102_1_hw7_soln

# 102_1_hw7_soln - 1 EE102 Fall Quarter 2011 Systems and...

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1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #7 Due: Not Handed In 1. Practical Signal Reconstruction If we have a samples of a bandlimited signal sampled at the Nyquist rate, we can per- fectly reconstruct the signal by passing the sequence of impulses through a lowpass filter. However, the impulse train is not a realizable signal. In this problem we will look at the case where we approximate the impulses by short rectangular pulses, and see what effect this has on the reconstructed signal. The ideal sampled signal, sampled at an interval of T , is 2T -T -3T -2T T 0 3T t Ideal Sampled Signal ¯ f ( t ) = n = - f ( nT ) δ ( t - nT ) If we approximate the impulses with rects, the signal looks like 2T -T -3T -2T T 0 3T t Approximate Sampled Signal ˜ f ( t ) = n = - f ( nT ) p ( t - nT ) where the rect we are using to approximate the impulses is -T T 0 t p ( t ) = rect ( 4 t / T ) (a) Write an expression for ˜ f ( t ) in terms of ¯ f ( t ) , using the definition of p ( t ) . Solution: The approximate sampled signal is the ideal sampled signal convolved with the rect we are using to approximate the impulse ˜ f ( t ) = ¯ f ( t ) * rect(4 t/T ) .

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2 (b) Find an expression for ˜ F ( ) in terms of ¯ F ( ) .
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