signals and systems · 2019. 9. 12. · signals and systems fall 2003 lecture #6 23 september 2003...
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Signals and SystemsFall 2003
Lecture #623 September 2003
1. CT Fourier series reprise, properties, and examples2. DT Fourier series3. DT Fourier series examples and
differences with CTFS
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CT Fourier Series Pairs
Skip it in future for shorthand
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Another (important!) example: Periodic Impulse Train
— All components have:(1) the same amplitude,
&(2) the same phase.
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(A few of the) Properties of CT Fourier Series
• Linearity
Introduces a linear phase shift ∝ to
• Conjugate Symmetry
• Time shift
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Example: Shift by half period
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• Parseval’s Relation
Energy is the same whether measured in the time-domain or the frequency-domain
• Multiplication Property
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Periodic Convolution
x(t), y(t) periodic with period T
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Periodic Convolution (continued)
Periodic convolution: Integrate over any one period (e.g. -T/2 to T/2)
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Periodic Convolution (continued) Facts
1) z(t) is periodic with period T (why?)
2) Doesn’t matter what period over which we choose to integrate:
3) Periodic
convolutionin time
Multiplicationin frequency!
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Fourier Series Representation of DT Periodic Signals
• x[n] - periodic with fundamental period N, fundamental frequency
• Only ejω n which are periodic with period N will appear in the FS
• So we could just use• However, it is often useful to allow the choice of N consecutive
values of k to be arbitrary.
⇓
• There are only N distinct signals of this form
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DT Fourier Series Representation
= Sum over any N consecutive values of kk =<N >∑
— This is a finite series
{ak} - Fourier (series) coefficients
Questions:
1) What DT periodic signals have such a representation?
2) How do we find ak?
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Answer to Question #1:
Any DT periodic signal has a Fourier series representation
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A More Direct Way to Solve for ak
Finite geometric series
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So, from
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DT Fourier Series Pair
Note: It is convenient to think of ak as being defined for allintegers k. So:
1) ak+N = ak — Special property of DT Fourier Coefficients.
2) We only use N consecutive values of ak in the synthesis equation. (Since x[n] is periodic, it is specified by N numbers, either in the time or frequency domain)
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Example #1: Sum of a pair of sinusoids
01/21/2
ejπ/4/2e-jπ/4/200
a-1+16 = a-1 = 1/2
a2+4×16 = a2 = ejπ/4/2
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Example #2: DT Square Wave
Using n = m - N1
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Example #2: DT Square wave (continued)
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Convergence Issues for DT Fourier Series:Not an issue, since all series are finite sums.
Properties of DT Fourier Series: Lots, just as with CT Fourier Series
Example: