1. The document provides an overview of Fourier analysis techniques including Fourier series, Fourier transforms, and their applications to signal representation and analysis. 2. Key concepts covered include representing periodic and aperiodic signals in the time and frequency domains, properties of linear and time-invariant systems, Parseval's theorem relating signal energy in the time and frequency domains, and the Fourier transforms of basic functions like impulses and complex exponentials. 3. The document establishes essential mathematical foundations for further study of analog and digital communications techniques that involve signal processing and transmission in the frequency domain.

1. TELE3113 Analogue & Digital
Communications
Review of Fourier Transform
p. 1

2. Signal Representation
s(t) = A sin(2π fo t +φo ) or A sin(ωo t +φo )
Time-domain: waveform
A: Amplitude
Time (seconds) f : Frequency (Hz) (ω=2πf)
φ : Phase (radian or degrees)
Period (seconds)
S(f)
Frequency-domain: spectrum
fo Frequency (Hz)
p. 2

3. Energy and Power of Signals
For an arbitrary signal f(t), the total energy normalized to unit
resistance is defined as
∆ T
E = lim ∫ f (t ) 2 dt joules,
T →∞ −T
and the average power normalized to unit resistance is defined as
∆ 1 T
P = lim
T → ∞ 2T ∫
−T
f (t ) 2 dt watts ,
• Note: if 0 < E < ∞ (finite) P = 0.
• When will 0 < P < ∞ happen?
p. 3

4. Periodic Signal
A signal f(t) is periodic if and only if
f (t + T0 ) = f (t ) for all t (*)
where the constant T0 is the period.
The smallest value of T0 such that equation (*) is satisfied is
referred to as the fundamental period, and is hereafter simply
referred to as the period.
Any signal not satisfying equation (*) is called aperiodic.
p. 4

5. Deterministic & Random Signals
Deterministic signal can be modeled as a completely specified
function of time.
Example
f (t ) = A cos( ω 0 t + θ )
Random signal cannot be completely specified as a function of
time and must be modeled probabilistically.
p. 5

6. System
Mathematically, a system is a rule used for assigning a function g(t)
(the output) to a function f(t) (the input); that is,
g(t) = h{ f(t) }
where h{•} is the rule or we call the impulse response.
f(t) h(t) g(t)
For two systems connected in cascade, the output of the first system
forms the input to second, thus forming a new overall system:
g(t) = h2 { h1 [ f(t) ] } = h{ f(t) }
p. 6

7. Linear System
If a system is linear then superposition applies; that is, if
g1(t) = h{ f1(t) }, and g2(t) = h{ f2(t) }
then
h{ a1 f1(t) + a2 f2(t) } = a1 g1(t) + a2 g2(t) (*)
where a1, a2 are constants. A system is linear if it satisfies
Eq. (*); any system not meeting these requirement is nonlinear.
p. 7

8. Time-Invariant and Time-Varying
A system is time-invariant if a time shift in the input results
in a corresponding time shift in the output so that
g (t − t 0 ) = h{ f (t − t 0 )} for any t 0 .
The output of a time-invariant system depends on time differences and
not on absolute values of time.
Any system not meeting this requirement is said to be time-varying.
p. 8

9. Fourier Series
A periodic function of time s(t) with a fundamental period of T0 can be
represented as an infinite sum of sinusoidal waveforms. Such
summation, a Fourier series, may be written as:
∞
2 π nt ∞ 2 πnt
s (t ) = A0 + ∑ An cos + ∑ B n sin , (1)
n =1 T0 n =1 T0
where the average value of s(t), A0 is given by
1 T20
A0 =
T0 ∫− T20 s (t ) dt , (2)
while
2 T0
2 π nt
An = ∫ (3)
2
T0
s (t ) cos dt ,
T0 − 2 T0
and
2 T0
2 π nt
Bn = ∫
2
T0
s (t ) sin dt . (4)
T0 − 2 T0
p. 9

10. Fourier Series
An alternative form of representing the Fourier series is
∞
 2 πnt 
s (t ) = C 0 + ∑ C n cos 
 − φn 

(5)
n =1  T0 
where
C0 = A0 , (6)
2 2
Cn = An + B n , (7)
B
φ n = tan −1 n . (8)
An
The Fourier series of a periodic function is thus seen to consist of a
summation of harmonics of a fundamental frequency f0 = 1/T0.
The coefficients Cn are called spectral amplitudes, which represent the
amplitude of the spectral component Cn cos(2πnf0t − φn) at frequency
nf0.
p. 10

11. Fourier Series
The exponential form of the Fourier series is used extensively in
communication theory. This form is given by
∞ j 2 π nt
s (t ) = ∑S
n = −∞
n e T0
, (9)
where
1 T0
−
j 2 π nt
(10)
Sn = ∫ s (t ) e dt
2 T0
T0
T0 − 2
Note that Sn and S−n are complex conjugate of one another, that is
S n = S −n .
*
(11)
These are related to the Cn by
C n − jφ n (12)
S0 = C0 , Sn = e .
2
p. 11

