1. TELE3113 Analogue and Digital
Communications
Review of Fourier Transform
Wei Zhang
w.zhang@unsw.edu.au
School of Electrical Engineering and Telecommunications
The University of New South Wales

2. 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(−j2πf t)dt.
−∞
The inverse Fourier transform is given by
∞
g(t) = G(f ) exp(j2πf t)df.
−∞
We call G(f ) and g(t) as the Fourier-transform pair, denoted by
g(t) ⇔ G(f ).
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3. 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).
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4. Rectangular Pulse (1)
Define rectangular function of unit amplitude and unit duration
centered at t = 0 as

 1, −1 ≤ t ≤ 1
2 2
rect(t) =
 0, t < − 1 or t > 1
2 2
Then, a rectangular pulse of duration T and amplitude A, as
t
shown in Figure, can be expressed as g(t) = A rect( T ).
g (t )
A
t
−T /2 0 T /2 TELE3113 - Review of Fourier Transform. July 29, 2009. – p.3/1

5. Rectangular Pulse (2)
The FT of a rectangular pulse of duration T and amplitude A is
t
Arect ⇔ AT sinc(f T )
T
sin(πλ)
where sinc(·) denotes the sinc function as sinc(λ) = πλ .
1
0.8
0.6
0.4
sinc(λ)
0.2
0
−0.2
−0.4
−3 −2 −1 0 1 2 3 TELE3113 - Review of Fourier Transform. July 29, 2009. – p.4/1
λ

6. Properties of FT (1)
Linearity Property:
If g(t) ⇔ G(f ), then
c1 g1 (t) + c2 g2 (t) ⇔ c1 G1 (f ) + c2 G2 (f ).
Dilation Property:
If g(t) ⇔ G(f ), then
1 f
g(at) ⇔ G .
|a| a
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7. Properties of FT (2)
Conjugation Rule:
If g(t) ⇔ G(f ), then
g ∗ (t) ⇔ G∗ (−f ).
Duality Property:
If g(t) ⇔ G(f ), then
G(t) ⇔ g(−f ).
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8. Properties of FT (3)
Time Shifting Property:
If g(t) ⇔ G(f ), then
g(t − t0 ) ⇔ G(f ) exp(−j2πf t0 ).
Frequency Shifting Property:
If g(t) ⇔ G(f ), then
exp(j2πfc t)g(t) ⇔ G(f − fc ).
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9. Properties of FT (4)
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 .
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10. Properties of FT (5)
Correlation Theorem:
Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then
∞
g1 (τ )g2 (t − τ )dτ ⇔ G1 (f )G∗ (f ).
∗
2
−∞
Rayleigh’s Energy Theorem:
Let g1 (t) ⇔ G1 (f ) and g2 (t) ⇔ G2 (f ). Then
∞ ∞
|g(t)|2 dt = |G(f )|2 df.
−∞ −∞
Note that in the above formula, it is “=”, not “⇔”.
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11. 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.
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12. 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.
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13. 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.
3 dB Bandwidth of LP signal
1
0.9
−3 dB
0.8
0.7
BW
0.6
0.5
0.4
0.3
0.2
0.1
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
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.12/1

14. 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, 
+∞, x = 0
δ(x) =
0, x=0
and which is also constrained to satisfy the identity
∞
δ(x) dx = 1.
−∞
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15. Applications of δ Function
dc Signal:
1 ⇔ δ(f ).
Complex Exponential Function:
exp(j2πfc t) ⇔ δ(f − fc ).
Sinusoidal Function:
1
cos(2πfc t) ⇔ [δ(f − fc ) + δ(f + fc )].
2
1
sin(2πfc t) ⇔ [δ(f − fc ) − δ(f + fc )].
2j
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16. 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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