ABSTRACT
The global positioning system (GPS) is a typical illustration of what signals and systems are all about GPS is a satellite-based navigation system made up of a network of 24 satellites placed into orbit by the US Department of Defense GPS was originally designed for military use, but in the 1980s, the government made the system available for civilian use
The 24 satellites that make up the GPS space segment are orbiting the earth about 12,000 miles above us These satellites travel at speeds of roughly 7,000 miles/h GPS satellites transmit two low power radio signals The signals travel by line of sight, meaning they will pass through clouds, glass, and plastic but will not go through most solid objects, such as buildings and mountains
A GPS signal contains three different bits of information-a pseudorandom code, ephemeris data, and almanac data The pseudorandom code identifies which satellite is transmitting information You can view this number on your GPS unit’s satellite page Ephemeris data contains important information about the status of the
satellite, current date, and time This part of the signal is essential for determining a position The almanac data tells the GPS receiver where each GPS satellite should be at any time throughout the day Each satellite transmits almanac data showing the orbital information for that satellite and for every other satellite in the system
A GPS receiver calculates its position by precisely timing the signals sent by GPS satellites The receiver uses the messages it receives to determine the transit time of each message and computes the distance to each satellite These distances along with the satellites’ locations are used to compute the position of the receiver
The idea of signals and systems arises in different disciplines such as science, engineering, economics, politics, and medicine Scientists, mathematicians, financial analysts, cardiologists, and engineers all use the concepts of systems and signals because they are the foundation on which we build many things in our daily lives Typical examples of systems include radio and television, telephone networks, radar systems, computer networks, wireless communication, military surveillance systems, and satellite communication systems The theory of signals and systems provides a solid foundation for control systems, communication systems, power systems, and networking, to name a few
Our objective in this book is to present an introductory, yet comprehensive, treatment of signals and systems with an emphasis on computing using MATLAB® The knowledge of a broad range of signals and systems is of practical value in describing human experience It is also important because engineers must be familiar with signal and system concepts and apply the knowledge to analyze some specific signals and systems they will deal with in their professional life
In this chapter, we begin by discussing some of the basic concepts in signals and systems We introduce the mathematical representations of signals, their properties, and applications We also discuss different systems and how the material covered in this chapter is used in some applications We finally learn how to use MATLAB to process signals
To avoid any misconception, it is expedient that we define at the outset what we mean by signals and systems
A signal x(t) is a set of data or function of time that represents a variable of interest
A signal typically contains information about the nature of a phenomenon Examples of signals include the atmospheric temperature, humidity, human voice, television images, a dog’s bark, and birdsongs More generally, a signal may be a function of more than one independent variable (time) For example, pictures are signals that depend on two independent variables (horizontal and vertical positions)
and may be regarded as two-dimensional signals However, in this text, we will consider only one-dimensional signals with time as the independent variable
A system is a collection of devices that operate on input signal x(t) (or excitation) to produce an output signal y(t) (or response)
A system may also be regarded as a mathematical model of a physical process that relates the input signal to the output signal Examples of systems include electric circuits, computer programs, the stock market, weather, and the human body A system may have several mathematical models or representations The variables in the mathematical model are described as signals, which may be current, voltage, or displacement In electrical systems, signals are often represented as currents and voltages In mechanical systems, they are often represented as temperatures, forces, and velocities In hydraulic systems, signals may be displacements and pressures
Figure 11 illustrates the block diagram of a single-input single-output system We classify the signals that enter the system as input signals, while the signals produced by the system as outputs For example, we may regard voltages and currents as functions of time in an electric circuit as signals, while the circuit itself is regarded as a system In engineering systems, signals may carry energy or information
