ECE 8443 – Pattern RecognitionECE 3163 – Signals and Systems

• Objectives:DerivationTransform PairsResponse of LTI SystemsTransforms of Periodic SignalsExamples

• Resources:Wiki: The Fourier TransformCNX: DerivationMIT 6.003: Lecture 8 Wikibooks: Fourier Transform TablesRBF: Image Transforms (Advanced)

• URL: .../publications/courses/ece_3163/lectures/current/lecture_10.ppt• MP3: .../publications/courses/ece_3163/lectures/current/lecture_10.mp3

LECTURE 10: THE FOURIER TRANSFORM

ECE 3163: Lecture 10, Slide 2

Motivation• We have introduced a Fourier Series for analyzing periodic signals. What

about aperiodic signals? (e.g., a pulse instead of a pulse train)

• We can view an aperiodic signal as the limit of a periodic signal as T .

• The harmonic components are spaced apart.

• As T , 0 0, then k0 becomes continuous.

• The Fourier Series becomes the Fourier Transform.

T 2

0

1

01

0

110

sin2,,

)/(2sin2sin

TTc

kTfixedTTk

TTkkTk

c

k

k

ECE 3163: Lecture 10, Slide 3

Derivation of Analysis Equation• Assume x(t) has a finite duration.

• Define as a periodic extensionof x(t):

• As

• Recall our Fourier series pair:

• Since x(t) and are identical over this interval:

• As

2

22)(

)(~ Ttperiodic

TtTtxtx

)(~ tx

)()(~, txtxT

k

tjkkectx 0)(~

2/

2/

0)(~1 T

T

tjkk dtetxT

c

2/

2/

2/

2/

00 )(1)(~1 T

T

tjkT

T

tjkk dtetx

Tdtetx

Tc

)(~ tx

0, kT

dtetxT

jX tj )(1)(

ECE 3163: Lecture 10, Slide 4

Derivation of the Synthesis Equation

• Recall:

• We can substitute for ck the sampled value of :

• As

and we arrive at our Fourier Transform pair:

• Note the presence of the eigenfunction:

• Also note the symmetry of these equations (e.g., integrals over time and frequency, change in the sign of the exponential, difference in scale factors).

22)()(~ 0

TtTforectxtxk

tjkk

k

tjk

k

tjk

ejX

ejXT

txtx

0

0

)(21

))(1()()(~

00

0

)( jX

000 ,,0, kdT

k

dtetxT

jX

dejXtx

tj

tj

)(1)(

)(21)( (synthesis)

(analysis)

tj

js

st ee

ECE 3163: Lecture 10, Slide 5

Frequency Response of a CT LTI System• Recall that the impulse response of

a CT system, h(t), defines the properties of that system.

• We apply the Fourier Transform toobtain the system’s frequency response:

except that now this is valid for finite duration (energy) signals as well as periodic signals!

• How does this relate to what you have learned in circuit theory?

CT LTI)()(jXtx

)()(jYty

)()( jHth

CT LTItje tjejH )()()( jHth

dtethT

jH

dejHth

tj

tj

)(1)(

)(21)(

ECE 3163: Lecture 10, Slide 6

Existence of the Fourier Transform• Under what conditions does this transform exist?

x(t) can be infinite duration but must satisfy these conditions:

ECE 3163: Lecture 10, Slide 7

Example: Impulse Function

ECE 3163: Lecture 10, Slide 8

Example: Exponential Function

ECE 3163: Lecture 10, Slide 9

Example: Square Pulse

ECE 3163: Lecture 10, Slide 10

Example: Gaussian Pulse

ECE 3163: Lecture 10, Slide 11

CT Fourier Transforms of Periodic Signals

ECE 3163: Lecture 10, Slide 12

Example: Cosine Function

ECE 3163: Lecture 10, Slide 13

Example: Periodic Pulse Train

ECE 3163: Lecture 10, Slide 14

• Motivated the derivation of the CT Fourier Transform by starting with the Fourier Series and increasing the period: T

• Derived the analysis and synthesis equations (Fourier Transform pairs).

• Applied the Fourier Transform to CT LTI systems and showed that we can obtain the frequency response of an LTI system by taking the Fourier Transform of its impulse response.

• Discussed the conditions under which the Fourier Transform exists. Demonstrated that it can be applied to periodic signals and infinite duration signals as well as finite duration signals.

• Worked several examples of important finite duration signals.

• Introduced the Fourier Transform of a periodic signal.

• Applied this to a cosinewave and a pulse train.

Summary