1 Introduction CHAPTER 1.6 Elementary Signals ★ Exponential Signals (1.31) B and a are real parameters 1.Decaying exponential, for which a < 0 2.Growing exponential, for which a > 0 Figure 1.28 (p. 34) (a) Decaying exponential form of continuous-time signal. (b) Growing exponential form of continuous-time signal.

2 Introduction CHAPTER Fig Ex. Lossy capacitor: Fig Figure 1.29 (p. 35) Lossy capacitor, with the loss represented by shunt resistance R. KVL Eq.: (1.32) (1.33) RC = Time constant Discrete-time case: (1.34) where Fig. 1.30

3 Introduction CHAPTER Figure 1.30 (p. 35) (a) Decaying exponential form of discrete-time signal. (b) Growing exponential form of discrete-time signal.

4 ★ Sinusoidal Signals (1.35) where periodicity ◆ Continuous-time case: Fig Introduction CHAPTER

5 Introduction Figure 1.31 (p. 36) (a) Sinusoidal signal A cos(  t + Φ) with phase Φ = +  /6 radians. (b) Sinusoidal signal A sin (  t + Φ) with phase Φ = +  /6 radians.

6 Introduction CHAPTER Fig Ex. Generation of a sinusoidal signal  Fig Figure 1.32 (p. 37) Parallel LC circuit, assuming that the inductor L and capacitor C are both ideal. Circuit Eq.: (1.36) (1.37) where (1.38) Natural angular frequency of oscillation of the circuit ◆ Discrete-time case : (1.40) (1.39) Periodic condition: or Ex. A discrete-time sinusoidal signal: A = 1,  = 0, and N = 12. Fig (1.41)

7 Introduction CHAPTER Figure 1.33 (p. 38) Discrete-time sinusoidal signal.

8 Introduction CHAPTER Example 1.7 Discrete-Time Sinusoidal Signal A pair of sinusoidal signals with a common angular frequency is defined by and (a)Both x 1 [n] and x 2 [n] are periodic. Find their common fundamental period. (b)Express the composite sinusoidal signal In the form y[n] = Acos(  n +  ), and evaluate the amplitude A and phase . <Sol.> (a) Angular frequency of both x 1 [n] and x 2 [n]: This can be only for m = 5, 10, 15, …, which results in N = 2, 4, 6, … (b) Trigonometric identity: x 1 [n] + x 2 [n] with the above equation to obtain that Let  = 5 , then compare

9 Introduction CHAPTER  =   / 6 Accordingly, we may express y[n] as

10 ★ Relation Between Sinusoidal and Complex Exponential Signals 1. Euler’s identity: (1.41) Complex exponential signal: (1.42) (1.43) (1.35) ◇ Continuous-time signal in terms of sine function: (1.44) (1.45)

11 Introduction CHAPTER (1.46)(1.47) and 3. Two-dimensional representation of the complex exponential e j  n for  =  /4 and n = 0, 1, 2, …, 7. Fig : Fig Projection on real axis: cos(  n); Projection on imaginary axis: sin(  n) Figure 1.34 (p. 41) Complex plane, showing eight points uniformly distributed on the unit circle. 2. Discrete-time case:

12 Introduction CHAPTER ★ Exponential Damped Sinusoidal Signals (1.48) Example for A = 60, Fig.1.35  = 6, and  = 0: Fig Figure 1.35 (p. 41) Exponentially damped sinusoidal signal Ae  a t sin(  t), with A = 60 and  = 6.

13 Introduction CHAPTER Ex. Generation of an exponential damped sinusoidal signal Fig  Fig Figure 1.36 (p. 42) Parallel LRC, circuit, with inductor L, capacitor C, and resistor R all assumed to be ideal. Circuit Eq.: (1.49) (1.50) where (1.51) Comparing Eq. (1.50) and (1.48), we have

14 ◆ Discrete-time case: (1.52) ★ Step Function ◆ Discrete-time case: (1.53) Figure 1.37 (p. 43) Discrete-time version of step function of unit amplitude. Fig x[n]x[n] n 11 22 33 1 ◆ Continuous-time case: (1.54) Figure 1.38 (p. 44) Continuous-time version of the unit-step function of unit amplitude.

15 Introduction CHAPTER Example 1.8 Rectangular Pulse Fig (a). Consider the rectangular pulse x(t) shown in Fig (a). This pulse has an amplitude A and duration of 1 second. Express x(t) as a weighted sum of two step functions. <Sol.> 1. Rectangular pulse x(t): (1.55) (1.56)

16 Introduction CHAPTER Figure 1.39 (p. 44) (a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x 1 (t) and x 2 (t), respectively. Note that x(t) = x 1 (t) – x 2 (t).

