ABSTRACT
An electrical filter is a system that can be used to modify, reshape, or manipulate the frequency spectrum of an electrical signal according to some prescribed requirements. For example, a filter may be used to amplify or attenuate a range of frequency components, reject or isolate one specific frequency component, and so on. The applications of electrical filters are numerous, for example,
. To eliminate signal contamination such as noise in communication systems
. To separate relevant from irrelevant frequency components
. To detect signals in radios and TV’s
. To demodulate signals
. To bandlimit signals before sampling
. To convert sampled signals into continuous-time signals
. To improve the quality of audio equipment, e.g., loudspeakers
. In time-division to frequency-division multiplex systems
. In speech synthesis
. In the equalization of transmission lines and cables
. In the design of artificial cochleas
Typically, an electrical filter receives an input signal or excitation and produces an output signal or response. The frequency spectrum of the output signal is related to that of the input by some rule of correspondence. Depending on the type of input, output, and internal operating signals, three general types of filters can be identified, namely, continuous-time, sampled-data, and discrete-time filters. A continuous-time signal is one that is defined at each and every instant of time. It can be represented
by a function x(t) whose domain is a range of numbers (t1, t2), where 1 t1 and t21. A sampleddata or impulse-modulated signal is one that is defined in terms of an infinite summation of continuoustime impulses (see Ref. [1, Chapter 6]). It can be represented by a function
x^(t) ¼ X1
n¼1 x(nT)d(t nT)
where d(t) is the impulse function. The value of the signal at any instant in the range nT< t< (nþ 1)T is zero. The frequency spectrum of a continuous-time or sampled-data signal is given by the Fourier transform.* A discrete-time signal is one that is defined at discrete instants of time. It can be represented by a
function x(nT), where T is a constant and n is an integer in the range (n1, n2) such that 1 n1 and n21. The value of the signal at any instant in the range nT< t< (nþ 1)T can be zero, constant, or undefined depending on the application. The frequency spectrum in this case is obtained by evaluating the z transform on the unit circle jzj ¼ 1 of the z plane. Depending on the format of the input, output, and internal operating signals, filters can be classified
either as analog or digital filters. In analog filters the operating signals are varying voltages and currents, whereas in digital filters they are encoded in some binary format. Continuous-time and sampled-data filters are always analog filters. However, discrete-time filters can be analog or digital. Analog filters can be classified on the basis of their constituent components as
. Passive RLC filters
. Crystal filters
. Mechanical filters
. Microwave filters
. Active RC filters
. Switched-capacitor filters
Passive RLC filters comprise resistors, inductors, and capacitors. Crystal filters are made of piezoelectric resonators that can be modeled by resonant circuits. Mechanical filters are made of mechanical resonators. Microwave filters consist of microwave resonators and cavities that can be represented by resonant circuits. Active RC filters comprise resistors, capacitors, and amplifiers; in these filters, the performance of resonant circuits is simulated through the use of feedback or by supplying energy to a passive circuit. Switched-capacitor filters comprise resistors, capacitors, amplifiers, and switches. These are discrete-time filters that operate like active filters but through the use of switches the capacitance values can be kept very small. As a result, switched-capacitor filters are amenable to VLSI implementation. This section provides an introduction to the characteristics of analog filters. Their basic characteriza-
tion in terms of a differential equation is reviewed in Section 1.2 and by applying the Laplace transform, an algebraic equation is deduced that leads to the s-domain representation of a filter. The representation of analog filters in terms of the transfer function is then developed. Using the transfer function, one can
obtain the time-domain response of a filter to an arbitrary excitation, as shown in Section 1.3. Some important time-domain responses, i.e., the impulse and step responses, are examined. Certain filter parameters related to the step response, namely, the overshoot, delay time, and rise time, are then considered. The response of a filter to a sinusoidal excitation is examined in Section 1.4 and is then used to deduce the basic frequency-domain representations of a filter, namely, its frequency response and loss characteristic. Some idealized filter characteristics are then identified and the differences between idealized and practical filters are delineated in Section 1.5. Practical filters tend to introduce signal degradation through amplitude and=or delay distortion. The causes of these types of distortion are examined in Section 1.6. In Section 1.7, certain special classes of filters, e.g., minimum-phase and allpass filters, are identified and their applications mentioned. This chapter concludes with a review of the design process and the tasks that need to be undertaken to translate a set of filter specifications into a working prototype.
