SPFirst-L07EV

Copyright © Monash University 2009

Signal Processing

First

Lecture 7Fourier Series & Spectrum

1


• Assume that i(t) is the a current signal going through a 1 resistor then the power is:

• Note that it does not depend on freq. and phase but only on the amplitude

Power of a sine wave

2

0

0 0

2

21

2

0[ cos( )] [ ]2[ cos( )] 0

T

T

AP A t dt W

A


• Using the Fourier Series x(t) can be written as the sum of many harmonics each contributing

• Proof see lecture recording

Power of a periodic signal

3

222

01 2

nn

n

AP A a


Proof:

4

222

01 2

nn

n

AP A a

0(2 / )0 0

1( ) cos(2 ) j k T t

k k kk

x t A A kf t a e

0 0

0 00 0

2 *1 1( ) ( ) ( )T T

T TP x t dt x t x t dt

0

0 0

0 0

(2 / ) (2 / )*1T

j k T t j l T tk lTP a e a e dt


Proof:

5

222

01 2

nn

n

AP A a

0

0 0

0 0

(2 / ) (2 / )* 1T

j k T t j T tk T

kP a a e e dt

2*k k kP a a a

k

kdtee

T

TktTjtTj

1

01 0

00

0

)/2()/2(

0


Proof:

6

222

01 2

nn

n

AP A a

22 2 2 2 2 2 2*0 0 0

1 1 12k k k k k

k k kP a a a a a a a a

12k ka X

is the phasor of 0cos(2 )k kA kf t kjk kX A e

2 2 21 14 4k k ka X A 2 2

0 0a A


Problem

7

Approximate a sine wave using three delayed square waves

0 0 1 2( ) sin(2 ) ( ) ( ) ( )s t f t x t x t x t

1t

01

t

T1

1t

T2

0 ( )x t

1( )x t

2 ( )x t


Problem

8

02( ) j k f tk

k

x t a e

0 02 20

1 1( ) sin(2 )2 2

j k f t j k f ts t f t e ej j

0 0 0 1 1 2 2( ) sin(2 ) ( ) ( ) ( )s t f t a x t a x t a x t

The square waves have a positive DC component we need to allow for more degrees of freedom


Problem

9

I which sense can we say the approximation is good?

SPFirst-L07EV

Documents

Transcript of SPFirst-L07EV