Introduction. Figure 1.1 (p. 2) Block diagram representation of a system. 1.3 Overview of Specific Systems Communication systems

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1 1.1 What is a signal? A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon. 1.2 What is a system? A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. 1.3 Overview of Specific Systems Communication systems Figure 1.1 (p. 2) Block diagram representation of a system. Elements of a communication system Fig Analog communication system: modulator + channel + demodulator 1

2 Figure 1.2 (p. 3) Elements of a communication system. The transmitter changes the message signal into a form suitable for transmission over the channel. The receiver processes the channel output (i.e., the received signal) to produce an estimate of the message signal. Modulation: 2. Digital communication system: sampling + quantization + coding transmitter channel receiver Two basic modes of communication: 1. Broadcasting 2. Point-to to-point communication Radio, television Telephone, deep-space communication Fig

3 Figure 1.3 (p. 5) (a) Snapshot of Pathfinder exploring the surface of Mars. (b) The 70-meter (230-foot) diameter antenna located at Canberra, Australia. The surface of the 70-meter reflector must remain accurate within a fraction of the signal s wavelength. (Courtesy of Jet Propulsion Laboratory.) 3

4 1.3.2 Control systems Figure 1.4 (p. 7) Block diagram of a feedback control system. The controller drives the plant, whose disturbed output drives the sensor(s). The resulting feedback signal is subtracted from the reference input to produce an error signal e(t), which, in turn, drives the controller. The feedback loop is thereby closed. Reasons for using control system: 1. Response, 2. Robustness Closed-loop loop control system: Fig Single-input, single-output (SISO) system 2. Multiple-input, multiple-output (MIMO) system Controller: digital computer (Fig. 1.5.) 4

5 Figure 1.5 (p. 8) NASA space shuttle launch. (Courtesy of NASA.) 5

6 1.3.3 Microelectromechanical Systems (MEMS( MEMS) Structure of lateral capacitive accelerometers: Fig (a). Figure 1.6a (p. 8) Structure of lateral capacitive accelerometers. (Taken from Yazdi et al., Proc. IEEE,, 1998) 6

7 SEM view of Analog Device s ADXLO5 surface- micromachined polysilicon accelerometer: Fig (b). Figure 1.6b (p. 9) SEM view of Analog Device s s ADXLO5 surface- micromachined polysilicon accelerometer. (Taken from Yazdi et al., Proc. IEEE,, 1998) 7

8 1.3.4 Remote Sensing Remote sensing is defined as the process of acquiring information about an object of interest without being in physical contact with it 1. Acquisition of information = detecting and measuring the changes that the 2. Types of remote sensor: Radar sensor Infrared sensor Visible and near-infrared sensor X-ray sensor Synthetic-aperture radar (SAR) Satisfactory operation High resolution object imposes on the field surrounding it. See Fig. 1.7 Ex. A stereo pair of SAR acquired from earth orbit with Shuttle Imaging Radar (SIR-B) 8

9 Figure 1.7 (p. 11) Perspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B). (Courtesy of Jet Propulsion Laboratory.) 9

10 1.3.5 Biomedical Signal Processing Morphological types of nerve cells: Fig Figure 1.8 (p. 12) Morphological types of nerve cells (neurons) identifiable in monkey cerebral cortex, based on studies of primary somatic sensory and motor cortices. (Reproduced from E. R. Kande,, J. H. Schwartz, and T. M. Jessel, Principles of Neural Science, 3d ed., 1991; courtesy of Appleton and Lange.) 10

11 Important examples of biological signal: 1. Electrocardiogram (ECG) 2. Electroencephalogram (EEG) Fig Figure 1.9 (p. 13) The traces shown in (a), (b), and (c) are three examples of EEG signals recorded from the hippocampus of a rat. Neurobiological studies suggest that the hippocampus plays a key role in certain aspects of learning and memory. 11

12 Measurement artifacts: 1. Instrumental artifacts 2. Biological artifacts 3. Analysis artifacts Auditory System Figure 1.10 (p. 14) (a) In this diagram, the basilar membrane in the cochlea is depicted as if it were uncoiled and stretched out flat; the base and apex refer to the cochlea, but the remarks stiff region and flexible region refer to the basilar membrane. (b) This diagram illustrates the traveling waves along the basilar membrane, showing their envelopes induced by incoming sound at three different frequencies. 12

13 The ear has three main parts: 1. Outer ear: collection of sound 2. Middle ear: acoustic impedance match between the air and cochlear fluid Conveying the variations of the tympanic membrane (eardrum) 3. Inner ear: mechanical variations electrochemical or neural signal Basilar membrane: Traveling wave Fig Analog Versus Digital Signal Processing Digital approach has two advantages over analog approach: 1. Flexibility 2. Repeatability 1.4 Classification of Signals 1. Continuous-time and discrete-time signals Parentheses ( ) Continuous-time signals: x(t) Fig Discrete-time signals: xn [ ] xnt ( s ), n 0, 1, 2,... = = ± ± (1.1) where t = nt s Fig Brackets [ ] 13

14 Figure 1.11 (p. 17) Continuous-time signal. Figure 1.12 (p. 17) (a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal x[n]. 14

15 2. Even and odd signals Even signals: x( t) = x( t) for all t (1.2) Odd signals: x( t) = x( t) for all t (1.3) Symmetric about vertical axis Example 1.1 Antisymmetric about origin Consider the signal π t sin, T t T xt () = T 0, otherwise Is the signal x(t) an even or an odd function of time? <Sol.> π t sin, T t T x( t) = T 0, otherwise π t sin, T t T = T 0, otherwise = xt ( ) for all t odd function 15