12. Fourier Series
Amplitude Spectra (Line Spectra)
Fig.(a)
Cn
Note that except S0 = C0, each
0 fo 2fo 3fo 4fo 5fo 6fo (n-1) fo nfo spectral line in Fig. (a) at frequency f
is replaced by the two spectral lines in
Fig. (b), each with half amplitude,
Fig.(b) one at frequency f and one at
|Sn|
frequency - f.
••• •••
-nfo -(n-1)fo ••• - 6fo0-5fo -4fo -3fo -2fo -fo 0 fo 2fo 3fo 4fo 5fo 6fo ••• (n-1) fo nfo
p. 12

13. Fourier Series : Example
Consider a unitary square wave defined by The Bn coefficients are given by
1, 0 < t < 0.5 2 T0
2πnt
Bn = ∫
2
x(t ) =  T0
x(t ) sin dt
T0 −2 T0
− 1, 0.5 < t < 1
= 2 ∫ x(t ) sin (2πnt )dt
1
and periodically extended outside this interval. 0
The average value is zero, so = 2 ∫ sin (2πnt )dt + 2 ∫ − sin (2πnt )dt
0.5 1
0 0.5
A0 = 0.  cos(2πnt ) cos(2πnt ) 
1 0.5
= 2 − + 
Recall that 2 T0
2πnt 
 2πn 0 2πn 0.5 
An = ∫
2
x(t ) cos dt
T0
T0
−2 T0 2
= (1 − cos nπ)
πn
= 2 ∫ x(t ) cos(2πnt )dt
1
0
which results in
= 2 ∫ cos(2πnt )dt + 2 ∫ − cos(2πnt )dt
0.5 1
 4
0 0.5
 , n is odd
Bn =  nπ
 sin (2πnt ) sin (2πnt ) 
0.5 1
= 2 −  0,
 n is even

 2πn 0 2πn 0.5 
=0
Thus all An coefficients are zero.
p. 13

14. Fourier Series : Example
The Fourier series of a square wave of unitary amplitude with odd symmetry is
therefore
4 1 1
x (t ) = (sin 2 πt + sin 6 πt + sin 10 πt + K)
π 3 5
1st term 1st + 2nd terms 1st + 2nd + 3rd terms
Sum up to the 6th term
p. 14

15. Fourier Transform
Representation of an Aperiodic Function
Consider an aperiodic function f(t)
To represent this function as a sum of exponential functions over
the entire interval (-∞, ∞), we construct a new periodic function
fT(t) with period T.
By letting T→∞,
lim f T (t ) = f (t ) (13)
T →∞
p. 15

16. Fourier Transform
The new function fT(t) can be represented by an exponential
Fourier series, which is written as
∞
f T (t ) = ∑ Fn e jn ω 0 t ,
n = −∞
(14)
where
1 T /2
(15)
Fn =
T ∫−T / 2
f T (t ) e − jn ω 0 t dt
and ω0 = 2π / T .
p. 16

17. Fourier Transform
For the sake of clear presentation, we set
∆ ∆
ω n = nω 0 , F ( ω n ) = TF n , (16)
Thus, Eq.(14) and (15) become
∞
1
f T (t ) = ∑T
n = −∞
F ( ω n ) e jω n t , (17)
T /2
(18)
F (ω n ) = ∫−T / 2
f T (t ) e − jω n t dt .
The spacing between adjacent lines in the line stream of fT(t)
is
∆ω = 2π / T . (19)
p. 17

18. Fourier Transform
Using this relation for T, we get
∞
∆ω
f T (t ) = ∑
n = −∞
F (ω n )e jω n t
2π
. (20)
As T becomes very large, ∆ω becomes smaller and the spectrum
becomes denser.
In the limit T → ∞, the discrete lines in the spectrum of fT(t) merge
and the frequency spectrum becomes continuous.
Therefore, 1 ∞
lim f T (t ) = lim
T →∞ T →∞ 2π
∑
n = −∞
F ( ω n ) e jω n t ∆ ω (21)
becomes 1 ∞
2 π ∫− ∞
f (t ) = F ( ω ) e jω t d ω (22)
p. 18

19. Fourier Transform
In a similar way, Eq. (18) becomes
∞
F (ω) = ∫−∞
f (t ) e − jω t dt . (23)
Eq. (22) and (23) are commonly referred to as the
Fourier transform pair.
Fourier Transform
∞
F (ω ) = ∫
−∞
f (t ) e − jω t dt
Inverse Fourier Transform
1 ∞
2 π ∫− ∞
f (t ) = F ( ω ) e jω t d ω
p. 19

20. Spectral Density Function
F(ω): The spectral density function of f(t).
Fig. 3.2
A unit gate function Its spectral density graph
sin( ω / 2 )
Sa ( ω / 2 ) =
ω/2
p. 20

