Chapter 10 fourier analysis of signals
using discrete fourier transform
10.1 fourier analysis of signals using the DFT
10.2 DFT analysis of sinusoidal signals
10.3 the time-dependent fourier transform
10.1 fourier analysis of signals using the DFT
Figure 10.1
true frequency spectrum
frequency response of
antialiasing filter
error caused by ideal filter
error caused by
quantization and aliasing
frequency spectrum of
window sequence
error caused by windowing
in time domain and
spectral sampling
Figure 10.2
  2 / N
   / T  2f s / N
f   / 2  f s / N
10.2 DFT analysis of sinusoidal signals
10.2.1 the effect of windowing
10.2.2 the effect of spectral sampling
10.2.1 the effect of windowing
X (e j )  V (e j )
Before windowing:
x[n]  A0 cos(0n  0 )  A1 cos(1n  1)

A0 j 0 j 0 n A0  j 0  j 0 n A1 j1 j1n A1  j1  j1n
e e
 e
e
 e e
 e
e
2
2
2
2
A0 j 0
A
e  (  0 )  0 e j 0  (  0 )
2
2
A
A
 1 e j1 (  1 )  1 e j1 (  1 )
2
2
X ( e j ) 
A0 j 0
A
e  (  0 )  0 e j 0  (  0 )
2
2
A
A
 1 e j1 (  1 )  1 e j1 (  1 )
2
2
X ( e j ) 
After windowing:
v[n]  x[n]w[n]
A0
A
w[n]e j 0 e j 0 n  0 w[n]e  j 0 e  j 0 n
2
2
A
A
 1 w[n]e j1 e j1n  1 w[n]e j1 e  j1n
2
2

A0 j 0
A
e W (e j (  0 ) n )  0 e j 0W (e j (  0 ) n )
2
2
A
A
 1 e j1W (e j ( 1 ) n )  1 e j1W (e j ( 1 ) n )
2
2
V ( e j  )  X ( e j  ) * W ( e j ) 
EXAMPLE
Example 10.3
after windowing
| X ( e j ) |
before windowing
effects of windowing：
(1) spectral lines are broadened
to the mainlobe of window
frequency spectrum
(2) produce sidelobe, its
attenuation is equal to sidelobe
attenuation of window
frequency spectrum
Figure 10.3(a)(b)
broadened spectral lines
lead to:
difficult to confirm the
frequency；
reduced resolution
mainlobe leads to ：
produce false signal；
flood small signal.
leakage:
（magnitudes of two
frequency components
interact each other）
Figure 10.3(c)(d)(e)
EXAMPLE
example10.8 :
2
x[n]  (cos(
n)
14
4
 0.75 cos(
n))
15
w[n]  kaiser ( L  32
~ 64,   5.48)
V (e j )
L  32
L=32
L=42
L  42
L  54
L=54
conservative definition
（exercise10.7）：
resolution=
main-lobe width
L  64
L=64
Conclusion: increase L can increase resolution
Figure 10.10
EXAMPLE
changing the shape of window
can change resolution
2
v[n]  (3.5 * cos(
n)
14
2
 3.5 * 0.75 cos(
n))
25
wR [n] : red
whanning[n] : blue
L  32
Conclusion: shape of window has effect on frequency resolution
Determin window’s shape and length
（1） for kaiser windows:
0.12438( Asl  6.3)


  0.76609( Asl  13.26) 0.4  0.09834( Asl  13.26)

0

24 ( Asl  12)
L
1
155 ml
 ml : main  lobe width
Asl : relative side  lobe level
（2）for blackman window:
look up the table
60  Asl  120
13.26  Asl  60
Asl  13.26
10.2.2 the effect of spectral sampling
V (e j )  V [k ]
EXAMPLE
example10.4 : x[n]  (cos(
Before sampling
4
 0.75 cos(
n)) R64 [n]
15
w[n]' s length L  64
can not reach peak value
After sampling
N=128
N=64
Figure 10.5(a)(b)(f)
2
n)
14
EXAMPLE
example10.5 : x[n]  (cos(
2
n)) R64 [n]
8
w[n]' s length L  64
2
n
16
 0.75 cos(
V [k ]  X (e j )  V (e j )
N=64N=64
only reach the peak
value and zero values
Figure 10.6
N=128N=128
conclusion : increase N
can improve effect
Figure 10.7
N=32
N=32
EXAMPLE
example10.7 :
N=64
N=64
N=128
2
x[n]  (cos(
n)
14
4
 0.75 cos(
n))
15
w[n]  kaiser ( L  32,   5.48)
N  32 ~ 1024
N=128
N=1024
N=1024
Conclusion: increase N can’t
increase resolution
Figure 10.9
EXAMPLE
analyze effects of window length
to DFT using MATLAB
f (t )  cos( 2f1t )  cos( 2f 2t ), f1  2 Hz , f1  2.5 Hz , f s  64 Hz ,
0  n  63
 f (t ) |t  nT
f1[n]  
,T  1/ fs
64  n  128
 0
f 2 [n]  f (t ) |t  nT
0  n  128
do 128 points DFT, respective ly, and compare their difference .
L1=64;
L2=128; N=128; T=1/64
n1=0:L1-1;
x1=cos(2*pi*2*n1*T)+ cos(2*pi*2.5*n1*T)
n2=0:L2-1;
x2=cos(2*pi*2*n2*T)+ cos(2*pi*2.5*n2*T)
k=0:N-1;
X1=fft(x1,N);
X2=fft(x2,N)
stem(k,abs(X1));
hold on;
stem(k,abs(X2),’r.’);
consideration：how to get DFS of periodic sequence with period N using
windowing DFT？
10.3 the time-dependent fourier transform
1.definition
X [n,  ) 

 jm
x
[
n

m
]
w
[
m
]
e

m  
1
x[n  m]w[n] 
2

2
0
X [n,  )e jmd
x[n]  cos(0n2 )
Figure 10.11
Figure 10.12
summary
10.1 fourier analysis of signals using the DFT
10.2 DFT analysis of sinusoidal signals
conclusion: effects of windowing and spectral sampling
10.3 the time-dependent fourier transform
apply the conclusion in 10.2 in analysis of general signals
requirements：
effects of windowing and sampling to DFT line;
concept of frequency resolution，relationship
among the shape and length of window , and the points of
DFT;
DFT analysis of sinusoidal signals .
exercises and experiment
10.21 10.31(a)-(c)
experiment 31
34 and 35（draw them together）
36
37（B）
39（A）