1. BYST CPE200-W2003: Introduction 1 CPE200 Signals and Systems Chapter 1: Introduction

2. BYST CPE200-W2003: Introduction 2 Signals and Systems To study and analyze some physical phenomena, we need to determine the mean allowing us to understand and describe such phenomena in a systematic way. In practice, we can accomplish this goal by describing a physical phenomenon as a mathematical function called “Signal”. 1. 1 We encounter so many signals in our daily life which are generated by natural means. For example, the speech or the sound coming into our ears is a mechanical signal representing the air pressure. The picture that we see is a light signal representing the intensity of light. All electrical devices are associated with voltage and current signals.

3. BYST CPE200-W2003: Introduction 3 Definition 1: Signal is a pattern of variations of a measurable quantity that is a function of one or more independent variables such as time (t) and space (x and y). Signal can be represented mathematically as: S(t) = at S(t) = at 2 + bt + c S(x,y) = ax + by + cxy S(t) = Asin(  t+  ) (1.1) Where a, b, c, A, , and  are constant values. The function which is used to describe a signal is called the representation of the signal.

4. BYST CPE200-W2003: Introduction 4 There are 3 components to signal theory: Typically, a signal carries information about the behavior or nature of the phenomenon. Modeling: A process to determine a representation of the signal. Analysis: A process to extract information carried by the signal (Signal Processing). Design: A process to synthesize a physical process that is described by the signal. Examples of signals can be shown as following:

5. BYST CPE200-W2003: Introduction 5 0200400600800100012001400160018002000 -0.5 0 0.5 1 1.5 x 10 4 A Speech Signal A speech signal is a mechanical signal representing the air pressure and carries voice information. Amplitude Time A Speech Waveform Figure 1.1 Speech waveform.

6. BYST CPE200-W2003: Introduction 6 A Digital Image (2-D Signal) A digital image is a light signal representing the light intensity and carries visual information. The intensity I(x,y) at any location of a digital image is a function of two spatial independent variables (x and y). Figure 1.2 Digital image (2-D signal).

7. BYST CPE200-W2003: Introduction 7 A continuous-time (c-t) signal is a signal that is present for all instants in time or space. 1.1.1 Types of Signals Continuous-Time ( c-t or analog ) Signal Time Amplitude 1. Figure 1.3 Continuous-time (c-t) signal.

8. BYST CPE200-W2003: Introduction 8 A discrete-time (d-t) signal is a signal that is present only at certain specific values of time or space. 2. Discrete-Time (d-t) Signal Amplitude Time Figure 1.4 Discrete-time (d-t) signal.

9. BYST CPE200-W2003: Introduction 9 A digital signal is a discrete-time signal having a set of discrete values. 3. Digital Signal Amplitude Time Figure 1.5 A digital signal

10. BYST CPE200-W2003: Introduction 10 Typically, we will denote continuous-time signals with the continuous-time independent variable, t. We will also enclose the independent variable in parentheses ( ). For example, x(t), y(t), o(t), etc. In the case of discrete-time signals, we typically denote them with the discrete-time independent variable, n and enclose the independent variable in brackets [ ]. For example, x[n], y[n], o[n], etc.

11. BYST CPE200-W2003: Introduction 11 System is a physical device that performs an operation on a signal. We can consider a system as anything that takes an input signal, operates on it, and produces an output signal. System  x(t)y(t) Input (or Excitation) signal Output (or Response) signal Let x(t) and y(t) be the input and output signals, respectively, of a system. The system can be represented mathematically as: y(t) =  [x(t)] (1.2) 1.1.2 Systems Figure 1.6A block diagram representation of a continuous-time system.

12. BYST CPE200-W2003: Introduction 12 Examples of Systems RC Circuit x(t) = i(t) R C + - V c (t) = y(t) Mass-Spring-Damper System M K D f(t) y(t) c dy(t) dt 1 R +y(t)= x(t) Figure 1.7RC Network. Figure 1.8Mass-Spring-Damper System.