16 2t x() t = e cost CHAPTER Even-odd decomposition of x(t): xt () = x() t + x() t where e o x ( t) = x ( t) e e x ( t) = x ( t) o x( t) = x ( t) + x ( t) e o o = x ( t) x ( t) e 1 xe = x t + x t 2 [ () ( )] 1 xo = x t x t 2 [ () ( )] o (1.4) (1.5) Example 1.2 Find the even and odd components of the signal 2t x() t = e cost <Sol.> 2t x( t) = e cos( t) Even component: 2t = e cos( t) 1 2t 2t xe() t = ( e cost+ e cos) t 2 = cosh(2 t) cost Odd component: 1 2t 2t xo( t) = ( e cost e cos t) = sinh(2 t) cost 2 16

17 Conjugate symmetric: A complex-valued signal x(t) is said to be conjugate symmetric if Let x( t) = x ( t) (1.6) xt () = at () + jbt () x * () t = a() t jb() t a( t) + jb( t) = a() t jb() t a( t) = a( t) b( t) = b( t) 3. Periodic and nonperiodic signals (Continuous-Time Case) Periodic signals: x() t = x( t+ T) for all t (1.7) T = T0, 2 T0, 3 T0,... and T = T0 Fundamental period Fundamental frequency: 1 f = (1.8) T Angular frequency: 2π ω = 2π f = (1.9) T Refer to Fig Problem 1-2 Figure 1.13 (p. 20) (a) One example of continuoustime signal. (b) Another example of a continuous-time signal. 17

18 Example of periodic and nonperiodic signals: Fig Figure 1.14 (p. 21) (a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T 1. Periodic and nonperiodic signals (Discrete-Time Case) x[ n] = x[ n+ N] for integer n (1.10) Fundamental frequency of x[n]: 2π Ω= (1.11) N N = positive integer 18

19 Figure 1.15 (p. 21) Triangular wave alternative between 1 and +1 for Problem 1.3. Example of periodic and nonperiodic signals: Fig and Fig Figure 1.16 (p. 22) Discrete-time square wave alternative between 1 and

20 Figure 1.17 (p. 22) Aperiodic discrete-time signal consisting of three nonzero samples. 4. Deterministic signals and random signals A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time. Figure 1.13 ~ Figure 1.17 A random signal is a signal about which there is uncertainty before it occurs. Figure Energy signals and power signals Instantaneous power: v 2 () t If R = 1 Ω and x(t) represents a current or a voltage, pt () = (1.12) then the instantaneous power is R 2 2 p() t = x () t (1.14) p() t = Ri () t (1.13) 20

21 The total energy of the continuous-time signal x(t) is T lim T ( ) ( ) T 2 (1.15) E = x t dt = x t dt Time-averaged, or average, power is 1 P= x t dt T 2 2 lim T ( ) T T 2 1 T 2 2 P= T x () t dt T 2 (1.16) For periodic signal, the time-averaged power is (1.17) Discrete-time case: Total energy of x[n]: E = n= x 2 [ n] 1 P= x n N 2 lim [ ] n 2N n= N 1 = N Energy signal: If and only if the total energy of the signal satisfies the condition 0 < E < Power signal: N 1 2 P x n n= 0 (1.18) Average power of x[n]: If and only if the average power of the signal satisfies the condition 0 < P < [ ] (1.19) (1.20) 21

22 1.5 Basic Operations on Signals Operations Performed on dependent Variables c = scaling factor Amplitude scaling: x(t) y() t = cx() t (1.21) Discrete-time case: x[n] y[ n] = cx[ n] Addition: Performed by amplifier y() t = x1() t + x2() t (1.22) Discrete-time case: yn [ ] = x1[ n] + x2[ n] Multiplication: y() t = x () t x () t (1.23) Ex. AM modulation 1 2 y[ n] = x[ n] x [ n] 1 2 Differentiation: d d yt () = xt () (1.24) Inductor: vt () = L it () (1.25) dt dt Integration: t yt () = x( τ ) dτ (1.26) Figure 1.18 (p. 26) Inductor with current i(t), inducing voltage v(t) across its terminals. 22

23 1 t Capacitor: vt () = i( τ ) dτ C (1.27) Figure 1.19 (p. 27) Capacitor with Operations Performed on voltage v(t) across independent Variables its terminals, Time scaling: inducing current i(t). a >1 compressed y() t = x( at) 0 < a < 1 expanded Fig Figure 1.20 (p. 27) Time-scaling operation; (a) continuous-time signal x(t), (b) version of x(t) compressed by a factor of 2, and (c) version of x(t) expanded by a factor of 2. 23

24 Discrete-time case: y[ n] = x[ kn], k > 0 k = integer Some values lost! Figure 1.21 (p. 28) Effect of time scaling on a discrete-time signal: (a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with some values of the original x[n] lost as a result of the compression. Reflection: yt () = x( t) The signal y(t) represents a reflected version of x(t) about t = 0. Ex. 1-3 Consider the triangular pulse x(t) shown in Fig. 1-22(a). Find the reflected version of x(t) about the amplitude axis (i.e., the origin). <Sol.> Fig.1-22(b) 22(b). 24

25 Figure 1.22 (p. 28) Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin. xt () = 0 for t< T and t> T yt ( ) = 0 for t> T and t< T Time shifting: yt () = xt ( t0) Ex. 1-4 Time Shifting: Fig Figure 1.23 (p. 29) Time-shifting operation: (a) continuoustime signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin; and (b) timeshifted version of x(t) by 2 time shifts. t 0 > 0 shift toward right t 0 < 0 shift toward left 25

26 Discrete-time case: y[ n] = x[ n m] where m is a positive or negative integer Precedence Rule for Time Shifting and Time Scaling 1. Combination of time shifting and time scaling: yt () = xat ( b) y(0) = x( b) b y( ) x(0) a = (1.28) (1.29) (1.30) 2. Operation order: To achieve Eq. (1.28), 1st step: time shifting 2nd step: time scaling vt () = xt ( b) yt () = vat ( ) = xat ( b) Ex. 1-5 Precedence Rule for Continuous-Time Signal Consider the rectangular pulse x(t) depicted in Fig. 1-24(a). 1 Find y(t)=x(2t + 3). <Sol.> Case 1: Fig Shifting first, then scaling Case 2: Fig Scaling first, then shifting yt () = vt ( + 3) = x(2( t+ 3)) x(2t+ 3) 26

