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M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 1 Fourier Series and Fourier Transform Complex exponentials Complex version of Fourier Series Time Shifting, Magnitude, Phase Fourier Transform Copyright © 2007 by M.H. Perrott All rights reserved.

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M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 1

Fourier Seriesand

Fourier Transform• Complex exponentials• Complex version of Fourier Series• Time Shifting, Magnitude, Phase• Fourier Transform

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 2

The Complex Exponential as a Vector

• Euler’s Identity:

Note:

• Consider I and Q as the real and imaginary parts– As explained later, in communication systems, I stands

for in-phase and Q for quadrature• As t increases, vector rotates counterclockwise

– We consider ejwt to have positive frequency

e jωt

I

Q

cos(ωt)

sin(ωt)

ωt

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 3

The Concept of Negative Frequency

Note:

• As t increases, vector rotates clockwise– We consider e-jwt to have negative frequency

• Note: A-jB is the complex conjugate of A+jB– So, e-jwt is the complex conjugate of ejwt

e-jωt

I

Q

cos(ωt)

-sin(ωt)

−ωt

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 4

Note:

• As t increases, the addition of positive and negative frequency complex exponentials leads to a cosine wave– Note that the resulting cosine wave is purely real and

considered to have a positive frequency

e-jωt

I

Q

ejωt

2cos(ωt)

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 5

Subtract Positive and Negative Frequencies

Note:

• As t increases, the subtraction of positive and negative frequency complex exponentials leads to a sine wave– Note that the resulting sine wave is purely imaginary and

considered to have a positive frequency

-e-jωt

I

Q

ejωt

2sin(ωt)

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 6

Fourier Series

• The Fourier Series is compactly defined using complex exponentials

• Where:

t

T

x(t)

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 7

From The Previous Lecture

• The Fourier Series can also be written in terms of cosines and sines:

t

T

x(t)

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 8

Compare Fourier Definitions• Let us assume the following:

• Then:

• So:

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 9

Square Wave Example

t

T

T/2

x(t)

A

-A

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 10

Graphical View of Fourier Series• As in previous lecture, we can plot Fourier Series

coefficients– Note that we now have positive and negative values of n

• Square wave example:

2Aπ

2A

-2Aπ

-2A

nn

BnAn

1 3 5 7 9

-9 -7 -5 -3 -1

1 3 5 7 9

-9 -7 -5 -3 -1

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 11

2Aπ

2A

-2Aπ

-2A

ff

BfAf

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

• A given Fourier coefficient, ,represents the weight corresponding to frequency nwo

• It is often convenient to index in frequency (Hz)

Indexing in Frequency

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 12

The Impact of a Time (Phase) Shift

• Consider shifting a signal x(t) in time by Td

t

T/4

x(t)

T

T/4

A

-A

t

T

T/2

x(t)

A

-A

• Define:

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 13

Square Wave Example of Time Shift

• To simplify, note that except for odd n

t

T/4

x(t)

T

T/4

A

-A

t

T

T/2

x(t)

A

-A

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 14

2Aπ

f

f

Bf

Af

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

2Aπ

2A

-2Aπ

-2A

f

f

Bf

Af

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

Graphical View of Fourier Series

t

T/4

x(t)

T

T/4

A

-A

t

T

T/2

x(t)

A

-A

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 15

Magnitude and Phase• We often want to ignore the issue of time (phase)

shifts when using Fourier analysis– Unfortunately, we have seen that the An and Bn

coefficients are very sensitive to time (phase) shifts

• The Fourier coefficients can also be represented in term of magnitude and phase

• where:

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 16

Graphical View of Magnitude and Phase

t

T/4

x(t)

T

T/4

A

-A

t

T

T/2

x(t)

A

-A

f-3

T

-5

T

-1

T

-7

T

-9

T

2Aπ

2A

f

f

Xf

Φf

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

2Aπ

2A

Φf

2Aπ

2A

f

Xf

1

T

-3

T

-5

T

-1

T

3

T

5

T

7

T

9

T

-7

T

-9

T

2Aπ

2A

-π/2

π/2

1

T

3

T

5

T

7

T

9

T

π

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 17

Does Time Shifting Impact Magnitude?• Consider a waveform x(t) along with its Fourier

Series

• We showed that the impact of time (phase) shifting x(t) on its Fourier Series is

• We therefore see that time (phase) shifting does not impact the Fourier Series magnitude

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 18

Parseval’s Theorem• The squared magnitude of the Fourier Series

coefficients indicates power at corresponding frequencies– Power is defined as:

Note:* meanscomplexconjugate

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 19

The Fourier Transform• The Fourier Series deals with periodic signals

• The Fourier Transform deals with non-periodicsignals

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 20

Fourier Transform Example

• Note that x(t) is not periodic• Calculation of Fourier Transform:

t

x(t)

T

A

-T

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 21

Graphical View of Fourier Transform

t

x(t)

T

A

-T

This is calleda sinc function

X(f)2TA

2T1

2T-1

f

M.H. Perrott © 2007 Fourier Series and Fourier Transform, Slide 22

Summary• The Fourier Series can be formulated in terms of

complex exponentials– Allows convenient mathematical form– Introduces concept of positive and negative frequencies

• The Fourier Series coefficients can be expressed in terms of magnitude and phase– Magnitude is independent of time (phase) shifts of x(t)– The magnitude squared of a given Fourier Series coefficient

corresponds to the power present at the corresponding frequency

• The Fourier Transform was briefly introduced– Will be used to explain modulation and filtering in the

upcoming lectures– We will provide an intuitive comparison of Fourier Series

and Fourier Transform in a few weeks …