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Convolution Properties
DSP for Scientists
Department of Physics
University of Houston
Properties of Delta Function
◗
δ [n]:
Identity for Convolution
x[n] ∗ δ [n] = x[n]
◗ x[n] ∗ kδ [n] = kx[n]
◗ x[n] ∗ δ [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]
◗ Then
◗
◗
a[n]
c[n]
y[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
◗ a[n]
b[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]
◗ If L 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
◗
If
x[n]
h[n]
Linear
System
y[n]
Same Linear
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 - 1 b[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 (MA
Model)
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: δ [n]
Output: h[n]
For Causal system, h[n] = 0, n < 0
h[0] = ah[-1] + δ [0] = 1
h[1] = ah[0] + δ [1] = a
…
h[n] = anu[n]
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Anti-causal
◗
◗
◗
◗
◗
◗
◗
Input: x[n] = δ [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] - δ [1]) / a = 0
h[-1] = (h[0] - δ [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
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1 .5
1
0 .5
0
-0 . 5
-1
-1 . 5
-2
-2 5
-2 0
-1 5
-1 0
-5
0
5
1 0
1 5
2 0
2 5
1
0 .8
0 .6
0 .4
0 .2
0
-0 . 2
-0 . 4
-0 . 6
-0 . 8
-1
-1 5
-1 0
-5
0
5
1 0
1 5
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Correlation Results
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-5
-4
-3
-2
-1
0
1
2
3
4
5
8
6
4
2
0
-2
-4
-6
0
200
400
600
800
1000
1200
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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
1 40
1 20
0.5
1 00
0.4
80
60
0.3
40
0.2
20
0
0.1
-20
0
-40
-60
0
50
1 00
150
20 0
25 0
3 00
-0.1
-1 0
3 50
-8
-6
-4
-2
0
2
4
6
8
10
1 40
1 20
1 00
80
60
40
20
0
-20
-40
-60
0
50
1 00
150
20 0
25 0
3 00
3 50
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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
◗
◗
∑n g[n] = 0
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Example: Highpass
0.6
1 40
1 20
0.4
1 00
80
0.2
60
40
0
20
-0.2
0
-20
-0.4
-40
-60
0
50
1 00
150
20 0
25 0
3 00
3 50
-0.6
-4
-3
-2
25 0
3 00
3 50
-1
0
1
2
3
4
8
6
4
2
0
-2
-4
-6
-8
0
50
1 00
150
20 0
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Delta Function
x[n] ∗ δ[n] = x[n]
• Do not Change Original Signal
• Delta function:
All-Pass filter
•
•
Further Change: Definition (Low-pass,
High-pass, All-pass, Band-pass …)
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