2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal.

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2 2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 2

3 Time Scaling. Figure 2.4 Time scaling of a signal Classification of Signals A continuous-time signal is a signal x(t) for which the independent variable t takes real numbers. A discrete-time signal, denoted by x[n], is a signal for which the independent variable n takes its values in the set of integers. 3

4 Figure 2.5 Examples of discrete-time and continuous-time signals. Example Example Figure 2.6 Sinusoidal signal. Figure 2.7 Discrete-time sinusoidal signal. 4

5 Real and Complex Signals. In communications, complex signals are usually used to model signals that convey amplitude and phase information. Example The signal is a complex signal. Its real part is and its imaginary part is We could equivalently describe this signal in terms of its modulus and phase. and its phase is 5

6 Deterministic and Random Signals. In a deterministic signal at any time instant t, the value of x(t) is given as a real or a complex number. In a random (or stochastic) signal at any given time instant t, x(t) is a random variable, i.e., it is defined by a probability density function. 6

7 Periodic and Nonperiodic Signals. For discrete-time periodic signals, we have for all integers n, and a positive integer N0 Example The signals and are examples of real and complex periodic signals. The period of both signals is. The signal Figure 2.9 The unit-step signal 7

8 Causal and Noncausal Signals. A signal x(t) is called causal if for all t < 0, we have x(t) = 0; otherwise, the signal is noncausal. Equivalently, a discrete-time signal is a causal signal if it is identically equal to zero for n < 0. Example The signal is a causal signal. Its graph is shown in Figure Figure 2.10 An example of a causal signal. 8

9 Even and Odd Signals. The signal x(t) is even if and only if, for all t, and is odd if and only if, for all t, Figure 2.11 Examples of even and odd signals. 9

10 Example The sinusoidal signal is generally neither even nor odd. However, the special cases θ = 0 and θ = ±π/2 correspond to even and odd signals, respectively. In general, Since is even and is odd, we conclude that and Example From Figure 2.8, we can see that for and, the real part and the magnitude are even and the imaginary part and the phase are odd. 10

11 Energy-Type and Power-Type Signals. For any signal x(t), the energy content of the signal is defined by The power content is defined by A signal x(t) is an energy-type signal if and only if power-type signal if and only if Px satisfies is finite. A signal is a 11

12 Example Find the energy in the signal described by Solution We have Therefore, this signal is an energy-type signal. 12

13 Example The energy content of is Therefore, this signal is not an energy-type signal. However, the power of this signal is Hence, x(t) is a power-type signal and its power is 13

14 Example For any periodic signal with period, the energy is 14

15 2.1.3 Some Important Signals and Their Properties The Sinusoidal Signal. The sinusoidal signal is defined by where the parameters A, f0, and θ are the amplitude, frequency, and phase of the signal. A sinusoidal signal is periodic with the period. The Complex Exponential Signal. The complex exponential signal is defined by. Again A, f0, and θ are the amplitude, frequency, and phase of the signal. 15

16 The Unit-Step Signal. The unit step multiplied by any signal produces a causal version of the signal. Note that for positive a, we have Figure 2.9 The unit-step signal. The Rectangular Pulse. This signal is defined as Figure 2.13 The rectangular pulse 16

17 Example Plot the signal The Triangular Signal. This signal is defined as Example To plot 17

18 The Sinc Signal. The sinc signal is defined as From this figure, we can see that the sinc signal achieves its maximum of 1 at t = 0. The zeros of the sinc signal are at t = ±1,±2,±3,

19 The Sign or the Signum Signal. The sign or the signum signal denotes the sign of the independent variable t and is defined by The Impulse or Delta Signal. Figure 2.20 The impulse signal. 19

20 and The following properties are derived from the definition of the impulse signal: 20

21 4. For any φ(t) continuous at t0, 5. For all a 0, 6. The result of the convolution of any signal with the impulse signal is the signal itself: Also, 21

22 7. The unit-step signal is the integral of the impulse signal, and the impulse signal is the generalized derivative of the unit-step signal, i.e., and 22

23 2.1.4 Classification of Systems A system is an entity that is excited by an input signal and, as a result of this excitation, produces an output signal. Figure 2.21 A system with an input and output 23

24 Discrete-Time and Continuous-Time Systems. A discrete-time system accepts discrete-time signals as the input and produces discrete-time signals at the output. For a continuous-time system, both input and output signals are continuous-time signals. Example This system is a discrete-time differentiator. Linear and Nonlinear Systems. Linear systems are systems for which the superposition property is satisfied, i.e., the system s response to a linear combination of the inputs is the linear combination of the responses to the corresponding inputs. 24

