CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS
3.0.
Introduction
3.1.
A Historical Perspective
3.2.
The Response of LTI Systems to Complex Exponentials
3.3.
Fourier Series Representation of Continuous-Time
Periodic Signals
3.3.1. Linear Combination of harmonically Related Complex
Exponentials
3.3.2. Determination of the Fourier Series Representation of a
Continuous-Time Periodic Signal
3.4.
Convergence of Fourier Series
3.5.
Properties of Continuous-Time Fourier Series
3.5.1. Linearity
3.5.2. Time Shifting
3.5.3. Time Reversal
3.5.4. Time Scaling
3.5.5. Multiplication
3.5.6. Conjugation and Conjugate Symmetry
3.5.7. Parseval’s Relation for Continuous-Time Periodic Signals
3.5.8. Summary of Properties of the Continuous-Time Fourier Series
3.5.9. Examples
3.6.
Fourier Series Representation of Discrete-Time Periodic
Signals
3.6.1. Linear Combination of harmonically Related Complex
Exponentials
3.6.2. Determination of the Fourier Series Representation of a Periodic
Signal
3.7.
Properties of Discrete-Time Fourier Series
3.7.1. Multiplication
3.7.2. First Difference
3.7.3. Parseval’s Relation for Discrete-Time Periodic Signals
3.7.4. Examples
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
3.8.
Fourier Series and LTI Systems
3.9.
Filtering
3.9.1. Frequency-Shaping Filters
3.9.2. Frequency-Selective Filters
3.10.
Examples of Continuous-Time Filters Described by
Differential Equations
3.10.1.
A Simple RC Lowpass Filter
3.10.2.
A Simple RC Highpass Filter
3.11.
Examples of Discrete-Time Filters Described by Difference
Equations
3.11.1.
First-order Recursive Discrete-Time Filters
3.11.2.
Non-recursive Discrete-Time Filters
3.12.
Summary
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
Page 3
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
Page 4
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
Page 5
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
Page 6
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
Page 8
College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
k=n
k≠n
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
a0 =0
n
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
3.5.
Lecture No. 6
Properties of Continuous-Time Fourier Series
+∞
+∞
𝑥(𝑡) = ∑ 𝑎𝑘 𝑒 𝑗𝑘𝑤0𝑡 = ∑ 𝑎𝑘 𝑒 𝑗𝑘(2𝜋/𝑇)𝑡
𝑘=−∞
𝑎𝑘 =
Synthesis Equation
(3.38)
Analysis Equation
(3.39)
𝑘=−∞
1
1
∫ 𝑥(𝑡)𝑒 −𝑗𝑘𝑤0 𝑡 𝑑𝑡 = ∫ 𝑥(𝑡)𝑒 −𝑗𝑘(2𝜋/𝑇)𝑡 𝑑𝑡
𝑇 𝑇
𝑇 𝑇
TABLE 3.1 PROPERTIES OF CONTINUOUS-TIME FOURIER SERIES
Property
Periodic Signal
Fourier Series Coefficients
periodic with period T and
𝑥(𝑡)
}
𝑦(𝑡) fundamental frequency 𝜔0 =2𝜋/𝑇
𝑎𝑘
𝑏𝑘
Linearity
𝐴𝑥(𝑡) + 𝐵𝑦(𝑡)
𝐴𝑎𝑘 + 𝐵𝑏𝑘
Time Shifting
𝑥(𝑡 − 𝑡0 )
𝑎𝑘 𝑒 −𝑗𝑘𝑤0𝑡0 = 𝑎𝑘 𝑒 −𝑗𝑘(2𝜋/𝑇)𝑡0
Frequency Shifting
𝑒 𝑗𝑀𝑤0𝑡 𝑥(𝑡) = 𝑒 𝑗𝑀(2𝜋/𝑇)𝑡 𝑥(𝑡)
𝑎𝑘−𝑀
Conjugation
𝑥 ∗ (𝑡)
∗
𝑎−𝑘
Time Reversal
𝑥(−𝑡)
𝑎−𝑘
Time Scaling
𝑥(𝛼𝑡), 𝛼 > 0 (periodic with period 𝑇/𝛼)
𝑎𝑘
Periodic Convolution
𝑇𝑎𝑘 𝑏𝑘
∫ 𝑥(𝜏)𝑦(𝑡 − 𝜏)𝑑𝜏
𝑇
Multiplication
𝑥(𝑡)𝑦(𝑡)
Differentiation
𝑑𝑥(𝑡)
𝑑𝑡
+∞
∑ 𝑎𝑙 𝑏𝑘−𝑙
𝑙=−∞
Conjugate Symmetry for
Real Signals
𝑥(𝑡) real
Real and Even Signals
𝑥(𝑡) real and even
2𝜋
𝑎
𝑇 𝑘
1
1
(
) 𝑎𝑘 = (
)𝑎
𝑗𝑘𝜔0
𝑗𝑘(2𝜋/𝑇) 𝑘
∗
𝑎𝑘 = 𝑎−𝑘
}
ℛℯ{𝑎𝑘 = ℛℯ{𝑎−𝑘 }
ℐ𝓂{𝑎𝑘 } = −ℐ𝓂{𝑎−𝑘 }
|𝑎𝑘 | = |𝑎−𝑘 |
∢ 𝑎𝑘 = −∢ 𝑎−𝑘
𝑎𝑘 real and even
Real and Odd Signals
𝑥(𝑡) real and odd
𝑎𝑘 purely imaginary and odd
Even-Odd Decomposition
of real Signals
𝑥 (𝑡) = ℰ𝓋{𝑥(𝑡)},
{ 𝑒
𝑥𝑜 (𝑡) = 𝒪𝒹{𝑥(𝑡)},
Integration
𝑡
∫ 𝑥(𝑡)𝑑𝑡
−∞
𝑗𝑘𝜔0 𝑎𝑘 = 𝑗𝑘
(finite valued and
periodic only if 𝑎0 = 0)
𝑥(𝑡) real
𝑥(𝑡) real
ℛℯ{𝑎𝑘 }
𝑗 ℐ𝓂{𝑎𝑘 }
Parseval’s Relation for Periodic Signals
+∞
𝟏
∫ |𝒙(𝒕)|𝟐 𝒅𝒕 = ∑ |𝒂𝒌 |𝟐
𝑻 𝑻
𝒌=−∞
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
3.7.