27 Figure 1.24 (p. 31) The proper order in which the operations of time scaling and time shifting should be applied in the case of the continuous-time signal of Example 1.5. (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2. 27

28 Figure 1.25 (p. 31) The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)). Ex. 1-6 Precedence Rule for Discrete-Time Signal A discrete-time signal is defined by 1, n = 1, 2 xn [ ] = 1, n= 1, 2 0, n= 0 and n > 2 Find y[n] = x[2x + 3]. 28

29 <Sol.> See Fig Figure 1.27 (p. 33) The proper order of applying the operations of time scaling and time shifting for the case of a discrete-time signal. (a) Discrete-time signal x[n], antisymmetric about the origin. (b) Intermediate signal v(n) obtained by shifting x[n] to the left by 3 samples. (c) Discrete-time signal y[n] resulting from the compression of v[n] by a factor of 2, as a result of which two samples of the original x[n], located at n = 2, +2, are lost. 29

30 1.6 Elementary Signals B and a are real parameters at Exponential Signals x() t = Be (1.31) 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. 30

31 Ex. Lossy capacitor: Fig KVL Eq.: d RC vt () + vt () = 0 (1.32) dt vt () Ve t/( RC) = 0 (1.33) Discrete-time case: n xn [ ] = Br (1.34) where r = e α Fig Sinusoidal Signals Continuous-time case: xt () = Acos( ωt+ φ) (1.35) where T 2π = ω periodicity RC = Time constant Fig Figure 1.29 (p. 35) Lossy capacitor, with the loss represented by shunt resistance R. x( t+ T) = Acos( ω( t+ T) + φ) = Acos( ωt+ ωt + φ) = Acos( ωt+ 2 π + φ) = Acos( ωt+ φ) = xt ( ) 31

32 Figure 1.30 (p. 35) (a) Decaying exponential form of discrete-time signal. (b) Growing exponential form of discrete-time signal. 32

33 Figure 1.31 (p. 36) (a) Sinusoidal signal A cos(ω t + Φ) with phase Φ = +π/6 radians. (b) Sinusoidal signal A sin (ω t + Φ) with phase Φ = +π/6 radians. 33

34 Ex. Generation of a sinusoidal signal Fig Circuit Eq.: 2 d LC v() t + v() t = 0 2 dt (1.36) vt () = Vcos( ω t), t 0 (1.37) where ω 0 = (1.38) LC Natural angular frequency of oscillation of the circuit Discrete-time case : xn [ ] = Acos( Ω n+ φ) Periodic condition: Ω N = 2π m or (1.39) xn [ + N] = Acos( Ω n+ω N+ φ) 2π m Ω= N radians/cycle, integer m, Figure 1.32 (p. 37) Parallel LC circuit, assuming that the inductor L and capacitor C are both ideal. (1.40) N (1.41) Ex. A discrete-time sinusoidal signal: A = 1, φ = 0, and N = 12. Fig

35 Figure 1.33 (p. 38) Discrete-time sinusoidal signal. 35

36 Example 1.7 Discrete-Time Sinusoidal Signal A pair of sinusoidal signals with a common angular frequency is defined by x 1 [ n] = sin[5 π n] and x 2 [ n] = 3cos[5 π n] (a) Both x 1 [n] and x 2 [n] are periodic. Find their common fundamental period. (b) Express the composite sinusoidal signal yn [ ] = x[ n] + x[ n] 1 2 In the form y[n] = Acos(Ωn + φ), and evaluate the amplitude A and phase φ. <Sol.> (a) Angular frequency of both x 1 [n] and x 2 [n]: Ω= 5 π radians/cycle N 2π m 2π m 2m = = = Ω 5π 5 This can be only for m = 5, 10, 15,, which results in N = 2, 4, 6, (b) Trigonometric identity: Acos( Ω n+ φ) = Acos( Ωn)cos( φ) Asin( Ωn)sin( φ) Let Ω = 5π, then compare x 1 [n] + x 2 [n] with the above equation to obtain that 36

37 Asin( φ) = 1 and Acos( φ) = 3 sin( φ) amplitude of x1[ n] 1 tan( φ) = = = cos( φ) amplitude of x [ n] 3 Asin( φ ) = 1 1 A = = 2 sin / 6 ( π ) yn [ ] 2cos 5π n π = 6 2 Accordingly, we may express y[n] as φ = π / Relation Between Sinusoidal and Complex Exponential Signals jθ 1. Euler s identity: e = cosθ + jsinθ (1.41) j Complex exponential signal: B= Ae φ (1.42) xt () = Acos( ωt+ φ) (1.35) j t Acos( ωt+ φ) = Re{ Be ω } (1.42) Be = = jωt Ae Ae jφ e jωt j( φ+ ωt) = Acos( ωt+ φ) + jasin( ωt+ φ) 37

38 Continuous-time time signal in terms of sine function: xt () = Asin( ωt+ φ) (1.44) j t Asin( ωt+ φ) = Im{ Be ω } (1.45) 2. Discrete-time case: jωn Acos( Ω n+ φ) = Re{ Be } (1.46) and jωn Asin( Ω n+ φ) = Im{ Be } (1.47) 3. Two-dimensional representation of the complex exponential e j Ω n for Ω = π/4 and n = 0, 1, 2,, 7. : Fig Projection on real axis: cos(ωn); Projection on imaginary axis: sin(ωn) Figure 1.34 (p. 41) Complex plane, showing eight points uniformly distributed on the unit circle. π /4 π /4 38

39 1.6.4 Exponential Damped Sinusoidal Signals αt xt () = Ae sin( ωt+ φ), α > 0 (1.48) Example for A = 60, α = 6, and φ = 0: Fig Figure 1.35 (p. 41) Exponentially damped sinusoidal signal Ae at sin(ωt), with A = 60 and α = 6. 39

