Chapter 2 Systems
A system is a function that transforms signals in the sense that a system takes as input a signal and produces as output another signal. A wide range of things we usually think of as engineering systems can be interpreted in this way. Systems range from being quite simple, to being extremely complicated.
On the simple end of the spectrum, we can think of a single resistor as a system. If the current i(t) through the resistor is interpreted as the input, and the voltage drop v(t) across the resistor is thought of as the output, Ohm’s law tells us that there is a simple linear relationship v(t) = Ri(t) between the input and the output.
On the other end of the spectrum, we can think of a smartphone as a very complex system, with multiple inputs—the audio signal picked up by the microphone, the video signal picked up by the camera, the touch-screen signal which could be thought of as a value of capacitance for each cell of the screen that changes with time, as well as various electromagnetic signals coming from mobile phone towers, wifi access points, etc. The multiple outputs include an audio signal, a video signal, and various electromagnetic signals. Writing down a precise mathematical formula for a smartphone’s behaviour seems very daunting indeed!
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Because of the complexity of systems, an important way to think about them is to abstract away all the complicated details, and focus on one aspect of the system. For instance, let’s think about the system that has input a sound wave near the microphone of one smartphone, and has output a sound wave near the speaker of another smartphone, while a call is happening between the two phones. This can be modeled as a fairly simple system. Roughly, the output is a slightly delayed, slightly distorted copy of the input. This model completely ignores all the physics of sound propagation, the computations in the phones, the electromagnetic propagation between phones and phone towers, the sending of IP packets over the internet between the towers, etc, etc. Despite this, as long as we can model the delay and the distortion well, we will obtain a good model of the overall system.
In this unit we often ignore the internal workings of a system, so we can focus on the relationships between its inputs and outputs, rather than the mechanism that causes the input to be transformed into the output. An electrical circuit, or a mechanical system, or a biological process, might all have similar input-output behaviour. Abstracting away the details helps us think about the essential behaviour of the system without worrying about the underlying physics/chemistry/biology. It also allows us to reason about what happens when we connect smaller systems to form larger systems. This is very important because complex engineering systems are usually designed by building them out of simpler subsystems.
In this chapter we introduce basic notation and definitions related to systems. On the way we will see examples of the simplest systems. We also see ways to combine simple systems to make more complex systems. We also discuss special properties that some systems have. It is useful to keep track of special properties, because these can greatly simplify the task of analysing and designing systems.
Figure 2.1: On the left is a rectifier circuit. On the right is a block diagram representing the system with input x and output y.
2.1 Systems 2.1.1 Notation
Systems can be complicated. Describing them precisely using mathematics can also be compli- cated. In this unit we will use a number of different ways of describing systems, some of which are briefly discussed here.
Function notation
The most abstract way to describe a system is to use function notation. This emphasises the relationship between the inputs and outputs of a system.
• If S denotes a system, and x is the input signal we write the output signal as S(x) (or sometimes just Sx). With this notation, the input at time t is the number x(t), and the output at time t is the number S(x)(t) (or (Sx)(t)).
• Sometimes it is easier to introduce a new letter to describe the output. If x is the input signal, let y = Sx denote the output signal. Then the output at time t is just y(t).
Function notation for systems is useful for describing properties of systems. However, it is not so easy to describe complicated systems that are built up from simpler systems using this notation.
Block diagrams
Systems that are built up from simpler sub-systems can be described using block diagrams. These specify how inputs and outputs of subsystems are connected. Block diagrams abstract away the details of how the subsystems work and focuse on the interconnections. Anything we describe with block diagrams we can also describe using function notation, but the function notation gets messy very quickly. For example, if we are interested in the relationship between the voltage x(t) and the voltage y(t) in the circuit shown on the left in Figure 2.1, we might abstract away the details of the circuit and represent this system with the block diagram shown on the right in Figure 2.1.
Other descriptions of systems
Systems with special properties (such as the class of linear time-invariant systems, described later) can be described in a number of other ways. These will be introduced throughout the unit. Some descriptions make it easy to understand the properties of the system. Other descriptions make it easy to implement the system in a computer, or as a circuit. As such, there is not a single ‘best’ way to describe a system.
