Leo Lam © 2010-2012

Signals and Systems

EE235Lecture 22

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Today’s menu

• Fourier Series (periodic signals)

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It’s here!

Solve

Given

Solve

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Reminder from last week

4

• We want to write periodic signals as a series:

• And dn:

• Need T and w0 , the rest is mechanical

00 0( ) 2 /jn t

nn

x t d e T

T

tjnn dtetfT

d 0)(1

Leo Lam © 2010-2012

Harmonic Series

5

• Example (your turn):

• Write it in an exponential series:

• d0=-5, d2=d-2=1, d3=1/2j, d-3=-1/2j, d4=1

4( ) 5 2cos(2 ) sin(3 ) j tx t t t e

(0)( ) (2)( ) ( 2)( )

(3)( ) ( 3)( ) (4)( )

1( ) 5 2

2

1(1)

2

j t j t j t

j t j t j t

x t e e e

e e ej

0

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Harmonic Series

6

• Graphically:

(zoomed out in time)

One period: t1 to t2

All time

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Harmonic Series (example)

7

• Example with d(t) (a “delta train”):

• Write it in an exponential series:

• Signal is periodic: only need to do one period• The rest just repeats in time

t

T

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Harmonic Series (example)

8

• One period:

• Turn it to: • Fundamental frequency:• Coefficients:

tT

*

All basis function equally weighted and real! No phase shift!

Complex conj.

Leo Lam © 2010-2012

Harmonic Series (example)

9

• From:

• To:

• Width between “spikes” is:

tT

Fourier spectra

0

1/T

w

Time domain

Frequency domain

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Exponential Fourier Series: formulas

10

• Analysis: Breaking signal down to building blocks:

• Synthesis: Creating signals from building blocks

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Example: Shifted delta-train

11

• A shifted “delta-train”

• In this form:• For one period:

• Find dn:

timeT 0 T/2

*

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Example: Shifted delta-train

12

• A shifted “delta-train”

• Find dn:

timeT 0 T/2

Complex coefficient!

Leo Lam © 2010-2011

Example: Shifted delta-train

13

• A shifted “delta-train”

• Now as a series in exponentials:

timeT 0 T/2

0

Same magnitude; add phase!

Phase of Fourier spectraw

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Example: Shifted delta-train

14

• A shifted “delta-train”• Now as a series in exponentials:

0Phase

0

1/TMagnitude (same as non-shifted)

Leo Lam © 2010-2011

Example: Sped up delta-train

15

• Sped-up by 2, what does it do?

• Fundamental frequency doubled

• dn remains the same (why?)• For one period:

timeT/2 0 m=1 2 3

Tdtet

Td

T

Ttjn

n

1)(

14

4

0

Great news: we can be lazy!

Leo Lam © 2010-2011

Lazy ways: re-using Fourier Series

16

• Standard notation: “ ” means “a given periodic signal has Fourier series coefficients ”

• Given , find where is a new signal based on

• Addition, time-scaling, shift, reversal etc.• Direct correlation: Look up table!• Textbook Ch. 3.1 & everywhere online:

http://saturn.ece.ndsu.nodak.edu/ecewiki/images/3/3d/Ece343_Fourier_series.pdf

kdtx )(

kd)(tx

kdtx )( kdtx )( )(tx)(tx