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Convolution PropertiesDSP for Scientists
Department of Physics
University of Houston
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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 ofConvolution (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]
◗ 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] 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 - 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 (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: 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 withitself many times, a Gaussian will beproduced.
◗ 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
2
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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
2
4
6
8
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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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
20
40
60
80
100
120
140
0 50 100 150 200 250 300 350
-60
-40
-20
0
20
40
60
80
100
120
140
-10 -8 -6 -4 -2 0 2 4 6 8 10-0.1
0
0.1
0.2
0.3
0.4
0.5
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High-Pass Filter
◗ Filter g[n]:
◗ Remove the Average Value of Signal(Direct Current Components), OnlyPreserve the Quick Undulation Terms
◗ ∑n g[n] = 0
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Example: Highpass
0 50 100 150 200 250 300 350-8
-6
-4
-2
0
2
4
6
8
0 50 100 150 200 250 300 350
-60
-40
-20
0
20
40
60
80
100
120
140
-4 -3 -2 -1 0 1 2 3 4-0.6
-0.4
-0.2
0
0.2
0.4
0.6
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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 …)