Convolution Theorem: Difference between revisions

Latest revision as of 01:55, 13 October 2006

Convolution Theorem is as follows

$ℱ^{- 1} [X (f) H (f)] = x (t) \times h (t) = \int_{- \infty}^{\infty} x (λ) h (t - λ) d λ$
$\int_{- \infty}^{\infty} (\int_{- \infty}^{\infty} x (λ) h (t - λ) d λ) e^{- j 2 π f t} d t$
$\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (\int_{- \infty}^{\infty} X (f^{″}) e^{j 2 π f λ} d f \int_{- \infty}^{\infty} H (f^{'}) e^{j 2 π f^{'} (t - λ)} d f^{'}) e^{- j 2 π f t} d t d λ$
$\int_{- \infty}^{\infty} X (f^{″}) \int_{- \infty}^{\infty} H (f^{'}) \int_{- \infty}^{\infty} e^{j 2 π (f^{'} - f) t} d t \int_{- \infty}^{\infty} e^{j 2 π (f^{″} - f^{'})} d λ d f^{'} d f^{″}$
$\int_{- \infty}^{\infty} X (f^{″}) \int_{- \infty}^{\infty} H (f^{'}) δ (f - f^{'}) δ (f^{″} - f^{'}) d f^{'} d f^{″}$
$\int_{- \infty}^{\infty} X (f^{'}) H (f^{'}) δ (f - f^{'}) d f^{'}$
$X (f) H (f)$

$x (t) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x (λ) e^{- j 2 π f λ} d λ e^{j 2 π f t} d f$

Switching the order of integration

$x (t) = \int_{- \infty}^{\infty} x (λ) (\int_{- \infty}^{\infty} e^{j 2 π f (t - λ)} d f) d λ$

Taking note of the fact that the inner integral simplifies to $\int_{- \infty}^{\infty} e^{j 2 π f (t - λ) t} d f = δ (t - λ) = δ (λ - t)$

$\bar{x} = \sum_{i} (\bar{x} - \hat{a_{i}}) \hat{a_{i}} = \sum_{i} (\sum_{j} x_{j} a_{i j}) \hat{a_{i}}$

Work in progress
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 Convolution Theorem is as follows
+*<math>\mathcal{F}^{-1}\left[ X(f)H(f)\right] = x(t)\times h(t) = \int_{-\infty}^{\infty}x(\lambda)h(t-\lambda)\,d\lambda</math>
+*<math>\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}x(\lambda)h(t-\lambda)\,d\lambda\right)e^{-j2\pi ft}\,dt</math>
+*<math>\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}X(f'')e^{j2\pi f \lambda}\,df\int_{-\infty}^{\infty}H(f')e^{j2 \pi f'(t-\lambda)}\,df'\right) e^{-j2 \pi f t}\,dt\,d\lambda </math>
+*<math>\int_{-\infty}^{\infty}X(f'')\int_{-\infty}^{\infty}H(f')\int_{-\infty}^{\infty} e^{j2\pi (f'-f)t} \, dt \int_{-\infty}^{\infty} e^{j2\pi (f''-f')}\,d \lambda \, df' \, {df''}</math>
+*<math>\int_{-\infty}^{\infty}X(f'')\int_{-\infty}^{\infty}H(f')\delta(f-f') \delta(f''-f') \, df' \, {df''}</math>
+*<math>\int_{-\infty}^{\infty}X(f')H(f')\delta(f-f')\, {df'}</math>
+*<math>X(f)H(f)\,</math>
 *<math> x(t) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(\lambda)e^{-j2\pi f \lambda}\,d\lambda \,e^{j2\pi f t}\,df</math>