There are many ways of classifying signals: continuous-time or discrete-time, periodic or nonperiodic, energy or power, analog or digital, random or nonrandom, real or complex, etc
A signal x(t) that is defined at all instants of time is known as a continuous-time signal
A continuous-time signal takes a value at every instant of time t
An example of a continuous-time signal x(t) is shown in Figure 12a
A discrete-time signal is defined only at particular instants of time
A discrete-time signal is usually identified as a sequence of numbers, denoted by x[n], where n is an integer It may represent a phenomenon for which the
independent variable n is inherently discrete An example of discrete-time signal x[n] is shown in Figure 12b
Since time is naturally continuous, most physical systems are continuous-time systems Discrete-time signals are often obtained from continuous-time signals through sampling
As typically shown in Figure 13, the continuous-time signal x(t) in Figure 13a is sampled uniformly with sampling period T to produce the discrete-time signal x[n] in Figure 13b We simplify notation by letting x(kT)≜x[k] A discrete-time signal
is equally spaced in time with sampling period T Thus, discrete-time signals are samples of continuous-time signals, or they may exist naturally
A periodic continuous-time signal satisfies
x t x t nT( )= +( ) (11)
where n is an integer T is the period of the signal
A periodic signal is one that repeats itself every T seconds
A popular example of a periodic signal is the sinusoid
x t A t t( ) sin( ),= + - ¥ < < ¥w q (12) where
A is the amplitude of the signal ω( = 2πf = 2π/T) is the angular frequency in radians per second θ is the phase in radians
Another example of a periodic (nonsinusoidal) continuous-time signal is shown in Figure 14a Any signal that does not satisfy the periodicity condition in Equation 11
is called a nonperiodic signal We will be dealing with nonperiodic signals (step functions, ramp functions, rectangular functions, etc) later
A discrete-time signal x[n] is periodic with period N if it satisfies
x n x n Néë ùû = +éë ùû (13)
The discrete-time sine and cosine signals are given in terms of complex exponential signals as
x n n e ej n j n[ ] cos( ) ( )= = + -w w w1
2 (14)
y n n j e e
j n j n[ ] sin( ) ( )= = - -w w w1 2
(15)
Another example of a periodic discrete-time signal is shown in Figure 14b
If a continuous-time signal x(t) can assume any value in the range −∞.< t < ∞, then it is called an analog signal Although all analog signals are continuous-time signals, not all continuous-time signals are analog signals If a discrete-time signal assumes only finite values, then it is called a digital signal
An analog signal is a continuous-time signal in which the variation with time is analogous (or proportional) to some physical phenomena
A digital signal is a discrete-time signal that can have a finite number of values (usually binary)
A digital signal can assume only a finite number of values The difference between analog and digital signals is that analog is a continuous electrical signal, whereas digital is a discrete electrical signal We live in an analog world and most signals are analog Although some signals are inherently digital, most digital signals are obtained from analog signals by sampling or an analog-to-digital converter (ADC)
For example, an analog signal is taken straight from the microphone and recorded into a tape in its original form The signal from the microphone is an analog signal, and therefore the signal on the tape is analog as well Since data is sent using variable currents in an analog system, it is very difficult to remove noise and signal distortions during the transmission For this reason, analog signals cannot perform highquality data transmission On the other hand, digital signals use binary data strings (0 and 1) to reproduce data being transmitted
For continuous-time signal x(t), the normalized energy E of x(t) (assuming x(t) is real) is
E x t dt= -¥
ò ( )2 (16)
If x(t) is complex valued, Equation 16 can be generalized:
E x t dt= -¥
ò | ( ) |2 (17)
where |x(t)| is the magnitude of x(t) The normalized power P for real x(t) is
P T
x t dt T
= ®¥
- òlim ( )
(18)
This can be generalized for complex value x(t) as
P T
x t dt T
= ®¥
- òlim | ( ) |
(19)
Similarly, for a discrete-time signal x[n], the normalized energy E of x[n] is
E x n n
= =-¥
å | [ ] |2 (110)
while the normalized power P is
P N
x n N
= +®¥
=- ålim | [ ] |12 1 2 (111)
Based on the definitions of E and P in Equations 16 through 111, we define the following:
A signal x(t) or x[n] is an energy signal if and only if 0 < E < ∞ and consequently P = 0
A signal x(t) or x[n] is a power signal if and only if 0 < P < ∞ and consequently E = ∞
If a signal is a power signal, then it cannot be an energy signal or vice versa; power and energy signals are mutually exclusive A signal may be neither a power nor an energy signal if the conditions in Equations 16 through 111 are not met Almost all periodic functions of practical interest are power signals
By definition, a signal is even if
x t x t( ) = -( ) (112)
