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I am reading The Scientist and Engineer's Guide to Digital Signal Processing by S.W. Smith. In Chapter 9: Applications of the DFT in the 3rd paragraph he writes "kernel (impulse response)"

What does it mean? What is meant by kernel? Is it same as impulse response?

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A kernel is a more general concept, but an impulse response is a special case of a kernel. One usage of the term kernel is to describe an integral transform:

$$y(t)=\int_{-\infty}^{\infty}x(\tau)K(\tau,t)d\tau\tag{1}$$

The function $K(\tau,t)$ is called the kernel of the integral transform.

If you compare $(1)$ to the convolution

$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\tag{2}$$

then you see that the kernel of the convolution $(2)$ is given by

$$K(\tau,t)=h(t-\tau)\tag{3}$$

Note that the term kernel is also used to describe linear transformations in the discrete domain:

$$y[n]=\sum_{m=-\infty}^{\infty}x[m]K[m,n]\tag{4}$$

Discrete-time convolution is again a special case of $(4)$ with $K[m,n]=h[n-m]$.

In signal processing we regularly use linear transforms such as $(1)$ and $(4)$. The most well-known examples apart from convolution are the Fourier transform, the Laplace transform, the $\mathcal{Z}$-transform, and the Hilbert transform (which is in fact just a convolution).

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  • $\begingroup$ What do you mean here " The most well-known examples apart from convolution are the Fourier transform, the Laplace transform, the Z-transform, and the Hilbert transform (which is in fact just a convolution) " ?? those all mentioned examples also use kernel?? $\endgroup$ – engr Jul 4 '20 at 18:23
  • $\begingroup$ @engr: Yes, they are all examples of linear transforms that can be written in the forms (1) or (4) with a specific kernel. $\endgroup$ – Matt L. Jul 4 '20 at 18:25

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