Full-rectify
2.1.2 Examples of simple systems
In what follows, we introduce a number of simple systems that can be combined to create more complicated systems.
One of the simplest systems is an adder. It has two inputs and one output and is often denoted with a block diagrams similar to the following:
y = x1 + x2
x1 + y=x1+x2 x2
In function notation, we could define a system Add with two inputs and one output as Add(x1, x2) = x1 + x2.
This means that if y = Add(x1, x2) then y(t) = x1(t) + x2(t) for all t. Gain (or amplitude scaling)
For a number K, the system GainK scales its input by K, i.e., GainK(x) = Kx.
This is often denoted with a block diagram of the form:
x(t) if x(t) ≥ 0
y(t) = 0 if x(t) < 0 for all times t.
Rectifiers
Examples of systems that have simple mathematical descriptions are half- and full-wave rec- tifiers. If y = Half-rectify(x), so that y is the output of a half-wave rectifier with input x,
Similarly if y = Full-rectify(x), so that y is the output of a full-wave rectifier with input x, then y(t) = |x(t)| for all times t
Time-shifting systems
A delay system is one for which the output is a copy of the input delayed by a certain amount of time. An advance system is one for which the output is a copy of the input but shifted forward in time by a certain amount. Together these are called time-shifting systems. We can always think of advance systems as delay systems with a negative delay, so we will only formally define delay systems.
DT delay A discrete-time delay system has output that is a copy of the input but delayed by a given number of time-steps. More precisely, if N is an integer and y = DelayN (x) then
y[n] = x[n − N] for all times n.
If N is positive this corresponds to delaying by N samples. If N is negative, it corresponds to advancing by |N| samples.
A discrete-time unit delay is another name for Delay1, the DT system that delays its input by one time-step. It is particularly useful because any other DT delay system can be built out of unit delay systems.
CT delay A continuous-time delay system has output that is a copy of the input but delayed by a given amount of time. More precisely, if τ is a real number and y = Delayτ (x) then
y(t) = x(t − τ) for all times t.
If τ is positive then y is the same as x but delayed by τ seconds. If τ is negative then y is the
same as x but advanced in time by |τ| seconds.
Example 2.1
A DT signal x is shown in the upper part of the figure. The output y = Delay2(x) is shown in the middle of the figure and y = Delay−2(x) is shown at the bottom of the figure.
The plot of y = Delay2(x) is the same as the plot of x but shifted to the right by 2 time units. The plot of y = Delay−2(x) is the same as the plot of x by shifted to the left by 2 time units.
Example 2.2
A CT signal x is shown in the upper part of the figure. The output y = Delay2(x) is shown in the lower part of the figure.
The plot of y is the same as the plot of x but shifted to the right by 2 time units.
Integrator
The integrator is a continuous-time system. Its output at time t is the integral of the input for
alltimesτ uptotimet,i.e.,
y(t) = x(τ) dτ.
An integrator is often represented with a block-diagram of the form:
Example 2.3
A 1F capacitor can be thought of as an integrator system. If we take the input to be the
current i through the capacitor, and the output to be the voltage vacross the capacitor,
then the relationship between the voltage and the current is v(t) = t i(τ) dτ. −∞
Differentiator
The differentiator is a continuous-time system. Its output at time t is the derivative of the input at time t, i.e.,
y(t) = dx = lim x(t)−x(t−τ). dtτ→0 τ
A differentiator is often represented with a block-diagram of the form:
2.1.3 Composition of systems
We often use block diagrams to describe a system as an interconnection of subsystems. Each subsystem is represented by a block (usually a rectangle) with arrows entering the block (indi- cating inputs) and arrows leaving the block (indicating outputs). By connecting the outputs of one block to the inputs of another, we can define a larger composite system.
Series composition
If S1 and S2 are two systems, each with a single input and a single output, we can form the series composition of S1 and S2. This is a new system with input being the input of S1, and output being the output of S2. The systems are connected so that the output of S1 is also the input of S2. The block diagram of the series composition is
x y = S2S1x
• This series composition is only valid if the output of S1 is allowable as the input of S2. For example, if the output of S1 is a discrete-time signal, S2 must accept discrete-time signals as inputs.