40 Ex. Generation of an exponential damped sinusoidal signal Fig d 1 1 t Circuit Eq.: C v() t + v() t + v( τ) dτ 0 dt R L = (1.49) vt () = Ve cos( ω t) t 0 t/(2 CR) 0 0 (1.50) 1 1 ω 0 = 2 2 LC 4C R Comparing Eq. (1.50) and (1.48), we have where (1.51) R> L/(4 C) A= V, α = 1/(2 CR), ω = ω, and φ = π /2 0 0 Discrete-time case: n xn [ ] = Br sin[ Ω n+ φ] (1.52) Step Function Discrete-time case: un [ ] { 1, n 0 0, n< 0 = (1.53) Fig Figure 1.37 (p. 43) Discrete-time version of step function of unit amplitude L t v( τ ) dτ Figure 1.36 (p. 42) Parallel LRC, circuit, with inductor L, capacitor C, and resistor R all assumed to be ideal. 1 0 x[n] n

41 Continuous-time case: { 1, t > 0 ut () = 0, t < 0 (1.54) Figure 1.38 (p. 44) Continuous-time version of the unit-step function of unit amplitude. Example 1.8 Rectangular Pulse 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.> A, 0 t< Rectangular pulse x(t): xt () (1.55) 1 1 xt () = Au t+ Au t 2 2 = 0, t > 0.5 (1.56) Example 1.9 RC Circuit Find the response v(t) of RC circuit shown in Fig (a). <Sol.> 41

42 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). 42

43 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: v (0) = 0 2. Final value: 3. Complete solution: 0 v( ) = V t/( RC) ( ) vt () = V 1 e ut () (1.57) 0 43

44 1.6.6 Impulse Function Discrete-time case: { 1, n = 0 δ [ n] = (1.58) 0, n 0 Figure 1.41 (p. 46) Discrete-time form of impulse. Fig δ(t) aδ(t) Figure 1.41 (p. 46) Discrete-time form of impulse. 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. 44

45 Continuous-time case: δ () t = 0 for t 0 (1.59) Dirac delta function δ () tdt= 1 (1.60) 1. As the duration decreases, the rectangular pulse approximates the impulse more closely. Fig Mathematical relation between impulse and rectangular pulse function: δ () t = lim xδ () t (1.61) 1. x Δ (t): even function of t, Δ = duration. Δ 0 Fig (a). 3. δ(t) is the derivative of u(t): d δ () t = u() t (1.62) dt 2. x Δ (t): Unit area. 4. u(t) is the integral of δ(t): ut () = δ ( τ) dτ (1.63) Example 1.10 RC Circuit (Continued) For the RC circuit shown in Fig (a), determine the current i (t) that flows through the capacitor for t 0. t 45

46 <Sol.> Figure 1.43 (p. 47) (a) Series circuit consisting of a capacitor, a dc voltage source, and a switch; the switch is closed at time t = 0. (b) Equivalent circuit, replacing the action of the switch with a step function u(t). 1. Voltage across the capacitor: vt () = Vut () 0 2. Current flowing through capacitor: dv() t it () C dt du() t i() t = CV = CV δ () t dt =

47 Properties of impulse function: 1. Even function: δ ( t) = δ ( t) (1.64) 2. Sifting property: x() t δ ( t t0) dt = x( t0) (1.65) 3. Time-scaling property: <p.f.> 1 δ( at) = δ( t), a > 0 (1.66) a Fig Rectangular pulse approximation: δ ( at) = lim xδ ( at) (1.67) Δ 0 2. Unit area pulse: Fig. 1.44(a). Time scaling: Fig. 1.44(b). Area = 1/a Restoring unit area ax Δ (at) 1 lim x ( at) ( t) 0 a δ Δ = (1.68) Δ Ex. RLC circuit driven by impulsive source: Fig For Fig (a) Fig (a), the voltage across the capacitor at time t = 0 + is + 1 V I t dt C I C = 0 0δ () = (1.69) 47

48 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. 48

49 1.6.7 Derivatives of The Impulse 1. Doublet: (1) 1 δ ( t) = lim ( δ( t+δ/2) δ( t Δ/2) ) Δ 0 Δ 2. Fundamental property of the doublet: δ (1) () tdt= 0 (1.71) d dt 3. Second derivative of impulse: (1) f() t δ ( t t0) dt = f() t t = t (1.72) 0 (1.70) 2 (1) (1) (1) ( /2) ( /2) () d δ () lim t +Δ δ t t t Δ δ = δ = 2 t dt Δ 0 Δ Ramp Function Problem 1.24 d f t t t dt f t dt 2 (2) () δ ( 0) = () 2 t = t (1.73) 1. Continuous-time case: t, t 0 rt () = (1.74) or rt () = tut () (1.75) Fig , t < 0 d f t t t dt f t dt n ( n) () δ ( 0) = () n t= t

50 2. Discrete-time case: n, n 0 rn [ ] = (1.76) 0, n < 0 or rn [ ] = nun [ ] (1.77) Figure 1.46 (p. 51) Ramp function of unit slope. Fig Figure 1.47 (p. 52) Discrete-time version of the ramp function. 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: it = 0 () Iut () x[n] n

51 2. Capacitor voltage: 1 t vt () = i( τ ) dτ C 1 t vt () = Iu 0 ( ) d C τ τ 0 for t < 0 = I0 t for t > 1 C = = I0 tu t C I0 rt C ( ) ( ) 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). 51

52 1.7 Systems Viewed as Interconnections 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: yt () = H{ xt ()} (1.78) 2. Discrete-time case: yn [ ] = H{ xn [ ]} (1.79) Figure 1.49 (p. 53) Fig (a) and (b). Block diagram representation of operator H for (a) continuous time and (b) discrete time. 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 1 yn [ ] = ( xn [ ] + xn [ 1] + xn [ 2]) 3 Formulate the operator H for this system; hence, develop a block diagram representation for it. 52

