Chapter 1 Signals
A signal is a function of an independent variable (usually time). A signal usually represents the variation of a physical quantity, such as voltage, force, position or light intensity as the independent variable changes. Some signals represent more abstract quantities, such as the price of a commodity over time, or the sequence of numbers stored in a particular register of a computer over time. Most broadly, signals carry information.
In this chapter we introduce basic notation and definitions related to signals. On the way we will see examples of the simplest signals. We also see ways to combine simple signals to make more complex signals, and discuss special properties that some signals have. It is useful to keep track of special properties, because these can greatly simplify the task of analysing and designing signals.
1.1 Signals
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A signal that depends on M independent variables is called M-dimensional. The speech signal shown in Figure 1.1 is a 1-dimensional signal. An image can be thought of as a 2-dimensional signal representing the variation in light intensity over a region. Most 1-dimensional signals we encounter in this unit are functions of time. One-dimensional signals could also be functions of other quantities. For example, we might be interested in the variation of air pressure with altitude above a particular point on earth. Despite this, for convenience when talking about 1-dimensional signals we will always refer to the independent variable as time.
Figure 1.1: A speech signal.
1.1.1 Continuous- and discrete-time signals
We classify signals as either continuous-time (CT) or discrete-time (DT) depending on whether the time variable takes (continuous) real values or (discrete) integer values. We generally use units of seconds for continuous time, and samples for discrete time.
Continuous-time (CT) signals
• We use the notation x for the entire signal. (Other lowercase letters may also be used.)
• For a real number t, the notation x(t) means the value of the signal at the continuous
• Signals x representing physical quantities usually take real values, i.e., x(t) is a real number
for every t.
• Often it is useful to consider signals x that take complex values. Then we can write
x(t) = xR(t) + xI (t)j where xR and xI are real-valued CT signals and j2 = −1. We describe CT signals using either
• (for real valued signals) a plot of the signal value (vertical axis) vs time (horizontal axis), as above;
• a rule, i.e., a mathematical expression that tells us the value of the signal for each time.
For instance
cos(2πt/5) if t ≥ 0
x(t) = 0 describes the signal shown below.
Discrete-time (DT) signals
• We use the notation x for the entire signal. (Other lowercase letters may also be used.) Another term for a DT signal is a sequence.
• For an integer n, the notation x[n] means the value of the signal at the discrete time n. For example, in the figure above, x[1] = 0. Note the use of square brackets around the time-index for discrete-time signals.
• Signals x representing physical quantities usually take real values, i.e., x[n] is a real number for every n. Sometimes the values a signal takes are quantised, i.e., the signal only takes discrete values (such as in the figure above). In this unit we largely ignore quantisation, because doing so makes things simpler and is often a good enough approximation.
• Often it is useful to consider signals x that take complex values. Then we can write x[n] = xR[n] + xI [n]j where xR and xI are real-valued DT signals and j2 = −1.
We describe DT signals using either
• a stem plot of the signal value (vertical axis) vs time (horizontal axis), as above;
• a rule, i.e., a mathematical expression that tells us the value of the signal for each time.
For instance
10 cos(πn/8 + 1/2) if n ≥ 0
x[n] = 0 describes the signal shown below.
Figure 1.2: Above is a CT signal x for 0 ≤ t ≤ 0.4. Below is the DT signal xs[n] := x(0.02n) obtained by sampling x with sampling period Ts = 0.02.
1.1.2 Sampling
Often signals corresponding to physical quantities are continuous-time signals. Computer sys- tems usually operate in discrete time. We can create a discrete-time signal from a continuous- time signal by choosing a sequence …,t−1,t0,t1,… of times and sampling the CT signal at those times. More precisely, if the continuous-time signal is x, the sampled signal xs is
xs[n] := x(tn) for all integers n.
Usually the sample times are equally spaced. The time between samples is then called the
sampling interval or sampling period, Ts. Then
xs[n] := x(nTs) for all integers n.
Figure 1.2 shows a CT signal x and a DT signal xs obtained by sampling x with sampling period Ts = 0.02.
1.1.3 Basic operations on signals
We can add and multiply real or complex numbers. Since our signals take values that are real or complex numbers, we can add signals and multiply them.
