hz cutoff, we loose signal in the range 800 Hz < |f| < 1400 Hz. So the low-passed signal cannot be reconstructed perfectly.
b) Now assume that the sampling from Part a) is downsampled by a factor 2. Can one process this downsampled signal in such a way a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how. If you answered no, explain why not.
- Answer: No. The same reason as part (a).
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