Digital Signal Processing

Digital Signal Processing

http://kom.aau.dk/~zt/cources/DSP/

Exercises of Lecture 7 (MM7)

1.

A continuous time system, with impulse response

h

_c

(t )

is given by the system function

2

)

2

( ) (

b a s

a s s

H

_c

+ +

= +

Determine

H

₁

( z )

for a discrete-time system such that

h

₁

( n ) = h

_c

( nT )

(using the impulse invariance method)

2.

The system function of a discrete system is

1 4 . 0 1

2 .

0

1

1 1

) 2

(

_{− − − −}

− −

= −

z e z

e z

H

.

Assume that this system was designed by the impulse invariance method with

T

_d

= 2

, i.e.

) 2 ( 2 )

( n h n

h =

_c . Determine

2 . 0 5 . 0 1 . 0 ) 1

( − +

= +

s s s

H

_c

3.

A discrete-time filter with system function H(z) is designed by transforming a continuous-time filter with system function

H

_c

(s )

. It is desired that

0

( )

0

)

(

_Ω=

=

= H j Ω e

H

^j _c

ω ω

What is the requirement to

H

_c

(s )

in order to obtain this (using the impulse invariance method) ?

4.

Suppose we are given an ideal lowpass discrete-time filter with the frequency response:

⎩ ⎨

⎧

≤

<

= <

π ω π

π

ω

ω

4 / , 0

4 / ,

) 1 ( e

^j

H

What is the frequency response

H

₁

( e

^jω

)

for a system having the impulse response

] 2 [ ]

1

[ n h n

h =

?

5. An analog 2. order Butterworth lowpass filter is given by the transfer function:

2 2

2

2 )

(

c c

c c

s s s

H + Ω + Ω

= Ω

s

c

= 2 π 800 rad / Ω

Ω

c

is the 3 dB cutoff frequency of the analog filter.

A digital filter, H(z), is designed to approximate the analog filter, using the bilinear transformation and having the 3 dB cutoff frequency 3

ω

_c

= π

a) Determine the sample frequency given the information about Ω

c

and ω

_c

b) Use the bilinear transformation to design H(z)

Thanks Borge Lindberg for providing part of the exercises and solutions.