13. BYST CPE200-W2003: Introduction 13 D = damping coefficient M = mass K = spring constant f(t) = force y(t) = displacement of the mass M d 2 y(t) dt 2 dy(t) dt +D+Ky(t)= f(t) Communication System Transmitter Message Signal Channel Transmitted Signal Receiver Received Signal Received Message Signal Figure 1.9Element of the communication system.

14. BYST CPE200-W2003: Introduction 14 Signal processing is a method to extract useful information carried by the signal. Analog Signal Processor C-T input signal C-T output signal There are two types of signal processing systems: analog signal processing and digital signal processing. 1.1.3 Signal Processing 1. Analog signal processing system. This is a system that processes the input signal directly on its analog form. Figure 1.10A block diagram representation of an analog signal processing system.

15. BYST CPE200-W2003: Introduction 15 2. Digital signal processing system. The digital signal processing system, on the other hand, is a system that processes the input signal on its digital form. Fig. 1.11 shows a block diagram of the digital signal processing system. To perform the processing digitally, the digital signal processing system requires two additional steps. The first step is the step to convert a continuous-time signal into a discrete-time discrete valued (digital) signal which is called the analog-to-digital (A/D) conversion. A digital signal produced from the digital signal processor is converted back to continuous-time form by the other additional step called the digital-to-analog (D/A) conversion.

16. BYST CPE200-W2003: Introduction 16 Digital Signal Processor C-T input signal C-T output signal Analog-to- Digital Converter Digital-to- Analog Converter Digital signal Figure 1.11A block diagram representation of a digital signal processing system.

17. BYST CPE200-W2003: Introduction 17 Analog-to-Digital Conversion 1. 2 The digital signal is generated from the analog (continuous-time) signal using these following two-step process: Analog Signal Discrete-Time Signal Digital Signal 1. Sampling Process 2. Quantization Process A sampling process is the process to sample a continuous-time (c-t) signal at a certain period of time called the sampling interval. Figure 1.12A two-step process to generate a digital signal from an analog signal.

18. BYST CPE200-W2003: Introduction 18 Thus, the sampling process will convert a c- t signal into a discrete-time (d-t) signal. Where T = the sampling period i.e. (1.3) F s = the sampling frequency = 1/T n = 0, ±1, ±2,… x a (t) F s = 1/T Sampler x(n) = x a (nT) Analog Signal Discrete-time Signal Figure 1.13 Periodic sampling of an analog signal.

19. BYST CPE200-W2003: Introduction 19 A quantization process is the process to round up the values of the d-t signal to a finite set of possible values. Thus, the quantization process will convert a d-t continuous-valued signal into a d-t discrete- valued (digital) signal. A digitization process is the process to convert a analog signal into an encoded digital signal. This process is usually called analog-to-digital (A/D) conversion and is illustrated in this Fig. 1.14. 1.2.1 Digitization

20. BYST CPE200-W2003: Introduction 20 Sampler Quantizer Encoder Figure 1.14A block diagram of a digitization process. Analog signal Discrete-time continuous-valued signal Discrete-time discrete-valued (digital) signal Encoded-digital signal

21. BYST CPE200-W2003: Introduction 21 Digital Sound Recording Sampling Quantizing Encoding An Example of Digital Signal Processing

22. BYST CPE200-W2003: Introduction 22 The Concept of Frequency in Continuous-Time and Discrete-Time Signals 1. 3 Acos(  ) x a (t) TpTp x a (t) = Acos(  t +  ) An analog sinusoidal signal x a (t) can be represented as: t (1.4) Figure 1.15 An analog sinusoidal signal. Where

23. BYST CPE200-W2003: Introduction 23 = T p = the fundamental period (sec) 1 F The analog sinusoidal signal described by Eq. 1.3 has these following properties: 1. Periodical: i.e. 2. Distinction: C-t sinusoidal signals with different frequencies are themselves distinct. x a (t + T p ) = x a (t) 3. The rate of oscillation of the signal will be increased if the frequency F is increased.  = the angular frequency (rad/sec) = 2  F F = the frequency (cycles/sec)  = the phase (1.5)

