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CHAPTER 1 Signals and Systems EKT 232 1 Signals and Systems. 1.1 What is a Signal ? 1.2 Classification of a Signals. 1.2.1 Continuous-Time and Discrete-Time Signals 1.2.2 Even and Odd Signals. 1.2.3 Periodic and Non-periodic Signals. 1.2.4 Deterministic and Random Signals. 1.2.5 Energy and Power Signals. 1.3 Basic Operation of the Signal. 1.4 Elementary Signals. 1.4.1 Exponential Signals. 1.4.2 Sinusoidal Signal. 1.4.3 Sinusoidal and Complex Exponential Signals. 1.4.4 Exponential Damped Sinusoidal Signals. 1.4.5 Step Function. 1.4.6 Impulse Function. 1.4.7 Ramped Function. 2 Cont’d… 1.5 What is a System ? 1.5.1 System Block Diagram. 1.6 Properties of the System. 1.6.1 Stability. 1.6.2 Memory. 1.6.3 Causality. 1.6.4 Invertability. 1.6.5 Time Invariance. 1.6.6 Linearity. 3 1.1 What is a Signal ? A common form of human communication; (i) use of speech signal, face to face or telephone channel. (ii) use of visual, signal taking the form of images of people or objects around us. Real life example of signals; (i) Doctor listening to the heartbeat, blood pressure and temperature of the patient. These indicate the state of health of the patient. (ii) Daily fluctuations in the price of stock market will convey an information on the how the share for a company is doing. (iii) Weather forecast provides information on the temperature, humidity, and the speed and direction of the prevailing wind. 4 Cont’d… By definition, signal is a function of one or more variable, which conveys information on the nature of a physical phenomenon. A function of time representing a physical or mathematical quantities. e.g. : Velocity, e.g. : Velocity, acceleration of a car, voltage/current of a circuit. An example of signal; the electrical activity of the heart recorded with electrodes on the surface of the chest — the electrocardiogram (ECG or EKG) in the figure below. 5 1.2 Classifications of a Signal. There are five types of signals; (i) Continuous-Time and Discrete-Time Signals (ii) Even and Odd Signals. (iii) Periodic and Non-periodic Signals. (iv) Deterministic and Random Signals. (v) Energy and Power Signals. 6 1.2.1 Continuous-Time and Discrete-Time Signals. Continuous-Time (CT) Signals Continuous-Time (CT) Signals are functions whose amplitude or value varies continuously with time, x(t). The symbol t denotes time for continuous-time signal and ( ) used to denote continuous-time value quantities. Example, speed of car, converting acoustic or light wave into electrical signal and microphone converts variation in sound pressure into correspond variation in voltage and current. Figure 1.1: Continuous-Time Signal. 7 Cont’d… Discrete-Time Signals Discrete-Time Signals are function of discrete variable, i.e. they are defined only at discrete instants of time. xn xnTs , n 0,1,2,.... It is often derived from continuous-time signal by sampling at uniform rate. Ts denotes sampling period and n denotes integer. The symbol n denotes time for discrete time signal and [. ] is used to denote discrete-value quantities. Example: the value of stock at the end of the month. Figure 1.3: Discrete-Time Signal. 8 1.2.2 Even and Odd Signals. A continuous-time signal x(t) is said to be an even signal if xt xt for all t The signal x(t) is said to be an odd signal if xt xt for all t In summary, an even signal are symmetric about the vertical axis (time origin) whereas an odd signal are antisymetric about the origin. Figure 1.4: Even Signal Figure 1.5: Odd Signal. 9 Cont’d… 10 Example 1.1: Even and Odd Signals. Find the even and odd components of each of the following signals: (a) x(t) = 4cos(3πt) Answer: ge(t) = 4cos(3πt) go(t) = 0 11 1.2.3 Periodic and Non-Periodic Signals. Periodic Signal. A periodic signal x(t) is a function of time that satisfies the condition xt xt T for all t, where T is a positive constant. The smallest value of T that satisfy the definition is called a period. 12 Figure 1.6: Aperiodic Signal. Figure 1.7: Periodic Signal. 