Sherna at 08:55, 13 October 2006

2006-10-13T08:55:51Z

← Older revision		Revision as of 01:55, 13 October 2006
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	Convolution Theorem is as follows		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>		*<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>

Sherna at 06:56, 13 October 2006

2006-10-13T06:56:59Z

New page

Convolution Theorem is as follows

*<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>
Switching the order of integration
*<math> x(t) = \int_{-\infty}^{\infty}x(\lambda) \left(\int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)}\, df\right)\,d\lambda</math>
Taking note of the fact that the inner integral simplifies to <math>\int_{-\infty}^{\infty}e^{j2\pi f(t-\lambda)t}\,df = \delta(t-\lambda) = \delta(\lambda - t)</math>
*<math> \bar{x} = \sum_i (\bar{x}-\hat{a_i})\hat{a_i} = \sum_i \left(\sum_j x_ja_{ij}\right)\hat{a_i}</math>

Work in progress
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Convolution Theorem - Revision history

Sherna at 08:55, 13 October 2006

Sherna at 06:56, 13 October 2006