24. BYST CPE200-W2003: Introduction 24 x(n) = Acos(  n +  ) n Figure 1.16 A discrete-time sinusoidal signal. x[n] = Acos(  n +  ) A discrete-time sinusoidal signal x(n) can be represented as: (1.6)  = the angular frequency (rad/sample) Where = 2  f

25. BYST CPE200-W2003: Introduction 25 f = the frequency (samples/sec)  = the phase The discrete-time sinusoidal signal described by Eq. 1.6 has these following properties: 1. x[n + N] = x[n] for all n A discrete-time sinusoid is periodic only if the frequency of the d-t signal is a rational number. By definition, a d-t signal x(n) is periodic with period N (N>0) if an only if (1.7) The smallest possible N satisfying the above condition is called “fundamental period” of the d-t signal.

26. BYST CPE200-W2003: Introduction 26 = 2  f 0 Eq. 1.8 will be equal to Eq. 1.6 only if sin   N = 0 and cos   N = 1 which can be satisfied if and only if: If x[n] = Acos(   n +  ), x[n+N] will be defined as: x[n+N] = Acos(    n+N) +  ) = A[cos(   n+  )cos    - sin(   n+  )sin    )] (1.8) Where   = the fundamental angular frequency f 0 = the fundamental frequency

27. BYST CPE200-W2003: Introduction 27 For a d-t sinusoidal x[n] with frequency f 0 to be periodic, this following condition must exist for any integer k: f 0 = k N (1.9) i.e.  0 = 2k2k N or   N = 2  k Where k = any integer number

28. BYST CPE200-W2003: Introduction 28 2. Non-distinction: Discrete-time sinusoids that have frequencies separated by an integer multiple of 2  are identical. Hence the frequency range for d-t sinusoids is finite with duration 2 . Usually, the range: 3. At  =  or -  (f=0.5 or -0.5), the highest frequency in a discrete-time signal is obtained. is used and it is called the fundamental range.

29. BYST CPE200-W2003: Introduction 29 To establish the relationship between the frequency F (or  ) of analog signals and the frequency f (or  ) of d-t signals, we start from considering an analog sinusoidal signal expressed as shown in Eq. (1.4): x a (t) = Acos(  t +  ) When we sample this analog signal at a rate of F s samples per second, the d-t signal x(n) can be expressed as follows: x a (nT) x(n) = Acos(  nT +  ) (1.10) By comparing Eq. (1.10) with Eq. (1.4), the relationship between the angular frequency of a d-t signal  and the angular frequency of an analog signal  can be expressed as shown in Eq. (1.11):

30. BYST CPE200-W2003: Introduction 30 (1.11)  =  T (1.12) f = F/F s or The conversion from the analog frequency to the digital frequency or the conversion from the digital frequency to the analog frequency can be summarized as illustrated in Table 1.1. From Table 1.1, we can notice that the basic difference between c-t and d-t signals is in their range of frequency values. C-T signals have the infinite frequency range. D-T signals, on the other hand, have the finite frequency range. Thus there is a possibility that more than one analog frequencies are mapped into the same digital frequency.

31. BYST CPE200-W2003: Introduction 31 Table 1.1 Relations among frequency variables C-T SignalsD-T Signals  = 2  F  = 2  f  =  T and f = F/F s  =  T and F = fF s - <  < - < F < < _ -- < _  -1/2 < _ < _ < _ f 1/2 < _  -  -F s /2 < _ < _ F F s /2

32. BYST CPE200-W2003: Introduction 32 The highest frequency in c-t signal depends on the sampling frequency. Since the highest frequency in a d-t signal is: (1.14) (1.13)  max =  or f max = 1/2, the corresponding highest values of analog frequency must be: (1.15) or

