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ELEN 4610:Analog Communications
Chapter #2: Fourier Representation of Signals and Systems
Matlab and Simulink Tutorialhttp://www.mathworks.com/academia/student_center/tutorials
Prof. Caroline González
Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.
Ch 2-1
Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.
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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
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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.
3Haykin, S., and M. Moher, Introduction to Analog & Digital
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.
4Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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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 integrableor the g(t) is an energy-like signal.
( )
( ) ∞<
∞<
∫
∫∞
∞−
∞
∞−
dttg
dttg
2
5Haykin, 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
( ) ( ) ( )
( )( )
is the continuous amplitude spectrum
is the continuous phase spectrum
j fG f G f e
where
G f
f
θ
θ
=
For a real-value function g(t) the FT has the following characteristics
( ) ( )( ) ( )( ) ( )
*G f G f
G f G f
f fθ θ
− =
− =
− = −6
Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.
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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 symmetricwith 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
7Haykin, S., and M. Moher, Introduction to Analog & Digital
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
8Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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Rectangular Pulse Amplitude Spectrum
-8 -6 -4 -2 0 2 4 6 80
0.5
1
1.5
2
Time
Amplitude
Spectrum of a Rectangular Pulse
-1.5 -1 -0.5 0 0.5 1 1.5-5
0
5
10
frequency
Amplitude Spectrum
9Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
Decaying Exponent Spectrum
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
Time
Amplitude
Decaying Exponential Pulse
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
frequency
Magnitude
Amplitude Spectrum of Decaying Exponential Pulse
-3 -2 -1 0 1 2 3-100
-50
0
50
100
frequency
Phase in degrees
Phase Spectrum of Decaying Exponential Pulse
10Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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Rising Exponent Spectrum
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10
0.5
1
Time
Amplitude
Rising Exponential Pulse
-3 -2 -1 0 1 2 30
0.5
frequency
Magnitude
Amplitude Spectrum of Rising Exponential Pulse
-3 -2 -1 0 1 2 3-100
0
100
frequency
Phase in degrees Phase Spectrum of Rising Exponential Pulse
11Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
Properties of the FT
Linearity
Dilation
Conjugation
Duality
( ) ( ) ( ) ( )fGcfGctgctgc 22112211 +⇔+
( )
⇔a
fG
aatg
1
( ) ( )fGtg −⇔ **
( ) ( ) ( ) ( )fgtGfGtg −⇔⇔ then , If
12Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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Properties of the FT
Time Shifting
Frequency Shifting
Differentiation
Integration
( ) ( ) 02
0
tfjefGttg
⋅⋅−⇔− π
( ) ( )ctfjffGtge c −⇔⋅⋅π2
( ) ( ) ( )fGfjtgdt
d n
n
n
⋅⋅⇔ π2
( ) ( )∫∞− ⋅
⇔t
fGfj
dgπ
ττ2
1
13Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
Properties of the FT
Area under g(t)
Area under G(f)
Modulation Theorem
Rayleigh’s Energy Theorem
( ) ( )0Gdttg =∫∞
∞−
( ) ( )∫∞
∞−
= dffGg 0
( ) ( ) ( ) ( ) λλλ dfGGtgtg −⇔ ∫∞
∞−2121
( ) ( )∫ ∫∞
∞−
∞
∞−
= dffGdttg22
14Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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Properties of the FT
15Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
FT Theorems
16Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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FT Properties Examples
17Haykin, S., and M. Moher, Introduction to Analog & Digital
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 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 Communications, 2nd ed., Wiley, 2007.
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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 of the peak value.
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2/1
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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 Communications, 2nd ed., Wiley, 2007.
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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:
( ) ∞<
∫−
∞→
T
TT
dttgT
2
2
1lim
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Dirac Delta Function
This can be accomplished with the use of the Dirac Delta function.
( )
( )
( ) ( ) ( )
( ) 1
1
0 ,00
00
=ℑ
=−
=
≠=
∫
∫∞
∞−
∞
∞−
t
tgdttttg
dtt
t
δ
δ
δ
δ
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(Sifting Property)
Applications of the Delta Function
DC signal
Complex Exponential
Sinusoidal Functions
( )fδ⇔1
( )ctfjffe c −⇔⋅⋅ δπ2
( ) ( ) ( )[ ]ccc fffftf ++−⇔⋅⋅ δδπ2
12cos
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( ) ( ) ( )[ ]ccc ffffj
tf +−−⇔⋅⋅ δδπ2
12sin
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Applications of the Delta Function
Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.
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Dirac Delta Function Examples
26Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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Fourier Transform of Periodic Signal
Using the Fourier series, a periodicsignal 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.
Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.
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o
oT
f1
=
FT of Periodic Signals
( ) ( ) ( ) ( ) ( )∑∑∞
−∞=
∞
−∞=
⋅−⋅=⇔−=n
o
m
fnffnGffXmTtgtx 000 δ
T0 T0t1 t1+T0
t
g(t)x(t)
( ) ( )
( ) ( ) ( ) ( )( )∑
∑∞
=
∞
−∞=
⋅⋅⋅
⋅∠+⋅⋅⋅⋅⋅⋅+⋅=
⋅⋅=
1
000
2
00
2cos20
0
n
oo
n
tfnj
fnGtfnfnGfGftx
efnGftx
π
π
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Fourier Series: Example 1
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.5
0
0.5
1
1.5
time (s)
Amplitude
Periodic Waveform
29Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
Fourier Series Example 2
-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5
-1
-0.5
0
0.5
1
1.5
time (s)
Amplitude
Periodic Waveform
30Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
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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 1Ω(normalized).
The PSD (energy) (in Watts / Hz) of a signal x(t) is
( ) 2fXSx =
31Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.
Power Spectral Density (PSD)
The average power (normalized) (in Watts) is
Parseval’s Theorem(Periodic Signals)
Haykin, S., and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., Wiley, 2007.
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( )2
0∑∞
−∞=
⋅=n
ave fnXP
( )dffSP xave ∫∞
∞−
=
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Examples 1 and 2 (PSD)
-1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4Example 1 PSD
Frequency (Hz)
PSD Sx
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4Example 2 PSD
Frequency (Hz)
PSD Sx
Pave = 0.4833 W
Pave=0.6464 W
33Haykin, S., and M. Moher, Introduction to Analog & Digital
Communications, 2nd ed., Wiley, 2007.