A function is even if its plot is symmetrical about the vertical axis; that is, the signal for t < 0 is the mirror image of the signal for t > 0 Examples of even signals are cos t, t2, and t4
By definition, a signal is odd if
x t x t( ) = - -( ) (113)
The plot of an odd function is antisymmetrical about the vertical axis Examples of odd functions are t, t3, and sin t
An even signal x(t) is one for which x(t) = x(−t) and an odd signal y(t) is one for which y(t)= −y(−t)
Any signal x(t) can be represented as the sum of even and odd signals as
x t x t x te o( ) ( ) ( )= + (114)
where xe(t) is the even part xo(t) is the odd part
Replacing t with −t in Equation 114 and incorporating Equations 112 and 113, we get
x t x t x t x t x te o e o( ) ( ) ( ) ( ) ( )- = - + - = - (115)
Adding Equations 114 and 115 and dividing by 2,
x t x t x te( ) ( ) ( )= + -éë ùû
1 2
(116)
Subtracting Equation 115 from Equation 114 and dividing by 2,
x t x t x to( ) ( ) ( )= - -éë ùû
1 2
(117)
Thus,
x t x t x t x t x t
( ) ( ) ( ) ( ) ( )= + - + - - 2 2
(118)
Equation 116 shows that the two signals are added and scaled in magnitude to produce the even signal xe(t), while Equation 117 indicates that x(−t) is subtracted from x(t) and the result is amplitude-scaled by 05 to produce the odd signal xo(t) For a discrete-time signal x[n], we can construct the even and odd parts using Equations 116 and 117
Note the following properties of even and odd functions:
1 The product of two even functions is also an even function 2 The product of two odd functions is an even function 3 The product of an even function and an odd function is an odd function 4 The sum (or difference) of two even functions is also an even function 5 The sum (or difference) of two odd functions is an odd function 6 The sum (or difference) of an even function and an odd function is neither
even nor odd
Each of these properties can be proved using Equations 112 and 113
Example 1.1
Example 1.2
Example 1.3
We now consider some simple, standard continuous-time signals These include the unit step function u(t), the unit ramp function r(t), and the impulse function δ(t) These three functions are called singularity functions A singularity function is one that is discontinuous or has discontinuous derivatives In addition to the singularity functions, we will also consider the unit rectangular function Π(t), the unit triangular function Λ(t), the sinusoidal function, and the exponential function
The unit step function u(t), also known as Heaviside unit function, is defined as
u t t
t ( ) ,
,
= > <
ì í î
1 0 0 0
(119)
The unit step function u(t) is 0 for negative values of t and 1 for positive values of t
The unit step function is shown in Figure 19 We should note that u(t) is discontinuous at t = 0; it is undefined at t = 0
We use the step function to represent an abrupt change, like the changes that occur in the circuits of control systems and digital computers It is used in representing a signal that is zero up to some time and finite thereafter For example, the signal
x t
t t
t ( ) cos ,
,
= ³ <
ì í î
w 0 0 0
(120)
can be written concisely as
x t u t t( ) = ( )cosw (121)
As another example, when we turn the key in the ignition system of a car, we are actually introducing a step voltage, the battery voltage Given a continuous-time signal x(t), the product x(t)u(t) is given by
x t u t
x t t
t ( ) ( ) ( ),
,
= > <
ì í î
0 0 0
(122)
This is illustrated in Figure 110 From this, we conclude that the multiplication of a signal x(t) with the unit step function u(t) eliminates the values of x(t) for t < 0
The derivative of the unit step function u(t) is the unit impulse function δ(t), which we write as
d( ) ( ) , ,
t d dt
u t t
t = =
¹ =
ì í î
0 0 0undefined
(123)
The unit impulse function, also known as the delta function, is shown in Figure 111
The unit impulse function δ(t) is zero everywhere, except at t = 0, where it is undefined
The unit impulse may be regarded as an applied or resulting shock It may be visualized as a very short duration pulse of unit area This may be expressed mathematically as
d( )t dt = -
(124)
where t = 0− denotes the time just before t = 0 and t = 0+ is the time just after t = 0 For this reason, it is customary to write 1 (denoting unit area) beside the arrow that is used to symbolize the unit impulse function, as shown in Figure 112 For example, an impulse function 10δ(t) has an area of 10 Figure 112 shows the impulse functions 5δ(t + 2), 10δ(t), and −4δ(t − 3)
To better understand the delta function, consider the function shown in Figure 113a The function g(t) becomes a unit step as Δ→0
u t g t( ) lim ( )=
D®0 (125)
The derivative g′(t) of g(t) is shown in Figure 113b, where we notice that as Δ→0, g′(t) becomes the unit impulse function
d( ) lim ( )t g t= ¢
D®0 (126)
Thus, we can consider the delta function δ(t) as a large spike or impulse at the origin To illustrate how the impulse function affects other functions, let us evaluate the
integral
f t t t dto a
( ) ( )d -ò (127)
where a < to < b Since δ(t−to) = 0 except at t = to, the integrand is zero except at to Thus,
f t t t dt f t t t dt f t t t dt f to a
( ) ( ) ( ) ( ) ( ) ( ) ( )d d d-= - = - =ò ò ò
or
f t t t dt f to a
o( ) ( ) ( )d - =ò (128)
