Introduction to Signals and Systems

Speak into a phone and your voice becomes a fluctuating voltage. Record a heartbeat and it becomes an ECG trace. Photograph a scene and it becomes a grid of brightness values. Watch a stock ticker and it becomes a string of numbers marching forward in time. These look like completely different things — sound, biology, light, money — yet an engineer treats them all with the same mathematics. Signals and systems is that shared language: a unified framework for describing the quantities that carry information and the processes that act on them.

The subject has two nouns in its name, and they belong together. A signal is the thing that carries information — the voice, the trace, the ticker. A system is anything that takes a signal in and puts a signal out — an amplifier, an equaliser, a control loop, a communication channel, a compression algorithm. Almost every task in electrical and electronics engineering can be phrased as: given this input signal and this system, what is the output? — or its inverse, what system will turn this input into the output I want?

One idea runs underneath all of it, and it is worth fixing in your mind before anything else. A system is a mapping from an input signal to an output signal. Filtering, modulation, feedback, sampling, coding — every topic in this book is a special case of studying that one mapping. Read it into every chapter that follows and the subject organises itself.

A signal is a function of one or more independent variables that conveys information about the behaviour of some physical phenomenon. In this course the independent variable is almost always time, and the signal is a single real (occasionally complex) number that varies with it. We write a continuous-time signal as \(x(t)\), read "x of t," where \(t\) ranges over the real line.

The idea is broader than time, though. An image is a signal of two spatial variables, \(f(x,y)\); a video adds time, \(f(x,y,t)\). Some signals are vectors — the three-axis output of an accelerometer — and some are complex-valued, a convenience we will lean on heavily once Fourier analysis arrives. For now, hold the simplest and most important picture: a signal is a function that assigns a value to every instant of time.

A system is any process that operates on one or more input signals to produce one or more output signals. It is, in other words, a transformation — an operator — that turns an input signal \(x\) into an output signal \(y\). We capture this with an operator symbol \(T\) and the compact notation \(y(t)=T\{x(t)\}\), and we picture it as a block with the input flowing in on the left and the output leaving on the right.

Systems can be physical or abstract. An RC circuit is a system: apply a voltage and it returns a smoothed version of it. A microphone-and-amplifier chain is a system. So is a purely computational one — a moving-average filter that replaces each sample of a data stream with the average of its neighbours, or the software that decodes a video. The block does not care whether it is made of resistors or lines of code; what matters is the rule connecting input to output.

Signals split into two great families, and the whole course is organised around the pair. A continuous-time (CT) signal is defined at every instant: its independent variable \(t\) runs over the real line, and we write it \(x(t)\) with round brackets. A discrete-time (DT) signal is defined only at integer instants: its variable \(n\) takes integer values, and we write it \(x[n]\) with square brackets. The bracket style is a deliberate, permanent signal of which world you are in — keep it precise from the first page.

The two families are intimately linked. A discrete-time signal very often arises by sampling a continuous-time one — reading \(x(t)\) at evenly spaced instants \(t = nT_s\) to form \(x[n] = x(nT_s)\), where \(T_s\) is the sampling period. This is exactly what an analog-to-digital converter does inside every phone, camera and instrument. When and how a sampled sequence faithfully represents the original — and when it does not — is the sampling theorem of Part 6, one of the pivotal results of the whole subject.

How "big" is a signal? A single number like a peak value is misleading — a brief tall spike and a long low hum can matter very differently. The two measures that behave well are energy and power, borrowed from the physical case where a voltage \(x(t)\) across a 1 Ω resistor dissipates instantaneous power \(x^2(t)\). Total energy is that quantity integrated over all time:

Some signals — a sinusoid that runs forever, a constant DC level — have infinite energy simply because they never stop. For these the useful measure is the average power, the energy per unit time in the limit of a long window:

These two measures cleanly split most signals of interest into two classes. An energy signal has finite, non-zero energy — and therefore zero average power, because a finite amount of energy spread over infinite time averages to nothing. A power signal has finite, non-zero average power — and therefore infinite energy. A signal is one or the other, never both; transient, decaying signals tend to be energy signals, while everlasting periodic signals are power signals.

Return now to the block diagram, because the operator view is where the real power of this subject lives. Writing \(y=T\{x\}\) invites a decisive question: what kind of operator is \(T\)? A handful of properties, studied in detail in Chapter 5, decide how hard or easy a system is to analyse, and every one of them is a yes-or-no question you can ask of the input-output rule.

When a system is both linear and time-invariant — an LTI system — something remarkable happens: its entire behaviour is captured by a single signal, its response to one sharp impulse, and its output to any input follows by an operation called convolution. LTI systems are the crown jewels of the course, the reason Chapters 6–9 exist, and the foundation on which every transform method is built. Most real amplifiers, filters and circuits are LTI to an excellent approximation, which is exactly why the theory is so useful.

If you take one idea from this chapter into the rest of the book, take this one. Real signals are complicated, and pushing a complicated signal directly through a system is hard. So we never do. Instead we follow a three-step strategy that recurs, in one costume or another, in almost every chapter that follows.

First, decompose the input into a sum of simple building-block signals \(\varphi_k\), each weighted by a coefficient \(c_k\). Second, find how the system responds to each individual building block — a far easier problem, because the blocks are chosen to be the ones the system handles cleanly. Third, because the systems we care about are linear, superpose those individual responses to assemble the full output.

The genius is in the choice of building block, and that choice is the difference between the great methods of the course. Choose shifted impulses and superposition becomes convolution (Part 2). Choose sinusoids or complex exponentials and it becomes the Fourier series and Fourier transform (Parts 3–4). Broaden those exponentials and you arrive at the Laplace and Z-transforms (Parts 5, 7). Four transforms, one strategy underneath them all.

The framework is abstract on purpose, which is why it reaches so far. Wherever information is carried by a changing quantity and a process reshapes it, signals and systems is the language in use.

For each item, first identify which idea it tests — the signal/system definitions, the CT/DT distinction, the energy/power measures, the operator properties, or the decompose-respond-superpose strategy — then apply the matching rule. Difficulty rises down the list.