53 <Sol.> 1. Discrete-time-shift operator S k : Fig Shifts the input x[n] by k time units to produce an output equal to x[n k]. 2. Overall operator H for the moving-average system: 1 (1 2 ) H = + S + S 3 Fig Figure 1.50 (p. 54) Discrete-time-shift operator S k, operating on the discretetime signal x[n] to produce x[n k]. Fig (a): cascade form; Fig (b): parallel form. 1.8 Properties of Systems Stability 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. 2. The operator H is BIBO stable if the output signal y(t) satisfies the condition y() t M < for all t (1.80) y whenever the input signals x(t) satisfy the condition x() t M < for all t (1.81) x Both M x and M y represent some finite positive number 53

54 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. 54

55 One famous example of an unstable system: 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.) 55

56 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: x[ n] M < for all n 2. Input-output relation: 1 yn [ ] = ( xn [ ] + xn [ 1] + xn [ 2] ) 3 1 yn [ ] = xn [ ] + xn [ 1] + xn [ 2] 3 1 xn [ ] + xn [ 1] + xn [ 2] 3 1 ( Mx + Mx + Mx) 3 = M x ( ) x The moving-average system is stable. 56

57 Example 1.14 Unstable system Consider a discrete-time system whose input-output relation is defined by yn [ ] = r n xn [ ] where r > 1. Show that this system is unstable. <p.f.> 1. Assume that: x[ n] M < for all n 2. We find that n n yn [ ] = r xn [ ] = r. xn [ ] x With r > 1, the multiplying factor r n diverges for increasing n Memory The system is unstable. 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. 57

58 Ex.: Resistor Ex.: Inductor Ex.: Moving-average system 1 it () = vt () R Memoryless! 1 t it () = v( τ ) dτ L Memory! 1 yn [ ] = ( xn [ ] + xn [ 1] + xn [ 2]) Memory! 3 Ex.: A system described by the input-output relation y n 2 [ ] x [ n] = 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. 58

59 Ex.: Moving-average system 1 yn [ ] = ( xn [ ] + xn [ 1] + xn [ 2]) Causal! 3 Ex.: Moving-average system 1 yn [ ] = ( xn [ + 1] + xn [ ] + xn [ 1]) Noncausal! 3 A causal system must 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. 1. Continuous-time system: Fig x(t) = input; y(t) = output H = first system operator; H inv = second system operator 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. 59

60 2. Output of the second system: { } { } { } { } H inv y() t = H inv H x() t = H inv H x() t 3. Condition for invertible system: inv H H = I (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 { } y t x t t S x t t0 () = ( ) = () 0 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: t0 t0 t0 t0 t0 S y t = S S x t = S S x t { ()} { { ()}} { ()} 2. Invertibility condition: t 0 t0 0 S S = I t S Time shift of t 0 60

61 Example 1.16 Non-Invertible System Show that a square-law system described by the input-output relation y t x t 2 () = () 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. 61

62 1. Continuous-time system: y () t = H{ x ()} t Input signal x 1 (t) is shifted in time by t 0 seconds: t0 x () t = x ( t t ) = S { x ()} t S t 0 = operator of a time shift equal to t Output of system H: y () t = H{ x ( t t )} = = H S HS t0 { { x ()}} t t { x ()} t (1.83) 4. For Fig (b), the output of system H is y 1 (t t 0 ): t0 y1( t t0) = S { y1()} t t0 = S { H{ x ()}} t = 1 t0 S H x t { ()} 1 (1.84) t0 t0 5. Condition for time-invariant system: HS = S H (1.85) 62

63 y 1 (t) = i(t) Example 1.17 Inductor The inductor shown in figure is described by the input-output relation: 1 t x 1 (t) = v(t) y1() t = x1( τ ) dτ L where L is the inductance. Show that the inductor so described is time invariant. <Sol.> 1. Let x 1 (t) x 1 (t t 0 ) Response y 2 (t) of the inductor to x 1 (t t 0 ) is 1 t y2() t = x1( τ t0) dτ L (A) 2. Let y 1 (t t 0 ) = the original output of the inductor, shifted by t 0 seconds: 1 t t0 y1( t t 0) = x1( τ ) dτ L τ ' = τ t (B) 3. Changing variables: 0 (A) 1 t t0 y2() t = x1( τ ') dτ ' L Inductor is time invariant. 63

64 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 y () t = x ()/ t R() t 1 1 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 y () t = 2 x1( t t0) Rt () y 1 (t) = i(t) x 1 (t) = v(t) 2. Let y 1 (t t 0 ) = the original output of the thermistor due to x 1 (t), shifted by t 0 seconds: y ( t t ) = 1 0 x1( t t0) Rt ( t) 0 3. Since R(t) R(t t 0 ) y ( t t ) y ( t) for t 0 Time variant! 64

65 1.8.6 Linearity A system is said to be linear in terms of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two properties of superposition and homogeneity: 1. Superposition: x() t () = x1 t y t = 1 = x2 t y t = 2 x() t () 2. Homogeneity: () y () t () y () t x() t y() t ax() t ay() t x() t = x () t + x () t 1 2 y() t = y () t + y () t 1 2 a = constant factor Linearity of continuous-time system 1. Operator H represent the continuous-tome system. 2. Input: x N 1 (t), x 2 (t),, x N (t) input signal; a 1, a 2,, a N x() t = a x () t (1.86) Corresponding weighted factor i= 1 3. Output: i i N yt () = H{ xt ()} = H{ ax()} t (1.87) i= 1 i i 65

66 N yt () = ay() t (1.88) i= 1 i i where y ( t) = H{ x ( t)}, i = 1, 2,..., N. (1.89) i i 4. Commutation and Linearity: yt () = H{ ax()} t = = N i= 1 N i= 1 N i= 1 i ah{ x( t)} i i i ay() t i i Superposition and homogeneity (1.90) Fig Linearity of discrete-time system Same results, see Example Example 1.19 Linear Discrete-Time system Consider a discrete-time system described by the input-output relation yn [ ] = nxn [ ] Show that this system is linear. 66