• The order in which systems are composed in series matters! In general S1 followed by S2 is a different system than S2 followed by S1.
Example 2.4
Suppose that S1 = Half-rectify is a half-wave rectifier that passes through positive
values unchanged, and sets negative values to zero. Suppose that S2 = Gain−1 is
the system that multiplies its input by −1. If x(t) = 1 + cos 2πt (which is always T
non-negative), the series composition y = Gain−1(Half-rectify x) = −Half-rectify(x) produces output
y(t)=−x(t)=−1+cos2πt forallt. T
On the other hand, with the same input, but composing the systems in series in the opposite order, i.e., y = Half-rectify(Gain−1 x) = Half-rectify(−x), produces output
y(t)=0 forallt
because −x(t) is always less than or equal to zero. Clearly the system behaviour is different depending on the order in which we apply the two systems!
Parallel composition
Another common way to compose two systems S1 and S2 is via parallel composition:
y = S1x + S2x
A very important type of interconnection of systems is one in which (a modification) of the output of a system is fed back into the input of that system. Any interconnection with this property is called a feedback interconnection. An example of a feedback interconnection of two systems S1 and S2 is shown below
Example 2.5
Consider the system S, with input x and output y = Sx, such that y[n] = x[n − 1] + 2x[n] for all n. The following block diagram shows how to express S as a parallel composition of a gain block and a unit delay block:
2.1.4 Feedback composition
Summary of Section 2.1
We model systems as functions that associate an output signal to any input signal. This is a very general model that includes very complicated systems. One way to model simple systems is to start with ‘atomic’ examples (like gains, delays, etc) and combine (or compose) them to create more complicated systems. We often use block diagrams to illustrate the way more complicated systems are built out of simpler systems.
2.2 Properties of systems
2.2.1 Systems with and without memory
A system is memoryless if the output at time t depends only on the input at time t (and not on future or past inputs). For example, the adder, gain, and rectifier systems are all examples of memoryless systems.
If a system is not memoryless we say that it has memory. Examples of systems with memory include delay systems, advance systems, integrators, and differentiators.
2.2.2 Causal and non-causal systems
A system is causal if the output at time t depends only on the current input and past inputs (but not on future inputs). Every memoryless system is causal. Other examples of causal systems are delay systems and integrators.
If a system is not causal we call it non-causal. An example of a non-causal system is a DT unit advance, since the output at time n depends on the input at time n + 1, which is in the future.
When we are designing systems that operate in real-time, we usually need to restrict to causal systems. On the other hand, a non-causal systems may make sense for off-line audio or video post-processing.
2.2.3 Linear and non-linear systems
A system is linear if it satisfies the following two properties (additivity and homogeneity). Any system that does not have both of these properties is called non-linear.
Additivity/superposition property
A system is additive if, when we add two input signals and then apply the system, the result is the same as applying the system to the two inputs separately and then adding up the results afterwards. More formally, S is additive if, for all inputs x1 and x2,
S(x1 + x2) = Sx1 + Sx2.
We can also think of this in terms of block diagrams. A system S has the additivity property
if the following block diagrams represent the same two-input, single-output system:
S(x1 + x2)
Homogeneity/scaling property
A system is homogeneous if, when we scale an input signal by a number and then apply the system, the result is the same as applying the system to the input signal and then scaling the result afterwards. More formally, S is homogeneous if, for all inputs x, and all (complex) gains α,
S(αx) = αSx.
We can also think of this in terms of block diagrams. A system S is homogeneous if the following block diagrams represent the same signal-input single-output system:
The following general facts hold for linear systems: (try to convince yourself why they are true)
• Any gain system is linear.
• Any time-shifting system is linear. Any time-scaling system is linear.
• Integrator and differentiator systems are linear.
• The series composition of two linear systems is linear.
• The parallel composition of two linear systems is linear.
• Iftheinputtoalinearsystemisx(t)=0forallt,thentheoutputisy(t)=0forallt. This follows from the homogeneity property.
