Download - Convolution Properties
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Convolution PropertiesDSP for Scientists
Department of PhysicsUniversity of Houston
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Properties of Delta Function
[n]: Identity for Convolution
x[n] [n] =x[n]
x[n] k[n] = kx[n]
x[n] [n + s] =x[n + s]
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Mathematical Properties of
Convolution (Linear System)
Commutative: a[n] b[n] = b[n] a[n]
a[n] b[n] y[n]
Then b[n] a[n] y[n]
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Properties of Convolution
Associative:{a[n] b[n]} c[n] = a[n] {b[n] c[n]}
If
a[n] b[n] c[n] y[n]
Then a[n] b[n] c[n] y[n]
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Properties of Convolution
Distributive
a[n]b[n] + a[n]c[n] = a[n]{b[n] + c[n]}
If b[n]
a[n] + y[n]
c[n]
Then
a[n] b[n]+c[n] y[n]
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Properties of Convolution
Transference: between Input & Output
Suppose x[n] * h[n] =y[n] IfL is a linear system,
x1[n] =L{x[n]}, y1[n] =L{y[n]} Then
x1[n] h[n]= y1[n]
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Continue
Ifx[n] h[n] y[n]
Linear Same Linear
System System
Then
x1[n] h[n] y1[n]
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Special Convolution Cases
Auto-Regression (AR) Model
y[n] = k = 0, M - 1.h[k]x[n - k]
For Example: y[n] =x[n] -x[n - 1] (first difference)
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Special Convolution Cases
Moving Average (MA) Model
y[n] = b[0]x[n] + k= 1,M- 1b[k]y[n - k]
For Example: y[n] =x[n] +y[n - 1] (Running Sum)
AR and MA are Inverse to Each Other
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Example
For One-order Difference Equation (MAModel)
y[n] = ay[n - 1] +x[n]
Find the Impulse Response, if the system is
(a) Causal
(b) Anti-causal
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Causal System Solution
Input: [n] Output: h[n] For Causal system, h[n] = 0, n < 0
h[0] = ah[-1] +[0] = 1
h[1] = ah[0] + [1] = a
h[n] = anu[n]
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Anti-causal
Input:x[n] = [n] Output:y[n] = h[n] For Anti-Causal system, h[n] = 0, n > 0
y[n - 1] = (y[n] -x[n]) /a h[0] = (h[1] - [1]) /a = 0
h[-1] = (h[0] -[0])/ a = - a
-1
h[- n] = - a-n h[n] = -anu[-n - 1]
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Central Limit Theorem
If a pulse-like signal is convoluted with
itself many times, a Gaussian will be
produced.
a[n] 0
a[n] a[n] a[n] a[n] = ???
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Central Limit Theorem
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Correlation !!!
Cross-Correlation a[n] b[-n] = c[n]
Auto-Correlation:
a[n] a[-n] = c[n] Optimal Signal Detector (Not Restoration)
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Correlation Detector
- 1 5 - 1 0 - 5 0 5 1 0 1 5
- 1
- 0 . 8
- 0 . 6
- 0 . 4
- 0 . 2
0
0 . 2
0 . 4
0 . 6
0 . 8
1
- 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0 2 5- 2
- 1 . 5
- 1
- 0 . 5
0
0 . 5
1
1 . 5
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Correlation Results
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0
-6
-4
-2
0
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-5 -4 -3 -2 -1 0 1 2 3 4 5-0.6
-0.4
-0.2
0
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Low-Pass Filter
Filter h[n]:
Cutoff the high-frequency components(undulation, pitches), smooth the signal
h[n] 0, (- 1)n h[n] = 0, n = -,
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Example: Lowpass
0 50 100 150 200 250 300 350-60
-40
-20
0
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10 0
12 0
14 0
0 50 100 150 200 250 300 350
-60
-40
-20
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10 0
12 0
14 0
-10 -8 -6 -4 -2 0 2 4 6 8 10-0.1
0
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High-Pass Filter
Filter g[n]:
Remove the Average Value of Signal
(Direct Current Components), Only
Preserve the Quick Undulation Terms
ng[n] = 0
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Example: Highpass
0 50 100 150 200 250 300 350-8
-6
-4
-2
0
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0 50 100 150 200 250 300 350
-60
-40
-20
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10 0
12 0
14 0
-4 -3 -2 -1 0 1 2 3 4
-0.6
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0
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Delta Function
x[n] [n] =x[n]
Do not Change Original Signal Delta function: All-Pass filter
Further Change: Definition (Low-pass,
High-pass, All-pass, Band-pass )