Circuits, Systems and Signal Processing

In this chapter, I shall tell you all you wanted to know about signals and systems but were afraid to ask your teacher. Starting with the definition of linear systems, some elementary signals are introduced, which is followed by the notion of time-invariant systems. Signals and systems are then coupled through impulse response, convolution and response to exponential signals. This naturally leads to Fourier series representation of periodic signals and consequently, signal representation in the frequency domain in terms of amplitude and phase spectra. Does this sound difficult? I shall simplify it to the extent possible, do not worry. Linear system response to periodic signals, discussed next, is then easy to understand. To handle non-periodic signals, Fourier transform is introduced by viewing a non-periodic function as the limiting case of a periodic one, and its application to linear system analysis is illustrated. The concepts of energy and power signals and the corresponding spectral densities are then introduced.

Some fundamental issues relating to this mysterious but fascinating impulse function in continuous as well as discrete time domain are discussed first. These are: definition and relation to the unit step function, dimension of impulse response, solution of differential and difference equations involving the impulse function and Fourier transform of the unit step function. Conceptual understanding is emphasized at every point. This is very important, particularly for beginners like you. So read the chapter carefully and grasp the contents. This will help you to understand the later course on signals and systems, control, DSP, etc.

Here, we introduce state variables, which were the hot topics in the middle of 1960s. Later, they have seeped into and made deep impact on signals and systems, circuit theory, controls, etc. You better get familiar with them and make friends with them as early as possible.

In the first part of this discussion on state variables, which hopefully you have grasped, we presented the basic concepts of state variables and state equations, and some methods for solution of the latter. In this second and concluding part, we dwell upon the properties and evaluation of the fundamental matrix. An appendix on matrix algebra is also included. Several examples have been given to illustrate the techniques. This is the last part, be assured!

This chapter aims to simplify partial fraction expansion with repeated poles––presented here are some techniques which should make this topic considerably easier.

If you have followed the last chapter carefully, this one would be a cakewalk! The two discussions are similar except for the variables. A very simple method is given for finding the residues at repeated poles of a rational function in z−1. Compared to the multiple differentiation formula given in most textbooks, and several other alternatives, this method appears to be the simplest and the most elegant. It requires only a long division preceded by a small amount of processing of the given function.

Is it simpler? In most cases, it is. Remember the difficulty you faced by working solely in the time domain, in the previous chapter (Chap. 2), in solving a differential equation with impulsive excitation? Except for these odd cases, time domain analysis is usually simpler. In this chapter, we discuss how linear circuits can be completely analysed without using Laplace or Fourier transforms. Is this analysis simpler than that using transform techniques? You should judge for yourself to realize.

As compared to the conventional approach of trial solutions for solving the differential equation governing the transient response of RLC networks, we present here a different approach which is totally analytical. We also show that the three cases of damping, viz. overdamping, critical damping and underdamping, can be dealt with in a unified manner from the general solution. Won’t you appreciate my innovations? Please do and encourage me.

How can a paradox be deceptive? What appears to be an obvious conclusion may not be correct after all! This is illustrated with the help of a differential amplifier circuit, whose gain is actually half of what it appears to be. This paradox is indeed deceptive. See for yourself and decide if you wish to agree or not.

An initial value problem is posed and solved in a systematic way, illustrating the fact that what meets the eye may not be the truth!

In this chapter, we discuss the basic concepts of resonance in electrical circuits, and its characterization, and illustrate its application by an example. Several problems have been added at the end for the students to work out. Do work them out.

It is shown that the simple single-tuned circuit is capable of performing a variety of filtering functions and that it can be analyzed graphically for obtaining the relevant performance parameters.

Following a review of the various alternative methods available for analyzing the parallel-T RC network, we present yet another conceptually elegant method. This discussion illustrates the famous saying of Ramakrishna Paramhansa: As many religions, as many ways. Don’t just grab one method; learn all of them and decide for yourself which one you find to be the simplest.

