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TELE3113 Analogue and DigitalCommunications

Review of Fourier Transform

Wei Zhang

[email protected]

School of Electrical Engineering and Telecommunications

The University of New South Wales

Fourier Transform

Let g(t) denote a nonperiodic deterministic signal, the Fourier

transform (FT) of the signal g(t) is given by

G(f) =

∫

∞

−∞

g(t) exp(−j2πft)dt.

The inverse Fourier transform is given by

g(t) =

∫

∞

−∞

G(f) exp(j2πft)df.

We call G(f) and g(t) as the Fourier-transform pair, denoted by

g(t) ⇔ G(f).

TELE3113 - Review of Fourier Transform. July 29, 2009. – p.1/15

Spectrum

The FT G(f) is a complex function of frequency f , so it can be

expressed as

G(f) = |G(f)| exp(jθ(f)),

where

|G(f)| is called the amplitude spectrum of g(t);

θ(f) is called the phase spectrum of g(t).


Rectangular Pulse (1)

Define rectangular function of unit amplitude and unit duration

centered at t = 0 as

rect(t) =

1, −12 ≤ t ≤ 1

2

0, t < −12 or t > 1

2

Then, a rectangular pulse of duration T and amplitude A, as

shown in Figure, can be expressed as g(t) = A rect( t

T).

)(tg

A

2/T 2/T−

t

0 TELE3113 - Review of Fourier Transform. July 29, 2009. – p.3/15

Rectangular Pulse (2)

The FT of a rectangular pulse of duration T and amplitude A is

Arect

(

t

T

)

⇔ AT sinc(fT )

where sinc(·) denotes the sinc function as sinc(λ) = sin(πλ)πλ

.

−3 −2 −1 0 1 2 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

λ

sinc

(λ)


Properties of FT (1)

Linearity Property:

If g(t) ⇔ G(f), then

c1g1(t) + c2g2(t) ⇔ c1G1(f) + c2G2(f).

Dilation Property:


g(at) ⇔ 1

|a|G(

f

a

)

.



Conjugation Rule:


g∗(t) ⇔ G∗(−f).

Duality Property:


G(t) ⇔ g(−f).



Time Shifting Property:


g(t − t0) ⇔ G(f) exp(−j2πft0).

Frequency Shifting Property:


exp(j2πfct)g(t) ⇔ G(f − fc).



Modulation Theorem:

Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then

g1(t)g2(t) ⇔ G1(f) ? G2(f),

where G1(f) ? G2(f) =∫

∞

−∞G1(λ)G2(f − λ)dλ.

Convolution Theorem:

Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then

g1(t) ? g2(t) ⇔ G1(f)G2(f),

where g1(t) ? g2(t) =∫

∞

−∞g1(τ)g2(t − τ)dτ .



Correlation Theorem:

Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then∫

∞

−∞

g1(τ)g∗2(t − τ)dτ ⇔ G1(f)G∗

2(f).

Rayleigh’s Energy Theorem:

Let g1(t) ⇔ G1(f) and g2(t) ⇔ G2(f). Then∫

∞

−∞

|g(t)|2dt =

∫

∞

−∞

|G(f)|2df.

Note that in the above formula, it is “=”, not “⇔”.


LP versus BP

Low-pass (LP) signal: Its significant spectral content is

centered around the origin f = 0.

Band-pass (BP) signal: Its significant spectral content is

centered around ±fc, where fc is a constant frequency.


Bandwidth

Definition of bandwidth (BW):

For LP signal, the BW is one half the total width of the main

spectral lobe.

For BP signal, the BW is the width of the main lobe for

positive frequencies.


3-dB Bandwidth

3-dB BW of the LP signal: the separation between zero

frequency and the positive frequency at which the amplitude

spectrum drops to 1/√

2 of the peak value at zero frequency.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

13 dB Bandwidth of LP signal

−3 dB

BW

3-dB BW of the BP signal: the separation between the two

frequencies at which the amplitude spectrum drops to 1/√

2

of the peak value at fc. TELE3113 - Review of Fourier Transform. July 29, 2009. – p.12/15

Dirac Delta Function

The Dirac delta can be loosely thought of as a function on the

real line which is zero everywhere except at the origin, where it

is infinite,

δ(x) =

+∞, x = 0

0, x 6= 0

and which is also constrained to satisfy the identity∫

∞

−∞

δ(x) dx = 1.


Applications of δ Function

dc Signal:

1 ⇔ δ(f).

Complex Exponential Function:

exp(j2πfct) ⇔ δ(f − fc).

Sinusoidal Function:

cos(2πfct) ⇔ 1

2[δ(f − fc) + δ(f + fc)].

sin(2πfct) ⇔ 1

2j[δ(f − fc) − δ(f + fc)].


Reference

All the proofs of the properties of FT are available in

Chapter 2 of the book

Introduction to Analog & Digital Communications, 2nd Ed.

by Simon Haykin and Michael Moher.


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