EE210_F11_HW4

# EE210_F11_HW4 - EE210-2 Fall 2011 Prof Essam Marouf Linear...

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EE210-2, Fall 201 1 Prof. Essam Marouf Linear Systems Theory (Transform Theory & Applications) ---------------------------------------------------------------------------------------------------------- Homework #4 To be Completed by Wednesday October 1 9 , 201 1 -------------------------------------------------------- ---- See Bracewell, Ch. 8 ------------------------- 1- Determine which of the real signals x(t ) depicted on next page, if any, have Fourier transforms X(f ) that satisfy each of the following conditions: a) Re { X ( f )} = 0, b) Im { X ( f )} = 0 c) The phase of X ( f ) has a linear term of the form 2 π af , where a is a real constant d) X ( f ) df = 0 −∞ e) f X ( f ) df −∞ = 0 f) X ( f ) is periodic g) X ( f ) decays as | f | k , k 2, as f →∞ . You need not evaluate X ( f ) explicitly to answer these questions. ----------------------------------------------------------------------------------------------------------- 2- a) Samples of a speech signal may be modeled as a random variable x with probability density function f ( x ) = ce α | x | , where c and are constants. Use a two-domain approach to determine c and if f ( x ) is to have unit area and unit variance. b) Derive the mean < x > and variance σ x 2 of the wave-packet & x ( t ) = exp[ ( t / W ) 2 ] cos(2 f 0 t ) ----------------------------------------------------------------------------------------------------------- 3- a) Show that the convolution of x ( t ) = (1 + t 2 ) 1 with itself yields a new signal y ( t ) which has a 6 dB width that is twice as large as that of

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## This note was uploaded on 11/28/2011 for the course EE 210 taught by Professor Moriarty,e during the Fall '08 term at San Jose State.

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EE210_F11_HW4 - EE210-2 Fall 2011 Prof Essam Marouf Linear...

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