Signals

Continuous and discrete time, energy and power, periodicity, even and odd parts, and the three transformations of time

Paris — 9 April 1860

Édouard-Léon Scott de Martinville sings "Au clair de la lune" into a horn. At its narrow end a membrane drives a bristle, and the bristle scratches a wavy line into soot on a turning drum. Scott called the machine a phonautograph. He built it as a laboratory instrument for studying sound by eye: the waveforms of speech, the envelope of a note. Playing the line back as sound was never part of the plan. In 2008, researchers scanned his tracings optically, and scientists at the Lawrence Berkeley National Laboratory turned the scans into audio. The first version was 10 seconds of a woman or a child singing at an ordinary tempo. Then the team found that they had misread a reference frequency on the paper and doubled the playback speed. The real recording is 20 seconds long: a man, probably Scott himself, singing very slowly (Phonautograph; Scott de Martinville). The trace never changed. What changed was how its time axis was read: a factor of two in time turned a man into a child. This lesson is about that axis.

A signal is a quantity that varies and carries information: the pressure at a microphone, the voltage on a strain gauge, the temperature of a turbine blade, the brightness along one row of a photograph, a country's daily case count. Mathematically it is simply a function. That is the most useful sentence in this course: everything you know about functions — shifting a graph, integrating its square, splitting it into even and odd parts — becomes a tool for sound, images, radar echoes and heartbeats.

Two kinds of time run through the subject. A continuous-time signal

x(t)

has a value at every real instant

t

. A discrete-time signal

x[n]

has values only at integer indices

n

: the samples of a recording, the readings of a daily log, the pixels of a scan line. Brackets for integers, parentheses for reals — the notation tells you which world you are in.

💡Continuous time, discrete time, digital

Continuous time: $x(t)$ , $t \in \mathbb{R}$ — a microphone voltage, an accelerometer output.
Discrete time: $x[n]$ , $n \in \mathbb{Z}$ — often samples $x[n] = x(nT)$ of a continuous signal taken every $T$ seconds (Unit 5), but some signals are discrete by nature: monthly rainfall, a stock's closing prices.
Digital: discrete in time and in amplitude — each sample rounded to one of finitely many levels (16 bits on a CD: 65 536 levels). Rounding is a small, deliberate error we treat as noise.

How big is a signal? For most physical signals,

|x|^2

is proportional to a power: a voltage across a resistor dissipates

v^2/R

, a pressure wave carries

p^2/(\rho c)

per unit area. So we measure a signal's size by its energy — the integral of

|x|^2

over all time — or, for a signal that never dies out, by its average power.

E = \int_{-\infty}^{\infty} |x(t)|^2\,dt, \qquad P = \lim_{T\to\infty} \frac{1}{2T}\int_{-T}^{T} |x(t)|^2\,dt, \qquad E = \sum_{n=-\infty}^{\infty} |x[n]|^2,\quad P = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2

Energy and average power, in continuous and discrete time. An energy signal has finite E (and P = 0); a power signal has finite, non-zero P (and E = ∞).

📐Energy and power of three signals

$x_1(t) = e^{-2t}u(t)$ , a decaying exponential switched on at $t = 0$ : $E = \int_0^\infty e^{-4t}\,dt = \tfrac14$ . An energy signal.
$x_2(t) = 3\cos(\omega_0 t)$ : it never dies out, so $E = \infty$ . Over one period the average of $\cos^2$ is $\tfrac12$ , so $P = \tfrac{9}{2} = 4.5$ . A power signal — every sinusoid of amplitude $A$ has $P = A^2/2$ .
$x_3[n] = (\tfrac12)^n u[n]$ : $E = \sum_{n=0}^{\infty} (\tfrac14)^n = \dfrac{1}{1 - \frac14} = \tfrac43$ , a geometric series.

The ramp

t\,u(t)

is neither: its energy over

[-T, T]

T^3/3

and its average power

T^2/6

, and both grow without bound.

Periodic signals repeat:

x(t + T) = x(t)

for every

t

. The smallest such

T > 0

is the fundamental period, and

\omega_0 = 2\pi/T

the fundamental frequency. A sum of periodic signals is periodic only if the periods fit together: some common multiple of all of them must exist.

Discrete time adds a surprise.

x[n+N] = x[n]

needs an integer

N

, so

\cos(\Omega_0 n)

is periodic only when

\Omega_0 N = 2\pi m

for integers

m

and

N

— that is, when

\Omega_0/2\pi

is a rational number.

\cos(\pi n/4)

repeats every 8 samples;

\cos(3\pi n/8)

every 16; but

\cos(2n)

, whose

\Omega_0/2\pi = 1/\pi

is irrational, never repeats exactly, although its samples lie on a perfectly periodic curve.

Derivation

When is a sum of two sinusoids periodic?

