Signals and Systems - Electrical Engineering

680 C H A P T E R 11: Introduction to the Design of Discrete Filters

FIGURE 11.20

(a) Elliptic band-pass filter and (b)

high-pass filter using cheby2: (clockwise

for each side from top left) poles and

zeros, magnitude, phase frequency

responses, and loss.

− 1

−0.5

0

0.5

1

Imaginary part

− 1 01

Real part

Loss (dB)

ω/π

0 0.5 1

0

40

20

60

80

100

− 1

−0.5

0

0.5

1

− 1 01

0 0.5 1

0

40

20

60

80

100

Real part

Imaginary part

Loss (dB)

ω/π

(b)

(a)

Magnitude

0 0.2 0.4 0.6 0.8

ω/π

0

0.2

0.4

0.6

0.8

1

Phase (rad)

ω/π

0 0.2 0.4 0.6 0.8

0

5

− 5

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8

0

4

2

6

Magnitude

ω/π

Phase (rad)

ω/π

0

0.2

0.4

0.6

0.8

1

Consider the stability of this filter, and determine if the phase of this filter is linear and what type

of filter it is.

Solution

The impulse responseh[n] is absolutely summable given its finite lengthM; thus the filter is BIBO

stable. Indeed, the apparent pole atz=1, which would make the filter unstable, is canceled by a

zero also atz=1 (notice thatH( 1 )is 0/0, according to the final expression above, indicating that

a pole and a zero atz=1 exist, but also from the sumH( 1 )=1, so there are no poles atz=1).