365 Spectrum Analysis

8/12/2019 365 Spectrum Analysis

1/19

Slide 1

Spectrum Analysis Signals

Periodic Signals Fourier Series Representation of Periodic Signals

Frequency Spectra Amplitude and Phase Spectra of Signals

Signal Through Systems - a Frequency Spectrum Perspective

Non-periodic Signals - Fourier Transform

Random Signals - Power Spectral Density

sin( ) sin( ) cos( ) cos( )sin( ) cos( ) sin( ) sin( )

cos( ) cos( ) cos( ) sin( )sin( ) sin( ) ( ) cos( )

A B A B A B A A A A

A B A B A B A A A A

; sin2

;

; cos2

; cos


2/19

Slide 2

Signals

Signals can be categorized as:

Periodic Signals Non-Periodic Signals (well defined)

Random Signals

Would like to characterize signals in thefrequency domain!

Linear SystemG(jw)

Input Signal Output Signal

x(t) y(t)

Can be characterized by itsFrequency Response

Function

G(jw) = | G(jw)|ejArg[G(jw) ]


3/19

Slide 3

Periodic Signals Fourier SeriesAny periodic function x(t), of period T, can be represented by aninfinite series of sine and cosine functions of integer multiples of itsfundamental frequency w1 = 2/T .

x t A

A k t B k t

x t x t T T

k k

A x t dt

A x t k t dt

B x t k t dt

k kk

A

T

T

T

k T

T

k T

T

0 1 11

1

2

12

02

0

21

0

210

2

2

0

cos sin

:

:

( )

( )cos( )

( )sin( )

w w

w

w

w

w

where and rad/sec and

Amplitude of the DC component

the k harmonic frequency

Fourier Coefficients:

th


4/19

Slide 4

Fourier SeriesEx: triangle signal with

period Tsec.

Summation of Input Signals

-10

-8

-6

-4

-2

0

24

6

8

10

0 0.05 0.1 0.15 0.2

Time (sec)

Acceleration

x t A

k

A

k t

A

B

k

k

k

( )( )

cos( )

82 1

0

0

21

1

0

w

A

T 2T

Take the first six terms and letA = 10, T = 0.1:

x t t

t

t

t

t

t

( ) .105 cos( )

. cos( )

. cos( )

.165 cos( )

.1 cos( )

. cos( )

8 20

0 901 3 20

0 324 5 20

0 7 20

0 9 20

0 067 11 20


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Slide 5

Pointers for Calculating the Fourier Coefficients:

A0 /2 represents the average of the signal x(t). It contains the DC(zero frequency) component of the signal.

When calculating the Fourier coefficients, changing the integral limitswill not affect the results, as long as the integration covers one period ofthe signal.

Fourier Series

3

1 2t

1

42 t-1

A0

2 A0

2

-T/2

T/2

T3T/2

T

t

x(t)

AT

x t k t dt

Tx t k t dt

kT

T

T

T

T

2

2

0

2

2

22

( )cos( )

( )cos( )

T


6/19

Slide 6

Pointers for Calculating the Fourier Coefficients:

Odd functions (signals), for which x(t) = x(t), will only contain sineterms, i.e.Ak = 0, for k = 0, 1, 2, ... .

Even functions (signals), for which x(t) = x(t), will only contain cosineterms, i.e. Bk = 0, for k = 1, 2, ... .

Fourier Series

If sin

sin sin

x t x t x t B k t

t t

k

k

( ) ( ) ( ) ( )

( ) ( )

11

If x t x t x t

A

A k t

t t

kk( ) ( ) ( ) cos( )

cos( ) cos( )

0

112 w

T

T t

x(t)

0

A

T 2T0


7/19

Slide 7

Fourier SeriesEx: Half rectified sine wave

Symbolic:

Q: What is the period, T, of this signal?

What is theFundamental Frequency of thissignal?

Q: Expand the signal into its Fourier series.

Find the correspondingFourier Coefficients!

Numeric:

Let w 1 [rad/s], E = 1 [V].

time (t)

x(t)

E

/ //

x t E t t

t

( ) sin( )

ww

w

-

0

0 0


8/19

Slide 8

Fourier SeriesSymbolic:

Calculate Fourier Coefficients:

A0 :

Ak :

Numeric:

AT

x t dt E dtT

0 0 0

2 2

2

sin( t)( ) w

w

w

AT

x t k t dtkT

2

10 ( ) cos( )w


9/19

Slide 9

Fourier SeriesSymbolic:

Calculate Fourier Coefficients:

Bk :

Fourier Series Representation:

Numeric:

BT

x t k t dtkT

2

10

( ) sin( )w

x t E E

B

t

Et t

( )

2

22 4

1

13 1

15 3

sin

cos cos


10/19

Slide 10

Fourier SeriesEx:

Q: What is the period, T, of this signal?What is theFundamental Frequency of this signal?

Q: Expand the signal into its Fourier series.

time (t)

x(t)E

6

x t t t( ) 4 2 22 -

AT

x t d t

AT

x t k t dtk

B

x t t t t t

TT

k T

T kk

02

2

12

2

2

1

2 2 2

2 163

2 161 0

8

3

16

2

1

2

22

1

3

32

1

4

42

; WHY?

