2 Continuous-Time Signals

A continuous-time (CT) signal is a function of one or more independent variables conveying information about a physical phenomena. This lecture gives an introduction to continuous-time signals as functions. You learn how to characterize such signals in a number of ways and are introduced to two very important signals: the unit impulse and the complex exponential.

2.1 Signals as Functions

In order to reason about signals mathematically we need a representation or model. Signals are modeled as functions, mappings between sets \[ f: A \rightarrow B \] where \(A\) is a set called the domain and \(B\) is a set called the range.

The most basic classification of signals depends on the sets that makeup the domain and co-domain. We will be interested in two versions of the domain, the reals denoted \(\mathbb{R}\) and the integers denoted \(\mathbb{Z}\). We will be interested in two versions of the co-domain, the reals \(\mathbb{R}\) and the set of complex numbers \(\mathbb{C}\).

Note

Analog Signal: If the function \(f: \mathbb{R} \rightarrow \mathbb{R}\), we call this an analog or real, continuous-time signal, e.g. a voltage at time \(t \in \mathbb{R}\), \(v(t)\). We will write these as \(x(t)\), \(y(t)\), etc. The units of \(t\) are seconds. Figure 2.1 shows some graphical representations, i.e. plots.

Code

import matplotlib
import matplotlib.pyplot as plt
import numpy as np

t = np.linspace(-6, 6, 1000);
u = (t >= 0)
e = np.exp(-t)*u
s = np.sin(2*np.pi*t)
es = e*s

f = plt.figure()

plt.subplot(2, 2, 1)
plt.plot(np.linspace(-6, 0, 1000), np.zeros(1000), color='b')
plt.plot(np.linspace(0, 6, 1000), np.ones(1000), color='b')
plt.xlabel('$t$')
plt.ylabel('$x(t)$')
plt.title('$x(t) = u(t)$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.subplot(2, 2, 2)
plt.plot(t,e)
plt.xlabel('$t$')
plt.ylabel('$x(t)$')
plt.title('$x(t) = e^{-t}u(t)$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.subplot(2, 2, 3)
plt.plot(t,s)
plt.xlabel('$t$')
plt.ylabel('$x(t)$')
plt.title('$x(t) = sin(2\pi t)$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.subplot(2, 2, 4)
plt.plot(t,es)
plt.xlabel('$t$')
plt.ylabel('$x(t)$')
plt.title('$x(t) = e^{-t}sin(2\pi t)u(t)$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.tight_layout()

plt.show()

Figure 2.1: Example plots of analog signals.

Note

Real, Discrete-time Signal: If the function \(f: \mathbb{Z} \rightarrow \mathbb{R}\), we call this a real, discrete-time signal, e.g. the temperature every day at noon. We will write these as \(x[n]\), \(y[n]\), etc. Note \(n\) is dimensionless. Figure 2.2 shows some graphical representations.

Code

import matplotlib
import matplotlib.pyplot as plt
import numpy as np

n = np.linspace(-6, 6, 13);
u = (n >= 0)
e = np.exp(-n)*u
s = np.sin(n)
es = e*s

f = plt.figure()

plt.subplot(2, 2, 1)
plt.stem(n, u)
plt.xlabel('$n$')
plt.ylabel('$x[n]$')
plt.title('$x[n] = u[n]$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.subplot(2, 2, 2)
plt.stem(n,e)
plt.xlabel('$n$')
plt.ylabel('$x[n]$')
plt.title('$x[n] = e^{-n}u[n]$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.subplot(2, 2, 3)
plt.stem(n,s)
plt.xlabel('$n$')
plt.ylabel('$x[n]$')
plt.title('$x[n] = sin(n)$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.subplot(2, 2, 4)
plt.stem(n,es)
plt.xlabel('$n$')
plt.ylabel('$x[n]$')
plt.title('$x[n] = e^{-n}sin(n)u[n]$')
plt.autoscale(enable=True, axis='x', tight=True)

plt.tight_layout()
plt.show()

Figure 2.2: Example plots of analog signals.

Some other possibilities:

\(f: \mathbb{R} \rightarrow \mathbb{Z}\), digital, continuous-time signals, e.g. the output of a general purpose pin on a microcontroller
\(f: \mathbb{Z} \rightarrow \mathbb{Z}\), digital, discrete-time signals, e.g. the signal on a computer bus

The co-domain can also be complex.

\(f: \mathbb{R} \rightarrow \mathbb{C}\), complex-valued, continuous-time signals, e.g. \[ x(t) = e^{j\omega t} = \cos(\omega t) + j\sin(\omega t) \]
\(f: \mathbb{Z} \rightarrow \mathbb{C}\), complex-valued, discrete-time signals, e.g. \[ x[n] = e^{j\omega n} = \cos(\omega n) + j\sin(\omega n) \]

Since the domains \(\mathbb{R}\) and \(\mathbb{Z}\) are usually interpreted as time, we will call these time-domain signals. In the time-domain, when the co-domain is \(\mathbb{R}\) we call these real signals. All physical signals are real. However complex signals will become important when we discuss the frequency domain.

2.2 Primitive Models

We mathematically model signals by combining elementary/primitive functions, for example:

polynomials: \(x(t) = t\), \(x(t) = t^2\), etc.
transendental functions: \(x(t) = e^t\), \(x(t) = \sin(t)\), \(x(t) = \cos(t)\), etc.
piecewise functions, e.g. \[ x(t) = \left\{ \begin{array}{cl} f_1(t) & t < 0\\ f_2(t) & t \geq 0\\ \end{array}\right. \]

Note

Modeling a Switch: Consider a mathematical model of a switch, which moves positions at time \(t = 0\).

FIGURE HERE

We use this model so much we give it it’s own name and symbol: Unit Step, \(u(t)\)

\[ u(t) = \left\{ \begin{array}{cl} 0 & t < 0\\ 1 & t \geq 0\\ \end{array}\right. \] so a mathematical model of the switch circuit above would be \(x(t) = V u(t)\).