33. BYST CPE200-W2003: Introduction 33 The Sampling Theorem An analog signal x a (t) can be reconstructed from its sample values x a (nT) if the sampling rate 1/T is greater than twice the highest frequency F max presenting in x a (t). Definition:The sampling rate 2Fmax for an analog band-limited signal is referred to as the Nyquist rate. Therefore, any frequency above F s /2 or below -F s /2 results in samples that are identical to corresponding frequency in the range:

34. BYST CPE200-W2003: Introduction 34 Thus, we must have some knowledge about the frequency content of any given analog signal in order to sample it with an appropriate sampling rate. However, this detailed knowledge of the characteristics of such signals is normally not available prior to obtaining the signal. In fact, it is the information that we would like to extract in digital signal processing. From the sampling theorem, we must sample a c-t signal with Fs > 2Fmax to ensure that all the sinusoidal components in the c-t signal are mapped into corresponding d-t frequency components with frequency in the fundamental interval.

35. BYST CPE200-W2003: Introduction 35 Classification of Signals 1. 3 (1.16) (1.17) For a c-t signal, and Let x(t) be a c-t signal and x[n] be a d-t signal. The total energy, E, and power, P, of x(t)over an infinite time interval time interval can be determined as follows: 1.3.1 Energy and Power Signals

36. BYST CPE200-W2003: Introduction 36 (1.18) (1.19) and Similarly, E and P for a d-t signal can be defined as follows: A signal x(t)/x[n] is called an “energy signal” if and only if the total energy is finite which means the power is equal to zero. A signal x(t)/x[n] is called a “power signal” if and only if the total energy is infinite and the power is finite.

37. BYST CPE200-W2003: Introduction 37 Energy signal Power signal 0 i.e. EP 1.3.2 Periodic and Aperiodic Signals A signal x(t) is periodic with period T (T>0) if and only if: x(t+kT) = x(t)(1.20) Where k is any integer. The smallest value of T for which Eq. 1.20 holds is called the “fundamental period”.

38. BYST CPE200-W2003: Introduction 38 A signal x(t) that is not periodic will be called an “aperiodic” signal. On the other hand, if there is no value of N satisfying Eq. 1.21, the signal is called an “aperiodic” (non-periodic) signal. In case of a d-t signal, a signal x[n] is periodic with period N (N>0) if and only if: x[n+kN] = x[n] for all n(1.21) Where k is any integer. The smallest value of N for which Eq. 1.21 holds is called the “fundamental period”.

39. BYST CPE200-W2003: Introduction 39 A signal x(t) or x[n] is called a conjugate- symmetric signal if 1.3.3 Classification Based on Symmetry x(t) = x*(-t) x[n] = x*[-n] or(1.22) On the other hand, signal x(t) or x[n] is called a conjugate- antisymmetric signal if x(t) = -x*(-t) x[n] = -x*[-n] or(1.23)

40. BYST CPE200-W2003: Introduction 40 In the case of real-value signal, a conjugate- symmetric signal is called an “even signal” and a conjugate-antisymmetric signal is called an “odd signal”. Thus x(t) = x (-t) x[n] = x [-n] or(1.24) x(t) = -x(-t) x[n] = -x[-n] or(1.25) Even signal: Odd signal: From Eq. 1.25, x(0) = -x(0); therefore, an odd signal must be 0 at t=0 or n=0.

41. BYST CPE200-W2003: Introduction 41 Any complex signal x(t) or x[n] can be expressed as a sum of its conjugate- symmetric part x cs (t) or x cs [n] and its conjugate-antisymmetric part x ca (t) or x ca [n] as show in Eq. 1.26. x(t) = x cs (t) + x ca (t) x[n] = x cs [n] + x ca [n] or(1.26) or(1.27) Where x cs (t) = (x(t) + x*(-t)) 1 2 x cs [n] = (x[n] + x*[-n]) 1 2 and

