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Chapter 3 Linear Filters

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Chapter 3 Linear Filters Chapter 3 Linear Filters Mikael Olofsson — 2003–2007 A signal is a - normally time-varying - measure of some kind, and filters are devices that manipulate signals. We use mathematical descriptions to model our filters and signals. It should be emphasized that those mathematical descriptions are precisely that - models. How well such a model describes reality has to do with how detailed the model is, what frequencies we are interested in, or in what way input signals are limited. The same model can be good in one situation and bad in another, by which we mean how well it predicts the outcome of an experiment. Linear filters belong to the class of linear systems. However, before we can say anything about systems, we need to start our presentation with signals as functions of time. After that, we continue to describe systems and their properties in the time domain, followed by both signals and systems described in terms of frequencies. Last, standard characterizations of filters are given. 3.1 Signals First of all, a signal is represented as a function of time, where we usually let t denote time. In the practical situation, signals can be almost any physical measures, such as currents, voltages, preassures, positions, and so forth. In telecommunication systems, however, most often we are dealing with voltages and currents, but also electromagnetic phenomena like radio waves and light. Mostly, we are interested in signals that represent some information. In some situations, e.g. when recording sound, the signal can be that information in itself. In other situations, a signal can represent some particular information, e.g. a certain signal in a graphics chip in a computer can represent a blue pixel that is produced on its screen, while another signal in the same place may correspond to a red pixel. In yet other situations, a signal may not carry information at all, as is the case with 27 28 Chapter 3. Linear Filters various kinds of noise. In those cases, signals are normally described in probabilistic terms. The description given here, however, is for deterministic (non-random) signals. We will use a number of fundamental signals as building blocks to describe signals, such as the following. • Stationary sinusoidal signal : sin(2 πf 0t), with frequency f0 and period time T = 1 /f 0. j2πf 0t • The complex exponential signal : e , cos (2 πf 0t) + jsin (2 πf 0t). 0, t < 0, • The unit step : u(t) , 1, t ≥ 0. The complex exponential signal can be used to express sinusoidal signals as ej2πf 0t + e−j2πf 0t cos(2 πf t) = , 0 2 ej2πf 0t − e−j2πf 0t sin(2 πf t) = , 0 j2 using the Euler formulas. Shifting the unit step by τ seconds is simply 0, t<τ u(t − τ) = 1, t ≥ τ The unit step has a discontinuity in 0, so it is not differentiable in the normal sense. However, we can make it differentiable by introducing the unit impulse . It is a so called distribution, and is defined as follows. Definition 1 (Unit impulse) The unit impulse δ(t) is a function such that ∞ δ(t)x(t) dt = x(0) Z −∞ holds for any limited function x(t). It is possible to prove the following properties of the unit impulse. ∞, t = 0 , δ(t) = δ(t) = d u(t), 0, elsewhere , dt t δ(τ) dτ = u(t), δ (at ) = 1 δ(t), Z |a| −∞ ∞ δ(t − τ)x(τ) dτ = x(t), δ (t) = ∞ ej2πft df, Z −∞ −∞ R where at least the last property is rather tricky to prove. 3.2. Systems 29 x(t) System y(t) Figure 3.1: A system with input x(t) and output y(t). Usually, a condensed descrition of the system is given inside the rectangle representing the system. 3.2 Systems We are not only interested in signals. We are also interested in how signals are treated by various devices that observe those signals. Definition 2 (System) A system is a device with one or more input and output signals. Most systems that we will deal with have only one input and one output. That will therefore be understood if nothing else is stated. We usually illustrate a system as in Figure 3.1. The distinction between input signals and output signals is that systems observe input signals and produce output signals based on the observation. We assume that systems do not affect input signals at all. This of course may or may not be a good model. In practice, most (all) systems affect input signals aswell, at least to some extent. Systems can be in different initial states. We say that systems can have initial energy. For example, consider a simple system consisting of a rock, for which the force that we apply to the rock is the input and the position of the rock is its output. If the rock is elevated from the ground before we start observing the rock and then released, it will fall to the ground due to the initial energy it was given when it was elevated. The system is then said to have initial energy. If instead the rock is initially placed on the ground and released, it will stay there. The system is then said to be initially energy-free . The inital state of the rock is not only its initial position, but also its initial speed and direction. Or put in other words: An initially energy-free system is in a state such that the output will be constant (often zero) if the input is zero. What’s important to realize about initial states is that the output of a system depends on its input(s) as well as on its initial state. An example of an initially energy-free electrical system is a network consisting of capacitors and resistors, where the capacitors are not charged, i.e. over which the initial voltages are zero. The input and output signals of this system may be two voltages in the network. A mathematical example of an initially energy-free system is a system described by a linear differential equation with all initial values being 0. The electrical network mentioned above can be described by a linear differential equation, where the initial voltages over the capacitors are the initial conditions. For such systems we define the following. 30 Chapter 3. Linear Filters Definition 3 (Impulse response) Let the input to an initially energy-free system be the unit impulse, δ(t), and let h(t) denote the corresponding output. Then h(t) is referred to as the impulse response of that system. Definition 4 (Step response) Let the input to an initially energy-free system be the unit step, u(t), and let g(t) denote the corresponding output. Then g(t) is referred to as the step response of that system. So far, we have not stated anything specific about our systems. In order to say anything about the relation between inputs and outputs, we need to classify systems. Then we can say that for a system of a certain type, we have a certain relation between its input and its output. Definition 5 (Time-invariant system) Let x(t) be the input to an initially energy-free system, and let y(t) be the corresponding output. If y(t + τ) is the output corresponding to the input x(t + τ) for any x(t), t and τ, then the system is referred to as time-invariant. A system that is not time-invariant is referred to as time-varying. Put more simply: A time-invariant system is a system for which a time-shift of the input results in the same time-shift of the output. If we introduce the notation y(t) = H{ x(t)}, for the output from an initially energy-free system with input x(t), then the above can be described as y(t + τ) = H{ x(t + τ)} for all inputs x(t) and all time-shifts τ. Definition 6 (Linear system) Let x1(t) and x2(t) be input signals to a one-input initially energy-free system, and let y1(t) and y2(t) be the corresponding output signals. If y(t) = a1y1(t) + a2y2(t) is the output corresponding to the input x(t) = a1x1(t) + a2x2(t) for any x1(t), x2(t), a1 and a2, then the system is referred to as linear. A system that is not linear is referred to as non-linear. Using the notation H{ x(t)} for the output of the initially energy-free system given the input x(t), linearity can be described as H{ a1x1(t) + a2x2(t)} = a1H{ x1(t)} + a2H{ x2(t)}, which is supposed to hold for all inputs x1(t) and x2(t) as well as for all coefficients a1 and a2. Systems belonging to both the above mentioned classes are of special interest to us, and such systems have been given a special name. 3.2. Systems 31 C x(t) L R y(t) Figure 3.2: A passive linear filter. C2 R1 R2 x(t) C1 y(t) Figure 3.3: An active filter. Definition 7 (LTI system) A system that is both linear and time-invariant is referred to as an LTI system (Linear Time-Invariant). Examples of LTI systems include passive linear filters, also called RLMC circuits. Those are electrical networks built from resistances (R), inductances (L), mutual inductances (M) and/or capacitances (C). An example of a passive linear filter is given in Figure 3.2. Active filters are built around amplifiers, and those always have limited supply voltages, which limit the output.
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