Leo Lam © 2010-2013

Signals and Systems

EE235

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So stable

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Today’s menu

• Chocolates and cookies• Fourier Series (periodic signals)

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Exponential Fourier Series: formulas

4

• Analysis: Breaking signal down to building blocks:

• Synthesis: Creating signals from building blocks

Leo Lam © 2010-2013

Harmonic Series (example)

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• 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)

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• One period:

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

tT

*

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

Complex conj.

Leo Lam © 2010-2013

Harmonic Series (example)

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• From:

• To:

• Width between “spikes” is:

tT

Fourier spectra

0

1/T

w

Time domain

Frequency domain

Leo Lam © 2010-2013

Example: Shifted delta-train

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• A shifted “delta-train”

• In this form:• For one period:

• Find dn:

timeT 0 T/2

*

Leo Lam © 2010-2013

Example: Shifted delta-train

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• A shifted “delta-train”

• Find dn:

timeT 0 T/2

Complex coefficient!

Leo Lam © 2010-2013

Example: Shifted delta-train

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• A shifted “delta-train”

• Now as a series in exponentials:

timeT 0 T/2

0

Same magnitude; add phase!

Phase of Fourier spectraw

Leo Lam © 2010-2013

Example: Shifted delta-train

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

0Phase

0

1/TMagnitude (same as non-shifted)

Leo Lam © 2010-2013

Example: Sped up delta-train

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• 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!

The new T.

Leo Lam © 2010-2013

Lazy ways: re-using Fourier Series

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• 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

Leo Lam © 2010-2013

Graphical: Time scaling: Fourier Series

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• Example: Time scaling up (graphical)

• New signal based on f(t):

• Using the Synthesis equation:

Fourier spectra

0

tjn

nedtg 02)(

Twice as far apart as f(t)’s

Leo Lam © 2010-2013

Graphical: Time scaling: Fourier Series

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• Spectra change (time-scaling up):• f(t)

• g(t)=f(2t)

• Does it make intuitive sense?

0

1

0

1

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Graphical: Time scaling: Fourier Series

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• Spectra change (time scaling down):• f(t)

• g(t)=f(t/2) 0

1

0

1

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Fourier Series Table

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0

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

x t

y t ax t b

y t x at

y t x t

y t x t t

0 0

0 0

0 0

0,

,

k

y yk k

y yk k

yk k

jk tyk k

d

d ad k d ad b

d d a

d d

d d e

Added constant only affects DC term

Linear ops

Time scaleSame dk, scale w0reverse

reverse

Shift in time –t0 Add linear phase term –jkw0t0

• Fourier Series Properties:

Leo Lam © 2010-2013

Fourier Series: Fun examples

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• Rectified sinusoids

• Find its exponential Fourier Series:

t0

f(t) =|sin(t)|

Expand as exp., combine, integrate

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n

tjnen

tf 0

)41(

2)(

2

Leo Lam © 2010-2013

Fourier Series: Circuit Application

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• Rectified sinusoids

• Now we know:

• Circuit is an LTI system: • Find y(t)• Remember:

+-sin(t)

fullwaverectifier

y(t)f(t)

Where did this come from?

S

Find H(s)!

Leo Lam © 2010-2013

Fourier Series: Circuit Application

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• Finding H(s) for the LTI system:

• est is an eigenfunction, so• Therefore:• So:

)()( sHety stststst esHesHse )()(3

13

1)(

ssH

Shows how much an exponential gets amplified at different frequency s

Leo Lam © 2010-2013

Fourier Series: Circuit Application

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• Rectified sinusoids

• Now we know:

• LTI system: • Transfer function: • To frequency:

+-sin(t)

fullwaverectifier

y(t)f(t)

Leo Lam © 2010-2013

Fourier Series: Circuit Application

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• Rectified sinusoids

• Now we know:

• LTI system: • Transfer function:• System response:

+-sin(t)

fullwaverectifier

y(t)f(t)

Leo Lam © 2010-2013

Summary

• Fourier Series circuit example