42. BYST CPE200-W2003: Introduction 42 (1.28)or x ca (t) = (x(t) - x*(-t)) 1 2 x ca [n] = (x[n] - x*[-n]) 1 2 Similarly, any real signal x(t) or x[n] can be expressed as a sum of its even part x ev (t) or x ev [n] and its odd part x od (t) or x od [n] as show in Eq. 1.29. x(t) = x ev (t) + x od (t) x[n] = x ev [n] + x od [n] or(1.29) or(1.30) Where x ev (t) = (x(t) + x(-t)) 1 2 x ev [n] = (x[n] + x[-n]) 1 2

43. BYST CPE200-W2003: Introduction 43 (1.31)or x od (t) = (x(t) - x(-t)) 1 2 x od [n] = (x[n] - x[-n]) 1 2 and An example of the even signal is a sinusoidal signal expressed as the cosine function. A sinusoidal signal expressed as the sine function is an example of the odd signal. Basic Operation on Signals 1. 4 Typically, in the area of signal processing, it is required some basic operations to perform a simply process such as addition, multiplication, amplify, etc. on the signals. Note: Since all operations discussed later will perform the same results to c-t and d-t signals, only the d-t signals will be discussed or illustrated for convenience.

44. BYST CPE200-W2003: Introduction 44 1.4.1 A Signal Multiplier A signal multiplier will form the product of values of x 1 (t)/x 1 [n] and x 2 (t)/x 2 [n] signals at each instant as illustrated in Fig. 1.17. Figure 1.17 A signal multiplier. For a signal multiplier having a sinusoidal signal as one of its input, this operation will be called “modulation”. The device performs the modulation operation is known as a “modulator”.

45. BYST CPE200-W2003: Introduction 45 1.4.2 An Adder An adder will form the addition of values of x 1 (t)/x 1 [n] and x 2 (t)/x 2 [n] signals at each instant as illustrated in Fig. 1.18. Figure 1.18 An adder. An output y(t) of a scalar multiplier will be the result of multiplication a input signal x(t) with a scalar A as illustrated in Fig. 1.19. 1.4.3 A Scalar Multiplier

46. BYST CPE200-W2003: Introduction 46 Figure 1.19 A scalar Multiplier An operation that performs a time delaying and advancing on signal is known as “time- shifting” operation. 1.4.4 A Time Shifting Let x(t) or x[n] be an input signal and y(t) or y[n] be an output signal resulting from a time-shifting operation. y(t) and y[n] will be defined as follows: y(t) = x(t - t 0 ) where t 0 is the time shift and N is an integer (1.32) y[n] = x[n - N]

47. BYST CPE200-W2003: Introduction 47 t 0 > 0 Delaying (Shift to the right) t 0 < 0 Advancing (Shift to the left) N > 0 N < 0 1.4.5 A Time Reversal A time reversal, which sometimes is called the folding or the reflection operation, is the operation to replace the independent variable “t” or “n” by “-t” or “-n” as shown in Eq. 1.33. y(t) = x(-t) (1.33) y[n] = x[-n]

48. BYST CPE200-W2003: Introduction 48 Let TD and TR represent the time-shifting and time-reversal, respectively. Thus, we can represent TD and TR as shown in Eq. 1.34. TD k [x[n]] = x[n-k] k > 0 (1.34) TR[x[n]] = x[-n] The combination operation of TD and TR can be expressed as either Eq. 1.35 or Eq. 1.36. We can notice that the result of performing TR before TD (Eq. 1.35) is different from the result of performing TD before TR (Eq. 1.36). TD k {TR[x[n]]} = TD k {x[-n]} (1.35) = x[-n+k]

49. BYST CPE200-W2003: Introduction 49 TR{TD k [x[n]]} = TR{x[n-k]} (1.36) = x[-n-k] Figure 1.20 Graphical illustration of the different output signal generated by: (a) Time-delaying the input signal and then folding. (b) Folding the input signal and then time delaying.

