Filter Theory

To make the continuous world around us accessible to a computer that is only capable of processing discrete numbers, is it inevitable to convert a given signal into numbers. The first step is to transform the continuous-time signal into a discrete-time signal. This means measuring the signal value in a periodic manner over the domain of the function and storing the obtained values in a list. How many samples are necessary to represent a signal perfectly is determined by its frequency spectrum. If a function has a very complex characteristics, then more samples are needed to cover the information. Shannon's sampling theorem, also known as Nyquist criterion gives the general answer to the question of how many samples are necessary to completely represent the continuous function by its discrete samples. It states that the sampling frequency must be at least twice the maximum frequency in the sampled signal. If the sampling is done according to the Nyquist criterion, the original signal can be reconstructed by convolving with the $sinc$ function. If the sampling rate is smaller, aliasing artifacts occur. Figure 2.6 gives an overview of how sampling and reconstruction work. The left column represents the spatial domain and shows all the events that occur during the process, the right column does the same for the frequency domain. The first line introduces the band limited initial function Figure 2.6(a) and its spectrum Figure 2.6(b). In the second row the sampling $shah$ function Figure 2.6(c) and its Fourier transform pair, also a $shah$ function, but with reciprocal spacing as seen in Figure 2.6(d) are used to sample the input signal in spatial domain. The sampling is done in the spatial domain by multiplication of the input signal with the $shah$ function which gives Figure 2.6(e) as a result. Multiplication in spatial domain is equivalent to a convolution in frequency domain. Therefore Figure 2.6(f) is the result of convolving the spectrum of the input signal with the Fourier transform pair of the sampling $shah$ function. We now have the sampled data available. In order to reconstruct the original input function, the situation of the first row in Figure 2.6 has to be restored. This can be done by multiplying with a box function in the frequency domain to cut out the central replica of the spectrum (see Figure 2.6(h)). The Fourier transform pair to the box is the $sinc$ function (see Figure 2.6(g)). Multiplication with the box in frequency domain signifies convolution with the $sinc$ function in spatial domain, to restore the original function.

Figure 2.6: Sampling and reconstruction. The left column represents the spatial domain, whereas the right column displays the equivalent signals in the frequency domain. (a) initial function and (b) its frequency spectrum. (c) the sampling Shah-function and (d) its Fourier transform pair. (e) is the result after a multiplication of (a) and (c). Therefore (f) equals the convolution of (b) and (d). (g) the sinc function is a transform pair to (h) the box function. The multiplication of (f) with (h) creates (j) the spectrum of the original function. Hence a convolution of (e) with (g) leads to (i) the original function.

(a)	(b)
(c)	(d)
(e)	(f)
(g)	(h)
(i)	(j)

If the signal is sampled with a lower frequency, the replicas of the frequency spectrum move closer. At the point when the sample frequency gets bellow the Nyquist frequency the replicas begin to overlap, see Figure 2.7. The overlapping areas of the replicas are summed up, which makes a reconstruction of the original function impossible. This error introduces aliasing artefacts in the reconstruction process. An example for signal processing is the human voice, which has frequency components up to 20kHz. In order to follow the Nyquist criteria a sampling rate of 44.1kHz has been specified in the CD audio standard [11].

Figure 2.7: Sampling of a function below the Nyquist criteria. (a) and (b) show the same initial function and its spectrum as in Figure 2.6. (c) is the sampling Shah-function with a coarser spacing and (d) the Fourier transform pair which has a proportional finer spacing. The multiplication of (a) with (c) gives (e). The dual operation to the multiplication in spatial domain is the convolution in the frequency domain. Therefore the convolution of (b) with (d) gives (f). The closer spacing of the impulses in (d) leads to overlapping of the replicas of the spectrum in the frequency domain. The resulting function is indicated by a bold line in (f). The overlap of the replicas makes a perfect reconstruction of the initial function impossible. The introduced error is referred to as aliasing.

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