Classical Methods of Magnitude Approximation

Abstract

Electric wave filters, called simply as filters in this book, are circuits that are used to selectively filter out some of the frequency components of an input signal and transmit the remaining components as the output signal in such a way that the quality of information contained in the signal is improved. We consider continuous-time signals, one-dimensional (1-D) and two-dimensional (2-D) discrete-time signals as the input and output of filters and characterize them in the frequency domain by their appropriate Fourier Transform. For example, if x (t) and y(t) are the continuous-time, input and output signals of a linear, time-invariant, continuous-time filter, their Fourier Transforms X (jω) and Y(jω) are related by the equation Y(jω) = H(jω)X(jω) where H(jω) is the Fourier Transform of the unit impulse response h(t) of the continuous-time filter. The relation can also be expressed in terms of these complex valued functions as

$$\left| {Y(j\omega )} \right|{e^{j\varphi }} = \left[ {\left| {H(j\omega )} \right|{e^{j\theta }}} \right]\left[ {|X(j\omega )|{e^{j\phi }}} \right] = \left| {H(j\omega )} \right|\left| {X(j\omega )} \right|{e^{j(\theta + \phi )}}$$

(1.1)

We see that the magnitude of the output as a function of frequency is |H (jω)| times the magnitude |X (jω)| of the input signal as a function of the frequency and the phase angle of the output signal is the phase angle of the input increased by that of H(jω). Therefore by properly shaping the magnitude of H(jω) and the phase angle θ(jω) as a function of frequency, we can design the filter to transmit some frequencies and block out other frequencies as we decide.