50. BYST CPE200-W2003: Introduction 50 1.4.6 Time Scaling Let x(t) be a c-t signal and a d-t signal, respectively. The signal y(t) obtained by scaling the independent variable, time t, by a factor of a is defined by y(t) = x(at)(1.37) Compressing a > 1 Stretching a < 1 In the discrete time case, this operation is known as “sampling rate alteration”. Let x[n] be a d-t signal with a sampling rate of F x Hz. And y[n] be the d-t signal from altering the sampling rate F x Hz. To a new sampling rate F y Hz. The sampling rate alteration ration R can be expressed by

51. BYST CPE200-W2003: Introduction 51 (1.37) FyFy FxFx = R If R is greater than 1, the process is called “interpolation” process and the operation is call “up-sampling”. Thus, the up-sampling operation will increase the number of samples in the input signal. If R is less than 1, on the other hand, the process is called “decimation” process and the operation is call “down-sampling”. The number of samples in the input signal will be decreased after the down-sampling process. To perform an up-sampling operation by an integer factor of L (L>1), L-1 new samples between successive values of the input signal will be interpolated.

52. BYST CPE200-W2003: Introduction 52 This interpolation process can be accomplished in a various means. The easiest and very common mean to up- sampling the sequence is performed by adding L-1 zeros between successive values of the input signal x[n]. This up-sampling operation can be expressed by Eq. 1.38. (1.38) The decimation process by a factor of an integer M (M>1) can be performed by taking only every M th. Sample of the input signal. This results in a signal with a lower sampling rate. The down-sampling operation by a factor of M is expressed by Eq. 1.39 (1.39) x d [n] = x[Mn]

53. BYST CPE200-W2003: Introduction 53 (a) (b) Figure 1.21 Graphical illustration of the up-sampling operation. (a) the input signal. (b) Illustration of the output resulting from up-sampling the input in (a) by 4.

54. BYST CPE200-W2003: Introduction 54 (a) (b) Figure 1.22 Graphical illustration of the down-sampling operation. (a) the input signal. (b) Illustration of the output resulting from down-sampling the input in (a) by 4.

55. BYST CPE200-W2003: Introduction 55 Elementary Signals 1. 5 There are some elementary signals that are used in studying signals and systems. Such signals are exponential and sinusoidal signals (See section 1.3), the impulse function, the step function, and ramp function. A exponential signal is of the form: 1.5.1 Exponential Signals x(t) = Ce at (1.40) If C and a are complex numbers, a exponential signal x(t) defined by Eq. 1.40 will be called “complex exponential” signal. In the case that C and a are real, x(t) defined by Eq. 1.40 will be called “real exponential” signal.

56. BYST CPE200-W2003: Introduction 56 There are basically two types of exponential signals which determines by the value of “a”. Growing exponential a > 0 Decaying exponential a < 0 1.5.2 Relationship Between Sinusoidal and Complex Exponential Signals Consider the complex exponential containing “a” to be purely imaginary defined as follows: (1.41) whereand A is real. Using Euler’s identity, we can write the complex exponential in Eq. 1.41 in term of

57. BYST CPE200-W2003: Introduction 57 sinusoidal signals as: (1.42) (1.43) i.e. and where Re{ } and Im{ } denote the real and imaginary part of the complex number, respectively. The term Acos(   t +  ) can also represent as: (1.44)

58. BYST CPE200-W2003: Introduction 58 Similarly, in the d-t system, we can represent: (1.45) and 1.5.3 The D-T Unit Impulse and Unit Step Functions The d-t unit impulse (or unit sample) signal,  [n], plays the most important role in the representation of any d-t signals and in the analysis of a d-t system. This signal is defined as:

59. BYST CPE200-W2003: Introduction 59 The unit impulse signal is a signal that is zero everywhere, except at n=0 where its value is unity. (1.46) Any D-T signal is the sum of scaled and shifted unit impulses. = ++ + (1.47) 2 2 3 -101 -2 2

60. BYST CPE200-W2003: Introduction 60 The d-t unit step signal is denoted as u[n] and is defined as: (1.48) From Eq. 1.46 and Eq. 1.48, the relationship between the unit impulse signal and the unit step signal can be stated as follows: (1.49) (1.50)  [n] = u[n] - u[n-1] When x[n] in the Eq. 1.47 is equal to u[n], the Eq. 1.47 will be reduced to: (1.51) (running sum)