This shows that when a function is integrated with the impulse function, we obtain the value of the function at the point where the impulse occurs This is a useful property of the impulse function known as the sampling or sifting property The special case of Equation 128 is for to = 0 Then, Equation 128 becomes
f t t dt f( ) ( ) ( )d 0
ò = (129)
If the integration interval does not include the impulse function, then the integration in Equation 123 is zero For example,
d d( ) , ( )t dt t dt= = -¥
(130)
This and other properties of the impulse functions are presented in Table 11
Integrating the unit step function u(t) results in the unit ramp function r(t); we write
r t u d tu t t
( ) ( ) ( )= = -¥ ò l l (131)
or
r t t
t t ( ) ,
,
= < ³
ì í î
0 0 0 (132)
The unit ramp function is zero for negative values of t and has a unit slope for positive values of t
Figure 114 shows the unit ramp function In general, a ramp is a function that changes at a constant rate
We should keep in mind that the three singularity functions (step, impulse, and ramp) are related by differentiation as
d( ) ( ) , ( ) ( )t du t
dt u t
dr t dt
= = (133)
or by integration as
u t d r t u d t t
( ) ( ) , ( ) ( )= = -¥ -¥ ò òd l l l l (134)
The unit rectangular pulse function is defined as
P t
t t
t t t tæ
è ç
ö ø ÷ =
<ì í î
= - < <ì1 2
0 1 2 2 0
, | | / ,
, / / ,otherwise otherwiseíî
(135)
It is centered at the origin and has unit height and width τ Figure 115 shows the rectangular pulse function It is evident from the figure that we can express the rectangular pulse function in terms of the unit step function as
P t u t u t
t t tæ
è ç
ö ø ÷ = +
æ è ç
ö ø ÷ - -
æ è ç
ö ø ÷2 2
(136)
The rectangular function results from an on-off switching operation of a source It is used in extracting part of a signal
The unit triangular function is defined as
L t t
t
t t
t t
t t t
t t
t æ è ç
ö ø ÷ =
- - < <ì í ï
îï =
+ - < <
- < 1
1 0
1 0 | |
,
,
,
,
otherwise t <
ì
í
ï ï ï
î
ï ï ï
t
0, otherwise
(137)
This means that the unit triangular pulse is centered at the origin, has unit height, and width 2τ The triangular function is shown in Figure 116
Perhaps the most important and useful of all the standard signals is the sine wave or sinusoidal signal It can take the form of
x t t x t t( ) = ( ) =sin , cosw w (138)
or by combining both to form a complex exponential signal
x t e t j t j t( ) cos sin= = +w w w (139)
where ω is the frequency of the sinusoid We already came across sinusoids in Equation 12
The continuous-time exponential function is defined as
x t Ae at( ) = (140)
where A and a are constants, which may be complex in general If we assume that A and a are real, the exponential function is shown in Figure 117 for both positive and negative values of a The signal x(t) in Equation 139 is a complex exponential signal,
which will play a major role in our treatment of signals and systems One reason that causes the signal to arise in many applications is its rate of change
d dt
x t aAe ax tat( ) ( )= = (141)
This means that the derivative of x(t) at any time is proportional to its value at that time
Example 1.4
Example 1.5
+
Example 1.6
Here we consider some simple, standard discrete-time signals
Like the continuous-time unit step, we define the unit step sequence u[n] as
u n
n
n [ ] ,
,
= < ³
ì í î
0 0 1 0
(142)
As shown in Figure 124, u[n] is a sequence of 1s starting at the origin Notice that u[n] is defined at n = 0 unlike u(t), which is not defined at t = 0
In discrete time, we define unit impulse sequence as
d[ ] ,
,
n n
n =
¹ =
ì í î
0 0 1 0
(143)
This is illustrated in Figure 125 Notice that we do not have difficulties in defining δ[n] unlike δ(t) Some properties of the unit impulse sequence are listed in Table 12
The unit ramp sequence is defined as
r n
n n
n [ ] ,
,
= ³ <
ì í î
0 0 0
(144)
The sequence is shown in Figure 126 The relationships between unit impulse, unit step, and unit ramp sequences are
d[ ] [ ] [ ] ,
,
n u n u n n
n = - - =
= ¹
ì í î
1 1 0 0 0
(145)
u n m
[ ] [ ]= =-¥ å d (146)
u n r n r n[ ] [ ] [ ]= + -1 (147)
r n u m
[ ] [ ]= =-¥
(148)
The sinusoidal sequence or a discrete-time sinusoid is given by
x n A n
N Ae j n N[ ] cos Re /= +æ
è ç
ö ø ÷ = éë
ù û
where A is a positive real number and is the amplitude of the sequence N is the period θ is the phase n is an integer
A typical sinusoidal sequence for A = 1, N = 12, and θ = 0 is shown in Figure 127
If we sample a continuous-time exponential function x(t) = Ae−at with sampling period T, we obtain the sequence x[n] = Ae−anT = Aαn, with α = e−aT Thus, the exponential sequence is given by
x n A n[ ] = a (150)
where A and α are generally complex numbers n is an integer
A typical discrete-time exponential sequence is shown in Figure 128 For the signal shown in Figure 128, both A and α are real numbers
Example 1.7
Example 1.8
We have limited ourselves to considering signals with one independent variable time (t) or integer [n] We consider six basic operations or transformations on real function The first three operations have to do with time, while the second three deal with transformations in amplitude The combinations of these transformations make it possible to obtain complex signals from the basic, standard signals Signals that are produced by these transformations are useful in sonar, radar, signal processing, and communication systems
