This document summarizes Chapter 2 of the textbook "Introduction to Analog & Digital Communications" which covers the Fourier representation of signals and systems. The chapter introduces the Fourier transform and its properties, such as how it relates the frequency and time domains. It also defines the Fourier transform mathematically and covers important concepts like the power spectral density and Dirichlet's conditions. Examples of applying the Fourier transform to common signals like rectangular and exponential pulses are also presented.

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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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2. 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.
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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.
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Communications, 2nd ed., Wiley, 2007.
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3. 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
Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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 )
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Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
3

4. 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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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5. 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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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6. 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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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 )
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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7. Properties of the FT
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 )
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
Properties of the FT
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
−∞ −∞
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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8. Properties of the FT
Haykin, S., and M. Moher, Introduction to Analog & Digital
15
Communications, 2nd ed., Wiley, 2007.
FT Theorems
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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9. FT Properties
Examples
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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.
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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10. 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.
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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.
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10

11. 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.
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
Dirac Delta Function
(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 
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11

12. Dirac Delta Function
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
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12

13. Applications of the Delta
Function
Haykin, S., and M. Moher, Introduction to Analog & Digital
25
Communications, 2nd ed., Wiley, 2007.
Dirac Delta
Function
Examples
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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14. 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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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

15. 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)
Haykin, S., and M. Moher, Introduction to Analog & Digital
29
Communications, 2nd ed., Wiley, 2007.
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)
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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16. Power Spectral Density
(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
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
Power Spectral Density
(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
Haykin, S., and M. Moher, Introduction to Analog & Digital
32
Communications, 2nd ed., Wiley, 2007.
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17. 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)
Haykin, S., and M. Moher, Introduction to Analog & Digital
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Communications, 2nd ed., Wiley, 2007.
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