E4702 Fall 2004 Midterm Solutions

E4702 Fall 2004 Midterm Solutions - E4702 Midterm#2...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 2-3-2-1 1 2 3 0.2 0.4 0.6 0.8 1 f/W 2WP(f) raised consine triangular...
View Full Document

Page1 / 6

E4702 Fall 2004 Midterm Solutions - E4702 Midterm#2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online