Description of Systems | Signals and Systems - Electrical Engineering (EE) PDF Download

What is a system?

A signal is a mapping from a set of independent variable values (the domain) to a set of dependent variable values (the co-domain). A system is a mapping whose domain and co-domain are themselves sets of signals. In other words, a system accepts an input signal and produces a unique output signal for every possible input signal.

The study of signals and systems concerns two broad themes: the representation and transmission of information, and how that information causes a device or arrangement to respond. Formally, a signal is a time-varying (or independent-variable varying) quantity that conveys information, and a system is a collection of components or rules that transform input signals into output signals.

In signals and systems terminology, we say: corresponding to every possible input signal, a system produces an output signal. A system is therefore one level higher than a signal: a signal maps points in one set to points in another, while a system maps whole mappings (signals) to other mappings (signals).

As a simple example of a signal, consider the function f(t) = 2 · t, where the independent variable is time t and f(t) is the signal value at time t.

Graph for Signal

Illustrative example: the robot

Consider a line-following robot. Possible input signals are camera images (visual intensity as a function of space and time), tactile signals from the tyres, or magnetic field sensor readings. The robot (system) processes the chosen input and produces an output signal that directs steering-typically a voltage or digital command to motors. The environment largely determines the input; the system designer controls how that input is mapped into an output.

Common ways to represent systems

Systems are commonly described by one of the following representations:

• difference equations (discrete-time relationships),
• block diagrams (interconnection of subsystems),
• operator equations (differential or integral operator descriptions in continuous time).

We will study these representations in detail later, with worked examples and conversion between forms.

Examples of systems

Many physical devices are naturally viewed as systems by picking an appropriate input and output signal. Examples include:

• a loudspeaker: input is an electrical voltage as a function of time; output is an acoustic pressure (sound) as a function of time,
• a spring-mass system: input could be a time-varying longitudinal force; output could be the elongation as a function of time,
• a capacitance: depending on the chosen input and output variables the system description changes - for example, voltage as input and current as output defines a different mapping than charge as input and voltage as output.

Cathode Ray Oscilloscope

A cathode ray oscilloscope (CRO) converts an input voltage signal f(t) into a two-dimensional luminous trace on a screen. The input independent variable is time; the CRO's display is a space domain (x,y). Thus the output S(x,y) is a spatial distribution of luminosity while the input is a temporal waveform.

The CRO shows that input and output independent variables need not be the same: continuous-time inputs can produce spatial or discrete outputs, and vice-versa. For example, human speech is continuous in time, while a digital recording is discrete in time; a system (ADC) is responsible for converting between them.

These examples emphasise that when we identify a device as a system we do so by selecting particular input and output signals. A single physical device can be described as different systems depending on that choice.

System description

The system description specifies exactly how the input signal is transformed into the output signal. Some systems admit a closed-form description; others require tabulation or algorithmic descriptions.

For example, the continuous-time system defined by y(t) = x(t) + x(t - 1) is a closed-form description: given the input x(t) for all t, the output y(t) can be computed directly.

Explicit and implicit descriptions

An explicit description expresses the output purely in terms of input values (and possibly fixed parameters) at known arguments. When the description is explicit, the output at a given instant can be computed directly from the known input values. For example, y(t) = [x(t)]² + x(t - 5) is explicit.

An implicit description relates the input and output together in an equation where the output at an instant cannot be expressed directly without additional solution steps. For example, y(t) - y(t - 1) · x(t) = 1 is an implicit description. To find y(t) when x(t) is known, one typically needs initial conditions or to rearrange and solve the relation (analytically or numerically) to obtain an explicit form.

The mapping involved in systems

Recall that a signal maps each independent variable value to a dependent variable value. A system maps each entire input signal to an entire output signal. Consequently, the value of the output at a particular instant often depends on the input values at many instants, not only at that instant.

For example, consider a system whose description involves integration over past inputs. The output at time t = 5 may depend on input values for all times τ ≤ 5.

Systems where both input and output are continuous-time are called continuous-time systems. Systems where both are discrete-time are discrete-time systems. Systems that convert between discrete and continuous time (for example ADCs and DACs) are called hybrid systems.

Properties of systems

Important properties used to classify and analyse systems include the following:

1. Periodicity
2. Even and odd (symmetry)
3. Linearity
4. Time invariance
5. LTI (Linear Time-Invariant) systems
6. BIBO (Bounded Input Bounded Output) stability
7. Stability (general concepts)
8. Causality
9. Reflection (time reversal)

Periodicity

Periodicity is primarily a signal property. A continuous-time signal x(t) is periodic with period T > 0 if x(t + T) = x(t) for all t. A discrete-time signal x[n] is periodic with integer period N > 0 if x[n + N] = x[n] for all n. Systems themselves are not usually called periodic, but they can preserve or alter periodicity.

