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ELEC442-6601 chapter 4 examples, solution_2

# ELEC442-6601 chapter 4 examples, solution_2 - Sampling...

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Sampling Theorem Examples Concordia University 1 Question 1) Determine the condition on the sampling interval s T so that each ) ( t x is uniquely represented by the discrete-time sequence ) ( ] [ s nT x n x = . a) ) 4 sin( ) 2 sin( 3 ) cos( ) ( t t t t x π π π + + = b) t t t t t t t x π π π π π π ) 2 sin( ) 6 sin( 3 ) sin( ) 2 cos( ) ( + = c) The signal ) ( t x with FT given in the figure. Solution 1) Part a: The signal ) 4 sin( ) 2 sin( 3 ) cos( ) ( t t t t x π π π + + = has frequencies of π , π 2 and π 4 radians/sec. Therefore the maximum frequency of the signal is π 4 = Ω m and therefore the sampling frequency should be π 8 > Ω s . The sampling period is 4 1 < s T . Part b: The first term has a square spectrum which is extended up to frequency of π due to t t π π ) sin( and is shifted π 2 due to ) 2 cos( t π and therefore the overall spectrum has maximum of π π π 3 2 = + . The second term has a square spectrum which is extended up to frequency of π 2 due to t t π π ) 2 sin( and is shifted π 6 due to ) 2 sin( t π and therefore the overall spectrum has maximum of π π π 8 6 2 = + . Therefore π 8 = Ω m and 8 1 < s T . Part c:

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Sampling Theorem Examples Concordia University 2 The maximum frequency in this signal as shown in the frequency spectrum is about π 5 . 4 = Ω m radians/sec and therefore the sampling frequency should be π 9 > Ω s . The sampling period is 9 2 < s T Question 2) Consider the signal < Ω < - Ω = Ω otherwise j X 0 1000 1000 ) ( π π a) Clearly sketch the signal in the frequency domain include all relevant points and values. b) If the signal is fed into an ideal sampler with sampling frequency 3000π (rad/sec), clearly sketch the output of the sampler in the frequency domain include all relevant points and values. c) If the sampled signal is fed into an ideal low pass filter with a cut-off frequency of 1500π (rad/sec), clearly sketch the output of the filter in the frequency domain. In your drawing include all relevant points and values. d) If the ideal low pass filter had a cut-off frequency of 2500π (rad/sec), clearly sketch the output of the filter in the frequency domain. In your drawing include all relevant points and values.
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