HW_6(483)S09_Solns

# HW_6(483)S09_Solns - EE 483 Introduction to Digital Signal Processing Spring 2009 Homework#6 Solutions 6.1 The frequency response of the LTI

This preview shows pages 1–2. Sign up to view the full content.

1 EE 483 – Introduction to Digital Signal Processing Spring 2009 Homework #6 Solutions 6.1 The frequency response of the LTI discrete-time system is given by . e a e a e a e a ) e ( H m m m jm j ϖ + - - ϖ + - ϖ + - ϖ - ϖ + + + = ) 3 j( 1 ) 2 j( 2 ) 1 j( 2 1 There is no value of m that will make ) e ( H j ϖ a real function of ϖ . 6.2 ( 29 ] 1 [ ) ]( 0 [ ] 1 [ ) 1 ]( 0 [ ) ( 2 h e e h e e h e h e H j j j j j j + + + + + + = + + + + + + = - - - - ω ω ω ω ω ω ]). 1 [ cos ] 0 [ 2 ( h h e j + + + = - ω ω Thus, we require 1 1 7 0 0 2 7 0 = + = ] [ h ) . cos( ] [ h ) e ( H . j and . 0 ] 1 [ h ) 4 . 0 cos( ] 0 [ h 2 ) e ( H 4 . 0 j = + = Solving these two equations we get 3.2006 - = ] 0 [ h and . ] 1 [ h 5.896 = 6.3 α + α - = α + α + α + α - = α + α + α + α - = ϖ + ϖ - = ϖ + ϖ = ϖ 1 1 ) 1 )( 1 ( ) 1 )( 1 ( 1 2 1 ) 1 ( 4 1 ) cos( 1 ) ( cos 1 ) cos( 1 ) sin( 2 tan 2 2 2 2 2 2 2 2 2 c c c c c Because ϖ 2 tan 0 c for π ϖ c 0 , and , 2 tan 1 2 tan 1 ϖ + ϖ - = α c c 1 0 α . Therefore the pole is inside the unit circle which, since we assume the filter is causal, implies stability. 6.4 . ) 1 ( 1 2 1 2 1 ) ( 2 1 2 1 α + α + β - + β - α + = - - - - z z z z z H BS Thus, ) cos( ) 1 ( 2 ) 2 cos( 2 ) 1 ( 1 ) 2 cos( 2 ) cos( 8 4 2 2 1 ) ( 2 2 2 2 2 2 2 ϖ α + β - ϖ α + α + α + β + ϖ + ϖ β - β + α

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

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

## This note was uploaded on 02/02/2010 for the course EE 483 taught by Professor Mitra during the Fall '08 term at USC.

### Page1 / 4

HW_6(483)S09_Solns - EE 483 Introduction to Digital Signal Processing Spring 2009 Homework#6 Solutions 6.1 The frequency response of the LTI

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online