Convolution is a mathematical operation that combines two functions to produce a third function. It is a common technique used in signal processing and image processing to extract important information from a given input signal or image.

One of the simplest examples of convolution is the convolution of two signals. Consider two signals f(x) and g(x), where f(x) represents the input signal and g(x) represents the filter or kernel that is applied to the input signal. The convolution of these two signals is given by the following equation:

f(x) * g(x) = ∫ f(t)g(x-t) dt

In this equation, the asterisk symbol (*) denotes convolution, and the integral symbol represents the sum of the product of f(t) and g(x-t) over a given range of t. This operation can be visualized as sliding the kernel g(x) over the input signal f(x) and calculating the sum of the products of the two signals at each point.

For instance, if f(x) is a signal that represents a series of discrete time-steps and g(x) is a kernel that represents a low-pass filter, then the convolution of these two signals will produce a new signal that is the smoothed version of the original input signal. This is because the low-pass filter will suppress the high-frequency components of the input signal and retain only the low-frequency components, resulting in a smoothed output signal.

Another example of convolution is in image processing. In this case, the input signal is an image, and the kernel is a small matrix of numbers known as a filter. This filter is applied to the input image by sliding it over the entire image and computing the dot product of the filter and the overlapping image pixels at each position. This produces a new output image that has been transformed by the filter.

For instance, if the input image is a grayscale image of a face, and the kernel is a filter that detects edges, then the convolution of these two signals will produce a new output image that highlights the edges in the input image. This is because the edge detection filter will highlight the regions of the image where there is a sharp change in pixel intensity, which typically correspond to edges in the image.

In summary, convolution is a mathematical operation that combines two signals or functions to produce a third signal or function. It is a powerful tool that is widely used in signal and image processing to extract important information from given inputs.