61. BYST CPE200-W2003: Introduction 61 1.5.4 The C-T Unit Impulse and Unit Step Functions The c-t unit step signal is denoted as u(t) and is defined as: (1.52) Eq. 1.52 indicates that the c-t unit step u(t) is discontinuous at t=0. The c-t unit impulse, commonly denoted by  (t), is defined by the following pair of relations: (1.53) (1.54)

62. BYST CPE200-W2003: Introduction 62 Eq. 1.53 states that the c-t unit impulse  (t) is zero everywhere except at the origin and Eq. 1.54 states that the total area under  (t) is unity. The c-t unit impulse  (t) is also referred to as the Dirac delta function. Similar to the d-t case, u(t) the running integral of  (t) with respect to time t as illustrated by Eq. 1.55. Conversely,  (t) is the derivative of u(t) as illustrated by Eq. 1.56. (1.55) (running integral) (1.56)

63. BYST CPE200-W2003: Introduction 63 As we can see that the relationship between  (t) and u(t) is harder to visualize than the relationship between  [n] and u[n] because u(t) is discontinuous at t=0. One way to visualize this relationship is to consider an approximation to unit step u  (t) which slowly increases its values from 0 to 1 in a short time interval of length  as illustrate in Fig. 1.23. t 0  u  (t) 1 Figure 1.23 Continuous approximation to the unit step, u  (t). Thus (1.57)

64. BYST CPE200-W2003: Introduction 64 Unlike u(t), u  (t) is differentiable and the result is illustrate in Fig. 1.24. t 0    (t) 1  Figure 1.24 Derivative of u  (t). (1.58) Thus Note that we can consider   (t) is a short pulse having a unit area for any value of . As  is decreased, its amplitude is increased such that the area under the pulse is maintained constant at unity. Finally, if  is infinitesimal (  → 0), its amplitude will be

65. BYST CPE200-W2003: Introduction 65 i.e.  (t) has infinite amplitude but finite area. infinite. However, its area still remain at unity. Thus, we can consider  (t) as a pulse having infinitesimal width. (1.59) It is convention to graphically illustrate  (t) as an arrow having a unit area (Fig. 1.25). t 0   (t) 1 Figure 1.24 Derivative of u  (t). Area = 1 (Not an Amplitude) Area is concentrated at t=0.

66. BYST CPE200-W2003: Introduction 66 1.5.5 Ramp Functions The ramp signal is denoted as u r (t) and is defined as: (1.60) The ramp signal is similar to the unit step signal, except its value is equal to t when t is greater than or equal to 0. The ramp signal is considered as the integral of the unit step function u(t). Equivalently, we may defined the ramp signal as: u r (t) = tu(t) (1.61)

67. BYST CPE200-W2003: Introduction 67 (1.62 ) For a d-t ramp signal, or, equivalently, u r [n] = nu[n] (1.63) Basic System Properties 1. 6 The properties of a system describe the characteristic of the operator  (See 1.1.2) representing the system. In this section we will consider the most basic properties of systems such as static/dynamic, linearity, causality, time-invariance, and stability.

68. BYST CPE200-W2003: Introduction 68 1.6.1 Static and Dynamic System Static System: current A system that the output depends only on the current input. = memoryless Thus, a static system will be considered as a system without memory (memoryless). Dynamic System: current pastfuture A system that the output depends not only on the current input but also on the past or the future inputs. = memory Thus, a dynamic system will be considered as a system with memory. memory storage of energy Note that “memory”, in many physical systems, is referred to the storage of energy.

69. BYST CPE200-W2003: Introduction 69 1.6.2 Invertibility and Inverse System InputOutput 1 to 1 Mapping Invertible For each input, the system will produce a unique output. invertibleinput recovered existsinverse system A system is said to be invertible if the input of the system can be recovered from the system output. In other words, If a system is invertible, there exists an inverse system such that when cascaded with the original system, yields an output equal to the original input (Fig. 1.25).