• Addition: If x1 and x2 are continuous-time signals then y = x1 + x2 is also a continuous- time signal defined by
y(t) = x1(t) + x2(t) for all t.
• Multiplication by a number: If x is a continuous-time signal and α is a (real or complex)
number then y = αx is also a continuous-time signal defined by y(t) = αx(t) for all t.
• Multiplication: If x1 and x2 are continuous-time signals then y = x1x2 is also a continuous- time signal defined by
y(t) = x1(t)x2(t) for all t.
Very similar operations can be performed for discrete-time signals. The only difference is that we use slightly different notation.
Another operation we can perform on signals is time-shifting.
• If x is a continuous-time signal and τ is a real number we can define a new signal by
y(t)=x(t−τ) forallt.
The graph of y is the same as the graph of x but shifted to the right (or delayed) by τ.
• If x is a discrete-time signal and N is an integer we can define a new signal by y[n]=x[n−N] foralln.
The graph of y is the same as the graph of x but shifted to the right (or delayed) by N.
There are many other interesting operations we can perform on signals, many of which involve manipulating the time variable in some way. We can interpret these signal transformations as systems, and we discuss some of these in Chapter 2.
1.1.4 Fundamental signals
Signals, such as the speech signal shown in Figure 1.1 can be very complicated. A very powerful idea in the analysis of signals and systems is to rewrite a signal as a sum of carefully chosen ‘simpler’ signals. In this section we introduce some of the most important ‘simple’ signals.
The discrete-time unit step function is
u[n]= 0 ifn<0.
1 ifn≥0 The continuous-time unit step function is
1 t>0 u(t)= 0 t<0.
Note that we have chosen not to define u at t = 0. Some texts define u(0) = 1/2, others define u(0) = 1 for consistency with the discrete-time unit step function. The discrete-time and continuous-time unit step functions are shown in Figure 1.3.
Unit impulse
The discrete-time unit impulse (or Kronecker delta) is 1 ifn=0
δ[n]= 0 ifn̸=0.
The DT unit impulse is illustrated in Figure 1.4. The most important property of the unit
Figure 1.3: On the left is the discrete-time unit step function. On the right is the continuous- time unit step function.
Figure 1.4: On the left is the discrete-time unit impulse, also known as the Kronecker delta. On the right is the continuous-time unit impulse, also known as the Dirac delta.
impulse function is that if x is any DT signal,
x[n]δ[n]=x[0]. (1.1) n=−∞
The continuous-time unit impulse (or Dirac delta) is a CT analogue of the DT unit impulse, butisnotafunctionintheusualsense.1 Weusuallyillustratethecontinuous-timeunitimpulse with an upward arrow at t = 0, as in Figure 1.4, and denote it by δ. The best approach to defining the CT unit impulse is by the property that if x is any CT signal (that is continuous at t = 0 and sufficiently regular) then
x(t)δ(t) dt = x(0). (1.2)
Notice how (1.2) looks a lot like (1.1) but with the sum replaced by an integral2. There are some ways in which the CT unit impulse behaves like an ordinary function, and often it is very convenient to manipulate it as though it is a function. Whenever we do this later in the notes, we will need to be careful!
Unit complex exponentials
The CT unit complex exponential (or just complex exponential) signal is x(t) = ejω0t = cos(ω0t) + j sin(ω0t)
where the second equality is known as Euler’s formula. The parameter ω0 has units radi- ans/second. Note that the unit complex exponential puts together cos(ω0t) (as the real part) and sin(ω0t) (as the imaginary part), into one convenient package.
1It is possible to define in a rigorous way, but to do so very carefully requires mathematical sophistication that is well beyond the scope of this unit.
2To make this definition rigorous, for instance, we would need to define what we mean by this integral. 6
The DT unit complex exponential (or just complex exponential) signal is x[n] = ejω0n = cos(ω0n) + j sin(ω0n).
The parameter ω0 has units radians/sample. General complex exponential
Let s = σ0 +jω0 be a complex number in rectangular form. We can define a CT general complex exponential signal as
x(t) = est = e(σ0+jω0)t = eσ0t ejω0t = eσ0t(cos(ω0t) + j sin(ω0t)).
