22 Signals and Linear Systems Chapter 2 128 256 384 512 640 Figure 2.1 A sample speech Time,t(msec) waveform d.x(t-to) Figure 2.2 Time shifting of a signal the result is a shifted version ofx(t)by toto the right.If to is negative,then the result is a shift to the left by an amount equal to ltol.A plot of a signal shift for positive to is shown in Figure 2.2. Time Reversal.Time reversal,or flipping,of a signal results in flipping the signal around the vertical axis,or creating the mirror image of the plot with respect to the vertical axis.We can visualize this flipping of a signal as playing an audio tape in reverse.As a result,positive times are mapped as negative times and vice versa.In mathematical terms, time reversal of x(t)results in x(-t).Figure 2.3 shows this operation. Time Scaling.Time scaling of a signal results in a change in the time unit against which the signal is plotted.Time scaling results in either an expanded version of the signal (if the new time unit is a fraction of the original time unit)or a contracted version of the original signal(if the new time unit is a multiple of the original time unit).In general,time scaling is expressed as x(at)for some a 0.If a 1,then the result is an expanded
22 0 128 256 384 Time, t (msec) x(t) Figure 2.2 Time shifting of a signal. 512 Signals and Linear Systems Chapter 2 x(t to) to 640 Figure 2.1 A sample speech waveform. the result is a shifted version of x(t) by to to the right. If to is negative, then the result is a shift to the left by an amount equal to I to I. A plot of a signal shift for positive to is shown in Figure 2.2. Time Reversal. Time reversal, or flipping, of a signal results in flipping the signal around the vertical axis, or creating the mirror image of the plot with respect to the vertical axis. We can visualize this flipping of a signal as playing an audio tape in reverse. As a result, positive times are mapped as negative times and vice versa. In mathematical terms, time reversal of x (t) results in x(-t). Figure 2.3 shows this operation. Time Scaling. Time scaling of a signal results in a change in the time unit against which the signal is plotted. Time scaling results in either an expanded version of the signal (if the new time unit is a fraction of the original time unit) or a contracted version of the original signal (if the new time unit is a multiple of the original time unit). In general, time scaling is expressed as x(a t) for some a > 0. If a < 1, then the result is an expanded
Section 2.1 Basic Concepts 23 x() Figure 2.3 Time reversal of a signal. (at Figure 2.4 Time scaling of a signaL version of the original signal(such as a tape which is played at a slower speed than it was recorded).If a 1,theresult is a contracted form of the original signal (such as a tape that is played at a higher speed than it was recorded).The case of a >1 is shown in Figure 2.4. In general,we may have a combination of these operations.For instance,x(-2t)is a combination of flipping the signal and then contracting it by a factor of 2.Also,x(2t-3) is equal to x[2(t-1.5)],which is equivalent to contracting the signal by a factor of 2 and then shifting it to the right by 1.5. 2.1.2 Classification of Signals The classification of signals makes their study easier.Depending on the point of view, signals can be classified in a variety of ways.In this section,we present the most important ways to classify signals. Continuous-Time and Discrete-Time Signals.Based on the range of the inde- pendent variable,signals can be divided into two classes:continuous-time signals and discrete-time signals.A continuous-time signal is a signal x(t)for which the independent variable t takes real numbers.A discrete-time signal,denoted by x[n],is a signal for which the independent variable n takes its values in the set of integers. By sampling a continuous-time signal x(t)at time instants separated by To,we can define the discrete-time signal x[n]=x(nTo).Figure 2.5 shows examples of discrete-time and continuous-time signals. Example 2.1.1 Let x(t)=Acos(2πf6t+0) This is an example of a continuous-time signal called a sinusoidal signal.A sketch of this signal is given in Figure 2.6