Given a signal x(t), its time reversal is x(−t) The reversed signal x(−t) is obtained as a reflection of x(t) about the t = 0, that is, we perform a reflection about the vertical axis (Reflection about the horizontal axis results in −x(t)) An example of this is shown in Figure 131a In the figure, we get
x t
t
t t( ) ,
,
,
= - - < < - < <
ì
í ï
î ï
1 1 0 1 0 2
0 otherwise (151)
From this, we obtain x(−t) by replacing every t with −t in Equation 151 Hence,
x t
t
t t
t
t( ) ,
( ), ,
,
,- = - - < - <
- - < - < ì
í ï
î ï
= - < < +
1 1 0 1 0 2
1 0 1 1
otherwise - < <
ì
í ï
î ï
2 0 0
t
, otherwise (152)
as evident from Figure 131b
Time scaling involves the compression or expansion of a signal in time Given a continuous-time signal x(t), the time-scaled form of x(t) is x(at), where a is a constant The scaled signal x(at) will be compressed if |a| > 1 or expanded if |a| < 1; a negative value of a yields time reversal as well as compression or expansion Again, we can obtain x(at) from the signal in Equation 151 by replacing every t with at and simplifying We show an example in Figure 132 Notice that the time reversal discussed in Section 161 can be considered a special case of time scaling with the scaling factor a = −1
Given a continuous-time signal x(t), the time-shift form of x(t) is x(t − to), where to is a constant If to > 0, then the signal x(t − to) is delayed and the signal is shifted to the right relative to t = 0 When to < 0, the signal x(t − to) is an advanced replica of x(t), with x(t) shifted to the left Examples of x(t − 2) and x(t + 1) are shown in Figure 133
For any signal x(t), the transformation at +b on the independent variable can be performed as follows:
x at b x a t b
a +( )= +æ
è ç
ö ø ÷
æ
è ç
ö
ø ÷ (153)
where a and b are constants This involves two steps:
1 Scale by factor a If a is negative, reflect x(t) about the vertical axis 2 If b/a is negative, shift x(t) to the right If b/a is positive, shift x(t) to the left
The previous three transformations deal with the independent variable, t Equivalent transformations of the amplitude of the signal will now be discussed Given a signal x(t), amplitude transformations take the general form
y t Ax t B( ) = ( ) + (154)
where A and B are constants For example, let
y t x t( ) = - ( ) +2 4 (155)
We notice that A = −2 means amplitude reversal (−x(t) implies reflection about the horizontal axis) and amplitude scaling (|A| = 2) Also, B = 4 shifts vertically the amplitude of the signal
Example 1.9
Example 1.10
As mentioned in Section 12, a system may be regarded as a mathematical model of a physical process that relates the input signal x(t) to the output signal y(t) This relationship is illustrated in Figure 140a Although a system may have many input and output signals as shown in Figure 140b, we will focus our attention on the singleinput single-output case in this book
If x is the input signal and y is the output signal, they are related through a transformation
y x= T (156)
where T is an operator transforming x into y
A system is continuous-time if the input and output signals are continuous-time It is discrete-time if the input and output signals are discrete-time In continuoustime systems, time is measured continuously Some continuous-time systems are described by ordinary differential equations, algebraic equations (resistive circuits), polynomial equations (diodes), integral equations (op amp integrator circuits), etc For discrete-time systems, time is defined only at discrete instants and the systems are described by difference equation and any other way the input-output properties of the system may be specified Continuous-time and discrete-time systems are illustrated in Figure 141
The output of a system may depend on the present and past inputs A system that has this property is known as causal system A causal (or nonanticipatory) system is one whose output y(t) at the present time depends only on the present and past values of the input x(t)
A causal system is one whose present response does not depend on the future values of the input
Examples of causal systems are
y t x t( ) = -( )1 (157)
y n x néë ùû =
é ëê
ù ûú2
(158)
Every physical system is causal The motion of a car is causal since it does not expect the future actions of the driver Causality is a necessary condition for a system to be realized in the real world we live in Causal systems are physically realizable
If a system is not causal, it is said to be noncausal or anticipatory Typical examples of noncausal systems are
y t x t( ) = +( )2 (159)
y n x néë ùû = -éë ùû1 (160)
The system in Equation 159 is not causal because the output at, say t = 0, is equal to input at t =2 This is not physically realizable The same logic applies to the system in Equation 160
An ideal filter is noncausal and is not physically realizable; it cannot be built in practice It should be noted that causality is not often an essential constraint in applications in which the independent variable is not time, such as in image processing