If σ = 0 then we obtain a unit complex exponential signal. If ω0 = 0 then we obtain a CT real exponential signal x(t) = eωt.
Let α = rejω0 be a complex number in polar form. We can define a DT general complex exponential signal as
x[n] = αn = rnejω0n = rn(cos(ω0n) + j sin(ω0n)).
If r = 1 then we obtain a DT unit complex exponential signal. If ω0 = 0 then we obtain a DT
real exponential signal x[n] = rn.
Sinusoidal signals
A continuous-time sinusoidal signal has the form
x(t) = A cos(ω0t + θ) (1.3)
where A is the amplitude (a positive real number), ω0 has units radians/second, and θ it the phase (in radians). Using Euler’s formula, the sinusoidal signal in (1.3) can be expressed as
x(t) = A cos(ω0t + θ) = A Re{ej(ω0t+θ)} where Re denotes the real part of a complex signal.
A discrete-time sinusoidal signal has the form
x[n] = A cos(ω0n + θ) (1.4)
where A is the amplitude (a positive real number), ω0 has units radians/sample, and θ is the phase (in radians). Again, using Euler’s formula, the sinusoidal signal in (1.4) can be expressed as
x[n] = A cos(ω0n + θ) = A Re{ej(ω0n+θ)}.
Summary of Section 1.1
We model signals as functions that associate real or complex number with each point in time (or space). The signal as a whole describes the variation of a quantity (e.g., voltage, air pressure, position, light intensity) over time.
In this unit we use two models for time:
• discrete-time, indexed by whole numbers (useful for digital signals/systems)
• continuous-time, indexed by real numbers (useful for analogue signals/systems).
We can perform many transformations on signals, such as adding signals and multi- plying them by numbers. It is usually a good idea to write a signal as a sum of ‘simple’
signals (like impulses or complex exponentials). Much of this unit is about understanding the insights we can gain by doing so.
1.2 Properties of signals 1.2.1 Real and complex signals
A signal x is real if x(t) is a real number for all t. A CT signal x is complex if x(t) is a complex number for all t.
A general complex signal can be written in rectangular form as x(t) = xR(t) + jxI(t)
where xR and xI are real signals. We call xR(t) = Re{x(t)} the real part of x(t) and xI(t) = Im{x(t)} the imaginary part of x(t). Note that the imaginary part is actually a real signal!
A general complex signal can also be written in polar form as x(t) = r(t)ejθ(t)
r(t) = |x(t)| = [xR(t)]2 + [xI (t)]2 for all t
is the magnitude of x and θ(t) is the phase of x. There is ambiguity in the definition of the
phase since
for any integer k. To resolve this ambiguity, we often assume that the phase takes values in
ej(θ(t)+2πk) = ejθ(t) (−π, π]. An explicit formula for the phase is
−1 xI (t) tan
θ(t) = π − tan−1 xI (t) xR (t)
if xR(t) > 0
1.2.2 Periodic and non-periodic signals
A CT signal x has periodic with period T if there is a positive T for which
x(t) = x(t + T) for all times t. (1.5)
The figure below shows that if x is periodic with period T = T0, then it is also periodic with period 2T, or 3T, etc.
The fundamental period T0 of x is the smallest positive T for which (1.5) holds. A CT signal that is not periodic is called non-periodic or aperiodic.
if xR(t) < 0 and xI (t) > 0 −π+tan−1xI(t) ifxR(t)<0andxI(t)<0
The same definitions and formulas hold for discrete-time signals, with just a slight change
of notation.
Example 1.1
A unit complex exponential x(t) = ejω0t is periodic with fundamental period T0 = 2π .
This is because
x (t + 2π/|ω0|) = ejω0(t+2π/|ω0|) = ejω0t esign(ω0)j2π = ejω0t = x(t)
where we have used the definition
sign(ω0)= 0 = 0
ω 1 ifω>0 |ω0| −1 ifω0<0
for the second equality and the fact that ej2π = e−j2π = 1 for the third equality.
A DT signal x is periodic with period N if there is a positive integer N such that
x[n + N] = x[n] for all n. (1.6)
The fundamental period N0 of x is the smallest positive integer N such that (1.6) holds. The fundamental frequency is 2π/N0 where N0 is the fundamental period.