25 Time-Invariant and Time-Varying Systems. Figure 2.23 A time-invariant system. 25

26 Example The differentiator is a time-invariant system, since, is an example of a time- is Example The modulator, defined by varying system. The response of this system to which is not equal to. NOTE= The class of linear time-invariant (LTI) systems is particularly important. The response of these systems to inputs can be derived simply by finding the convolution of the input and the impulse response of the system. 26

27 Causal and Noncausal Systems. A system is causal if its output at any time t0 depends on the input at times prior to t0, i.e., A necessary and sufficient condition for an LTI system to be causal is that its impulse response h(t) must be a causal signal, i.e., for t < 0, we must have h(t) = 0. For noncausal systems, the value of the output at t0 also depends on the values of the input at times after t0 27

28 2.1.5 Analysis of LTI Systems in the Time Domain For this class of systems, the input output relationship is particularly simple and can be expressed in terms of the convolution integral. The impulse response h(t) of a system is the response of the system to a unit impulse input δ(t): The Convolution Integral. We will show that y(t ) can be expressed in terms of the input x(t) and the impulse response h(t) of the system. 28

29 Now, if we denote the response of the LTI system to the input x(t) by y(t ), we can write 29

30 Example Let a linear time-invariant system have the impulse response h(t). Assume this system has a complex exponential signal as input, i.e.,. The response to this input can be obtained by where 30

31 2.2 FOURIER SERIES In the next two sections we will develop another approach to analyzing LTI systems. The basic idea is to expand the input as a linear combination of some basic signals whose output can be easily obtained, and then to employ the linearity properties of the system to obtain the corresponding output. This approach is much easier than a direct computation of the convolution integral; at the same time, it provides better insight into the behavior of LTI systems. 31

32 2.2.1 Fourier Series and Its Properties Let the signal x(t) be a periodic signal with period T0. First, we need to determine whether the following Dirichlet conditions are satisfied: 1. x(t) is absolutely integrable over its period, i.e., 2. The number of maxima and minima of x(t) in each period is finite, 3. The number of discontinuities of x(t) in each period is finite. If these conditions are met, then x(t) can be expanded in terms of the complex exponential signals as where for some arbitrary α. 32

33 The quantity f0 = 1/T0 is called the fundamental frequency of the signal x(t). We observe that the frequencies of the complex exponential signals are multiples of this fundamental frequency. The nth multiple of f0 is called the nth harmonic. The Fourier-series expansion can be expressed in terms of the angular frequency by and 33

34 In general,. Thus, gives the magnitude of the nth harmonic and gives its phase. Figure 2.24 The discrete spectrum of x(t). 34

35 Example Let x(t) denote the periodic signal depicted in Figure 2.25 and described analytically by where τ is a given positive constant (pulse width). Determine the Fourier-series expansion for this signal. Figure 2.25 Periodic signal x(t) in Equation (2.2.6). 35

36 Solution We first observe that the period of the signal is T0 and where we have used the relation very simple and yields. Therefore,.For n = 0, the integration is A graph of these Fourier-series coefficients is shown in Figure

37 Figure 2.26 The discrete spectrum of the rectangular-pulse train. 37

38 Example Determine the Fourier-series expansion for the signal x(t) shown in Figure 2.27 and described by Figure 2.27 Signal x(t) in Equation (2.2.9) 38

39 Solution Since T0 = 2, it is convenient to choose α = 1/2. First, we note that for n = 0, we can easily find the integral to be zero; therefore x0 = 0. For n 0, we have From these values of xn, we have the following Fourier-series expansion for x(t): 39

40 Example Determine the Fourier-series representation of an impulse train denoted by and shown in Figure Figure 2.28 An impulse train. 40

41 Solution We have With these coefficients, we have the following expansion: 41

42 Fourier Series for Real Signals. If the signal x(t) is a real signal satisfying the conditions of the Fourier-series theorem, then there must be alternative ways to expand the signal. For real x(t), we have xn has even symmetry ( xn = x-n ) and has odd symmetry ( ) with respect to the n = 0 axis. An example of the discrete spectrum for a real signal is shown in Figure From, it follows that if we denote then therefore, for n 1, 42