Section 2.1 Basic Concepts x (t) x(t) x (-t) x(at) 23 Figure 2.3 Time reversal of a signal. Figure 2.4 Time scaling of a signal. version of the original signal (such as a tape which is played at a slower speed than it was recorded). If a > 1, the result is a contracted form of the original signal (such as a tape that is played at a higher speed than it was recorded). The case of a > 1 is shown in Figure 2.4. In general, we may have a combination of these operations. For instance, x(-2t) is a combination of pipping the signal and then contracting it by a factor of 2. Also, x (2t - 3) is equal to x [2(t - 1 .5) ], which is equivalent to contracting the signal by a factor of 2 and then shifting it to the rigqt by 1 .5. 2. 1 .2 Classification of Signals The classification of signals makes their study easier. Depending on the point of view, signals can be classified in a variety of ways. In this section, we present the most important ways to classify signals. Continuous-Time and Discrete-Time Signals. Based on the range of the independent variable, signals can be divided into two classes: continuous-time signals and discrete-time signals. A continuous-time signal is a signal x(t) for which the independent variable t takes real numbers. A discrete-time signal, denoted by x [n], is a signal for which the independent variable n takes its values in the set of integers. By sampling a continuous-time signal x(t) at time instants separated by To, we can define the discrete-time signal x [n] = x (nT0). Figure 2.5 shows examples of discrete-time and continuous-time signals. Example 2.1.1 Let x(t) =A cos(2nfot + 8). This is an example of a continuous-time signal called a sinusoidal signal. A sketch of this signal is given in Figure 2.6. •
24 Signals and Linear Systems Chapter 2 12 Figure 2.5 Examples of discrete-time and continuous-time signals. Figure 2.6 Sinusoidal signal. Example 2.1.2 Let x[n]=Acos(2fon+), where n(Z is the set of integers).A sketch of this discrete-time signal is given in Figure 2.7. ■ Real and Complex Signals.Signals are functions,and functions at a given value of their independent variable are just numbers,which can be either real or complex.A real signal takes its values in the set of real numbers,i.e.,x(t)ER.A complex signal takes its values in the set of complex numbers,i.e.,x(t)E C. In communications,complex signals are usually used to model signals that convey amplitude and phase information.Like complex numbers,a complex signal can be rep- resented by two real signals.These two real signals can be either the real and imaginary parts or the absolute value (or modulus or magnitude)and phase.A graph of a complex signal can be given by graphs in either of these representations.However,the magnitude and phase graphs are more widely used
24 -4 x(t) 3 2 x[n] -3 -2 -1 x(t) 2 3 1 &----1'----+�-l-�-+-�-+-�+-���..- -1 -2 -3 Example 2.1.2 Let ll Signals and Linear Systems Chapter 2 Figure 2.5 Examples of discrete-time and continuous-time signals. Figure 2.6 Sinusoidal signal. x [n] =A cos(2n:fon + e), where n E Z (Z is the set of integers). A sketch of this discrete-time signal is given in Figure 2.7. • Real and Complex Signals. Signals are functions, and functions at a given value of their independent variable are just numbers, which can be either real or complex. A real signal takes its values in the set of real numbers, i.e., x(t) ER A complex signal takes its values in the set of complex numbers, i.e., x (t) E C. In communications, complex signals are usually used to model signals that convey amplitude and phase information. Like complex numbers, a complex signal can be represented by two real signals. These two real signals can be either the real and imaginary parts or the absolute value (or modulus or magnitude) and phase. A graph of a complex signal can be given by graphs in either of these representations. However, the magnitude and phase graphs are more widely used
