10.1 fourier analysis of signals using the DFT10.2 DFT analysis of sinusoidal signals10.3 the time-dependent fourier transform

Chapter 10 fourier analysis of signals using discrete fourier transform

Figure 10.1

10.1 fourier analysis of signals using the DFT

Figure 10.2

true frequency spectrum

error caused by ideal filter

error caused by quantization and aliasing

error caused by windowing in time domain and spectral sampling

frequency response of antialiasing filter

frequency spectrum of window sequence

NffNfTN ss /2/ /2/ /2

10.2 DFT analysis of sinusoidal signals

10.2.1 the effect of windowing

10.2.2 the effect of spectral sampling

)cos()cos(][ 111000 nAnAnx

10.2.1 the effect of windowing

)()( jj eVeX

)(2

)(2

)(2

)(2

)(

11

11

00

00

11

00

jj

jjj

eA

eA

eA

eA

eX

njjnjjnjjnjj eeA

eeA

eeA

eeA

11110000

22221100

Before windowing:

][][][ nwnxnv

)(2

)(2

)(2

)(2

)(*)()(

)(1)(1

)(0)(0

1111

0000

njjnjj

njjnjjjjj

eWeA

eWeA

eWeA

eWeA

eWeXeV

After windowing:

njjnjj

njjnjj

eenwA

eenwA

eenwA

eenwA

1111

0000

][2

][2

][2

][2

11

00

)(2

)(2

)(2

)(2

)(

11

11

00

00

11

00

jj

jjj

eA

eA

eA

eA

eX

Figure 10.3(a)(b)

Example 10.3

EXAMPLE

|)(| jeX

before windowing

after 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(c)(d)(e)

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）

)48.5,64~

32(][

))15

4cos(75.0

)14

2(cos(][

:8.10

Lkaisernw

n

nnx

example

Conclusion: increase L can increase resolution Figure 10.10

EXAMPLE

42L

64L

32L

54L

L=32

L=42

L=54

L=64

conservative definition

（ exercise10.7）：resolution=

main-lobe width

)( jeV

32

:][

:][

))25

2cos(75.0*5.3

)14

2cos(*5.3(][

L

bluenw

rednw

n

nnv

hanning

R

EXAMPLE changing the shape of window can change resolution

Conclusion: shape of window has effect on frequency resolution

26.13

6026.13

12060

0

)26.13(09834.0)26.13(76609.0

)3.6(12438.04.0

sl

sl

sl

slsl

sl

A

A

A

AA

A

1155

)12(24

ml

slAL

Determin window’s shape and length

（ 2） for blackman window: look up the table

（ 1 ） for kaiser windows:

levellobesiderelativeA

widthlobemain

sl

ml

:

:

10.2.2 the effect of spectral sampling

][)( kVeV j

64]'[

][))15

4cos(75.0

)14

2(cos(][:4.10

64

Llengthsnw

nRn

nnxexample

EXAMPLE

Figure 10.5(a)(b)(f)

After sampling

Before sampling

can not reach peak value

N=64

N=128

N=64

64]'[

][))8

2cos(75.0

)16

2(cos(][:5.10

64

Llengthsnw

nRn

nnxexample

Figure 10.6

EXAMPLE

)()(][ jj eVeXkV only reach the peak

value and zero values

N=128

Figure 10.7

N=64

N=128

conclusion : increase N can improve effect

1024~32

)48.5,32(][

))15

4cos(75.0

)14

2(cos(][

:7.10

N

Lkaisernw

n

nnx

example

Figure 10.9

EXAMPLE

Conclusion: increase N can’t increase resolution

N=32

N=64

N=128

N=1024

N=32

N=64

N=128

N=1024

EXAMPLE

.difference their compare and ly,respective DFT, points 128 do

1280|)(][

/1,128640

630|)(][

,64,5.2,2),2cos()2cos()(

2

1

1121

ntfnf

fTn

ntfnf

HzfHzfHzftftftf

nTt

snTt

s

analyze effects of window length to DFT using MATLAB

L1=64; L2=128; N=128; T=1/64n1=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？

m

mjemwmnxnX ][][),[

10.3 the time-dependent fourier transform

1.definition

2

0),[

2

1][][ denXnwmnx mj

Figure 10.11

Figure 10.12

)cos(][ 20nnx

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）