Convolution Representation

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Convolution Representation

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Convolution Representation

Note that the output of the $k$th delay element in Fig. B.1 is$x(n-k)$, $k=0,1,2,\, where $x(n)$ is the input signalamplitude at time $n$. The output signal $y(n)$ is therefore

$\$\$\(B.1)
 $\$\(B.2)
 $\$\(B.3)
 $\$\(B.4)

where we have used the convolution operator “$\, defined in general for any two signals $x_1$, $x_2$ as

\

An FIR filter thus operates by convolving the input signal $x(n)$ with the filter’s impulse response $h(n)$.

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