( )

( ) cos( ) ( )

( ) cos cos cos cos

//

/

/


11/19

Slide 11

Fourier Series Different Fourier Series Representations:

x t A

A k t B k t

A

M k t

AM k t

M A B A M

B

AB M

A

B

k kk

k kk

k kk

k k k k k k

kk

kk k k

kk

k

01 1

1

0

11

01

1

2 2

1

1

2

2

2

cos sin

cos cosine series

sin sine series

cos( )

tan ; sin( )

tan

; A Bk k

2 2

Ak

Bk

qk


12/19

Slide 12

Fourier Series Fourier Series in Complex Form:

x t A

A k t B k t

e t j tt e e

t e e

e

k kk

j t jj t j t

j t j t

j t

0 1 11

12

12

2cos sin ( )

Recall cos sinsin

cos

Replace sine and cosine terms in ( ) by their representation to obtain:

( ) ( )( )

( )

x t C C T

x t dt

C A jB C A jB

C A B M

C B

A

A C

B C

kk

k T

T

k k k k k k

k k k k

kk

k

k k

k k

e ejk t jk t

1 11

1

2

1

2

2

2

2

2

12

2 2 12

1

where

Arg[ ] tan ;

Re

Im

( )

[ ]

[ ]


13/19

Slide 13

Fourier Series Summary

x t A

A k t B k t

AM k t M A B

B

A

AM k t M A B

A

B

C C A jB

k kk

k kk

k k k kk

k

k k k k k kk

kk

kk

k kejk t

01 1

1

01

1

2 2 1

01

2 2 1

1

2

2

2

1

2

1

cos sin

cos , tan

sin , tan

w w

w q q

w

w

;

;

; kk k k k

k kk

k k k k

k k k k

C A B M

C B

A

A M C

B M C

,

tan

cos( ) Re[ ]

sin( ) Im[ ]

12

2 2 12

1

2

2

Arg[ ]

q

q


14/19

Slide 14

Fourier SeriesEx: Write the following periodic signal in a

sine and cosine series form and plot itsmagnitude and phase vs frequency plot.

x t t t

t t

( ) sin( ) cos( )

sin( ) cos( )

4 24 3 24

5 72 12 72

Ex: Write the following periodic signal in acomplex Fourier series form and plot itsmagnitude and phase vs frequency plot.

x t t t

t t

( ) sin( ) cos( )

sin( ) cos( )

4 24 3 24

5 72 12 72


15/19

Slide 15

Another interpretation of the Fourier Series representation of periodicsignals is that the combination of the Fourier coefficients and theircorresponding harmonics characterizes the periodic signal.

A periodic signal can be represented by:

Time Domain:

Signal Amplitude vs Time

Frequency Domain:

Fourier coefficient Amplitude and Phase vs Frequency.

Frequency Spectra

x t A M k tk kk

0

112

cos w q

x(t)

Time (t)

Mk

0 w1 2w1 3w1 4w1 Frequency(w)

Amplitude Spectrumqk

0 w1 2w1 3w1 4w1 Frequency(w)

Phase Spectrum


16/19

Slide 16

Frequency SpectraEx: Plot the Amplitude and Phase Spectra of

the following signal:

x t t

t

t

( ) cos( )

co s( )

sin ( )

5 4 40

3 60

6 100

Ex: What is the fundamental frequency of thesignal in the previous example?

Ex: What is the output signal of a low passfilter if x(t) in the previous example is

passed through a passive low-pass filterwith gain 1 and time constant 0.1 sec?


17/19

Slide 17

Signals Through SystemsLinear System

G(jw)

Input Periodic Signal Output Signal

x(t) y(t)

Can be characterized by itsFrequency Response

Function

G(jw) = | G(jw)|ejArg[G(jw) ]

x t A

M k tk kk

0

11

2

cos w q

y t ?

y t

A

G j M G j k t G j

AM k t

k k k kk

k kk

k

OUTPUT

OUTPUT

k

OUTPUT

0

11

01

1

2 0

2

( ) ( ) cos( [ ( )])

cos( )

w w q w

w q

w

w

Arg


18/19

Slide 18

0 10 30 50 70 90 1000

0. 5

1

1. 5

2

2. 5

3

Frequency (Hz)

Amplitude(volt)

2. 5

0. 8

0. 50. 4

0. 3

0 10 30 50 70 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0. 2

0. 4

0. 6

0. 8

1

Frequency (Hz)

Phase(ra

d)

Signals Through SystemsEx:An inkjet nozzle "firing" signal has the following amplitude (Mk) and phase spectra (k).

(This is the spectrum generated from the sine with phase shift form of the Fourier Series.)

Write down the Fourier Series of this signal, xA(t).


19/19

Slide 19

Signals Through SystemsEx: If the signal xA(t), is to pass through a filter with the following frequency response function:

Write down the Fourier Series of the output signal, xB(t).

Plot the amplitude and phase spectrum of the output signal, xB(t).

G jj

( ).

ww

1

0 00318 1

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