analysis and processing a system may also be defined as any device (physical or otherwise) that performs an operation on a signal (modified from Proakis and Manolakis,
1988). In the present context it is therefore a signal processing device.
Conceptually the simplest way to consider a system is as an input-output device, which supplies at its output a modified version of the input (Lynn, 1982).
input output
LINEAR
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SYSTEM
Figure 4.8 "Black-box" representation of a linear system.
In general, systems are classified as either linear or non-linear based upon the general mathematical characteristics of the type of operation that they perform on the signal. The most important class of system for signal processing is the "discrete linear" system. The theory of discrete linear systems has been developed extensively in connection with electrical circuit theory, automatic control theory and the theory of mechanical vibration, as well as signal theory; and forms the basis for the signal processing operations used in this study.
A linear system may be represented schematically, as shown in Figure 4.8, by a "black box" that receives an input stimulus. The system then produces an output response; the relationship of the response to the input stimulus is the subject o f Linear systems theory. In general the relationship between input and output of a linear system is described by a "linear differential equation with constant coefficients" (it should be
noted that this does not imply a straight-line or "linear" relationship between input and output of the system). More simply; if the input consists of two signals and the output consists of the sum of the outputs produced by each signal separately, the system is linear. This seemingly trivial property, known as linear superposition, has the most profound implications for signal processing. It means in particular that it is possible to evaluate separately the response to each component of a signal (Lynn, 1982).
For the present purpose there are two other important features of a linear system. The first is given by its so-called eigenfunctions. An eigenfunction of a system is an input function which passes through the system unchanged except for a possible alteration in its amplitude. That is, the system has no effect on the function except perhaps to alter its "strength". The eigenfunctions of a linear system are complex exponentials which include the Fourier transform and Fourier series described above.
The final feature of importance for linear systems with two-dimensional input signals, is known as shift-invariance. This means that a spatial shift in the input produces a corresponding shift in the output but no other change.
4.5.1 Frequency response and impulse response
The effect of a linear shift-invariant system on a given input signal may be described in two basic ways: by its frequency response; or by its impulse response. The frequency response is the description of the operation of the system in terms of spatial frequencies in the Fourier domain, while the impulse response is the spatial domain description of the response of the system to a infinitely powerful point stimulus known as a delta function. The impulse response and the frequency response are a Fourier transform pair, that is, one is the transform of the other. Consequently, they are equivalent descriptions of the operation of a linear system shift invariant system.
If the impulse response or the frequency response and input to a linear system are known the output can be calculated in either the Fourier or spatial domains. In the Fourier domain the frequency response and the fourier transform of the input image are multiplied together to obtain the Fourier transform of the output. In the spatial domain the output of a linear system can be found by a convolution operation between the input signal and the impulse response of the system.
4.5.2 Spatial domain convolution.
Convolution between a sampled image and an impulse response is most easily described by the use of a moving "window" - the impulse response array - passing over the image to be processed (Figure 4.9). The impulse response array is a two-dimensional grid consisting of an odd number of cells. Each cell contains a weighting factor which is multiplied with the pixel over which it lies. Thus, for a 3 x 3 window, as shown in Figure 4.9 each of the nine cells will be multiplied with the corresponding pixels producing nine numbers at any given position of the window. For each position of the window these numbers are added together. This new value is then placed at the position of the central pixel below the window in the output image. The window is then moved by one pixel and the procedure repeated until the window has occupied all possible positions over the input image. Consequently, each pixel in the output image is the result of a weighted summation of pixels in the neighbourhood of the corresponding pixel in the input image.
It is usually assumed that the impulse response array has a "finite region of
Sum products