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Unformatted text preview: E4702 Midterm#2 solutions by Anmo Kim (ak2263@columbia.edu) 1. Consider the triangular spectrum shown below. 2 ( ) WP f 1.0 f W 2 2 (a) Does it satisfy Nyquists criterion for no intersymbol interference for the corresponding pulse of duration T b ? ( T b = 1 / 2 W ) Explain. (Solution)  4pts The Nyquist criterion for the distortionless baseband transmission in the absence of noise is as follows: X n = P ( f nR b ) = T b where R b = 1 /T b . The plotting of this signal can be done by locating the frequency shifted P ( f ) at every 2 W and summing all of them. 2 ( 2 ) W P f nW 1.0 f W 2 2 After the summation, it becomes a constant T b , which satisfies the Nyquist criterion. (b) What is the corresponding pulse? (Solution) 3pts The corresponding pulse is the inverse Fourier transform of the given spectrum. By taking the inverse Fourier transform of P ( f ) = 1 (2 W ) 2 (2 W  f  ), if 2 W f 2 W , we get 1 p ( t ) = Z  P ( f ) e j 2 ft df = 2 Z 2 W 1 (2 W ) 2 (2 W f )cos(2 ft ) df = 1 2 W 2 " (2 W f ) sin(2 ft ) 2 t 2 W + 1 2 t Z 2 W sin(2 ft ) df # = 1 4 W 2 t  1 2 t cos(2 ft ) 2 W = 1 cos(4 Wt ) 8 W 2 2 t 2 = sin(2 Wt ) 2 Wt 2 = sinc 2 (2 Wt ) = sinc 2 ( t/T b ) (c) Compare the above pulse with that obtained from a rectangular spectrum and a full raised cosine spectrum. Give advantages and disadvantages of each. (Solution) 3pts The signals obtained from the rectangular spectrum and a full raised cosine spectrum are p rect = sinc(2 Wt ) p cosine = sinc(4 Wt ) 1 16 W 2 t 2321 1 2 3 0.2 0.4 0.6 0.8 1 f/W 2WP(f) raised consine triangular...
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 Summer '08
 LAZANO

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