HW_Set_5_Solutions - Introduction to Signal Processing...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Introduction to Signal Processing Hayder Radha - 1 - ECE 366 Homework Set #5 Solutions ECE 366 – Introduction to Signal Processing Spring 2009 Michigan State University Department of Electrical and Computer Engineering [1] Problem 6.13-10 (a) From Exercise E6.1a in the book, we have: () 22 1 1( 1 ) 4 cos 3 n n x tn t n π = =+ . We also have, ( ) 2 x tt = for 11 t −<<+ with a period of 0 2 T = and hence a fundamental frequency 0 ω = . We know that this is a power signal with a power x P that can be evaluated using the standard formula for average power. For a periodic signal, we have: 0 2 0 1 x T Px t d t T = . Parseval’s theorem states that the signal power x P can be evaluated using the Fourier series amplitudes: 0 1 1 2 x n n PC C = . Here, we need to show that both of the above two expressions for x P lead to the same value.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Introduction to Signal Processing Hayder Radha - 2 - ECE 366 Now, using the standard power expression: () 0 1 2 22 0 1 111 25 x T Px t d t t d t T + === ∫∫ . Now, using the second expression for x P : 2 0 11 1( 1 ) 4 23 2 n xn nn PC C n π ∞∞ == ⎛⎞⎛ ⎛⎞ =+ = + ⎜⎟⎜ ⎜⎟ ⎝⎠ ⎝⎠⎝ ∑∑ . Hence, we have: 2 2 42 4 4 1 ) 1 6 1 8 32 9 n x P ππ = + . Now using the expression given by the problem statement regarding the summation of the infinite series: 4 4 1 1 90 n n = = , leads to: 4 44 4 1 1 18 1 0 8 9 9 90 90 x n P n = + = Hence, the power evaluation based on the Fourier series terms leads to: 18 1 90 5 x P = = . Hence, both expressions for the power x P lead to the same value; and the Parseval’s theorem is verified for this example.
Background image of page 2
Introduction to Signal Processing
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

Page1 / 7

HW_Set_5_Solutions - Introduction to Signal Processing...

This preview shows document pages 1 - 4. Sign up to view the full document.

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