Let

x(t) = \cos(\omega_1 t) + \cos(\omega_2 t)

with periods

T_1 = 2\pi/\omega_1

and

T_2 = 2\pi/\omega_2

. A period

T

of the sum must bring both terms back, so

T = k_1 T_1 = k_2 T_2

for positive integers

k_1

k_2

. Then

\frac{T_1}{T_2} = \frac{k_2}{k_1} = \frac{\omega_2}{\omega_1},

a ratio of integers. So the sum is periodic exactly when

\omega_1/\omega_2

is rational. If

T_1/T_2 = p/q

in lowest terms, the smallest common multiple is

T = q\,T_1 = p\,T_2

Example:

\cos(3\pi t/4) + \sin(\pi t/6)

has

T_1 = 8/3

and

T_2 = 12

. Their ratio is

(8/3)/12 = 2/9

, so

T = 9 T_1 = 2 T_2 = 24

. For

\cos t + \cos(\sqrt2\, t)

the ratio is irrational: the sum never repeats.

💡Even and odd parts

Every signal splits uniquely into an even part and an odd part:

x_e(t) = \tfrac12\left[x(t) + x(-t)\right], \qquad x_o(t) = \tfrac12\left[x(t) - x(-t)\right], \qquad x = x_e + x_o .

For a real signal the energies add,

E = E_e + E_o

, because the cross term

\int x_e x_o\,dt

integrates an odd function over all time and vanishes. A signal that starts at

t = 0

(a causal signal) has

E_e = E_o = E/2

. The split matters later: the Fourier transform of the even part is real, that of the odd part imaginary.

The three transformations of time. We can change a signal by changing the variable it depends on:

Shift: $x(t - t_0)$ is $x$ delayed by $t_0$ if $t_0 > 0$ — every feature arrives $t_0$ later. (An echo is a delayed copy.)
Reversal: $x(-t)$ plays the signal backwards — the mirror image about $t = 0$ .
Scaling: $x(at)$ with $a > 1$ compresses the signal in time: it lasts $1/a$ as long and every frequency in it is $a$ times higher. With $0 < a < 1$ it stretches and drops in pitch. The phonautogram played at double speed was $x(2t)$ : half as long, an octave higher — a man turned into a child.

Scaling also changes the energy: substituting

\tau = at

in the energy integral gives

E\{x(at)\} = E\{x\}/|a|

. Compressing a signal by 2 halves its energy, because it lasts half as long at the same amplitude.

📐Reading x(2t − 3)

Let

x(t)

be a pulse that is non-zero for

0 \le t < 2

. Where is

y(t) = x(2t - 3)

non-zero? Solve the inequality for

t

0 \le 2t - 3 < 2 \quad\Longleftrightarrow\quad 1.5 \le t < 2.5 .

y

is the pulse compressed to half its length and moved to start at

t = 1.5

. The order of operations is the usual trap. Write

x(2t-3) = x\big(2(t - 1.5)\big)

: compress by 2 first, then delay by 1.5. Or delay by 3 first,

x(t - 3)

, then replace

t

2t

. Delaying by 3 after compressing would put the pulse at

[3, 4)

— wrong. When in doubt, solve the inequality.

The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point.

– Claude Shannon, A Mathematical Theory of Communication (1948), quoted in Information theory

💡Discrete-time scaling throws samples away

In discrete time,

x[2n]

keeps only the even-numbered samples: the odd ones are gone for good, so the operation cannot be undone. And

x[n/2]

is not even defined for odd

n

— to stretch a sequence you must invent the in-between samples. Both operations, decimation and interpolation, are real tools with real dangers, and Lesson 5-3 does them properly.

🕰Recorded in 1860, heard in 2008

Scott patented the phonautograph in 1857, two decades before Edison's phonograph. He meant people to read the tracings, as one reads writing; that a machine could play them back was something he never imagined. The 2008 playback made his "Au clair de la lune" the oldest known intelligible recording of a human voice — and its first, double-speed version became a lesson in how much depends on the time axis (Phonautograph).

💡Where you will use this

Energy and power come back as Parseval's relation (Lessons 3-2 and 4-2), as the noise power under a spectrum (Lesson 6-3) and as signal-to-noise ratios (Lesson 10-3). Time shifts are the core of convolution (Unit 2) and become phase in the frequency domain (Lesson 4-2). Time scaling is the time–frequency trade-off of Lesson 4-2, and discrete-time scaling is the decimation and interpolation of Lesson 5-3.

📚Further reading

Alan V. Oppenheim and Alan S. Willsky, Signals and Systems (2nd ed., Prentice Hall, 1997), chapter 1 — signals, transformations of the independent variable, periodicity.
B. P. Lathi and Roger Green, Linear Systems and Signals (3rd ed., Oxford University Press, 2018), chapter 1 — energy, power and signal operations with worked examples.
Phonautograph on Wikipedia — the instrument, the tracings and the 2008 playback.

↓But there is a problem

We can now describe a signal's size, its rhythm and its symmetry, and move it about in time. But to analyse systems, we need building blocks — signals out of which every other signal can be assembled. Two will do the job. One is a signal that is zero at every instant but one, and still has area one. The other is a signal no instrument can record, because it has an imaginary part. The first does not even exist as a function — and engineers used it for decades before mathematicians could say what it was.