• Complex exponentials • Complex version of Fourier Series • Time PDF

Fourier Series and Fourier Transform Slide 3. The Concept of Negative Frequency. Note: • As t increases

Lecture 11 The Fourier transform

The Fourier transform we'll be interested in signals defined for all t the Fourier transform of a signal f is the function. F(?) = ?. ?. ?? f(t)e.

Untitled

Find the Fourier transform of the casual exponential f(t) = Ae?¤? u(t) and plot its rule shows that sinc(0) = 1. W1-4. F(w). -. -jot dt. -e. -jwt.

Fourier Series and Fourier Transform

(1) by sin (mw0t) and integrate over one period Example 4: Find the trigonometric Fourier series for the periodic signal x(t). ... jw [e?jwT. 2 ?ejwT.

Fourier Transforms - 2

Fourier series is a way to represent a function as the Fourier Transform can be performed on aperiodic ... x(t) = = = = 0 X (jw) e jwt dw.

a) Determine the function f(t) whose Fourier transform is shown in

T)e¯jwr f(t)e-jwt et dt = f(t)e-jºdt = 10. t = f(t)dt; Problem 3.2.1 a) Find the Fourier transform for of the raised cosine pulse signal defined by:.

? ? ? ? ? ? ?

On the other hand the Fourier transform of a one-sided decaying exponential function is: 0. )( )( > = ? a and tuetx at jw a dt e dt etue. wX tjwa jwt.

Chapter One : Fourier Series and Fourier Transform

2020?3?4? Example 5: Plot the spectrum for the gate function shown. Solution: X(w) = ? x(t)e?jwt dt = ? e?jwt dt.

Appendix B: Fourier Transform

The Fourier Transform (FT) is widely used in audio signal analysis and B.1 Properties of the Fourier Transformation ... ? /i(r)e-jWT dr ? X(jw).

Tutorial 6 - Fourier Analysis Made Easy Part 2

Complex representation of Fourier series e wt i jwt = + cos sin wt. (1). Bertrand Russell called this equation “the most beautiful profound and subtle

Fourier Series and Fourier Transform - MIT

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

6.082 Spring 2007Fourier 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.

6.082 Spring 2007Fourier 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 tincreases, vector rotates counterclockwise -We consider e jwt to have positive frequency e jωt I Q cos(ωt)sin(ωt) ωt

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 3

The Concept of Negative Frequency

Note: •As tincreases, 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 e jwt e -jωt I Q cos(ωt) -sin(ωt) -ωt

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 4

Add Positive and Negative Frequencies

Note: •As tincreases, 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 e jωt

2cos(ωt)

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 5

Subtract Positive and Negative Frequencies

Note: •As tincreases, 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 e jωt

2sin(ωt)

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 6

Fourier Series

•The Fourier Series is compactly defined using complex exponentials •Where: t T x(t)

6.082 Spring 2007Fourier 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)

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 8

Compare Fourier Definitions

•Let us assume the following: •Then: •So:

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 9

Square Wave Example

t T T/2 x(t) A -A

6.082 Spring 2007Fourier 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: 2A 2A 3π -2A -2A 3π n n B n A n 13579
-9 -7 -5 -3 -113579 -9 -7 -5 -3 -1

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 11

2A 2A 3π -2A -2A 3π f f B f A f 1T -3 T-5 T-1 T3 T5 T7 T9 T -7 T-9 T1 T -3 T-5 T-1 T3 T5 T7 T9 T -7 T-9 T

Indexing in Frequency

•A given Fourier coefficient, ,represents the weight corresponding to frequency nw o •It is often convenient to index in frequency (Hz)

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 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/4A -A t T T/2 x(t) A -A •Define: •Which leads to:

6.082 Spring 2007Fourier 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/4A -A t T T/2 x(t) A -A

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 14

2A f f B f A f 1T -3 T-5 T-1 T3 T5 T7 T9 T -7 T-9 T1 T-3 T -5 T-1 T3 T 5 T7 T 9 T-7 T -9 T 2A 2A 3π -2A -2A 3π f f B f A f 1T -3 T-5 T-1 T3 T5 T7 T9 T -7 T-9 T1 T -3 T-5 T-1 T3 T5 T7 T9 T -7 T-9 T

Graphical View of Fourier Series

t T/4 x(t) T T/4A -A t T T/2 x(t) A -A

6.082 Spring 2007Fourier 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 A n and B n coefficients are very sensitive to time (phase) shifts •The Fourier coefficients can also be represented in term of magnitude and phase •where:

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 16

Graphical View of Magnitude and Phase

t T/4 x(t) T T/4A -A t T T/2 x(t) A -A f -3 T-5 T-1 T-7 T-9 T 2A 2A 3π f f X f f 1T-3 T-5 T-1 T3 T5 T7 T9 T-7 T-9 T 1 T -3 T-5 T-1 T3 T5 T7 T9 T -7 T-9 T 2A 2A 3π f 2A 2A 3π f X f 1T-3 T-5 T-1 T3 T5 T7 T9 T-7 T-9 T 2A 2A 3π -π/2

π/2

1T3 T5 T7 T9 T

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 17

Does Time Shifting Impact Magnitude?

•Consider a waveform x(t) along with its Fourier

Series

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[PDF] • Complex exponentials • Complex version of Fourier Series • Time

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 1

Fourier Series

Fourier Transform

Copyright © 2007 by M.H. Perrott

All rights reserved.

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 2

The Complex Exponential as a Vector

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 3

The Concept of Negative Frequency

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 4

Add Positive and Negative Frequencies

2cos(ωt)

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 5

Subtract Positive and Negative Frequencies

2sin(ωt)

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 6

Fourier Series

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 7

From The Previous Lecture

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 8

Compare Fourier Definitions

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 9

Square Wave Example

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 10

Graphical View of Fourier Series

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 11

Indexing in Frequency

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 12

The Impact of a Time (Phase) Shift

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 13

Square Wave Example of Time Shift

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 14

Graphical View of Fourier Series

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 15

Magnitude and Phase

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 16

Graphical View of Magnitude and Phase

π/2

6.082 Spring 2007Fourier Series and Fourier Transform, Slide 17

Does Time Shifting Impact Magnitude?

Series