Linearity is the property of the system describing a linear relationship between input (cause) and output (effect) The property is a combination of both homogeneity ( scaling) property and the additivity property
The homogeneity property requires that if the input is multiplied by any constant k, then the output is multiplied by the same constant, that is,
T kx ky{ } = (161)
The additivity property requires that the response to a sum of inputs is the sum of the responses to each input applied separately If Tx1 = y1 and Tx2 = y2, then
T x x y y1 +{ } = +2 1 2 (162)
We can combine Equations 161 and 162 as
T k x k x k y k y1 1 2 2 1 2+{ } = +1 2 (163)
where k1 and k2 are constants
A linear system meets the two conditions in Equations 161 and 163, that is, a system is linear if it satisfies homogeneity and additivity properties
A linear system may be regarded as one in which all the interrelationships among the quantities involved are expressed by linear equations, which may be algebraic, differential, or integral A very important consequence of linearity is that superposition principle applies If an input consists of the weighted sum of several signals, then the output is the weighted sum of the responses of the system to each of those signals
An example of a linear system is a circuit that contains resistors, capacitors, and inductors, since these are linear elements Devices such as rectifiers, diodes, and saturating magnetic devices are nonlinear A system having even one such device is treated as nonlinear Examples of nonlinear systems are
y x= sin (164)
y x= 3 (165)
When one or more parameters of the system vary with time, the system is said to be time-varying
A time-varying system is one whose parameters vary with time
For continuous-time, time-varying systems are described by time-varying differential equations For discrete-time, time-varying systems are described by timevarying difference equations If all system parameters are constant with time, the system is said to be time-invariant or fixed
In a time-invariant system, a time shift (advance or delay) in the input signal leads to the time shift in the output signal
For a continuous-time system, the system is time-invariant when
T x t y t{ ( )} ( )- = -t t (166)
where τ is a constant Consider the input x(t) of a linear system yielding an output y(t) For the system to be time-invariant, a shifted input x(t − τ) should result in a shifted output y(t − τ) Figure 142 shows an example of time-invariant system, while Figure 143 gives an example of time-varying system
For a discrete-time system, the system is time-invariant when
T x n m y n m-éë ùû{ } = -éë ùû (167)
for any integer m A continuous-time/discrete-time system that does not satisfy Equation 163 or 164 is time-varying Such a system has parameters that vary with time We will confine our efforts to linear time-invariant (LTI) systems in this book
When the output of a system depends on the past and/or future input, the system is said to have a memory For example, a system described by
y t x t x t x t( ) = ( ) -( ) + +( )– 1 2 (168)
has a memory since the output y(t) requires the past input x(t − 1), the current input x(t), and the future input x(t + 2)
If the output of a system does not depend on the past and/or future input, the system is memoryless (or instantaneous)
A memoryless system is one in which the current output depends only on the current input; it does not depend on the past or future inputs
A system with a memory is also called a dynamic system A memoryless system is called a static system. Many practical systems such as resistive networks, amplifiers, operational amplifiers, and diodes are usually modeled as memoryless systems
Example 1.11
Example 1.12
The information covered in this chapter is useful in many systems such as massspring-damper system, moving average filter, stock trading, electrical circuits, square-law device, and digital signal processing (DSP) We will briefly consider three simple applications related to what we have covered in this chapter These applications deal with electric circuits, square-law device, and DSP system
Electric circuits are mostly linear systems Consider the one shown Figure 144 Our objective is to find the system equation for it By applying Kirchhoff’s voltage law, we obtain
x t Ri t y t( ) = ( ) + ( ) (169)
But i t L
y d t
( ) ( )= -¥ò
1 l l Substituting this in Equation 169,
y t x t R L
y d t
( ) ( ) ( )= - -¥ ò l l (170)
Or we may differentiate this to get
dy t dt
R L
y t dx dt
( ) ( )+ = (171)
This is an explicit input-output relationship between x(t) (source voltage) and y(t) (inductor voltage)
A square-law device is a device where either current or voltage depends on the square of the other The input-output relationship is given by
y t x t( ) ( )= 2 (172)
Any system that is defined by Equation 172 is called a square-law device To realize this system, we use a signal amplifier, as shown in Figure 145 For this reason, the square-law detector is also called a signal multiplier. It can be built using operational amplifiers and diodes It is evident from Equation 172 that the system is nonlinear and memoryless A square-law device is often found in the receiver front end of radar and communication devices