2.2.4 Time-varying and time-invariant systems
Informally, a system is time-invariant if it transforms an input signal in the same way, no matter when the input signal is fed into the system. Another way to say this is that a system is time- invariant if delaying the input by an amount, and then applying the system, is the same as first applying the system and then delaying the output (by the same amount).
More formally, a CT system S is time-invariant if, for all time-shifts τ and all inputs x,
S(Delayτ x) = Delayτ (Sx).
We can also think of this in terms of block diagrams. A CT system S is time-invariant if, for all time-shifts τ, the following block diagrams represent the same single-input single-output system:
x S(Delayτ x)
x Delayτ (Sx)
A DT system S is time-invariant if, for all time-shifts N and all inputs x, S(DelayN x) = DelayN (Sx).
We can also think of this in terms of block diagrams. A DT system S is time-invariant if, for all time-shifts N, the following block diagrams represent the same single-input single-output system:
x S(DelayN x)
x DelayN (Sx)
Example 2.6
The DT delay-by-two system, Delay2, is time-invariant. To show this, let N be any time- shift and let x be any input signal. Then
Delay2(DelayN x) = Delay2+N (x) = DelayN (Delay2x).
This is perhaps more obvious in words—if we delay by N time units then delay by 2 time units, it is the same as first delaying by 2 time units and then by N time units. This is because both are the same as delaying by N + 2 time units.
The following general facts hold for time invariant systems:
• Any gain system is time-invariant.
• Any time-shifting system is time-invariant. Time-scaling systems are usually not time- invariant.
• Integrator and differentiator systems are time-invariant.
• The series composition of two time-invariant systems is time-invariant.
• The parallel composition of two time-invariant systems is time-invariant. A system that is not time-invariant is called time-varying.
Example 2.7
Consider a system in which the output is obtained by multiplying the input signal by sin(t). This is an example of a modulation system. In other words the output y is related to the input x by
y(t) = sin(t) × x(t) for all times t.
This is a time-varying system. To see why, let τ = π be a time shift. For any input signal
x, if we delay the output y by π time units, we get
(Delayπy)(t) = y(t − π) = sin(t − π) × x(t − π) = − sin(t) × x(t − π).
Yet if we delay the input by π and then consider the output of the system, we get sin(t) × (Delayπx)(t) = sin(t) × x(t − π).
These are not the same, so the system is time-varying.
2.2.5 BIBO stable systems
Intuitively, a system is stable if ‘small’ inputs don’t cause ‘large’ outputs. Perhaps the most crude way to make this precise is to require that bounded inputs produce bounded outputs. This is the notion of system stability we focus on in this unit.
BIBO stability
A system S is bounded-input bounded-output (BIBO) stable if S(x) is bounded whenever x is bounded.
Example 2.8
Consider the integrator system where the output y is related to the input x via
This system is not BIBO stable. To see this we need to find a bounded input that produces an unbounded output.
Let x(t) = u(t) be the unit step function. Then |x(t)| ≤ 1 for all t, and so x is bounded. On the other hand 0 if t ≤ 0
y(t)= 0t dτ=t ift>0.
The output signal y(t) = tu(t) is unbounded, establishing that the system is not BIBO stable.
If we want to show that a system is BIBO stable, it seems like we would have to check all possible bounded inputs, and see whether the corresponding output is bounded, an impossible task. For systems that are both linear and time-invariant (which we introduce next), the
situation is much simpler. Later in the unit In topics 4 and 5 will see ways to check that a linear and time-invariant system is BIBO stable without having look at all the inputs and their corresponding outputs.
Summary of Section 2.2
Systems with properties such as being memoryless, or linear, or time-invariant are often much simpler to understand than general systems, which can be hopelessly complicated. Recognizing and understanding these properties may help us to
• gain insights into how systems behave
• select appropriate tools to analyse systems.
Some properties (such as linearity and time invariance) are preserved when we compose systems with that property. This allows us to build more complicated systems with a desired property from simpler systems with that property.
2.3 Linear time-invariant (LTI) systems 2.3.1 Definition of LTI systems
A system is linear time-inv
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