Signal

A signal is any physical quantity that carries information. The voltage at a microphone terminal carries the information of the sound that hit it; the light intensity at a single cell on a camera sensor carries the brightness at that point in the scene; the reading on a thermometer carries the temperature. The signal is interesting not for itself, but for what it tells us — some quantity in the world has been encoded into a physical observable we can measure.

Signals in this course are studied as functions of time. We write $x(t)$ when time is continuous (defined at every real $t$) and $x[n]$ when time is discrete (defined at integer indices $n$). The brackets-vs-parentheses convention is universal in signal processing.

Continuous vs discrete, on both axes

Two binary distinctions give four kinds of signal. On the time axis, a continuous-time signal is defined for every real $t$; a discrete-time signal is defined only at isolated instants $t=0,\Delta t,2\Delta t,\dots$. On the value axis, a continuous-value signal can take any real value in some range; a discrete-value signal is restricted to a finite or countable set.

Crossing the two distinctions:

	Continuous time	Discrete time
Continuous value	Analog (e.g. microphone voltage)	Sampled-but-unquantized
Discrete value	Digital-like rectangular waveforms	Digital signals proper

A digital-to-analog converter outputs a staircase waveform (continuous time, discrete value). A computer’s internal bit stream is discrete time, discrete value. The output of a sample-and-hold circuit before quantization is discrete time, continuous value.

Continuous-time is about where the signal is defined

A square wave that jumps between $0$ and $1$ is still a continuous-time signal — it has a value at every real $t$. It is not, however, a continuous function in the calculus sense. “Continuous-time” describes the domain; “continuous as a function” describes the graph. We’ll work happily with continuous-time signals that have jumps.

Noise

Noise is a continuous-time, continuous-value signal that we cannot predict in detail — only statistically. See Noise (signal). Every real measurement is signal plus noise, and a large fraction of signal-processing theory is about separating one from the other.

Operating on signals

Signals can be added pointwise, multiplied pointwise, shifted and scaled in time and amplitude, differentiated, integrated. They are inputs and outputs of systems — the other half of the “signals and systems” framework.

In electronics (Electronics I)

In a microelectronics course the signal is concretely a voltage or current that carries information: a microphone’s voltage tracking sound pressure, an antenna’s current tracking a radio field, a sensor voltage tracking temperature. Processing that signal means filtering it (selecting frequency content), amplifying it (making it larger without distorting it), and coupling it between stages. The simplest sinusoidal signal $v(t)=V_p\sin (\omega t+\phi )$ is described by its amplitude, frequency, and phase.