43 Since x0 is real and given as we conclude that Figure 2.30 Discrete spectrum of a real-valued signal. 43

44 This relation, which only holds for real periodic signals, is called the trigonometric Fourier-series expansion. To obtain an and bn, we have therefore, Thus, we obtain 44

45 A third way exists to represent the Fourier-series expansion of a real signal. Note that Substituting (2.2.19) in (2.2.2), we have 45

46 In summary, for a real periodic signal x(t), we have three alternatives to represent the Fourier-series expansion where the corresponding coefficients are obtained from 46

47 Example Determine the sine and cosine coefficients in Example Solution We have previously seen that Therefore, and 47

48 Fourier-Series Expansion for Even and Odd Signals. For even x(t), we have Therefore, for even signals, the Fourier-series expansion has only cosine terms, i.e., we have For odd signals, 48

49 2.2.2 Response of LTI Systems to Periodic Signals if h(t) is the impulse response of the system, then from Example , we know that the response to the exponential is, where Now let us assume that x(t), the input to the LTI system, is periodic with period T0 and has a Fourier-series representation Then we have where 49

50 From this relation, we can draw the following conclusions: If the input to an LTI system is periodic with period T0, then the output is also periodic. (What is the period of the output?) The output has a Fourier-series expansion given by where This is equivalent to and 50

51 Example Let x(t) denote the signal shown in Figure 2.27, but set the period equal to T o = 10^( 5) seconds. This signal is passed through a filter with the frequency response depicted in Figure Determine the output of the filter. Figure 2.32 Frequency response of the filter 51

52 Solution We first start with the Fourier-series expansion of the input. This can be easily obtained as The frequency response 52

53 For higher frequencies, H(f) = 0. Therefore, we have or, equivalently, 53

54 2.2.3 Parseval s Relation Parseval s relation says that the power content of a periodic signal is the sum of the power contents of its components in the Fourier-series representation of that signal. Let us assume that the Fourier-series representation of the periodic signal x(t) is given by The formal statement of Parseval s relation is 54

55 If is used in Parseval 55

56 Example Determine the power contents of the input and output signals in Example Solution We have We could employ Parseval s relation to obtain the same result. To do this, we have Equating the two relations, we obtain To find the output power, we have 56

57 2.3.1 From Fourier Series to Fourier Transforms In this section, we will apply the Fourier series representation to nonperiodic signals. We will see that it is still possible to expand a nonperiodic signal in terms of complex exponentials. However, the resulting spectrum will no longer be discrete. In other words, the spectrum of nonperiodic signals covers a continuous range of frequencies. This result is the well-known Fourier transform given next. First, the signal x(t) must satisfy the following Dirichlet conditions: Then the Fourier transform (or Fourier integral) of x(t), defined by exists and the original signal can be obtained from its Fourier transform by 57

58 To denote that X(f ) is the Fourier transform of x(t), we frequently employ the following notation: To denote that x(t) is the inverse Fourier transform of X(f ), we use the following notation: Sometimes we use a shorthand for both relations: 58

59 If the variable in the Fourier transform is ω rather than f, then we have and and 59

60 Example Determine the Fourier transform of the signal shown in Figure Solution We have given in Equation (2.1.15) and Therefore, Figure 2.33 illustrates the Fourier-transform relationship for this signal. 60

61 Figure 2.33 п(t) and its Fourier transform. 61

62 Example Determine the Fourier transform of an impulse signal x(t) = δ(t). Solution The Fourier transform can be obtained by where we have used the sifting property of δ(t). This shows that all frequencies are present in the spectrum of δ(t) with unity magnitude and zero phase. The graphs of x(t) and its Fourier transform are given in Figure Similarly, from the relation we conclude that 62

63 Figure 2.34 Impulse signal and its spectrum. 63

64 Signal Bandwidth. The bandwidth of a signal represents the range of frequencies present in the signal. The higher the bandwidth, the larger the variations in the frequencies present. In general, we define the bandwidth of a real signal x(t) as the range of positive frequencies present in the signal. In order to find the bandwidth of x(t), we first find X(f ), which is the Fourier transform of x(t); then, we find the range of positive frequencies that X(f ) occupies. The bandwidth is BW = Wmax Wmin, where Wmax is the highest positive frequency present in X(f ) and Wmin is the lowest positive frequency present in X(f ). 64