67 Figure 1.56 (p. 64) The linearity property of a system. (a) The combined operation of amplitude scaling and summation precedes the operator H for multiple inputs. (b) The operator H precedes amplitude scaling for each input; the resulting outputs are summed to produce the overall output y(t). If these two configurations produce the same output y(t), the operator H is linear. <p.f.> N 1. Input: x[ n] = ax i i[ n] i= 1 2. Resulting output signal: 67

68 N N N where yi[ n] = nxi[ n] yn [ ] = n ax[ n] = anx[ n] = ay[ n] i i i i i i i= 1 i= 1 i= 1 Linear system! Example 1.20 Nonlinear Continuous-Time System Consider a continuous-time system described by the input-output relation yt () = xtxt () ( 1) Show that this system is nonlinear. <p.f.> N 1. Input: 2. Output: x() t = ax i i() t i= 1 N N N N yt () = ax() t ax( t 1) = aax() tx( t 1) i i j j i j i j i= 1 j= 1 i= 1 j= 1 Here we cannot write N yt () = ay() t Nonlinear system! i= 1 i i 68

69 Example 1.21 Impulse Response of RC Circuit For the RC circuit shown in Fig. 1.57, determine the impulse response y(t). <Sol.> 1. Recall: Unit step response t/ RC yt () = (1 e ) ut (), xt () = ut () (1.91) 2. Rectangular pulse input: Fig x(t) = x Δ (t) 1 Δ x1() t = u( t+ ) Δ 2 1 Δ x2() t = u( t ) 2 Δ 3. Response to the step functions x 1 (t) and x 2 (t): 1/Δ Δ/2 Δ/2 Figure 1.57 (p. 66) RC circuit for Example 1.20, in which we are given the capacitor voltage y(t) in response to the step input x(t) = y(t) and the requirement is to find y(t) in response to the unit-impulse input x(t) = δ(t). Figure 1.58 (p. 66) Rectangular pulse of unit area, which, in the limit, approaches a unit impulse as Δ 0. 69

70 y e u t Δ x t x t Δ 2 Δ 1 t+ /( RC) = +, () = 1() y e u t Δ x t x t Δ 2 Δ 1 t /( RC) =, () = 2() Next, recognizing that xδ () t = x () t x () t (( t+δ/2)/( RC) ) 1 ( ( t Δ/2)/( RC) ) yδ() t = (1 e ) u( t+δ/2) (1 e ) u( t Δ/2) Δ Δ 1 1 (( t+δ/2)/( RC) ) ( ( t Δ/2)/( RC) ) = ( ut ( +Δ/2) ut ( Δ/2)) ( e ut ( +Δ/2) e ) ut ( Δ/2)) Δ Δ i) δ(t) = the limiting form of the pulse x Δ (t): (1.92) δ () t = lim x () t Δ 0 Δ 70

71 ii) The derivative of a continuous function of time, say, z(t): d 1 Δ Δ zt ( ) = lim{ ( zt ( + ) zt ( ))} dt Δ 0 Δ 2 2 (1.92) yt () = lim y() t Δ 0 Δ d t/( RC) = δ ( t) ( e u( t)) dt t/( RC) d d t/( RC) = δ () t e u() t u() t ( e ) dt dt 1 = δ t e δ t + e u t x t = δ t RC t/( RC) t/( RC) () () (), () () Cancel each other! 1 t/( RC) yt () = e ut (), RC xt () = δ () t (1.93) 71

72 1.9 Noise Noise Unwanted signals 1. External sources of noise: atmospheric noise, galactic noise, and humanmade noise. 2. Internal sources of noise: spontaneous fluctuations of the current or voltage signal in electrical circuit. (electrical noise) Fig Thermal Noise Thermal noise arises from the random motion of electrons in a conductor. Two characteristics of thermal noise: 1. Time-averaged value: 2T = total observation interval of noise 1 T v= lim v( t) dt T 2T (1.94) T As T, v 0 Refer to Fig Time-average-squared value: k = Boltzmann s constant = v 2 1 T 2 = lim ( ) T 2T v t dt (1.95) 23 J/K T T abs = absolute temperature 2 2 As T, v = 4kT RΔ f volts (1.96) abs 72

73 Figure 1.60 (p. 68) Sample waveform of electrical noise generated by a thermionic diode with a heated cathode. Note that the timeaveraged value of the noise voltage displayed is approximately zero. 73

74 Thevenin s equvalent circuit: Fig. 1.61(a), Norton s equivalent circuit: Fig. 1.61(b). Noise voltage generator: 2 vt () = v Noise current generator: 2 1 T vt ( ) 2 i = lim ( ) dt T 2T T R (1.97) 2 = 4kT GΔf amps abs where G = 1/R = conductance [S]. Maximum power transfer theorem: the maximum Figure 1.61 (p. 70) possible power is transferred (a) Thévenin equivalent circuit of a noisy resistor. from a source of internal (b) Norton equivalent circuit of the same resistor. resistance R to a load of resistance R l when R = R l. Under matched condition, the available power is ktabs Δf watts 74

75 Two operating factor that affect available noise power: 1. The temperature at which the resistor is maintained. 2. The width of the frequency band over which the noise voltage across the resistor is measured Other Sources of Electrical Noise 1. Shot noise: the discrete nature of current flow electronic devices 2. Ex. Photodetector: 1) Electrons are emitted at random times, τ k, where < τ k < 2) Total current flowing through photodetector: xt () = ht ( τ k ) (1.98) where k = ht ( τ k ) is the current pulse generated at time τ k. 3. 1/f noise: The electrical noise whose time-averaged power at a given frequency is inversely proportional to the frequency. 75

