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Transcript of signal and system Lecture 10
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Signals and SystemsFall 2003
Lecture #10
7 October 2003
1. Examples of the DT Fourier Transform
2. Properties of the DT Fourier Transform3. The Convolution Property and its
Implications and Uses
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DT Fourier Transform Pair
Analysis Equation
FT
Synthesis Equation
Inverse FT
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Convergence Issues
Synthesis Equation: None, since integrating over a finite interval
Analysis Equation: Need conditions analogous to CTFT, e.g.
Absolutely summable
Finite energy
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More Examples
Infinite sum formula
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Still More
4) DT Rectangular pulse (Drawn forN1 = 2)
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5)
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DTFTs of Sums of Complex Exponentials
Recall CT result:
What about DT:
Note: The integration in the synthesis equation is over 2 period,
only needX(ej) in one 2 period. Thus,
a) We expect an impulse (of area 2) at = ob) ButX(ej) must be periodic with period 2
In fact
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DTFT of Periodic Signals
Linearity
of DTFT
DTFS
synthesis eq.
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Example #1: DT sine function
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Example #2: DT periodic impulse train
Also periodic impulse train in the frequency domain!
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Properties of the DT Fourier Transform
Different from CTFT
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More Properties
Example
Important implications in DT because of periodicity
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Still More Properties
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Insert two zeros
in this example
(k=3)
But we can slow a DT signal down by inserting zeros:
k an integer 1
x(k)[n] insert (k- 1) zeros between successive values
Yet Still More Properties
7) Time Expansion
Recall CT property:Time scale in CT is
infinitely fine
But in DT: x[n/2] makes no sense
x[2n] misses odd values ofx[n]
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Time Expansion (continued)
Stretched by a factor
ofkin time domain
-compressed by a factor
ofkin frequency domain
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Is There No End to These Properties?
8) Differentiation in Frequency
Total energy in
time domain
Total energy in
frequency domain
9) Parsevals Relation
Differentiationin frequency
Multiplicationby n
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Example #1:
The Convolution Property
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Example #2: Ideal Lowpass Filter
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Example #3: