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Fourier Series and Fourier Transform

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(1)

Fourier Series

and

Fourier Transform

Complex exponentials

Complex version of Fourier Series

Time Shifting, Magnitude, Phase

Fourier Transform

(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 increases, vector rotates

counterclockwise

e jωt I Q cos(ωt) sin(ωt) ωt

(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

(4)

Add Positive and Negative Frequencies

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

e-jωt

I Q

ejωt

(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

(6)

Fourier Series

The Fourier Series is compactly defined using complex exponentials

Where:

t T

(7)

From The Previous Lecture

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

t T

(8)

Compare Fourier Definitions

Let us assume the following:

Then:

(9)

Square Wave Example

t T T/2 x(t) A -A
(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 nSquare wave example:

2A π 2A-2An n Bn An 1 3 5 7 9 -9 -7 -5 -3 -1 1 3 5 7 9 -9 -7 -5 -3 -1

(11)

2A π 2A-2A -2Af f Bf Af 1 T -3 T -5 T -1T 3 T T5 7T 9T -7 T -9 T 1 T -3 T -5 T -1T 3 T 5T 7T T9 -7 T -9 T

Indexing in Frequency

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

nw

o
(12)

The Impact of a Time (Phase) Shift

Consider shifting a signal

x(t)

in time by

T

d

t T/4 x(t) T T/4 A -A t T T/2 x(t) A -A • Define:

(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

(14)

2A π f f Bf Af 1 T -3 T -5 T -1T 3 T 5T 7T 9T -7 T -9 T 1 T -3 T -5 T -1T 3 T 5 T 7 T 9 T -7 T -9 T 2A π 2Af f Bf Af 1 T -3 T -5 T -1T 3 T T5 7T T9 -7 T -9 T 1 T -3 T -5 T -1T 3 T T5 7T T9 -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

(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

(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 2A π 2Af f Xf Φf 1 T -3 T -5 T -1T 3T 5T 7T T9 -7 T -9 T 1 T 3T T5 7T T9 2A π 2A 3π Φf 2A π 2Af Xf 1 T -3 T -5 T -1T 3T 5T 7T T9 -7 T -9 T 2A π 2A 3π π/2 π
(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

(18)

Parseval’s Theorem

The squared magnitude of the Fourier Series coefficients indicates

power

at corresponding frequenciesPower is defined as: Note: * means complex conjugate
(19)

The Fourier Transform

The Fourier Series deals with

periodic

signals

The Fourier Transform deals with

non-periodic

(20)

Fourier Transform Example

Note that

x(t)

is

not

periodic

Calculation of Fourier Transform:

t x(t)

T A

(21)

X(j2πf)

2TA

1 -1

f

Graphical View of Fourier Transform

t x(t) T A -T This is called a

sinc

function
(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

References

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