76 1.10 Theme Example Differentiation and Integration: RC Circuits 1. Differentiator Sharpening of a pulse x(t) d y() t = x() t (1.99) dt 1) Simple RC circuit: Fig ) Input-output relation: 1 t v2() t + v2( τ) dτ v1() t RC = d 1 d v2() t + v2() t = v1() t (1.100) dt RC dt If RC (time constant) is small enough such that (1.100) is dominated by the second term v 2 (t)/rc, then 1 v t RC d dt 2() v1() t 2 1 differentiator y(t) Figure 1.62 (p. 71) Simple RC circuit with small time constant, used as an approximator to a differentiator. d v () t RC v () t for RC small (1.101) dt 76

77 Input: x(t) = RCv 1 (t); output: y(t) = v 2 (t) 2. Integrator smoothing of an input signal t x(t) y() t = x( τ ) dτ (1.102) 1) Simple RC circuit: Fig ) Input-output relation: d RC v2() t + v2() t = v1() t dt t RCv ( t) + v ( τ ) dτ = v ( τ) dτ If RC (time constant) is large enough such that (1.103) is dominated by the first term RCv 2 (t), then RCv2() t v1( τ ) dτ t t (1.103) 1 t v2() t v1( τ) dτ for large RC RC integrator y(t) Figure 1.63 (p. 72) Simple RC circuit with large time constant used as an approximator to an integrator. Input: x(t) = [1/(RC)v 1 (t)]; output: y(t) = v 2 (t) 77

78 MEMS Accelerometer 1. Model: second-order mass-damper-spring system Fig M = proof mass, K = effective spring constant, D = damping factor, x(t) =external acceleration, y(t) =displacement of proof mass, Md 2 y(t)/dt 2 = inertial force of proof mass, Ddy(t)/dt = damping force, Ky(t) = spring force. 2. Force Eq.: 2 d y() t dy() t Mx() t = M + D + Kyt () 2 dt dt 2 d y() t D dy() t K + + () () 2 yt = xt (1.105) dt M dt M 1) Natural frequency: K ω n = [rad/sec] (1.106) M 2) Quality factor: Figure 1.64 (p. 73) Mechanical lumped model of an accelerometer. KM Q = (1.107) D 78

79 (1.105) CHAPTER 2 d y t n 2 dt Q dt () ω dy() t + + ω y() t = x() t (1.108) Radar Range Measurement 1. A periodic sequence of radio frequency (RF) pulse: Fig n T 0 = duration [μsec], 1/T = repeated frequency, f c = RF frequency [MHz~GHz] Figure 1.65 (p. 74) Periodic train of rectangular FR pulses used for measuring molar ranges. The sinusoidal signal acts as a carrier. 79

80 2. Round-trip time = the time taken by a radar pulse to reach the target and for the echo from the target to come back to the radar. 2d τ = (1.109) c d = radar target range, c = light speed. 3. Two issues of concern in range measurement: 1) Range resolution: The duration T0 of the pulse places a lower limit on the shortest round-trip delay time that the radar can measure. Smallest target range: d min = ct 0 /2 [m] 2) Range ambiguity: The interpulse period T places an upper limit on the largest range that the radar can measure. Largest target range: d max = ct/2 [m] Moving-Average Systems 1. N-point moving-average system: Fig N 1 1 yn [ ] = xn [ k] (1.110) x(t) = input signal N k = 0 The value N determines the degree to which the system smooths the input data. 80

81 Figure 1.66a (p. 75) (a) Fluctuations in the closing stock price of Intel over a three-year period. 81

82 Figure 1.66b (p. 76) (b) Output of a four-point moving-average system. N = 4 case 82

83 Figure 1.66c (p. 76) (c) Output of an eight-point moving-average system. N = 8 case 83

84 2. For a general moving-average system, unequal weighting is applied to past values of the input: N = 1 yn [ ] = axn k [ k] (1.111) k = Multipath Communication Channels 1. Channel noise degrades the performance of a communication system. 2. Another source of degradation of channel: dispersive nature, i.e., the channel has memory. 3. For wireless system, the dispersive characteristics result from multipath propagation. Fig For a digital communication, multipath propagation manifests itself in the form of intersymbol interference (ISI). 4. Baseband model for multipath propagation: Tapped-delay line Fig p yt () = ωixt ( itdiff ) (1.112) T diff = smallest time difference i= 0 between different path 84

85 Figure 1.67 (p. 77) Example of multiple propagation paths in a wireless communication environment. 85

86 Figure 1.68 (p. 78) Tapped-delay-line model of a linear communication channel, assumed to be timeinvariant. 86

87 5. PT diff = the longest time delay of any significant path relative to the arrival of the signal. The coefficients w i are used to approximate the gain of each path. For P =1, then y t = ω0x t + ω1x t T diff () () ( ) ω 0 x(t) = direct path, ω 1 x(t T diff ) = single reflected path 6. Discrete-time case: p y[ n] = ωk x[ n k] (1.113) k = 0 For P =1, then yn [ ] = xn [ ] + axn [ 1] (1.114) Recursive Discrete-Time Computations 1. First-order recursive discrete-time filter: Fig yn [ ] = xn [ ] + ρ yn [ 1] (1.115) where ρ is a constant. x[n] = input, y[n] = output Linearly weighted moving-average system The term recursive signifies the dependence of the output signal on its own past values. 87

88 Figure 1.69 (p. 79) Block diagram of first-order recursive discrete-time filter. The operator S shifts the output signal y[n] by one sampling interval, producing y[n 1]. The feedback coefficient ρ determines the stability of the filter. Fig. 1.69: linear discrete-time feedback system. 2. Solution of Eq.(1.115): k yn [ ] = ρ xn [ k] k = 0 k yn [ ] = xn [ ] + ρ xn [ k] k = 1 (1.116) Setting k 1 = l, Eq.(1.117) becomes (1.117) 1+ l l [ ] = [ ] + ρ [ 1 ] = [ ] + ρ ρ [ 1 ] l= 0 l= 0 (1.118) yn xn n l xn n l r ρ 88