17 Introduction CHAPTER Figure 1.40 (p. 45) (a) Series RC circuit with a switch that is closed at time t = 0, thereby energizing the voltage source. (b) Equivalent circuit, using a step function to replace the action of the switch. 1. Initial value: (1.57) 2. Final value: 3. Complete solution: Example 1.9 RC Circuit Fig (a). Find the response v(t) of RC circuit shown in Fig (a). <Sol.>

18 Introduction CHAPTER ★ Impulse Function Figure 1.42 (p. 46) (a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero. Figure 1.41 (p. 46) Discrete-time form of impulse. ◆ Discrete-time case: (1.58) Fig Figure 1.41 (p. 46) Discrete-time form of impulse. (t)(t) a(t)a(t)

19 Introduction CHAPTER ◆ Continuous-time case: (1.59) (1.60) Dirac delta function 1. As the duration decreases, the rectangular pulse approximates the impulse more closely. Fig Mathematical relation between impulse and rectangular pulse function: (1.61) 1.x  (t): even function of t,  = duration. 2.x  (t): Unit area. Fig (a). 3.  (t) is the derivative of u(t): (1.62) 4. u(t) is the integral of  (t): (1.63)

20 Introduction CHAPTER <Sol.> 1. Voltage across the capacitor: 2. Current flowing through capacitor: Example 1.10 RC Circuit (Continued) Fig (a), For the RC circuit shown in Fig (a), determine the current i (t) that flows through the capacitor for t  0.

21 Introduction CHAPTER ◆ Properties of impulse function: 1. Even function: (1.64) 2. Sifting property: (1.65) 3. Time-scaling property: (1.66) <p.f.> (1.68) (1.69) Fig Rectangular pulse approximation: (1.67) Fig. 1.44(a). 2. Unit area pulse: Fig. 1.44(a). Fig. 1.44(b). Time scaling: Fig. 1.44(b). Area = 1/ a Restoring unit area ax(at)ax(at) Ex. RLC circuit driven by impulsive Fig source: Fig Fig (a) For Fig (a), the voltage across the capacitor at time t = 0 + is

22 Introduction CHAPTER Figure 1.44 (p. 48) Steps involved in proving the time-scaling property of the unit impulse. (a) Rectangular pulse xΔ(t) of amplitude 1/Δ and duration Δ, symmetric about the origin. (b) Pulse xΔ(t) compressed by factor a. (c) Amplitude scaling of the compressed pulse, restoring it to unit area. Figure 1.45 (p. 49) (a) Parallel LRC circuit driven by an impulsive current signal. (b) Series LRC circuit driven by an impulsive voltage signal.

23 Introduction CHAPTER ★ Derivatives of The Impulse 1. Doublet: (1.70) 2. Fundamental property of the doublet: (1.71) (1.72) (1.73) 3. Second derivative of impulse: ★ Ramp Function 1. Continuous-time case: Problem 1.24 (1.74) or (1.75) Fig. 1.46

24 Introduction CHAPTER 2. Discrete-time case: (1.76) Figure 1.46 (p. 51) Ramp function of unit slope. or (1.77) Figure 1.47 (p. 52) Discrete-time version of the ramp function. x[n]x[n] n 11 22 33 4 Fig Example 1.11 Parallel Circuit Consider the parallel circuit of Fig (a) involving a dc current source I 0 and an initially uncharged capacitor C. The switch across the capacitor is suddenly opened at time t = 0. Determine the current i(t) flowing through the capacitor and the voltage v(t) across it for t  0. <Sol.> 1. Capacitor current:

25 Introduction CHAPTER 2. Capacitor voltage: Figure 1.48 (p. 52) (a) Parallel circuit consisting of a current source, switch, and capacitor, the capacitor is initially assumed to be uncharged, and the switch is opened at time t = 0. (b) Equivalent circuit replacing the action of opening the switch with the step function u(t).

26 Introduction CHAPTER 1.7 Systems Viewed as Interconnections of Operations interconnection of operations A system may be viewed as an interconnection of operations that transforms an input signal into an output signal with properties different from those of the input signal. 1. Continuous-time case: 2. Discrete-time case: (1.78) (1.79) Figure 1.49 (p. 53) Block diagram representation of operator H for (a) continuous time and (b) discrete time. Fig (a) and (b). Example 1.12 Moving-average system Consider a discrete-time system whose output signal y[n] is the average of the three most recent values of the input signal x[n], that is Formulate the operator H for this system; hence, develop a block diagram representation for it.

27 Introduction CHAPTER <Sol.> Fig Discrete-time-shift operator S k : Fig Figure 1.50 (p. 54) Discrete-time-shift operator S k, operating on the discrete- time signal x[n] to produce x[n – k]. Shifts the input x[n] by k time units to produce an output equal to x[n  k]. H 2. Overall operator H for the moving-average system: Fig Fig (a): cascade form; Fig (b): parallel form.

