Ch2 rev[1]

ELEN 4610:
Analog Communications

Chapter #2:
Fourier Representation
of Signals and Systems
Prof. Caroline González

Matlab and Simulink Tutorial
http://www.mathworks.com/academia/student_center/tutorials

Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
Ch 2-1

In this chapter, we will study:
Definition of the Fourier Transform

Properties of the Fourier Transform

The Inverse Relationship between Time and
Frequency

Dirac Delta Function

Fourier Transform of Periodic Signals

Power Spectral Density

2

1

The Fourier Transform (FT)
The FT relates the frequency-domain
description of a signal to its time-domain
description.
− Determine the frequency content of a
continuous-time signal.
− Evaluates what happens to this
frequency content when the signal is
passed through a linear time-invariant
(LTI) system.
− A signal can only be strictly limited in the
time domain or the frequency domain,
but not both.
− Bandwidth is an important parameter in
describing the spectral content of a
signal and the frequency response of a
LTI filter.
3

Definition of the FT

Advantages of using frequency-domain analysis
− Resolution into eternal sinusoids presents the
behavior as the superposition of steady-state
effects.
− Usually the time-domain analysis involves
solving differential equations, but in the
frequency domain involves simple algebra
equations.
− Provides the frequency content of a signal.

4

2

Dirichlet’s Conditions
For the FT of a signal g(t) to exist, it is
sufficient, but not necessary, that g(t)
satisfies:
− The function g(t) is single-valued, with a
finite number of maxima and minima in
any finite time interval.
− The function g(t) has a finite number of
discontinuities in any finite time interval.
− The function g(t) is absolutely integrable
or the g(t) is an energy-like signal.
∞

∫ g (t )dt < ∞
−∞
∞ 2

∫ g (t )
−∞
dt < ∞
5

Continuous Spectrum
The FT is a complex function of
frequency so that
G ( f ) = G ( f ) e jθ ( f )
where
G ( f ) is the continuous amplitude spectrum
θ ( f ) is the continuous phase spectrum

For a real-value function g(t) the FT has the
following characteristics
G ( − f ) = G* ( f )
G (− f ) = G ( f )
θ ( − f ) = −θ ( f )
6

3

Continuous Spectrum
In conclusion
− The spectrum of a real-valued signal
exhibits conjugate symmetry.
The amplitude spectrum of a signal is
an even function of the frequency;
the amplitude spectrum is symmetric
with respect to the origin f=0.
The phase spectrum of a signal is an
odd function of the frequency; the
phase spectrum is antisymmetric with
respect to the origin f=0

7

Examples
Rectangular Pulse (Example 2.1)
− Matlab Demo

Decaying Exponential Pulse (Ex. 2)
− Matlab Demo

Rising Exponential Pulse (Ex. 2)
− Matlab Demo

Drill P2.1

8

4

Rectangular Pulse Amplitude
Spectrum
Spectrum of a Rectangular Pulse
2

1.5

Amplitude
1

0.5

0
-8 -6 -4 -2 0 2 4 6 8
Time

10
Amplitude Spectrum

5

0

-5
-1.5 -1 -0.5 0 0.5 1 1.5
frequency

9

Decaying Exponent Spectrum

Decaying Exponential Pulse
1

0.8
Amplitude

0.6

0.4

0.2

0
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
Time
Amplitude Spectrum of Decaying Exponential Pulse
0.8

0.6
Magnitude

0.4

0.2

0
-3 -2 -1 0 1 2 3
frequency
Phase Spectrum of Decaying Exponential Pulse
100
Phase in degrees

50

0

-50

-100
-3 -2 -1 0 1 2 3
frequency

10

5

Rising Exponent Spectrum
Rising Exponential Pulse
1

Amplitude
0.5

0
-4 -3.5 -3 -2.5 -2
-1.5 -1 -0.5 0 0.5 1
Time
Amplitude Spectrum of Rising Exponential Pulse
0.5
Magnitude

0
-3 -2 -1 0 1 2 3
frequency
Phase Spectrum of Rising Exponential Pulse
Phase in degrees

100

0

-100
-3 -2 -1 0 1 2 3
frequency

11

Properties of the FT
Linearity

c1 g1 (t ) + c2 g 2 (t ) ⇔ c1G1 ( f ) + c2G2 ( f )

Dilation 1 f
g (at ) ⇔ G 
a a
Conjugation

g * (t ) ⇔ G * (− f )
Duality

If g (t ) ⇔ G ( f ), then G (t ) ⇔ g (− f )
12

6

Time Shifting

g (t − t0 ) ⇔ G ( f )e − j 2π ⋅ f ⋅t0
Frequency Shifting

e j 2π ⋅ f c ⋅t g (t ) ⇔ G ( f − f c )
Differentiation
dn
n
{g (t )} ⇔ ( j 2π ⋅ f )n ⋅ G( f )
dt
Integration
t
1
∫ g (τ )dτ ⇔
−∞
j 2π ⋅ f
G( f )
13

Area under g(t) ∞

−∞
∫ g (t )dt = G (0 )

Area under G(f)
∞
g (0 ) = ∫ G ( f )df
−∞
Modulation
Theorem ∞
g1 (t )g 2 (t ) ⇔ ∫ G1 (λ )G2 ( f − λ )dλ
−∞

Rayleigh’s Energy ∞ ∞

∫ g (t ) ∫ G( f )
2 2
Theorem dt = df
−∞ −∞
14

7


15

FT Theorems

16

8

FT Properties
Examples

17

The Inverse Relationship
between Time and Frequency
The properties of the FT show that
the time-domain and frequency-
domain description of a signal are
inversely related to each other.
− If the time-domain description of a
signal is changed, the frequency-
domain description of the signal is
changed in an inverse manner, and
vice versa.
− A signal cannot be strictly limited in
both time and frequency.