Remarks:

• If a system is time-invariant and the input is periodic, the output is often periodic with the same fundamental period provided the system does not introduce uncontrolled growth or additional time-varying modulation.
• Passing a periodic signal through an LTI system produces an output that can be expressed by applying the system's frequency response to each harmonic component of the input (Fourier series approach).

Even and odd (symmetry)

A signal x(t) is even if x(-t) = x(t) for all t, and odd if x(-t) = -x(t) for all t. Any signal can be decomposed uniquely into an even part and an odd part.

Remarks for systems:

• Even/odd are signal symmetries; linear systems can be examined for how they treat these components separately because superposition applies.
• Some systems have symmetry properties (for example, an LTI system with an even impulse response h(t) produces an even output when given an even input), but such statements depend on the specific system description.

Linearity

A system S is linear if it satisfies the principle of superposition: for any signals x1 and x2 and any scalars a and b,

S{a x1 + b x2} = a S{x1} + b S{x2}.

Linearity is the conjunction of two properties: homogeneity (scaling) and additivity (superposition). Linear systems are important because they permit decomposition of signals into simpler components (for example, sinusoids or impulses), analysis on those components, and recombination of the responses.

Time invariance

A system S is time-invariant if a time shift in the input produces the same time shift in the output. Formally, if y(t) = S{x(t)}, then S{x(t - t0)} = y(t - t0) for every shift t0 and every input x(t). For discrete time, replace t by n and shifts by integers.

Time invariance means the system's behaviour does not explicitly depend on the absolute time origin.

LTI (Linear Time-Invariant) systems

An LTI system is both linear and time-invariant. LTI systems admit a powerful, compact description via the impulse response h. The impulse response is the output when the input is an impulse (delta) signal.

For continuous time, the input-output relation is convolution:

y(t) = ∫_-∞^∞ h(τ) x(t - τ) dτ.

For discrete time, the relation is a summation (discrete convolution):

y[n] = Σ_k=-∞^∞ h[k] x[n - k].

Key properties of convolution:

• Commutative: x * h = h * x.
• Associative: (x * h1) * h2 = x * (h1 * h2).
• Knowing h completely characterises the LTI system (together with initial conditions when relevant).

For LTI systems, many analysis tools (Fourier transform, Laplace transform, Z-transform) reduce system analysis to algebraic manipulation of transforms.

BIBO (Bounded Input Bounded Output) stability

A system is BIBO stable if every bounded input produces a bounded output. Formally, if |x(t)| ≤ Bx < ∞ for all t, then the output y(t) satisfies |y(t)| ≤ by />< ∞ for all t (for some finite by that may depend on bx but not on />

For LTI systems there is a convenient test:

• Continuous time: an LTI system is BIBO stable if and only if the impulse response h(t) is absolutely integrable, i.e. ∫_-∞^∞ |h(t)| dt < ∞.
• Discrete time: an LTI system is BIBO stable if and only if the impulse response h[n] is absolutely summable, i.e. Σ_n=-∞^∞ |h[n]| < ∞.

Stability (general concepts)

Beyond BIBO stability, other notions of stability appear in systems theory (e.g. internal stability, Lyapunov stability). For the signals and systems context of this chapter we focus mainly on BIBO stability as defined above. Internal modes or state variables that grow without bound indicate internal instability, which typically violates BIBO stability unless the unstable modes are not excited by the input.

Causality

A system is causal if the output at any time depends only on present and past input values, not on future inputs. Formally, if for any pair of inputs x1 and x2 that satisfy x1(τ) = x2(τ) for all τ ≤ t0, then the corresponding outputs y1 and y2 must satisfy y1(t0) = y2(t0).

For LTI systems this condition simplifies to the impulse response: the system is causal if and only if h(t) = 0 for t < 0 (continuous time) or h[n] = 0 for n < 0 (discrete time).

Reflection (time reversal)

Time reflection (time reversal) of a signal x(t) produces x(-t). Reflection is a useful operation in convolution and transform pairs. For example, convolution involves flipping one function in time and then sliding it over the other (e.g. x(t - τ) with τ as integration variable corresponds to time reversal when viewed from the kernel's perspective).

Reflection can change the causality or other attributes of a signal; for instance, reflecting a causal impulse response generally yields a non-causal function.

Remarks on hybrid systems (ADC/DAC)

Hybrid systems convert between continuous and discrete time. An analogue-to-digital converter (ADC) samples a continuous input and produces a discrete sequence; a digital-to-analogue converter (DAC) reconstructs a continuous signal from samples. These systems are described by sampling theory: sampling, quantisation, reconstruction filters, and their effects on bandwidth and stability must be considered.

Having introduced systems and their principal properties, subsequent chapters will develop mathematical tools (convolution, transforms, system functions) and worked examples to analyse, design and interpret system behaviour in electrical and electronic engineering contexts.