Statistical and Adaptive Signal Processing (2005 – Manolakis, Ingle & Kogon) | Complete Solutions Manual PDF

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, Chapter 2
Discrete-Time Signals and Systems

2.1 Sampling frequency Fs = 100 sam/sec

(a) Continuous-time signal xc (t) = 2 cos(40π t + π /3) has frequency of 20 Hz. Hence


40π n
x(n) = xc (t)|t =n/Fs = 2 cos + π /3
100
40π 2π
which implies that ω0 = 100
= 5
.
(b) Steady-state response yc,ss (t): Given that h(n) = 0.8n u(n), the frequency response function is

1
H (e j ω ) =
1 − 0.8e− j ω
Since ω0 = 2π /5, the system response at ω0 is
1
H (e j ω ) = = 0.9343 e− j 0.2517π
1 − 0.8e− j 2π/5
Hence yss (n) = 2(0.9343) cos(2π n/5 + π /3 − 0.2517π ), or

yc,ss(t) = 1. 868 6 cos(40π t + 0.585π )

(c) Any xc (t) that has the same digital frequency ω0 after sampling and the same phase shift as above will
have the same steady state response. Since Fs = 100 sam/sec, the two other frequencies are 120 and 220
Hz.

2.2 The discrete-time signal is

x(n) = A cos(ω0 n) wR (n)

where wR (n) is an N -point rectangular window.

(a) The DTFT of x(n) is determined as

X (e j ω ) = F [ A cos(ω0 n) wR (n)]
   
= (A/2) F e j ω0 wR (n) + (A/2) F e− j ω0 wR (n) (1)

Using the DTFT of wR (n) as
N−1
 sin(ωN/2)
F [wR (n)] = e− j ωn = e− j ω(N−1)/2 (2)
n=0
sin(ω/2)

and the fact that complex exponential causes a translation in the frequency domain (1) can be written
after a fair amount of algebra and trigonometry as

X (e j ω ) = X R (e j ω ) + j X I (e j ω )



1

,2 Statistical and Adaptive Signal Processing - Solution Manual


32−point DTFT (Real) 32−point DTFT (Imaginary)
20 15


15 10

5
10
0
5
−5

0
−10

−5 −15
−4 −2 0 2 4 −4 −2 0 2 4
Figure 2.2bc: Real and Imaginary DTFT and DFT Plots(ω0 = π/4)


where
A sin[(ω − ω0 )N/2]
X R (e j ω ) = cos[(ω − ω0 )(N − 1)/2]
2 sin[(ω − ω0 )/2]
A sin{[ω − (2π − ω0 )]N/2}
+ cos[(ω + ω0 )(N − 1)/2] (3)
2 sin{[ω − (2π − ω0 )]/2}
and
A sin[(ω − ω0 )N/2]
X R (e j ω ) = − sin[(ω − ω0 )(N − 1)/2]
2 sin[(ω − ω0 )/2]
A sin{[ω − (2π − ω0 )]N/2}
− sin[(ω + ω0 )(N − 1)/2] (4)
2 sin{[ω − (2π − ω0 )]/2}

(b) N = 32 and ω0 = π /4. The DTFT plots are shown in Figure 2.2bc.
(c) The DFT samples are shown in Figure 2.2bc.
(d) N = 32 and ω0 = 1.1π /4. The plots are shown in Figure 2.2d.
The added spectrum for the second case above (ω0 = 1.1π /4) is a result of the periodic extension of the
DFT. For a 32-point sequence, the end of each extension does not line up with the beginning of the next
extension. This results in sharp edges in the periodic extension, and added frequencies in the spectrum.

2.3 The sequence is x(n) = cos(πn/4), 0 ≤ n ≤ 15.

(a) The 16-point DFT is shown in the top-left plot of Figure 2.3.
(b) The 32-point DFT is shown in the top-right plot of Figure 2.3.
(c) The 64-point DFT is shown in the bottom plot of Figure 2.3.
(d) The zero padding results in a lower frequency sampling interval. Hence there are more terms in the DFT
representation. The shape of the DTFT continues to fill in as N increases from 16 to 64.

2.4 x(n) = {1, 2, 3, 4, 3, 2, 1}; h(n) = {−1, 0, 1}

, Statistical and Adaptive Signal Processing - Solution Manual 3




32−point DTFT (Real) 32−point DTFT (Imaginary)
20 15


15 10

5
10
0
5
−5

0
−10

−5 −15
−4 −2 0 2 4 −4 −2 0 2 4
Figure 2.2d: Real and Imaginary DTFT Plots (ω0 = 1.1π /4)




16−point DFT 32−point DFT

7 7

6 6

5 5

4 4

3 3

2 2

1 1

0 0
0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4
64−point DFT
8



6



4



2



0
0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4

Figure 2.3: The 16, 32, and 64-point DFTs of x(n) = cos(π n/4)