1 Fourier Representation of Signals and LTI Systems. CHAPTER 3 UniMAP

2 3.1 Introduction. 3.2 Complex Sinusoids and Frequency Response of LTI Systems. 3.3 Fourier Representation of Four Classes of Signals. 3.4 Discrete Time Periodic Signals: Discrete Time Fourier Series. 3.5 Continuous-Time Periodic Signals: The Fourier Series. 3.6 Discrete-Time Non periodic Signals: The Discrete-Time Fourier Transform. 3.7 Continuous-Time Non periodic Signals: The Fourier Transform. 3.8 Properties of Fourier Representation. 3.9 Linearity and Symmetry Properties Convolution Properties Differential and Integration Properties. 3.0 Fourier Representation of Signals and LTI Systems.

3   Representing signals as superposition's of complex sinusoids leads to a useful expression for the system output and provide a characterization of signals and systems.   Example, orchestra is a superposition of sounds generated by different equipment having different frequency range such as string, base, violin and ect.   Study of signals and systems using sinusoidal representation is termed as Fourier Analysis introduced by Joseph Fourier. 3.1 Introduction.

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Complex Sinusoidal and Frequency Response of LTI System.   The response of an LTI system to a sinusoidal input that lead to the characterization of the system behavior that is called frequency response of a system.   This characterization is obtained in terms of the impulse response by using convolution and a complex sinusoidal input signal.   Consider the output of LTI system with impulse response h[n] and the complex sinusoidal input x[n]=e j  n.

15 Cont’d…  Discrete-Time (DT)

16 Cont’d…  Continuous-Time (CT)

17 Hence, the output of a complex sinusoidal input to an LTI system is a complex sinusoid of the same frequency as the input, multiplied by the frequency response of the system. The complex scaling factor H(e jΩn ) is not a function of time n, but is only a function of frequency Ω and is termed the frequency response of the system.

18 Cont’d… Example: Frequency Response. The impulse response of a system is given as Find the expression for the frequency response, and plot the magnitude and phase response. Solution: Substitute h(t) into equation

19 Cont’d…Example The magnitude response is

20 Cont’d… Figure 3.1: Frequency response of the RC circuit (a) Magnitude response. (b) Phase response. The phase response is arg{H(j  )}= -arctan(  RC) .

Fourier Representation of Four Classes of Signals.   There are four distinct Fourier representations, class of signals; (i) Periodic Signals. (ii) Nonperiodic Signals. (iii) Discrete–Time Periodic Signals. (iv) Continuous-Time Periodic Signals.   Fourier series (FS) applies to continuous-time periodic signals.   Discrete Time Fourier series (DTFS) applies to discrete-time periodic signals.   Nonperiodic signals have Fourier Transform representation.   Discrete Time Fourier Transform applies to a signal that is discrete in time and non-periodic.

22 Time Property PeriodicNonperiodic Continuous (t) Fourier Series (FS) [Chapter 3.5] Fourier Transform (FT) [Chapter 3.7] Discrete (n)Discrete Time Fourier Series (DTFS) [Chapter 3.4] Discrete Time Fourier Transform (DTFT) [Chapter 3.6] Table 3.1: Relation between Time Properties of a Signal and the Appropriate Fourier Representation. Cont’d…

Periodic Signals: Fourier Series Representation  If x[n] is a discrete-time signal with fundamental period N, x[n] of DTFS is represents as where  o = 2π/N is the fundamental frequency of x[n]. The frequency of the kth sinusoid in the superposition is k  o

24  Similarly, if x(t) is a continuous-time signal with fundamental period T, x(t) of FS is represents as where   = 2  /T is the fundamental frequency of x[t]. The frequency of the kth sinusoid is k   and each sinusoid has a common period T.  A sinusoid whose frequency is an integer multiple of a fundamental frequency is said to be a harmonic of a sinusoid at the fundamental frequency. For example e jk   t is the kth harmonic of e j   t.  The variable k indexes the frequency of the sinusoids, A[k] is the function of frequency. Cont’d…

Nonperiodic Signals: Fourier- Transform Representation.  The Fourier Transform representations employ complex sinusoids having a continuum or range of frequencies.  The signal is represented as a weighted integral of complex sinusoids where the variable of integration is the sinusoid’s frequency.  Continuous-time sinusoids are used to represent continuous signal in FT. where X(j  )/(2  ) is the “weight” or coefficient applied to a sinusoid of frequency  in the FT representation

