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Unformatted text preview: ECE 342 Communication Theory Fall 2005, Solutions to Homework 4 Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/ ∼ jingli/teach email: [email protected] Problem 2.35 For no aliasing to occur we must sample at the Nyquist rate f s = 2 · 6000 samples/sec = 12000 samples/sec With a guard band of 2000 f s- 2 W = 2000 = ⇒ f s = 14000 The reconstruction filter should not pick-up frequencies of the images of the spectrum X ( f ). The nearest image spectrum is centered at f s and occupies the frequency band [ f s- W, f s + W ]. Thus the highest frequency of the reconstruction filter (= 10000) should satisfy 10000 ≤ f s- W = ⇒ f s ≥ 16000 For the value f s = 16000, K should be such that K · f s = 1 = ⇒ K = (16000)- 1 Problem 2.39 1) The linear interpolation system can be viewed as a linear filter where the sampled signal x ( t ) ∑ ∞ n =-∞ δ ( t- nT s ) is passed through the filter with impulse response h ( t ) = 1 + t T s- T s ≤ f ≤ 1- t T s ≤ f ≤ T s otherwise To see this write x 1 ( t ) = " x ( t ) ∞ X n =-∞ δ ( t- nT s ) # ? h ( t ) = ∞ X n =-∞ x ( nT s ) h ( t- nT s ) Comparing this with the interpolation formula in the interval [ nT s , ( n + 1) T s ] x 1 ( t ) = x ( nT s ) + t- nT s T s ( x (( n + 1) T s )- x ( nT s )) = x ( nT s ) 1- t- nT s T s + x (( n + 1) T s ) " 1 + t- ( n + 1) T s T s # = x ( nT s ) h ( t- nT s ) + x (( n + 1) T s ) h ( t- ( n + 1) T s ) 1 we observe that h ( t ) does not extend beyond [- T s , T s ] and in this interval its form should...
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- Fall '05
- Aliasing, TS, Howard Staunton, guard band, ts ts