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# Dt convolution

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# Dt convolution

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Convolution

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1. 1. Convolution of Discrete-Time Signals • Original Signals: x[n] = 5, 1, −2, 4, 0, 0, . . . h[n] = 1, 2, 3, 0, 0, 0, . . . • Convolution: y[n] = x[n] ∗ h[n] = ∞ i=−∞ x[i]h[n − i], −∞ < n < ∞ =    ∞ i=0 x[i]h[n − i], n ≥ 0 0, n < 0 – n = 0 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[0 − i] = . . . , 0, 3, 2, 1 y[0] = 5 · 1 = 5 – n = 1 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[1 − i] = . . . , 0, 0, 3, 2, 1 y[1] = 5 · 2 + 1 · 1 = 11 – n = 2 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[2 − i] = . . . , 0, 0, 0, 3, 2, 1 y[2] = 5 · 3 + 1 · 2 + (−2) · 1 = 15 – n = 3 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[3 − i] = . . . , 0, 0, 0, 0, 3, 2, 1 y[3] = 1 · 3 + (−2) · 2 + 4 · 1 = 3 – n = 4 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[4 − i] = . . . , 0, 0, 0, 0, 0, 3, 2, 1 y[4] = (−2) · 3 + 4 · 2 = 2 – n = 5 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[5 − i] = . . . , 0, 0, 0, 0, 0, 0, 3, 2, 1 y[5] = 4 · 3 = 12 – n > 5 (shown for n = 6) x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[n − i] = . . . , 0, 0, 0, 0, 0, 0, 0, 3, 2, 1 y[n] = 0, n > 5 • Output Signal: y[n] = 5, 11, 15, 3, 2, 12, 0, 0, . . .
2. 2. Convolution of Discrete-Time Signals • MATLAB: % Original signals n = 0:5; x = [5 1 -2 4 0 0]; h = [1 2 3 0 0 0]; y = conv(x, h); subplot(3,1,1); stem(n, x, ’k-’, ’filled’); grid; xlabel(’n’); ylabel(’x[n]’); subplot(3,1,2); stem(n, h, ’k-’, ’filled’); grid; xlabel(’n’); ylabel(’h[n]’); % Note: we cut trailing zeros off end of y before plotting subplot(3,1,3); stem(n, y(1:length(n)), ’k-’, ’filled’); grid; xlabel(’n’); ylabel(’y[n] = x[n] * h[n]’); 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −2 0 2 4 6 n x[n] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 n h[n] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 15 n y[n]=x[n]*h[n]

Convolution

#### Transcript

1. 1. Convolution of Discrete-Time Signals • Original Signals: x[n] = 5, 1, −2, 4, 0, 0, . . . h[n] = 1, 2, 3, 0, 0, 0, . . . • Convolution: y[n] = x[n] ∗ h[n] = ∞ i=−∞ x[i]h[n − i], −∞ < n < ∞ =    ∞ i=0 x[i]h[n − i], n ≥ 0 0, n < 0 – n = 0 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[0 − i] = . . . , 0, 3, 2, 1 y[0] = 5 · 1 = 5 – n = 1 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[1 − i] = . . . , 0, 0, 3, 2, 1 y[1] = 5 · 2 + 1 · 1 = 11 – n = 2 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[2 − i] = . . . , 0, 0, 0, 3, 2, 1 y[2] = 5 · 3 + 1 · 2 + (−2) · 1 = 15 – n = 3 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[3 − i] = . . . , 0, 0, 0, 0, 3, 2, 1 y[3] = 1 · 3 + (−2) · 2 + 4 · 1 = 3 – n = 4 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[4 − i] = . . . , 0, 0, 0, 0, 0, 3, 2, 1 y[4] = (−2) · 3 + 4 · 2 = 2 – n = 5 x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[5 − i] = . . . , 0, 0, 0, 0, 0, 0, 3, 2, 1 y[5] = 4 · 3 = 12 – n > 5 (shown for n = 6) x[i] = 5, 1, −2, 4, 0, 0, 0, . . . h[n − i] = . . . , 0, 0, 0, 0, 0, 0, 0, 3, 2, 1 y[n] = 0, n > 5 • Output Signal: y[n] = 5, 11, 15, 3, 2, 12, 0, 0, . . .
2. 2. Convolution of Discrete-Time Signals • MATLAB: % Original signals n = 0:5; x = [5 1 -2 4 0 0]; h = [1 2 3 0 0 0]; y = conv(x, h); subplot(3,1,1); stem(n, x, ’k-’, ’filled’); grid; xlabel(’n’); ylabel(’x[n]’); subplot(3,1,2); stem(n, h, ’k-’, ’filled’); grid; xlabel(’n’); ylabel(’h[n]’); % Note: we cut trailing zeros off end of y before plotting subplot(3,1,3); stem(n, y(1:length(n)), ’k-’, ’filled’); grid; xlabel(’n’); ylabel(’y[n] = x[n] * h[n]’); 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −2 0 2 4 6 n x[n] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 n h[n] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 5 10 15 n y[n]=x[n]*h[n]

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