Signals and Systems - Electrical Engineering

10.2 Discrete-Time Fourier Transform 593

FIGURE 10.6
Phase spectrum of
sinusoid in Gaussian
noise using magnitude
masking: (a) magnitude
response, (b) wrapped
phase and
(c) unwrapped phase.

− 1 −0.8 −0.6 −0.4 −0.2^0

(a)

(b)

(c)

0.2 0.4 0.6 0.8

− 1 −0.8 −0.6 −0.4 −0.2^0 0.2 0.4 0.6 0.8

− 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

0

20

40

− 2

0

2

θ(rad)

− 2

0

2

θ^1

(rad)

ω/π

|X

(e

jω

)|

nExample 10.11

Consider two FIR filters with impulse responses

h 1 [n]=

∑^9

k= 0

1

10

δ[n−k]

h 2 [n]=0.5δ[n−3]+1.1δ[n−4]+0.5δ[n−5]

Determine which of these filters has linear phase, and use the MATLAB functionunwrapto find

their unwrapped phase functions. Explain the results.

Solution

The transfer function of the filter withh 1 [n] is

H 1 (z)=

1

10

∑^9

n= 0

z−n=0.1

1 −z−^10

1 −z−^1

=0.1

z^10 − 1

z^9 (z− 1 )

=0.1

∏ 9

k= 1 (z−e

j 2 πk/ (^10) )
z^9
Because this filter has nine zeros on the unit circle, its phase is not continuous and it cannot be
unwrapped. The impulse response of the second filter is symmetric aboutn=4; thus its phase is