Lecture No. 6
Properties of Discrete-Time Fourier Series
𝑥[𝑛] = ∑ 𝑎𝑘 𝑒 𝑗𝑘𝑤0𝑛 = ∑ 𝑎𝑘 𝑒 𝑗𝑘(2𝜋/𝑁)𝑛
𝑘=〈𝑁〉
𝑎𝑘 =
Synthesis Equation
(3.38)
Analysis Equation
(3.39)
𝑘=〈𝑁〉
1
1
∑ 𝑥[𝑛]𝑒 −𝑗𝑘𝑤0𝑛 =
∑ 𝑥[𝑛]𝑒 −𝑗𝑘(2𝜋/𝑁)𝑛
𝑁
𝑁
𝑛=〈𝑁〉
𝑛=〈𝑁〉
TABLE 3.1 PROPERTIES OF DISCRETE-TIME FOURIER SERIES
Property
Periodic Signal
Fourier Series Coefficients
𝑥[𝑛]
periodic with period N and
}
𝑦[𝑛] fundamental frequency 𝜔0 =2𝜋/𝑁
𝑎𝑘 periodic with period N
𝑏𝑘 }
Linearity
𝐴𝑥[𝑛] + 𝐵𝑦[𝑛]
𝐴𝑎𝑘 + 𝐵𝑏𝑘
Time Shifting
𝑥[𝑛 − 𝑛0 ]
𝑎𝑘 𝑒 −𝑗𝑘𝑤0𝑛0 = 𝑎𝑘 𝑒 −𝑗𝑘(2𝜋/𝑁)𝑛0
Frequency Shifting
𝑗𝑀(
2𝜋
)𝑛
𝑁 𝑥[𝑛]
𝑎𝑘−𝑀
Conjugation
𝑒 𝑗𝑀𝑤0𝑛 𝑥[𝑛] = 𝑒
𝑥 ∗ [𝑛]
Time Reversal
𝑥[−𝑛]
𝑎−𝑘
Time Scaling
𝑥[𝑛⁄𝑚], if n is a multiple of 𝑚
𝑥𝑚 [𝑛] = {
0,
if 𝑛 is not a multiple of 𝑚
1
viewed as periodic
𝑎𝑘 (
)
𝑚
with period 𝑚𝑁
𝑁𝑎𝑘 𝑏𝑘
∗
𝑎−𝑘
Periodic Convolution
∑ 𝑥[𝑟]𝑦[𝑛 − 𝑟]
Multiplication
𝑛=〈𝑁〉
𝑥[𝑛]𝑦[𝑛]
∑ 𝑎𝑙 𝑏𝑘−𝑙
First Difference
Running Sum
𝑙=〈𝑁〉
(1 − 𝑒 𝑗𝑘(2𝜋⁄𝑁) )𝑎𝑘
𝑥[𝑛] − 𝑥[𝑛 − 1]
𝑛
∑ 𝑥[𝑘]
𝑘=−∞
(finite valued and
periodic only if 𝑎0 = 0)
1
(
) 𝑎𝑘
⁄𝑁)𝑛
𝑗𝑘(2𝜋
1−𝑒
Real and Even Signals
𝑥[𝑛] real and even
∗
𝑎𝑘 = 𝑎−𝑘
ℛℯ{𝑎𝑘 } = ℛℯ{𝑎−𝑘 }
ℐ𝓂{𝑎𝑘 } = −ℐ𝓂{𝑎−𝑘 }
|𝑎𝑘 | = |𝑎−𝑘 |
∢ 𝑎𝑘 = −∢ 𝑎−𝑘
𝑎𝑘 real and even
Real and Odd Signals
𝑥[𝑛] real and odd
𝑎𝑘 purely imaginary and odd
Even-Odd Decomposition
of real Signals
𝑥 [𝑛] = ℰ𝓋{𝑥[𝑛]},
{ 𝑒
𝑥𝑜 [𝑛] = 𝒪𝒹{𝑥[𝑛]},
Conjugate Symmetry for
Real Signals
𝑥[𝑛] real
𝑥[𝑛] real
𝑥[𝑛] real
ℛℯ{𝑎𝑘 }
𝑗 ℐ𝓂{𝑏𝑘 }
Parseval’s Relation for Periodic Signals
𝟏
∑ |𝒙[𝒏]|𝟐 = ∑ |𝒂𝒌 |𝟐
𝑵
𝒏=〈𝑵〉
𝒌=〈𝑵〉
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia
CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed
Lecture material courtesy of Dr. Anwar M. Mirza
Lecture No. 6
Department of Computer Engineering
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College of Computer & Information Sciences, King Saud University
Ar Riyadh, Kingdom of Saudi Arabia