Ps4sol

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: the frequency response: �� �� −j�t e−4|t| e−j�t dt h(t)e dt = H (j� ) = −� −� �0 �� −4(−t) −j�t = e e dt + e−4(t) e−j�t dt −� 0 �0 �� = e(4−j�)t dt + e(−4−j�t)t dt −� 0 �0 � � �� 1 1 (4−j� )t � (−4−j� )t � + = e e � � 4 − j� −4 − j � −� 0 1 1 = (1 − 0) + (0 − 1) (remember that e−�+ja = 0 for any real a) 4 − j� −4 − j � 1 1 4 + j � + 4 − j � = + = 4 − j� 4 + j� 16 + � 2 8 H (j� ) = . 16 + � 2 3 Next, we find the Fourier series coefficients of x(t), ak : ak � � 1 12 −jk�0 t = x(t) e dt = [2α (t) − α (t − 1)] e−jk�0 t dt T T 3 −1 ⎩ 1 � −jk�0 (0) 2e − e−jk�0 (1) = 3 2 1 −jk�0 , for allk. = −e 3 3 Now we are ready to find bk , the Fourier series representation of the output y (t): bk = ak H (jk0 ) (O & W, Section 3.8, p.226, and specifically eq.(3.124) ) � � �� 8 2 1 −jk�0 −e = 3 3 16 + (k�0 )2 �� 2� 8 2 − e−jk 3 = � ⎩ , for all k. 3 16 + k 2� 2 3 4 Pr...
View Full Document

This document was uploaded on 02/09/2014.

Ask a homework question - tutors are online