28 Introduction CHAPTER Figure 1.51 (p. 54) Two different (but equivalent) implementations of the moving-average system: (a) cascade form of implementation and (b) parallel form of implementation.

Properties of Systems ★ Stability BIBO 1. A system is said to be bounded-input, bounded-output (BIBO) stable if and only if every bounded input results in a bounded output. H 2. The operator H is BIBO stable if the output signal y(t) satisfies the condition (1.80) whenever the input signals x(t) satisfy the condition (1.81) Both M x and M y represent some finite positive number

30 Introduction CHAPTER Figure 1.52a (p. 56) Dramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, (a) Photograph showing the twisting motion of the bridge’s center span just before failure. (b) A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph. (Courtesy of the Smithsonian Institution.) One famous example of an unstable system:

31 Introduction CHAPTER Example 1.13 Moving-average system (continued) Show that the moving-average system described in Example 1.12 is BIBO stable. <p.f.> 1. Assume that: 2. Input-output relation: The moving-average system is stable.

32 Introduction CHAPTER Example 1.14 Unstable system Consider a discrete-time system whose input-output relation is defined by where r > 1. Show that this system is unstable. <p.f.> 1. Assume that: 2. We find that With r > 1, the multiplying factor r n diverges for increasing n. The system is unstable. ★ Memory A system is said to possess memory if its output signal depends on past or future values of the input signal. A system is said to possess memoryless if its output signal depends only on the present values of the input signal.

33 Introduction CHAPTER Ex.: Resistor Memoryless ! Ex.: InductorMemory ! Ex.: Moving-average system Memory ! Ex.: A system described by the input-output relation Memoryless ! ★ Causality A system is said to be causal if its present value of the output signal depends only on the present or past values of the input signal. A system is said to be noncausal if its output signal depends on one or more future values of the input signal.

34 Introduction CHAPTER Ex.: Moving-average system Causal ! Ex.: Moving-average system Noncausal ! real time  Causality is required for a systems to be capable of operating in real time. ★ Invertibility A system is said to be invertible if the input of the system can be recovered from the output. Figure 1.54 (p. 59) The notion of system invertibility. The second operator H inv is the inverse of the first operator H. Hence, the input x(t) is passed through the cascade correction of H and H inv completely unchanged. Fig Continuous-time system: Fig x(t) = input; y(t) = output H = first system operator; H inv = second system operator

35 Introduction CHAPTER 2. Output of the second system: 3. Condition for invertible system: (1.82) I = identity operator H inv = inverse operator Example 1.15 Inverse of System Consider the time-shift system described by the input-output relation where the operator S t 0 represents a time shift of t 0 seconds. Find the inverse of this system. <Sol.> 1. Inverse operator S  t 0 : 2. Invertibility condition: Time shift of  t 0

36 Introduction CHAPTER Example 1.16 Non-Invertible System Show that a square-law system described by the input-output relation is not invertible. <p.f.> Since the distinct inputs x(t) and  x(t) produce the same output y(t). Accordingly, the square-law system is not invertible. ★ Time Invariance A system is said to be time invariance if a time delay or time advance of the input signal leads to an identical time shift in the output signal.  A time-invariant system do not change with time. Figure 1.55 (p.61) The notion of time invariance. (a) Time-shift operator S t 0 preceding operator H. (b) Time-shift operator S t 0 following operator H. These two situations are equivalent, provided that H is time invariant.

37 Introduction CHAPTER 1. Continuous-time system: 2. Input signal x 1 (t) is shifted in time by t 0 seconds: S t 0 = operator of a time shift equal to t 0 3. Output of system H: (1.83) Fig (b), 4. For Fig (b), the output of system H is y 1 (t  t 0 ): (1.84) 5. Condition for time-invariant system: (1.85)

38 Introduction CHAPTER Example 1.17 Inductor x 1 (t) = v(t) y 1 (t) = i(t) The inductor shown in figure is described by the input-output relation: where L is the inductance. Show that the inductor so described is time invariant. <Sol.> 1. Letx1(t)x1(t) x 1 (t  t 0 ) Response y 2 (t) of the inductor to x 1 (t  t 0 ) is 2. Let y 1 (t  t 0 ) = the original output of the inductor, shifted by t 0 seconds: 3. Changing variables: (A) (B) (A)Inductor is time invariant.

39 Introduction CHAPTER Example 1.18 Thermistor Let R(t) denote the resistance of the thermistor, expressed as a function of time. We may express the input-output relation of the device as x 1 (t) = v(t) y 1 (t) = i(t) Show that the thermistor so described is time variant. <Sol.> 1. Let response y 2 (t) of the thermistor to x 1 (t  t 0 ) is 2. Let y 1 (t  t 0 ) = the original output of the thermistor due to x 1 (t), shifted by t 0 seconds: 3. Since R(t)  R(t  t 0 ) Time variant!