A simple analysis is presented for obtaining an expression for the transfer function and hence the selectivity, Q_T, of a general parallel-T resistance–capacitance network. The maximum value of Q_T obtainable by a suitable choice of elements is shown to be $$ \tfrac{1}{2} $$. A design procedure for approaching this maximum value is given.An expression for the selectivity, Q_A, of an amplifier using a general parallel-T resistance–capacitance network in the negative feedback line has been deduced and the advantages of having an increased Q_T explained. Parallel-T RC is an important network and you cannot do without it, if you wish to remain in circuit design.An expression has been given for estimating the departure from linearity of the amplitude response characteristic at a particular frequency. This is used to find an optimum value of Q_T for best performance of the network as an F.M. discriminator at low frequencies. It is shown that the required value of Q_T is very near its maximum value.

That on connecting a source in the primary circuit of a perfectly coupled transformer, the currents in both the primary and secondary coils may be discontinuous does not appear to have been widely discussed in the literature. In this discussion, we present an analysis of the general circuit and show that in general, the currents will be discontinuous, except for specific combinations of the initial currents in the two coils. Although unity coupling coefficient cannot be realized in practice, a perfectly coupled transformer is a useful concept in circuit analysis and synthesis, and the results presented here should be of interest to students as well as teachers of circuit theory.

An analytical solution is presented for the problem of charging a capacitor through a lamp, by assuming a polynomial relationship between the resistance of the lamp and the current flowing through it. The total energy dissipated in the lamp is also easily calculated thereby. An example of an available practical case is used to illustrate the theory.

It is shown that the semi-infinite and infinite resistive ladder networks composed of identical resistors can be conveniently analyzed by the use of difference equations or z-transforms. Explicit and simple expressions are obtained for the input resistance, node voltages and the resistance between two arbitrary nodes of the network.

Minimal realizations of an interesting third-order impedance function are discussed. The solution, based on an elegant algebraic identity, illustrates several basic concepts of driving point function synthesis.

Synthesis of an LC driving point function is one of the initial topics in the study of network synthesis. This chapter gives a practical example of application of such synthesis in the design of a notch filter for interference rejection in an ultra wide-band (UWB) system. The example can used to motivate students to learn network synthesis with all seriousness, and not merely as a matter of academic exercise.

A simple method is given for obtaining the transfer function of Butterworth filters of orders 1 to 6.

Given the specifications of a band-pass filter (BPF) or a band-stop filter (BSF), the same can be translated to those of a normalized low-pass filter (LPF) by frequency transformation. Once the latter is designed, one can realize the BPF/BSF by using the same transformation in a reverse manner. The process of translation to the normalized LPF is usually not explained in details in standard textbooks, and in some of them, the process has even been wrongly stated or illustrated. This chapter clarifies this important step in BPF/BSF design.

A general, nth order, the transfer function (TF) is derived, whose time-domain response approximates optimally that of an ideal differentiator, optimality criterion chosen being the maximization of the first n derivatives of the ramp response at t = 0+. It is shown that transformerless, passive, unbalanced realizability is ensured for n < 3, but for n > 3, the TF is unstable. For n = 3, the TF is not realizable, however, near optimum results can be obtained by perturbation of the pole locations. Optimum TFs are also derived for the additional constraint of inductorless realizability. It is shown that TFs for n ≥ 2 are not realizable. For all n, however, near optimum results can be achieved by small perturbations of the pole locations; this is illustrated in this chapter for n = 2. Network realizations, for a variety of cases, are also given.

This chapter presents the fundamentals of a bipolar junction transistor amplifier and includes the following aspects: choice of Q point, classes of operation, incremental equivalent circuit, frequency response, cascading, broadbanding and pulse testing. The emphasis is on understanding the fundamental, rather than rigorous analysis or elaborate design procedure.