21. Parseval’s Theorem
The energy delivered to a 1-ohm resistor is
∞ ∞
E= ∫ f (t ) dt = ∫ (24)
2
f (t ) f * (t ) dt .
−∞ −∞
Using Eq. (22) in (24), we get
∞ 1 ∞ *  1 ∞
E = ∫ f (t )  ∫ F (ω)e − jωt dω dt f (t ) = ∫− ∞ F (ω)e d ω
jω t
−∞
 2π − ∞  2π
1 ∞ *  ∞
F (ω) ∫ f (t )e − jωt dt  dω
2π ∫−∞
=
 −∞
 

1 ∞ * (25)
=
2π ∫−∞ F (ω) F (ω)dω.
Parseval’s Theorem:
∞ 1 ∞
∫ 2 π ∫− ∞
2 2
−∞
f (t ) dt = F ( ω) d ω. (26)
p. 21

22. Fourier Transform: Impulse Function
The unit impulse function satisfies
∞
∫ δ( x)dx = 1, (27)
−∞
∞ x = 0,
δ ( x) =  (28)
0 x ≠ 0.
Using the integral properties of the impulse function, the Fourier
transform of a unit impulse, δ(t), is
∞
ℑ{δ(t )} = ∫ δ(t )e − jωt dt = e j 0 = 1. (29)
−∞
If the impulse is time-shifted, we have
∞
ℑ{δ(t − t0 )} = ∫ δ(t − t0 )e − jωt dt = e − jωt0 . (30)
−∞
p. 22

23. Fourier Transform: Complex
Exponential Function
± jω t
The spectral density of e 0 will be concentrated at ±ω0.
1 ∞
ℑ {δ ( ω m ω 0 )} =
2 π ∫− ∞
−1
δ ( ω m ω 0 ) e jω t d ω
1 ± jω 0 t (31)
= e ,
2π
Taking the Fourier transform of both sides, we have
(32)
ℑℑ −1
2π
{
{δ ( ω m ω 0 ) } = 1 ℑ e ± j ω 0 t }
which gives
{ }
ℑ e ± j ω 0 t = 2πδ (ω m ω 0 ) (33)
p. 23

24. Fourier Transform: Sinusoidal Function
The sinusoidal signals cos ω0tand sin ωcan be written in terms of
0t
the complex exponentials.
Their Fourier transforms are given by
{
ℑ{cos ω 0 t } = ℑ 1 e jω 0 t + 1 e − jω 0 t
2 2
}
= πδ ( ω − ω 0 ) + πδ ( ω + ω 0 ),
(34)
ℑ{sin ω0t} = ℑ {1
2j e jω0t − 21j e − jω0t }
πδ(ω − ω0 ) − πδ(ω + ω0 )
= .
j
(35)
p. 24

25. Fourier Transform: Periodic Functions
We can express a function f(t) that is periodic with period T by its
exponential Fourier series
∞
f T (t ) = ∑ Fn e jn ω 0 t
n = −∞
where ω0 = 2π/T. (36)
Taking the Fourier transform, we have
 ∞ jnω0 t 
ℑ{ fT (t )} = ℑ ∑ Fn e 
e.g.
n = −∞ 
∑ F ℑ{e }
∞
jnω0t
= n A unit gate function Its Fourier transform
n = −∞
∞
= 2π ∑ Fn δ(ω − nω0 ).
n = −∞
(37) Line spectrum of f(t) Its spectral density graph
with period T p. 25

26. Time and Spectral Density Functions
p. 26

27. Selected Fourier Transform Pairs
p. 27

28. Properties of Fourier Transform
Linearity (Superposition) Time Shifting (Delay)
a1 f1 (t ) + a 2 f 2 (t ) ↔ a1 F1 ( ω ) + a 2 F2 ( ω ) f (t − t 0 ) ↔ F (ω ) e − jω t 0
Complex Conjugate Frequency Shifting (Modulation)
f * (t ) ↔ F * (−ω) f ( t ) e jω 0 t ↔ F ( ω − ω 0 )
Duality
Convolution
F (t ) ↔ 2 π f ( − ω ).
f1 (t ) ∗ f 2 (t ) ↔ F1 ( ω ) F2 ( ω )
Scaling
1  ω Multiplication
f (at ) ↔ F  for a ≠ 0.
a a
f1 (t ) f 2 (t ) ↔ F1 ( ω ) ∗ F2 ( ω )
Differentiation
dn
f (t ) ↔ ( jω) n F (ω)
dt n
p. 28

29. Properties of Fourier Transform
Duality F (t ) ↔ 2 π f ( − ω).
Scaling 1 ω
f ( at ) ↔ F  for a ≠ 0.
a a
p. 29

30. Properties of Fourier Transform
Frequency Shifting (Modulation)
jω 0 t
f (t ) e ↔ F (ω − ω 0 )
p. 30