Section 2.1 Basic Concepts 25 2.5 1.5 0.5 0 -0.5 -1 -1.5 -2 Figure 2.7 Discrete-time sinusoidal -2.5 signal. Example 2.1.3 The signal x(t)=Aei(2nf+0) is a complex signal.Its real part is x(t)=Acos(2πfot+8) and its imaginary part is x(t)=Asin(2πft+0), where we have used Euler's relation e=cos+jsin.We could equivalently describe this signal in terms of its modulus and phase.The absolute value of x(r)is Ix川=Vx好)+x)=IA, and its phase is Lx(t)=2rf1+0. Graphs of these functions are given in Figure 2.8. The real and complex components,as well as the modulus and phase of any complex signal,are represented by the following relations: x(t)=x(t)cos(Zx(t)), (2.1.1) xi(t)=x(t)I sin (Zx(t)), (2.1.2) lx(川=Vx(0+x), (2.1.3) x:() Lx(t)arctan (2.1.4) x(t)
Section 2.1 x[n] 2.5 2 1.5 0.5 Basic Concepts 25 Ot---'-rrrnrrr.><LLLJu..ui<TTrmT'-"u..uu..i..rrrnrrrr:'-LLlLLL1........,,----+- -0.5 -1 - 1.5 -2 -2.5 Figure 2.7 Discrete-time sinusoidal signal. Example 2.1.3 The signal x (t) = Aej(2nfor+o) is a complex signal. Its real part is x,(t) = A cos(2nfot + 8) and its imaginary part is xi(t) = A sin(2nf0t + 8), where we have used Euler's relation eN = cos ¢ + j sin ¢. We could equivalently describe this signal in terms of its modulus and phase. The absolute value of x (t) is and its phase is Lx(t) = 2nfot + 8. Graphs of these functions are given in Figure 2.8. • The real and complex components, as well as the modulus and phase of any complex signal, are represented by the following relations: x,(t) = Jx(t)J cos (Lx(t)) , Xi(t) = Jx(t)J sin (Lx(t)) , Jx(t)J = )x'f(t) + xi 2 (t), Xj(t) Lx(t) = arctan --. x,(t) (2.1.l) (2. 1.2) (2. 1 .3) (2. 1.4)
26 Signals and Linear Systems Chapter 2 品研 Figure 2.8 Real-imaginary and magnitude-phase graphs of the complex -4-20 exponential signal in Example 2.1.3. Deterministic and Random Signals.In a deterministic signal at any time instant t,the value of x(t)is given as a real or a complex number.In a random (or stochastic)sig- nal at any given time instant t,x(t)is a random variable;i.e.,it is defined by a probability density function. All of our previous examples were deterministic signals.Random signals are dis- cussed in Chapter 5. Periodic and Nonperiodic Signals.A periodic signal repeats in time;hence,it is sufficient to specify the signal in a basic interval called the period.More formally,a periodic signal is a signal x(t)that satisfies the property x(t+To)=x(t) for all t,and some positive real number To (called the period of the signal).For discrete- time periodic signals,we have x[n No]=x[n] for all integers n,and a positive integer No(called the period).A signal that does not satisfy the conditions of periodicity is called nonperiodic. Example 2.1.4 The signals x(t)=Acos(2πfot+0) and x(t)=Ae(2+) are examples of real and complex periodic signals.The period of both signals isToThe signal u-r( t之0 (2.1.5) t<0
26 lx(t)I IAI -4 -2 0 2 x/t) Lx(t) 4 t Signals and Linear Systems Chapter 2 Figure 2.8 Real-imaginary and magnitude-phase graphs of the complex exponential signal in Example 2.1.3. Deterministic and Random Signals. In a deterministic signal at any time instant t, the value of x (t) is given as a real or a complex number. In a random (or stochastic) signal at any given time instant t, x (t) is a random variable; i.e., it is defined by a probability density function. All of our previous examples were deterministic signals. Random signals are discussed in Chapter 5. Periodic and Nonperiodic Signals. A periodic signal repeats in time; hence, it is sufficient to specify the signal in a basic interval called the period. More formally, a periodic signal is a signal x (t) that satisfies the property x(t + To) = x(t) for all t, and some positive real number To (called the period of the signal). For discretetime periodic signals, we have x [n + No] = x [n] for all integers n, and a positive integer No (called the period). A signal that does not satisfy the conditions of periodicity is called non periodic. Example 2.1.4 The signals x (t) = A cos(2nfot + 8) and x(t) = Ae j(2rcfot+e) are examples of real and complex periodic signals. The period of both signals is To = To. The signal t 2: 0 t < 0 (2.1 .5)