0.4 Signal processing in processing: convolution and filtering  (Page 2/2)

To generalize the example [link] we can define the operation of convolution .

Convolution of two signals $h$ and $x$

$y(n)=\mathrm{h\; *\; x}(n)=\sum_{m=()}$∞ ∞ x m h n m

The operation of convolution can be fully understood by the explicit construction of some examples ofconvolution product. The module Discrete-Time Convolution gives the graphic construction of an examples and it offers pointers toother examples.

Properties

The properties of the convolution operation are well illustrated in themodule Properties of Convolution . The most interesting of such properties is the extension:

If $x(n)$ is extended over $M_1$ samples, and $h(n)$ is extended over $M_2$ samples, then the convolution product $y(n)$ is extended over $M_1+M_2-1$ samples.

Therefore, the signal convolution product is longer than both the input signal and the impulse response.

Another interesting property is the commutativity of the convolutionproduct, such that the input signal and the impulse response can change their roles without affecting the outputsignal.

Frequency response and filtering

The Fourier Transform of the impulse response is called Frequency Response and it is represented with $H(\omega )$ . The Fourier transform of the system output is obtained by multiplication of the Fourier transform of theinput with the frequency response, i.e., $Y(\omega )=H(\omega )X(\omega )$ .

The frequency response shapes, in a multiplicative fashion, the input-signal spectrum or, in other words, it performs some filtering by emphasizing some frequency components and attenuating some others. A filtering can alsooperate on the phases of the spectral components, by delaying them of different amounts.

Filtering can be performed in the time domain (or space domain), by the operation of convolution, or in the frequency domain by multiplication of the frequency response.

Causality

The notion of causality is quite intuitive and it corresponds to the experience of stimulating a system andgetting back a response only at future time instants. For discrete-time LTI systems, this happens when the impulseresponse is zero for negative (discrete) time instants. Causal LTI systems can produce with no appreciabledelay an output signal sample-by-sample , because the convolution operator acts only on present andpast values of the input signal. In [link] the impulse response is not causal, but this is not a problem because the whole input signal isalready available, and the filter can process the whole block of samples.

2d filtering

The notions of impulse response, convolution, frequency response, and filtering naturally extend from 1D to 2D, thusgiving the fundamental concepts of image processing.

Convolution of two 2D signals (images)

$y(m, n)=\mathrm{h\; *\; x}(m, n)=\sum_{k=()}$∞ ∞ l ∞ ∞ x k l h m k n l

As in the 1D case, the Fourier transform of the impulse response iscalled Frequency response and it is indicated by $H({\omega }_{\mathrm{X}}, {\omega }_{\mathrm{Y}})$ . The Fourier transform of the system output is obtained by Fourier-transforming the input and multiplying theresult by the frequency response. $Y({\omega }_{\mathrm{X}}, {\omega }_{\mathrm{Y}})=H({\omega }_{\mathrm{X}}, {\omega }_{\mathrm{Y}})X({\omega }_{\mathrm{X}}, {\omega }_{\mathrm{Y}})$ .