18

9

Bandwidth
Provides a measure of the extend of
the significant spectral content of
the signal for positive frequencies.
− A signal is low-pass if its significant
spectral content is centered around
the origin f = 0.
− A signal is band-pass if its significant
spectral content is centered around
±fc , where fc is a constant frequency.

19

Bandwidth
Null-to-null bandwidth
− when the spectrum of the signal is
symmetric with a main lobe bounded
by well-defined nulls (i.e. frequencies
at which the spectrum is zero), we
may use the main lobe for defining
the bandwidth of the signal.

3-dB bandwidth
− the separation (along the positive
frequency axis) between the two
frequencies at which the amplitude
spectrum of the signal drops to 1/ 2 of
the peak value.
20

10

Time-Bandwidth Product
The product of the signal’s duration
and its bandwidth is always a
constant.

(duration) X (bandwidth) = constant

The time-bandwidth product is
another manifestation of the inverse
relationship that exists between the
time-domain and frequency-domain
descriptions of a signal.

21

(Unit Impulse)
The theory of the FT is applicable to
only time functions that satisfy the
Dirichlet conditions, but it would be
helpful to extend the theory in two
ways
− To combine the theory of Fourier
series and FT, so that the Fourier
series may be treated as a special
case of the FT.
− To expand applicability of the FT to
include power signals (periodic
signals), signals that satisfy:
 1 T

∫ g (t ) dt  < ∞
2
lim 
T → ∞ 2T
 −T 
22

11

This can be accomplished with the use of
the Dirac Delta function.
δ (0 ) = 0 , t ≠ 0
∞

∫ δ (t )dt
−∞
=1

∞

∫ g (t )δ (t − t )dt
−∞
0 = g (t 0 ) (Sifting Property)

ℑ { (t )} = 1
δ

23

Applications of the Delta
Function
DC signal
1⇔ δ(f )
Complex
Exponential

e j 2π ⋅ f c ⋅t ⇔ δ ( f − f c )
Sinusoidal
Functions
1
cos(2π ⋅ f c ⋅ t ) ⇔ [δ ( f − f c ) + δ ( f + f c )]
2
1
sin (2π ⋅ f c ⋅ t ) ⇔ [δ ( f − f c ) − δ ( f + f c )]
2j
24

12

Applications of the Delta
Function

25

Dirac Delta
Function
Examples

26

13

Fourier Transform of
Periodic Signal
Using the Fourier series, a periodic
signal can be represented as a sum
of complex exponential or into an
infinite sum of sine and cosine
terms.

To denotes the period of the signal.

fo denotes the fundamental
frequency of the signal.
1
fo =
To
27

FT of Periodic Signals
∞ ∞
x(t ) = ∑ g (t − mT ) ⇔ X ( f ) = f ∑ G(n ⋅ f )δ ( f − n ⋅ f )
0 0 o 0
m = −∞ n = −∞

∞
x(t ) = f 0 ∑ G (n ⋅ f )⋅ e 0
j 2π ⋅n⋅ f 0 ⋅t

n = −∞
∞
x(t ) = f 0 ⋅ G (0 ) + 2 ⋅ f 0 ∑ G (n ⋅ f o ) ⋅ cos(2π ⋅ n ⋅ f 0 ⋅ t + ∠G (n ⋅ f o ))
n =1

x(t) g(t)

t
T0 T0
t1 t1+T0
28

14

Fourier Series: Example 1
Periodic Waveform
1.5

1

Amplitude
0.5

0

-0.5
-5 -4 -3 -2 -1 0 1 2 3 4 5
time (s)

29

Fourier Series Example 2
Periodic Waveform
1.5

1

0.5
Amplitude

0

-0.5

-1

-1.5
-5 -4 -3 -2 -1 0 1 2 3 4 5
time (s)

30

15

(PSD)
Parserval’s Theorem – relates the
energy associated with a time-
domain function of finite energy to
the Fourier transform of the
function. To calculate the PSD, it’s
necessary to assume a resistor of 1
(normalized).

The PSD (energy) (in Watts / Hz) of
a signal x(t) is

Sx = X ( f )
2

31

(PSD)
The average power (normalized) (in
Watts) is
∞
Pave = ∫ S ( f ) df
−∞
x

Parseval’s Theorem(Periodic Signals)
∞ 2

Pave = ∑ X (n ⋅ f )
n = −∞
0

32

16

Examples 1 and 2 (PSD)
Example 1 PSD
0.4

0.3 Pave = 0.4833 W

PSD Sx
0.2

0.1

0
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency (Hz)
Example 2 PSD
0.4

0.3 Pave=0.6464 W
PSD Sx

0.2

0.1

0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Frequency (Hz)

33

17

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