65 2.3.2 Basic Properties of the Fourier Transform Linearity. The Fourier-transform operation is linear. That is, if x 1 (t) and x 2 (t) are signals possessing Fourier transforms X1(f ) and X2(f ) respectively, the Fourier transform of αx 1 (t) + βx 2 (t) is αx1(f ) + βx2(f ), where α and β are two arbitrary (real or complex) scalars. This property is a direct result of the linearity of integration. Example Determine the Fourier transform of U-1(t), the unit-step signal. Solution Using the relation and the linearity theorem, we obtain 65

66 Duality. If then Example Determine the Fourier transform of sinc(t). Solution Noting that is an even signal and, therefore, that п ( f ) = п (f ), we can use the duality theorem to obtain 66

67 Shift in Time Domain. A shift of t0 in the time origin causes a phase shift of 2πft0 in the frequency domain. In other words, To see this, we start with the Fourier transform of x(t t0), namely, With a change of variable of u = t - t0, we obtain Note that a change in the time origin does not change the magnitude of the transform. It only introduces a phase shift linearly proportional to the time shift (or delay). 67

68 Example Determine the Fourier transform of the signal shown in Figure Solution We have By applying the shift theorem, we obtain Figure 2.37 Signal x(t). 68

69 Example Determine the Fourier transform of the impulse train Solution Applying the shift theorem, we have Therefore, Using Equation (2.2.14) and substituting f for t and 1/T0 for T0, we obtain 69

70 or This relation yields, The case of T0 = 1 is particularly interesting. For this case, we have That is, after substituting f for t, we find that the Fourier transform of is itself. 70

71 Scaling. For any real a 0, we have To see this, we note that and make the change in variable u = at. Then, where we have treated the cases a > 0 and a < 0 separately. 71

72 Example Determine the Fourier transform of the signal Solution Note that x(t) is a rectangular signal amplified by a factor of 3, expanded by a factor of 4, and then shifted to the right by 2. In other words,..using the linearity, time shift, and scaling properties, we have 72

73 Convolution. If the signals x(t) and y(t ) both possess Fourier transforms, then Example Determine the Fourier transform of the signal (t), shown in Figure Solution It is enough to note that theorem. Thus, we obtain Figure 2.15 and use the convolution 73

74 Example Determine the Fourier transform of the signal x(t)= shown in Figure 2.16 and discussed in Example ,which is Figure Solution Using scaling and linearity, we have 74

75 MODULATION: The Fourier transform of is.to show this relation,we have 75

76 Example Determine the Fourier transform of Solution Using the modulation theorem, we have Example Determine the Fourier transform of the signal Solution Using Euler s relation, we have Now using the linearity property and the result of Example , we have 76

77 Example Determine the Fourier transform of the signal Solution We have 77

78 Figure 2.38 Effect of modulation in both the time and frequency domain. 78

79 Example Determine the Fourier transform of the signal shown in Figure Figure 2.39 Signal x(t). 79

80 Solution Note that x(t) can be expressed as Therefore, where we have used the result of Example , with f0 = 1/2. 80

81 Parseval s Relation. If the Fourier transforms of the signals x(t) and y(t ) are denoted by X(f ) and Y(f ) respectively, then Note that if we let y(t ) = x(t), we obtain 81

82 Example Using Parseval s theorem, determine the values of the integral Solution We have we get Therefore, using Rayleigh s theorem, 82

83 Autocorrelation. The (time) autocorrelation function of the signal x(t) is denoted by Rx(τ ) and is defined by The autocorrelation theorem states that We note that By using the convolution theorem, the autocorrelation theorem follows easily. Differentiation. The Fourier transform of the derivative of a signal can be obtained from the relation Differentiation in Frequency Domain. We begin with 83

84 Example Determine the Fourier transform of the signal shown in Figure Figure 2.41 Signal x(t). Solution Obviously, theorem, we have. Therefore, by applying the differentiation 84

85 Figure 2.43 Signal e αt and its Fourier transform. 85

86 and we have already seen that By using the scaling theorem with α = 1, we obtain Hence, by the linearity property, we have 86

87 TABLE 2.1 TABLE OF FOURIER-TRANSFORM PAIRS Table 2.1 provides a collection of frequently used Fouriertransform pairs. Table 2.2 outlines the main properties of the Fourier transform. 87