Since most signals are analog in nature, an analog signal can be processed directly in its analog form, as shown in Figure 146 DSP provides an alternative
means of processing the analog signal, as shown in Figure 147 DSP techniques are replacing analog signal processing in many fields such as telecommunications; radar and sonar; digital control; speech, audio, and video processing; and biomedicine One main reason for preferring DSP is that, it allows programmability, which means that the same DSP hardware can be used for different applications
A DSP is a special microprocessor designed for DSP Most DSPs can be used to manipulate different kinds of information including sound, images, and video The goal of a DSP is to process real-world analog signals As illustrated in Figure 147, the first step is to convert the analog signal to digital signal using an ADC The ADC quantizes the sampled analog signal and codes the quantized signal into an acceptable format It is common for the sequence coming from the ADC to be converted back to analog to drive a motor, a stereo system, or whatever The conversion of the digital sequence to an analog signal requires a digital-to-analog converter (DAC) Most PCs include ADCs and DACs in their sound card In these modern days, DSP systems such as high-speed modems and cell phones are common
A brief introduction to MATLAB is given in Appendix C The reader is encouraged to read Appendix C before proceeding with this section
MATLAB provides built-in functions for unit step function u(t) and unit impulse function δ(t) The unit step function is called Heaviside or stepfun, while the impulse function is Dirac Heaviside(t) is zero when t < 0, 1 for t > 0 and 05 for t = 0 stepfun(t,t0) returns a vector of the same length at t with zeros for t < t0 and ones for t > t0 stepfun(n,n0) works the same way for discrete signals Dirac(t) is zero for all t, except t = 0, where it is infinite
MATLAB treats continuous-time signals differently from discrete-time signals We will treat them separately in the following two examples
Example 1.13
FIGURE 1.49 For Practice Problem 113
Example 1.14
1 A signal is a set of data or function that represents a variable of interest 2 A system is a device, process, or algorithm that operates on input signal to
yield an output signal 3 A signal may be classified in many ways: a A continuous-time signal x(t) has a value specified at all time, whereas a
discrete-time signal x[n] is specified only at a finite set of time instants b A periodic signal repeats itself after period T, that is, x(t + T) = x(t) An
aperiodic signal is one that is not periodic c An analog signal is one that can take on any value over a continuum;
a digital signal can only take a finite number of values d A signal x(t) or x[n] is an energy signal if and only if 0 < E < ∞ and con-
sequently P = 0, that is, it has finite energy A signal x(t) or x[n] is a power signal if and only if 0 < P < ∞ and consequently E = ∞; it has finite power
e A signal is even if x(t) = x(−t); it is odd if x(t) = −x(−t) 4 Examples of standard signals are the singularity functions, which include the
unit impulse δ(t) or δ[n], the unit step u(t) or u[n], and the unit ramp r(t) or r[n] The sifting property of the impulse function is
f t t t dt f to a
o( ) ( ) ( )d - =ò In addition to the singularity functions, we consider the unit rectangular func-
tion Π(t/τ), the unit triangular function Λ(t/τ), the sinusoidal function, and the exponential function
5 The following operations can be performed on a signal: a A signal x(−t) is the time reversal of x(t); it is obtained by reflecting x(t)
about t = 0 b The signal x(at) is compressed if a > 1 or expanded if a < 1 c A signal x(t − to) is delayed by to seconds, while the signal x(t + to) is
advanced by to seconds, assuming to > 0 d Amplitude transformations take the general form of y(t) = Ax(t) + B 6 A system may be classified in many ways: a A system is continuous-time if the input and output signals are
continuous-time; it is discrete-time if the input and output signals are discrete-time
b A system is causal if its present response does not depend on the future values of the input; it is noncausal otherwise
c A system is linear if it is characterized by the linearity property; it is nonlinear if the superposition does not hold
d A system is time-varying if its parameters change with time; it is timeinvariant if its parameters do not change with time
e A memoryless system is one in which the current output depends only on the current input; a system with memory is one whose output depends on the present and past or future inputs
7 Specific applications considered in this chapter are electric circuits, squarelaw device, and DSP system
8 MATLAB can be used to process continuous-time and discrete-time signals
1.1 Which of the following is not a signal? (a) Voltage, (b) blood, (c) torque, (d) pressure, (e) displacement 1.2 Any set of mathematical relationships between input and output variables con-