1.2.4 Deterministic and Random Signals. Deterministic Signal. A deterministic signal is a signal that has no uncertainty with respect to its value at any time. The deterministic signal can be modeled as completely specified function of time. Figure 1.8: Deterministic Signal; Square Wave. 13 Cont’d… Random Signal. A random signal is a signal about which there is uncertainty before it occurs. The signal may be viewed as belonging to an ensemble or a group of signals which each signal in the ensemble having a different waveform. The signal amplitude fluctuates between positive and negative in a randomly fashion. Example; noise generated by amplifier of a radio or television. Figure 1.9: Random Signal 14 Energy Signal and Power Signals. A signal with finite signal energy is called an energy signal. A signal with infinite signal energy and finite average signal power is called a power signal. 15 1.2.5 Energy Signal and Power Signals. Energy Signal. E 2 x n n A signal is refer to energy signal if and only if the total energy satisfy the condition; 0 E Power Signal. 1 P N N 1 x n 2 n 0 A signal is refer to power signal if and only if the average power of signal satisfy the condition; 16 0 P 1.3 Basic Operation of the Signals. 1.3.1 Time Scaling. 1.3.2 Reflection and Folding. 1.3.3 Time Shifting. 1.3.4 Precedence Rule for Time Shifting and Time Scaling. 17 1.3.1 Time Scaling. Time scaling refers to the multiplication of the variable by a real positive constant. yt xat If a > 1 the signal y(t) is a compressed version of x(t). If 0 < a < 1 the signal y(t) is an expanded version of x(t). Example: Figure 1.11: Time-scaling operation; 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. 18 Cont’d… In the discrete time, yn xkn, It is defined for integer value of k, k > 1. Figure below for k = 2, sample for n = +-1, Figure 1.12: 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. 19 1.3.2 Reflection and Folding. Let x(t) denote a continuous-time signal and y(t) is the signal obtained by replacing time t with –t; yt x t y(t) is the signal represents a refracted version of x(t) about t = 0. Two special cases for continuous and discrete-time signal; (i) Even signal; x(-t) = x(t) an even signal is same as reflected version. (ii) Odd signal; x(-t) = -x(t) an odd signal is the negative of its reflected version. 20 Example 1.2: Reflection. Given the triangular pulse x(t), find the reflected version of x(t) about the amplitude axis (origin). Solution: Replace the variable t with –t, so we get y(t) = x(-t) as in figure below. Figure 1.13: Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin x(t) = 0 for t < -T1 and t > T2. y(t) = 0 for t > T1 and t < -T2. . 21 1.3.3 Time Shifting. A time shift delay or advances the signal in time by a time interval +t0 or –t0, without changing its shape. y(t) = x(t - t0) If t0 > 0 the waveform of y(t) is obtained by shifting x(t) toward the right, relative to the tie axis. If t0 < 0, x(t) is shifted to the left. Example: Figure 1.14: Shift to the Left. Q: How does the x(t) signal looks like? Figure 1.15: Shift to the Right. 22 Example 1.3: Time Shifting. Given the rectangular pulse x(t) of unit amplitude and unit duration. Find y(t)=x (t - 2) Solution: t0 is equal to 2 time units. Shift x(t) to the right by 2 time units. (a) . Figure 1.16: Time-shifting operation: continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin; and (b) time-shifted version of x(t) by 2 time shifts. 23 1.3.4 Precedence Rule for Time Shifting and Time Scaling. Time shifting operation is performed first on x(t), which results in Time shift has replace t in x(t) by t - b. Time scaling operation is performed on v(t), replacing t by at and resulting in, y t vat y t xat b Example in real-life: Voice signal recorded on a tape recorder; (a > 1) tape is played faster than the recording rate, resulted in compression. (a < 1) tape is played slower than the recording rate, resulted 24 in expansion. Example 1.4: Continuous Signal. A CT signal is shown in Figure 1.17 below, sketch and label each of this signal; a) x(t -1) b) x(2t) c) x(-t) x(t) 2 t -1 3 Figure 1.17 25 Solution: (a) x(t -1) x(t) (b) x(2t) x(t-1) 2 2 t t 0 4 -1/2 3/2 (c) x(-t) x(-t) 2 t -3 1 26 Example 1.5: Continuous Signal. A continuous signal x(t) is shown in Figure 1.17a. Sketch and label each of the following signals. a) x(t)= u(t -1) b) x(t)= [u(t)-u(t-1)] c) x(t)= d(t - 3/2) Solution: Figure 1.17a (a) x(t)= u(t -1) (b) x(t)= [u(t)-u(t-1)] (c) x(t)=d(t - 3/2) 27 Example 1.5: Discrete Time Signal. A discrete-time signal x[n] is shown below, Sketch and label each of the following signal. (a) x[n – 2] (c.) x[-n+2] (b) x[2n] (d) x[-n] x[n] 4 2 0 1 2 3 n 28 Cont’d… (a) A discrete-time signal, x[n-2]. A delay by 2 x[n-2] 4 2 0 1 2 3 4 5 n 29 Cont’d… (b) A discrete-time signal, x[2n]. Down-sampling by a factor of 2. x(2n) 4 2 0 1 2 3 n 30 Cont’d… (c) A discrete-time signal, x[-n+2]. Time reversal and shifting x(-n+2) 4 2 -1 0 1 2 n 31 Cont’d… (d) A discrete-time signal, x[-n]. Time reversal x(-n) 4 2 -3 -2 -1 0 1 n 32 In Class Exercises . A continuous-time signal x(t) is shown below, Sketch and label each of the following signal (a) x(t – 2) (b) x(2t) (c.) x(t/2) (d) x(-t) x(t) 4 0 4 t 33 1.4.2 Sinusoidal Signals. A general form of sinusoidal signal is xt A coswot q where A is the amplitude, wo is the frequency in radian per second, and q is the phase angle in radians. Figure 1.20: Continuous-Time Sinusoidal signal A cos(ωt + Φ). 34 1.4.5 Step Function. The discrete-time version of the unit-step function is defined by, 1, un 0, n0 n0 Figure 1.24: Discrete–time of Step Function of Unit Amplitude. 35 The Unit Step Function 1 , t 0 u t 1/ 2 , t 0 0 , t 0 Precise Graph Commonly-Used Graph The product signal g(t)u(t) can be thought of as the signal g(t) “turned on” at time t = 0. 3/22/2016 Dr. Abid Yahya 36 The Unit Ramp Function t , t 0 t ramp t u d t u t 0 , t 0 3/22/2016 37 1.4.6 Impulse Function. The discrete-time version of the unit impulse is defined by, 1, n 0 d n 0, n 0 Figure 1.26: Discrete-Time form of Impulse. Figure 1.26 is a graphical description of the unit impulse d(t). The continuous-time version of the unit impulse is defined by the following pair, d n 0 for t0 d t dt 1 The d(t) is also refer as the Dirac Delta function. 38 Graphical Representation of the Impulse The impulse is not a function in the ordinary sense because its value at the time of its occurrence is not defined. It is represented graphically by a vertical arrow. Its strength is either written beside it or is represented by its length. 3/22/2016 39 The Unit Step and Unit Impulse As a approaches zero, g t approaches a unit step and g t approaches a unit impulse The unit step is the integral of the unit impulse and the unit impulse is the generalized derivative of the unit step 3/22/2016 40 Properties of the Impulse The Sampling Property g t d t t dt g t 0 0 The sampling property “extracts” the value of a function at a point. The Scaling Property 1 d a t t0 d t t0 a This property illustrates that the impulse is different from ordinary mathematical functions. 3/22/2016 41 The Unit Periodic Impulse The unit periodic impulse is defined by dT t d t nT , n an integer n The periodic impulse is a sum of infinitely many uniformly-spaced impulses. 3/22/2016 42 FEW Questions from Past Exam Paper , for Practice and Use as a Tutorial 43 Q.1 44 A.1 45 Q.2 46 A.2 47 A.2 48 Q.3 49 A.3 50 A.3 51 Q.4 52 Q.5 53 A.5 54 A.5 55 Q.6 56 Q.6 57 Q.6 58 Q.6 59 Q.6 60 Q.7 61 Q.7 62 Q.7 63 The Rectangle Function 1 , n N w rect N w n , N w 0 , N w an integer 0 , n N w 3/22/2016 Dr. Abid Yahya 64 Q.9 Find the signal energy of these signals. 65 66 1.5 What is a System ? A system can be viewed as an interconnection of operation that transfer an input signal into an output signal with properties different from those of the input signal. y(t) is the impulse response of the continuous-time system and y[t] is the impulse response of the discrete-time system. 67 Cont’d… Real life example of system; (i) In automatic speaker recognition system; the system is to extract the information from an incoming speech signal for the purpose of recognizing and identifying the speaker. (ii) In communication system; the system will transport the the information contained in the message over a communication channel and deliver that information to the destination. Figure 1.30: Elements of a communication system. Figure 1.31: Block diagram representation of a system. 