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Signal Theory

Signals and systems: energy, power, LTI systems and convolution

Classifying signals, energy and power, elementary signals, system properties, convolution with proof and graphical computation, causality and BIBO stability.

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A signal is a quantity that varies in time and carries information: the voltage from a microphone, the current in an antenna, the sequence of numbers read from a converter. A system is anything that turns one signal into another: a filter, an amplifier, a transmission channel, a program. Signal theory studies how to describe both and how to predict a system’s output. In this lesson we cover the basic definitions and the most important result: for linear time-invariant systems the output is a convolution.

Classifying signals

  • Continuous-time x(t) or discrete-time x[n], defined only at integer instants (for example the samples of a converter).
  • Deterministic, described by a formula, or random, of which only the statistical properties are known (noise, but also speech or transmitted data, which are not known in advance).
  • Periodic, with x(t + T) = x(t), or aperiodic.

Energy and power. The energy of a signal is

    \[ E = \int_{-\infty}^{+\infty} |x(t)|^2\,dt \]

(the name comes from the case of a voltage across a 1 Ω resistor). A rectangular pulse of amplitude A and duration T has energy A^2 T. Signals that last forever, like a sinusoid, have infinite energy: for them the average power is used,

    \[ P = \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2\,dt \]

For A\cos(2\pi f_0 t), P = A^2/2. A signal is an “energy signal” if E is finite, a “power signal” if P is finite.

Elementary signals

  • the step u(t), equal to 0 for t \lt 0 and 1 for t \gt 0: a switch closing;
  • the rectangular pulse \mathrm{rect}(t/T), of duration T;
  • the exponential e^{-t/\tau}u(t), the response of first-order circuits;
  • the sinc, \mathrm{sinc}(x) = \sin(\pi x)/(\pi x), which appears in every band-limited problem;
  • the Dirac delta \delta(t), a pulse of zero duration and unit area, with the sifting property \int x(\tau)\delta(t - \tau)\,d\tau = x(t) (Mathematical Methods lesson on the delta).

System properties

A system turns an input x(t) into an output y(t). The properties that matter:

  • linearity: the response to a x_1 + b x_2 is a y_1 + b y_2;
  • time invariance: if the input x(t) gives y(t), the delayed input x(t - t_0) gives y(t - t_0): the system does not change over time;
  • causality: the output at an instant depends only on the input up to that instant;
  • stability (BIBO, bounded input bounded output): every bounded input gives a bounded output.

Linear time-invariant (LTI) systems are the most important case: circuits with resistors, capacitors, inductors and linear amplifiers, digital filters, many transmission channels.

Convolution

The impulse response h(t) is the system’s output when the input is \delta(t).

Theorem. The output of an LTI system with impulse response h is the convolution of the input with h:

    \[ y(t) = (x * h)(t) = \int_{-\infty}^{+\infty} x(\tau)\,h(t - \tau)\,d\tau \]

Proof. By the sifting property, the input can be written as a “sum” of shifted, weighted impulses:

    \[ x(t) = \int x(\tau)\,\delta(t - \tau)\,d\tau \]

By time invariance, the response to \delta(t - \tau) is h(t - \tau). By linearity, the response to a combination (even a continuous one) of impulses is the same combination of the responses: y(t) = \int x(\tau)\,h(t - \tau)\,d\tau. ∎

The result is powerful: an LTI system is completely described by a single function, its impulse response.

Graphical computation. To find y at an instant t: flip h (h(-\tau)), shift it by t (h(t - \tau)), multiply it by x(\tau) and compute the area of the product. Sliding t gives the whole output.

Graphical computation of the convolution of a rectangular pulse with an exponential response
At the top the input, a rectangular pulse of duration T, and the impulse response flipped and shifted to the instant t; the area of the overlap is the value of the output at t. At the bottom the complete output: it rises while the pulse is present and then decays exponentially.

Example. An RC circuit with the output across the capacitor has h(t) = \frac{1}{RC}e^{-t/RC}u(t). With a rectangular pulse of duration T at the input, the convolution gives an exponential charge for 0 \lt t \lt T and an exponential discharge afterwards: exactly the curve in the figure, and it matches what one gets by solving the differential equation (Electrical Engineering lesson on transients).

Properties of convolution

  • commutative: x * h = h * x;
  • associative: two systems in cascade are equivalent to a single system with response h_1 * h_2, and the order does not matter;
  • distributive: two systems in parallel are equivalent to h_1 + h_2;
  • the identity element is the delta: x * \delta = x.

Causality and stability from the impulse response

Causality. An LTI system is causal if and only if h(t) = 0 for t \lt 0: in the convolution, h(t - \tau) must vanish for \tau \gt t, that is it must not “look at” future input.

Theorem. An LTI system is BIBO stable if and only if its impulse response is absolutely integrable:

    \[ \int_{-\infty}^{+\infty}|h(t)|\,dt \lt \infty \]

Proof. *Sufficiency:* if |x(t)| \le M for every t, then

    \[ |y(t)| \le \int |x(\tau)|\,|h(t - \tau)|\,d\tau \le M\int |h(s)|\,ds \]

which is finite. *Necessity:* if the integral diverges, choose the bounded input x(\tau) = \mathrm{sgn}\,h(-\tau); at t = 0 the output equals \int |h(-\tau)|\,d\tau = \infty. ∎

The RC circuit is stable (\int |h| = 1); an ideal integrator, with h(t) = u(t), is not: a step at the input produces a ramp that grows without bound.

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