Convolution Theorem

Convolution Theorem is as follows

• $x(t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(\lambda )e^{-j2\pi f\lambda }\,d\lambda \,e^{j2\pi ft}\,df$

Switching the order of integration

• $x(t)=\int _{-\infty }^{\infty }x(\lambda )\left(\int _{-\infty }^{\infty }e^{j2\pi f(t-\lambda )}\,df\right)\,d\lambda$

Taking note of the fact that the inner integral simplifies to $\int _{-\infty }^{\infty }e^{j2\pi f(t-\lambda )t}\,df=\delta (t-\lambda )=\delta (\lambda -t)$

• ${\bar {x}}=\sum _{i}({\bar {x}}-{\hat {a_{i}}}){\hat {a_{i}}}=\sum _{i}\left(\sum _{j}x_{j}a_{ij}\right){\hat {a_{i}}}$

Work in progress
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