89 y[ n] = x[ n] + ρ y[ n 1] 3. Three special cases (depending on ρ): 1) ρ = 1: (1.116) yn [ ] = xn [ k] (1.119) 2) ρ < 1: 3) ρ > 1: k = 0 Leaky accumulator Amplified accumulator 1.11 Exploring Concepts with MATLAB MATLAB Signal Processing Toolbox Accumulator 1. Time vector: Sampling interval T s of 1 ms on the interval from 0 to 1 s t = 0:.001:1; 2. Vector n: n = 0:1000; Periodic Signals 1. Square wave: A =amplitude, w0 = fundamental frequency, rho = duty cycle A*square(w0*t, rho); Stable in BIBO sense Unsatble in BIBO sense 89

90 Ex. Obtain the square wave shown in Fig (a) by using MATLAB. <Sol.> >> A = 1; >> w0 =10*pi; >> rho = 0.5; >> t = 0:.001:1; >> sq = A*square(w0*t, rho); >> plot (t, sq) >> axis([ ]) 2. Triangular wave: A =amplitude, w0 = fundamental frequency, w = width A*sawtooth(w0*t, w); Ex. Obtain the triangular wave shown in Fig by using MATLAB. <Sol.> >> A = 1; >> w0 =10*pi; >> w = 0.5; >> t = 0:0.001:1; >> tri = A*sawtooth(w0*t, w); >> plot (t, tri) 90

91 3. Discrete-time signal: stem(n, x); x = vector, n = discrete time vector Ex. Obtain the discrete-time square wave shown in Fig by using MATLAB. <Sol.> >> A = 1; >> omega =pi/4; >> n = -10:10; >> x = A*square(omega*n); >> stem(n, x) 4. Decaying exponential: B*exp(-a*t); Growing exponential: B*exp(a*t); Ex. Obtain the decaying exponential signal shown in Fig (a) by using MATLAB. <Sol.> >> B = 5; >> a = 6; >> t = 0:.001:1; >> x = B*exp(-a*t); % decaying exponential >> plot (t, x) 91

92 Ex. Obtain the growing exponential signal shown in Fig (b) by using MATLAB. <Sol.> >> B = 1; >> a = 5; >> t = 0:0.001:1; >> x = B*exp(a*t); % growing exponential >> plot (t, x) Ex. Obtain the decaying exponential sequence shown in Fig (a) by using MATLAB. <Sol.> >> B = 1; >> r = 0.85; >> n = -10:10; >> x = B*r.^n; % decaying exponential sequence >> stem (n, x) Sinusoidal Signals 1. Cosine signal: A*cos(w0*t + phi); 2. Sine signal: A*sin(w0*t + phi); A = amplitude, w0 = frequency, phi = phase angle 92

93 Ex. Obtain the sinusoidal signal shown in Fig (a) by using MATLAB. <Sol.> >> A = 4; >> w0 =20*pi; >> phi = pi/6; >> t = 0:.001:1; >> cosine = A*cos(w0*t + phi); >> plot (t, cosine) Ex. Obtain the discrete-time sinusoidal signal shown in Fig by using MATLAB. <Sol.> >> A = 1; >> omega =2*pi/12; % angular frequency >> n = -10:10; >> y = A*cos(omega*n); >> stem (n, y) Exponential Damped Sinusoidal Signals 1. Exponentially damped sinusoidal signal: at x() t = Asin( ω t+ φ) e MATLAB Format: A*sin(w0*t + phi).*exp(-a*t); 0.* element-by-element multiplication 93

94 Ex. Obtain the waveform shown in Fig by using MATLAB. <Sol.> >> A = 60; >> w0 =20*pi; >> phi = 0; >> a = 6; >> t = 0:.001:1; >> expsin = A*sin(w0*t + phi).*exp(-a*t); >> plot (t, expsin) Ex. Obtain the exponentially damped sinusoidal sequence shown in Fig by using MATLAB. <Sol.> >> A = 1; >> omega =2*pi/12; % angular frequency >> n = -10:10; >> y = A*cos(omega*n); >> r = 0.85; >> x = A*r.^n; % decaying exponential sequence >> z = x.*y; % elementwise multiplication >> stem (n, z) 94

95 Figure 1.70 (p. 84) Exponentially damped sinusoidal sequence. 95

96 Step, Impulse, and Ramp Functions MATLAB command: 1. M-by-N matrix of ones: ones (M, N) 2. M-by-N matrix of zeros: zeros (M, N) Unit amplitude step function: u = [zeros(1, 50), ones(1, 50)]; Discrete-time impulse: delta = [zeros(1, 49), 1, zeros(1, 49)]; Ramp sequence: ramp = 0:.1:10 Ex. Generate a rectangular pulse centered at origin on the interval [-1, 1]. <Sol.> >> t = -1:1/500:1; >> u1 = [zeros(1, 250), ones(1, 751)]; >> u2 = [zeros(1, 751), ones(1, 250)]; >> u = u1 u2; User-Defined Functions Defined Functions 1. Two types M-files exist: scripts and functions. Scripts, or script files automate long sequences of commands; functions, or function files, provide extensibility to MATLAB by allowing us to add new functions. 2. Procedure for establishing a function M-file: 96

97 1) It begins with a statement defining the function name, its input arguments, and its output arguments. 2) It also includes additional statements that compute the values to be returned. 3) The inputs may be scalars, vectors, or matrices. Ex. Obtain the rectangular pulse depicted in Fig (a) with the use of an M-file. <Sol.> >> function g = rect(x) >> g = zeros(size(x)); >> set1 = find(abs(x)<= 0.5); >> g(set1) = ones(size(set1)); 3. The new function rect.m can be used liked any other MATLAB function. Ex. To generate a rectangular pulse: >> t = -1:1/500:1; >> plot(t, rect(t)); 1. The function size returns a twoelement vector containing the row and column dimensions of a matrix. 2. The function find returns the indices of a vector or matrix that satisfy a prescribed relation Ex. find(abs(x)<= T) returns the indices of the vector x, where the absolute value of x is less than or equal to T. 97

98 98