It is shown that bias stability is the best with the four resistor circuit. A two-resistor BJT biasing circuit, which appears to be an attractive alternative to the familiar four resistor circuit, is shown to have serious limitations. It is also shown that even when augmented by one or two resistors, these limitations are only partially overcome and that the bias stability that can be achieved thereby is poorer than that of the four resistor circuit.

The familiar four resistor circuit for biasing a bipolar junction transistor (BJT) is generalized through simple reasoning, and transformed to yield a different topology. Three alternative four resistor circuits are derived as special cases of the transformed generalized circuit, which do not appear to have been widely known in the literature. A detailed and careful analysis reveals that the bias stability parameters of all alternative circuits are comparable to those of the conventional circuit. An illustrative example is included for demonstrating this fact.

It is shown that, contrary to popular belief, classical two-port network theory is adequate for an exact analysis of a general high-frequency transistor stage, including emitter feedback, almost by inspection.

Three possible circuits of transistor Wien bridge oscillator, derived from analogy with the corresponding vacuum tube circuit, are described. Approximate formulas for the frequency of oscillation and the voltage gain required for maintenance of oscillations are deduced. A practical circuit using two OC71 transistors is given. The frequency of oscillation is found to agree fairly well with that calculated from theory. The relative merits of the different forms have also been discussed.

Conventionally, in analysing sinusoidal oscillator circuits, one uses the Berkhausen’s criterion, viz. Aβ = 1 in a positive feedback amplifier whose gain without feedback is A and whose feedback factor is β. However, the identification of A and β poses problems because of mutual loading of the amplifier and the feedback network. A different approach is presented here which does not require such identification. The method is based on assuming a voltage at an arbitrary node and coming back to it through the feedback loop.

This chapter describes how a triangular wave is converted into a sine wave by using a piecewise linear transfer characteristic. A detailed analysis of the basic circuit is given, and its actual implementation in an available IC chip is briefly discussed.

A simple derivation is given for the dynamic output resistance of the Wilson current mirror, which forms a basic building block in many analog integrated circuits.

In this chapter, the basic concepts of digital signal processing will be introduced, leading to a mathematical description of a digital signal processor in terms of, first, a difference equation and, second, a z-domain transfer function. In the process, the effects of sampling and quantization will be briefly touched upon. Implementation of a processor by special purpose hardware and discrete Fourier transform technique will be discussed. The fast Fourier transform (FFT) will be introduced and several of its applications will be presented, along with the pitfalls and incorrect usage of the technique.

Here, we deal with the realizations of DSP’s DFT, FFT, application of FFT to compute convolution and correlation, and application of FFT to find the spectrum of a continuous signal.

The chapter deals with the derivation, design, limitations and realization of second-order digital band-pass (BP) and band-stop (BS) filters with independent control of the centre frequency and the bandwidth in the BP case, and rejection frequency and the difference between the pass-band edges in the BS case.

This chapter deals with the derivation of two canonic all-pass digital filter realizations, first proposed by Mitra and Hirano. In contrast to their derivation, which uses a three-pair approach, our derivation is much simpler because we use a two-pair approach, in which only four, instead of nine parameters have to be chosen.

A simple derivation is presented for the FIR lattice structure, based on the digital two-pair concept. Go ahead, read it and judge for yourself whether it is simple or not!

In FIR lattice synthesis, if at any but the last stage, a lattice parameter becomes ±1, then the synthesis fails. A linear phase transfer function is an example of this situation. This chapter, written in a tutorial style, is concerned with a simple solution to this problem, demonstrated through simple examples, rather than detailed mathematical analysis, some of which is available in (Dutta Roy in IEE Proc-Vis Image Signal Process 147:549–552, 2000 [1]).

An alternative derivation is given for the linear prediction FIR lattice structures with single-multiplier sections. As compared to the previous approaches, this method is believed to be conceptually simpler and more straightforward.

This chapter gives a formula for the exact number of non-trival multipliers required in the basic N-point FFT algorithms, where N is an integral power of 2. Now proceed further, but not too far!