stitutes a system (a) True, (b) false 1.3 Which of the following is not a system? (a) Automobile speed, (b) a circuit, (c) a camera, (d) a computer program 1.4 The exponential signal is (a) A power signal, (b) an energy signal, (c) neither a power nor an energy signal 1.5 The signals δ(t), u(t), and r(t) are known as (a) Transformation functions, (b) expansion functions, (c) singularity func-
tions, (d) basic functions 1.6 Which of these is not true? (a) t2δ(t) = 0 (b) cos(t)δ(t) = δ(t) (c) sin(t)δ(t−π) = δ(t−π)
(d) ( ) ( )2 1 12t t dt+ = -¥
1.7 Given the signal x(t), which of the following is true for signal x(2t + 8)? (a) x(t) shifted to the left by eight units (b) x(t) is compressed by a factor of 2 and then shifted left by four units (c) x(t) is expanded by a factor of 2 and then shifted right by four units (d) x(t) is reflected about the vertical axis and then shifted 1.8 A system is described by y(t) =cos x(t) + 2 sin x(t − 1) It is (a) Causal, (b) noncausal, (c) memoryless, (d) with memory 1.9 Causal systems are also referred to as physically realizable systems (a) True, (b) false 1.10 The impulse function in MATLAB is (a) Heaviside, (b) stepfun, (c) Dirac, (d) stem
Answers: 11(b), 12(a), 13(a), 14(c), 15(c), 16(c), 17(b), 18(a,d), 19(a), 110(c)
1.1 Show that the complex sinusoidal signal x t Ae j to( ) = w is periodic 1.2 Find the energy content of
x t e t
t
( ) , ,
= ³ <
ì í ï
îï
-3 0 0 0
and determine whether it is a power or energy signal 1.3 Let
x t e t tt( ) sin ,
,
= ³ì
í ï
îï
-2 0 0 otherwise
calculate the energy content of the signal 1.4 Find out if the following signals are power or energy signals or neither: (a) x(t) = 10 sin 2πt (b)y t u t u t( ) = ( ) -( )éë ùû2 4-(c) z(t) = r(t)–r(t−2) (d) h(t) = 10e2t
1.5 Repeat Problem 14 for the following signals: (a) x(t) = e−3tu(t) (b) x(t) = (e−2t + 1)u(t) (c) x(t) = t−1/2u(t−2)
1.6 Express the signals in Figure 152 in terms of the unit step function 1.7 Express the signals in Figure 153 in terms of the unit step and unit ramp
functions
1.8 If x t
t
t
t t
t
( )
,
,
,
,
=
< < <
+ < < >
ì
í ï ï
î ï ï
0 0 8 0 2
2 6 2 6 0 6
Express x(t) in terms of singularity functions 1.9 (a) Express x(t) in Figure 154 in terms of the unit step function (b) Sketch the derivative of x(t)
1.10 Express x(t) and y(t) in Figure 155 in terms of the step function 1.11 Using Euler relation, prove the following trig identities: (a) cos2t = ½(1 + cos 2t) (b) sin(x + y) = sin x cos y + cos x sin y (c) sin x sin y = ½ cos(x−y)−½ cos(x + y) 1.12 Evaluate the following:
(a) e t dtj t-
-ò w d( )3 (b)exp( ) ( )t t t dt2 2 1 2-+ -
(c)5 2d d p d( ) ( ) cos ( )t e t t t dtt+ +éë ùû -
(a)u d t
( )l l 1 ò
(b)r t dt( )-ò 1 0
(c)( ) ( )t t dt--ò 6 22 0
d
(d) 2 43
t t dt - ò -d( )
1.14 Evaluate the following derivatives:
(a) d dt
u t u t( ) ( )- +éë ùû1 1
(b) d dt
r t u t( ) ( )- -éë ùû6 2
(c) d dt
tu tsin ( )4 3-éë ùû
1.15 Sketch the following discrete-time signals: (a) x[n] = 4(−09)n, n ≥ 0 (b) x[n] = 2(12)n, n ≥ 0 (c) x[k] = (08)k sin(2πk/8), k ≥ 0
1.16 Sketch x n nn[ ] ( . ) ,
,
= - £ ³ì
í ï
îï
10 0 6 3 3 0 otherwise
1.17 Sketch the following discrete-time signals:
(a) x n n[ ] ,
,
= - £ £ì
í î
1 3 3 0 otherwise
(b)x t
n
n
n
n
( )
,
,
,
,
,
=
= - =
- = =
ì
í
ï ïï
î
ï ï ï
1 1 2 0 1 2
2 3 0 otherwise
(c) x[n] = 2 sin(n/4)cos(nπ/8) 1.18 Sketch the following signals: (a) x[n] = u[n−2] (b) y[n] = u[n + 2] (c) z[n] = u[n]–u[n−4] 1.19 Sketch the following discrete-time signals: (a) x[n] = x[n] + u[−n] (b) x[n] = u[3n] (c) x[n] = 10 sin(2πn/20) 1.20 Represent the signal in Figure 156 as a sum of impulses
1.21 Let x(t) = 10e4−3t Evaluate and simplify each of the following: (a) x(2) (b) x(1 − t)
(c) x(t/4 +1) (d) ½[x( jt) + x(−jt)] 1.22 Given the signal x(t) shown in Figure 157, sketch (a) x(t/2) and (b) x(1 + t/2)
1.23 If x t t t( ) ,
,
= + < <ì
í î
1 0 4 0 otherwise
, sketch (a) x(t) and (b) y(t) = 1 + x(2t)
1.24 If x(t) is the signal shown in Figure 158, sketch (a) x(t − 2), (b) x(3t), and (c) y(t) = 1 + 2x(t)
1.25 Sketch each of the following continuous-time signals and their derivatives: (a) x(t) = u(t)–u(t−1) and dx/dt (b) y(t) = u(t + 1)–2u(t) + u(t−1) and dy/dt (c) z t t u t u t( ) = +( ) -( )éë ùû1 1-and dz/dt 1.26 Given that x(t) = u(t)–u(t−1), sketch (a) y(t) = x(−t) (b) y(t) = x(3−t) (c) y(t) = x(t−3) 1.27 If x(t) = Π(t + 2) + Π(t−2), sketch (a) y(t) = x(t−2) (b) y(t) = x(t + 1)–x(t−1) 1.28 Sketch these signals:
(a) u t -( )éë ùû1 2/ (b) r[4−2t] (c) Π(−2t + 1) (d)L t -( )éë ùû1 3/
1.29 Given x(t) in Figure 159, sketch (a) y(t) = − x(t−1) (b) z(t) = 4x(t/2) (c) h(t) = x(2−t) 1.30 Given the two signals x1(t) and x2(t) in Figure 160, sketch the following: (a) y(t) = x1(t) + x2(t) (b) z(t) = 05x1(2t) + x2(t) (c) h(t) = x1(t−1)−x2(1−t) 1.31 Given the discrete-time signal in Figure 161, sketch the following signals: (a) y[n] = x[n−3] (b) z[n] = x[n]–x[n−1] 1.32 Consider the discrete-time signal in Figure 162 Sketch the following signals: (a) x[n]u[2−n] (b) x n u n u n[ ] +[ ] - [ ][ ]1 (c) x[n]δ[n−2]
1.33 Show that the system described by the differential equation
dy t dt
ty t x t( ) ( ) ( )+ =2
is linear
1.34 Show that the system described by the differential equation
dy t dt
y t x t( ) ( ) ( )+ + =4 1
is nonlinear 1.35 Which of the following systems can be classified as linear/nonlinear,
time-varying/time-invariant?