Unlike the CT case, understanding the periodicity properties of a DT unit complex expo- nential (or sinusoid) can be complicated.
Example 1.2
A unit complex exponential x[n] = ejωn is periodic with period N if and only if
N = m 2π for some positive integer m. (1.7)
The fundamental period N0 is the smallest positive N such that (1.7) holds. The following examples illustrate some of the subtleties in determining the period of DT unit complex exponentials.
• If x[n] = ejπn/2 then ω = π/2 rad/sample. Since ω=1
we see that this signal is periodic with fundamental period 4. Hence the fundamental frequency is π/2 rad/sample.
• If x[n] = ej4πn/5 then ω = 4π/5 rad/sample. Since ω=2
we see that this signal is periodic with fundamental period 5. Hence the fundamental
frequency is 2π/5 rad/sample which is different from ω.
• If x[n] = ejπn/√2 then ω = π/√2 rad/s. Since ω is not a rational number, it follows
that x is aperiodic (i.e., not periodic).
Periodicity is preserved by some operations but not others.
• The sum of two (CT or DT) periodic signals with the same period is always periodic (with the same period).
• If we multiply a (CT or DT) periodic signal by a number, the result is always periodic (with the same period).
• If we sample a periodic CT signal, the resulting DT signal is not always periodic.
• The sum of two periodic CT signals (with different periods) is not always periodic.
• The sum of two periodic DT signals (with possibly different periods) is always periodic.
1.2.3 Bounded signals
A CT signal is bounded if there is a positive number M such that |x(t)| ≤ M for all t. A DT signal is bounded if there is a positive number M such that |x[n]| ≤ M for all n. For a bounded CT signal, the smallest M such that |x(t)| ≤ M for all t is called the peak value of the signal. A similar definition holds for the peak value of a bounded DT signal. If a signal is not bounded we call it unbounded.
All practical physical signals are bounded, so what does an unbounded signal mean in practice? Generally, if a mathematical model predicts that a signal of interest is unbounded, it means something is going to go wrong (or “bang!”) in the corresponding practical system!
1.2.4 Even and odd signals
A signal is even if reversing time does not change the signal, i.e., x(−t) = x(t) for all continuous times t
x[−n] = x[n] for all discrete times n 10
Example 1.3
The CT signal x(t) = ejπt is periodic with fundamental period 2. If we sample it with sampling period 1/√2, the resulting DT signal x[n] = ejπn/√2 is not periodic.
Example 1.4
The sinusoid x(t) = A cos(ω0t + θ) is a bounded signal. In this case |x(t)| ≤ A for all t. The real exponential x(t) = et is an unbounded signal. In this case, as t keeps growing, |x(t)| keeps growing, and so is not bounded.
A signal is odd if reversing time changes the sign of the signal, i.e., x(−t) = −x(t) for all continuous times t
x[−n] = −x[n] for all discrete times n
Any signal can be written as the sum of an even signal and an odd signal as
xeven(t) = [x(t) + x(−t)]/2 xeven[n] = [x[n] + x[−n]]/2
xodd(t) = [x(t) − x(−t)]/2 xodd[n] = [x[n] − x[−n]]/2
for all continuous times t for all discrete times n
for all continuous times t for all discrete times n.
x(t) = xeven(t) + xodd(t) x[n] = xeven[n] + xodd[n]
Sometimes recognizing whether a signal is even or odd allows us to quickly recognize some of its properties.
Other useful facts about even and odd signals are
• The sum of two even signals is even.
• The sum of two odd signals is odd.
• The product of two even signals is even. The product of two odd signals is even. • The product of an even signal and an odd signal is odd.
1.3 Summary
Signals are usually modeled as functions in which the independent variable is time (or space, etc). We have seen a number of basic signals, and ways of building more complex signals from simpler ones by addition, multiplication by scalars, and time-shifting. Later in the unit we learn techniques to go in the opposite direction, to systematically decompose signals into simpler pieces, so they are easier to analyse and design. Some signals have special properties such as being periodic or bounded.
Example 1.5
If x is an odd signal then for any positive a, a
x(t) dt = 0. −a
Summary of Section 1.2
Some signals have special properties, such as being periodic or bounded or even. Recog- nizing these properties can help us choose an appropriate approach to analysing a signal.
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