88 TABLE 2.2 TABLE OF FOURIER-TRANSFORM PROPERTIES 88

89 2.3.4 Transmission over LTI Systems The convolution theorem is the basis for the analysis of LTI systems in the frequency domain. If we translate this relationship in the frequency domain using the convolution theorem, then X(f ), H(f ), and Y(f ) are the Fourier transforms of the input, system impulse response, and the output, respectively. Thus, Example Let the input to an LTI system be the signal and let the impulse response of the system be Determine the output signal. 89

90 Solution First, we transform the signals to the frequency domain. Thus, we obtain and Figure 2.44 shows X(f ) and H(f). Figure 2.44 Lowpass signal and lowpass filter. 90

91 To obtain the output in the frequency domain, we have From this result, we obtain 91

92 Frequency response of Filters Figure 2.45 Various filter types. 92

93 Figure db bandwidth of filters in Example

94 Example The magnitude of the transfer function of a filter is given by Determine the filter type and its 3 db bandwidth. Solution At f = 0, we have H(f) = 1 and H(f) decreases and tends to zero as f tends to infinity. Therefore, this is a lowpass filter. Since power is proportional to the square of the amplitude, the equation must be This yields f0 = ±10,000. Therefore, this is a lowpass filter with a 3 db bandwidth of 10 khz. A plot of H(f) is shown in Figure

95 Figure db bandwidth of filter in Example

96 2.5 POWER AND ENERGY The energy content of a (generally complex-valued) signal x(t) is defined as and the power content of a signal is A signal is energy-type if <, and it is power-type if 0 < Px <. A signal cannot be both power- and energy-type because Px = 0 for energy-type signals, and = for power-type signals. Practically all periodic signals are power-type and have power 96

97 2.5.1 Energy-Type Signals For an energy-type signal x(t), we define the autocorrelation function By setting τ = 0 in the definition of the autocorrelation function of x(t), we obtain its energy content, i.e., Parseval Theorem 97

98 Energy Spectral Density Since X(f) 2 df is the total energy of the signal, X(f) 2 presents the energy amont of the signal per frequency, so called energy spectral density (ESD) of x(t). Its unit is Joule/Hz. ESD is the Fourier Transform of the autocorrelation as 98

99 Energy of the LTI output If we pass the signal x(t) through a filter with the (generally complex) impulse response h(t) and frequency response H(f ), the output will be Y(f ) = X(f)H(f). where R y (τ ) = y(τ ) y ( τ) is the autocorrelation function of the output. The inverse Fourier transform of Y(f ) 2 is where (a) follows from the convolution property and (b) follows from the autocorrelation property. 99

100 To summarize, 1. For any energy-type signal x(t), we define the autocorrelation function 2. The energy spectral density of x(t), denoted by is the Fourier Transform of It is equal to 3. The energy content of x(t), is the value of the autocorrelation function evaluated at τ = 0 or, equivalently, the integral of the energy spectral density over all frequencies, i.e., 4. If x(t) is passed through a filter with the impulse response h(t) and the output is denoted by y(t ), we have 100

101 Example Determine the autocorrelation function, energy spectral density, and energy content of the signal Solution First we find the Fourier transform of x(t). From Table 2.1, we have Hence, and To find the energy content, we can simply find the value of the autocorrelation function at zero: 101

102 Example If the signal in the preceding example is passed through a filter with impulse response determine the autocorrelation function, the power spectral density, and the energy content of the signal at the output. Solution The frequency response of the filter is Therefore, Note that in the last step we used partial fraction expansion. From this result, and using Table 2.1, we obtain and 102

103 2.5.2 Power-Type Signals For the class of power-type signals, a similar development is possible. In this case, We define the time-average autocorrelation function of the power-type signal x(t) as Obviously, the power content of the signal can be obtained from We define S x (f ), the power-spectral density x(t), to be the Fourier transform of the time-average autocorrelation function: Now we can express the power content of the signal x( t) in terms of S x (f ) by noting that 103

104 If a power-type signal x(t) is passed through a filter with impulse response h(t), the output is and the time-average autocorrelation function for the output signal is Substituting for y(t ), we obtain This integral gives R Y τ = R X τ h τ h ( τ) Taking the Fourier transform of both sides of this equation, we obtain 104

105 Let us assume that the signal x(t) is a periodic signal with the period T 0 and has the Fourier-series coefficients {x n }. Thus, the power spectral density of a periodic signal is given by To find the power content of a periodic signal, we must integrate this relation over the whole frequency spectrum. When we do this, we obtain 105

106 If this periodic signal passes through an LTI system with the frequency response H(f ), the power spectral density of the output can be obtained by and the power content of the output signal is