70. BYST CPE200-W2003: Introduction 70 System  Inverse System  -1 x(t)y(t) Figure 1.25 A general invertible system. w(t) = x(t) inverse system  -1 inverse operator  -1 From Fig. 1.25, the second c-t system is called an “inverse system” and represented by the operator  -1. This operator is called the “inverse operator”. Note that, here,  -1 is not the reciprocal of the operator . i.e. In this case,

71. BYST CPE200-W2003: Introduction 71 1.6.3 Linearity Theorem linear A system is linear if and only if  {a 1 x 1 [n] + a 2 x 2 [n]} = a 1  {x 1 [n]} + a 2  {x 2 [n]} (1.64) where a 1 and a 2 = any constants. x 1 [n] and x 2 [n] = any input signals, superposition principle it satisfies the superposition principle defined by Eq 1.64: Let y[n] be the output of a d-t system when the input signal is x[n]. Thus y[n] =  {x[n]}.

72. BYST CPE200-W2003: Introduction 72 Additivity 2. Additivity property. Then, Eq. 1.64 simply states that a linear system requires two properties: ScalingMultiplicative 1. Scaling property (Multiplicative)  {ax[n]} = a  {x[n]} = ay[n] If the input signal x[n] is scaled by “a”, the output of a linear system will be scaled by the same factor. Thus, (1.65) This property can be simply demonstrated by Fig. 1.26 or Eq. 1.66 as follows:  {x 1 [n] + x 2 [n]} =  {x 1 [n]} +  {x 2 [n]} (1.66)

73. BYST CPE200-W2003: Introduction 73 The scaling and additivity properties of a linear system can be depicted by Fig. 1.26 as follows: x 1 [n] x 2 [n] a1a1 a2a2 + y[n]    x 1 [n] x 2 [n] a1a1 a2a2 + y’[n] Figure 1.26 Graphical representation of the superposition principle.

74. BYST CPE200-W2003: Introduction 74 1.6.4 Time Invariance Theorem time invariant A system  is time invariant or shift invariant or shift invariant if and only if y[n] =  {x[n]} implies that y[n-k] =  {x[n-k]}. time shift k for every input signal x[n] and every time shift k. where y[n] = the response of a system when the input is x[n], y[n,k] = the response of a system when the input is x[n-k], and (1.67)

75. BYST CPE200-W2003: Introduction 75 characteristics not a function of time Eq. 1.67 implies that the output (response) signal of a time-invariant system does not depend on the time when the input is applied, k. In the other words, the characteristics of a time-invariant system is not a function of time. time variant In contrast, the system is said to be time variant if its characteristics is a function of time. y[n-k] = y[n] which is delayed by k. = TD k {y[n]}. Therefore, if y[n,k] = y[n-k] for all k, Time-Invariant System

76. BYST CPE200-W2003: Introduction 76 y[n,k] ≠ y[n-k] for all k, If Time-Variant System 1.6.5 Causality causal A system is causal if the response at any time does not depend on values of the future inputs. Thus, in a causal system, the n 0 th (t 0 th) output signal depends only on input signals x[n] (x(t)) for n ≤ n 0 (t ≤ t 0 ). i.e. For a casual system, y[n] = F{x[n], x[n-1], x[n-2], …} or y(t) = F{x(t), x(t-1), x(t-2), …}

77. BYST CPE200-W2003: Introduction 77 1.6.6 Stability bounded input- bounded output (BIBO) stable A system is said to be bounded input- bounded output (BIBO) stable if and only if every bounded input results in a bounded output. x[n] is the bounded input if |x[n]| ≤ M x < ∞ for all n or |x(t)| ≤ M x < ∞ for all t (1.68) where M x = some finite positive numbers Therefore, whenever, the input signals satisfy Eq. 1.68, the output of BIBO stable system must satisfies the following condition:

78. BYST CPE200-W2003: Introduction 78 |y[n]| ≤ M y < ∞ for all n or |y(t)| ≤ M y < ∞ for all t (1.69) where M y = some finite positive numbers