68 Cont’d… By definition, a system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. A physical process or a mathematical model of the physical process that relates a set of input signals to yield another set of output signal. Process input signals to produce output signals System representation of the systems. 69 1.6 Properties of Systems. The properties of a system describe the characteristics of the operator H representing the system. Basic properties of the system; 1.6.1 Stability. 1.6.2 Memory. 1.6.3 Causality. 1.6.4 Invertibility. 1.6.5 Time Invariance. 1.6.6 Linearity. 70 1.6.1 Stability. A system is said to be bounded-input bounded-output (BIBO) stable if and only if all bounded inputs result in bounded outputs. The output of the system does not diverge if the input does not diverge. For the resistor, if i(t) is bounded then so is v(t), but for the capacitance this is not true. Consider i(t) = u(t) then v(t) = tu(t) which is unbounded. 71 1.6.2 Memory. A system is said to possess memory if its output signal depend on pass or future values of the input signal. Note that v(t) depends not just on i(t) at one point in time t. Therefore, the system that relates v to i exhibits memory. The system is said to be memoryless if its output signal depends only on the present value of the input signal. Example: The resistive divider network Therefore, vo(to) depends upon the value of vi(to) and not on vi(t) 72 for t = to. Example 1.6: Memory and Memoryless System. Below is the moving-average system described by the input-output relation. Does it has memory or not? (a) yn 1 xn xn 1 xn 2 3 (b) yn x 2 n Solution: (a) It has memory, the value of the output signal y[n] at time n depends on the present and two pass values of x[n]. (b) It is memoryless, because the value of the output signal y[n] depends only on the present value of the input signal x[n]. . 73 1.6.3 Causality. Causal. A system is said to be casual if the present value of the output signal depends only on the present or the past values of the input signal. The system cannot anticipate the input. Noncausal. In contrast, the output signal of a noncausal system depends on one or more future values of the input signal. 74 Causality Any system for which the response occurs only during or after the time in which the excitation is applied is called a causal system. All real physical systems are causal 3/22/2016 Dr. Abid Yahya 75 Example 1.7: Causal and Noncausal. 1 yn xn 1 xn xn 1 3 Causal or noncausal? Solution: Noncausal; the output signal y[n] depends on a future value of the input signal, x[n+1] yn 1 xn 1 xn xn 1 3 Causality is required for a system to be capable of operating in real time. . 76 1.6.4 Invertibility. A system is said to be invertible if the input of the system can be recovered from the output. H invyt H invH xt H invH xt Figure 1.32: The notion of system inevitability. The second operator Hinv is the inverse of the first operator H. Hence, the input x(t) is passed through the cascade correction of H and H-1 completely unchanged. 77 1.6.5 Time Invariance. A system is said to be time invariant if the time delay or time advance of the input signal leads to an identical time shift in the output signal. The Time invariance system responds identically no mater when the input signal is applied. HS t0 S t0 H Figure 1.33: (a) Time-shift operator St0 preceding operator H. (b) Time-shift operator St0 following operator H. These two situations are equivalent, provided that H is time invariant 78 1.6.6 Linearity. A system is said to be linear in term of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two properties. 1. Superposition The system is initially at rest. The input is x(t)=x1(t), the output y(t)=y1(t). So x(t)=x1(t)+x2(t) the corresponding output y(t)=y1(t)+y2(t). 2. Homogeneity/Scaling The system is initially at rest. Input x(t) result in y(t). The system exhibit the property of homogeneity if x(t) scaled by constant factor a result in output y(t) is scaled by exact constant a. 79 Example: 1. Determine whether the system 1. y[n] = nx[n-1]+2 is : (i) memoryless (ii) time invariant (iii) linear (iv) causal (v) stable 80 3. Determine whether the corresponding system is linear, time invariant or both. (a) y[n] = x2[n-2] (b) y[n] = Od{x(t)} 81