26  Discrete-time sinusoids are used to represent discrete time signal in DTFT.  It is unique only a 2  interval of frequency. The sinusoidal frequencies are within 2  interval. Cont’d…

Discrete Time Periodic Signals: Discrete Time Fourier Series.  The discrete time Fourier series representation; where x[n] is a periodic signal with period N and fundamental frequency   =2  /N.  The relationship of the above equation,

28  The X[k] is the frequency-domain representation of x[n], because each coefficient is associated with complex sinusoid of a different frequency.  DTFS is the only Fourier representation that can be numerically evaluated using computer, e.g. used in numerical signal analysis. Cont’d…

29 Example: Determining DTFS Coefficients. Find the frequency domain representation of the signal in Figure 3.2 below. Figure 3.2: Time Domain Signal. Solution: The signal has period N=5, so   =2  /5. Also the signal has odd symmetry, so we sum over n=-2to n=2 from equation Cont’d…

30 From the value of x{n} we get, From the equation, identify one period of the DTFS coefficient X[k], k=-2 to k=2, in the rectangular and polar coordinate as Cont’d…Example

31 Cont’d…Example Figure 3.3: Magnitude and phase of the DTFS coefficients for the signal in Fig The figure below shows the magnitude and phase of X[k] as a function of frequency index k.

32 Calculate X[k] using n=0 to n=4 for the limit of the sum. This expression is different from which we obtain from using n =-2 to n =2. However, note that, n=-2 to n=2 and n= 0 to n=4, yield equivalent expression for the DTFS coefficient. . Cont’d…Example

Continuous-Time Periodic Signals: The Fourier Series.  Continues time periodic signals are represented by Fourier series (FS). The fundamental period is T and fundamental frequency  o =2  /T.  X[k] is the FS coefficient of signal x(t).  Below is the relationship of the above equation,

34  The FS coefficient X[k] is known as frequency-domain representation of x(t).  The FS representation is often used in electrical engineering to analyze the effect of systems on periodic signals. Cont’d…

35 Cont’d… Example : Direct Calculation of FS Coefficients. Determine the Fs coefficients for the signal x(t) depicted in Figure 3.4. Figure 3.4: Time Domain Signal. Solution: The period of x(t) is T=2, so  0 =2  /2=  On the interval 0<=t<=2, one period of x(t) is expressed as x(t)=e -2t, so it yields

36 Since e -jk2  =1 Figure 3.5 shows the magnitude spectrum|X[k]| and the phase spectrum arg{X[k]}. . Cont’d…

37 Cont’d… Figure 3.5: Magnitude and Phase Spectra.

Discrete-Time Non periodic Signals: The Discrete-Time Fourier Transform.  The DTFT representation of time domain signal, DTFT  X[k] is the DTFT of the signal x[n].  Below is the relationship of the above equation,

39  The FT, X[j  ], describes the signal x[n] as a function of a sinusoidal frequency Ω and is termed as frequency-domain representation of the signal x(t).  The first equation is the inverse FT, where it maps the frequency domain representation X[j  ] back to time domain. Cont’d…

Continuous-Time Nonperiodic Signals: The Fourier Transform.  The Fourier transform (FT) is used to represent a continuous time nonperiodic signal as a superposition of complex sinusoids.  FT representation, where,  Below is the relationship of the above equation,

41 Cont’d… Example: FT of a Real Decaying Exponential. Find the Fourier Transform (FT) of x(t) =e -at u(t). Solution: The FT does not converge for a<=0, since x(t) is not absolutely integrable, that is

42 Cont’d…Example Converting to polar form, we find that the magnitude and phase of X(jw) are respectively given by This is shown in Figure 3.6 (b) and (c).

43 Cont’d…Example Figure 3.6: (a) Real time-domain exponential signal. (b) Magnitude spectrum. (c) Phase spectrum. .

Properties of Fourier Representation.  The four Fourier representations discussed in this chapter are summarized in Table 3.2.  Attached Appendix C is the comprehensive table of all properties. Table 3.2 : Fourier Representation.

45 Table 3.2: The Four Fourier Representations. Time Domain Periodic(t,n) Non periodic (t,n)CONTINUOUS (t) (t)NONPERIODIC(k,w) DISCRETE PERIODIC (k,  ) Discrete (k) Continuous (  ) Freq. Domain

Linearity and Symmetry Properties.  All four Fourier representations involve linear operations.  The linearity property is used to find Fourier representations of signals that are constructed as sums of signals whose representation are already known.

47 Figure 3.49 (p. 255) Representation of the periodic signal z(t) as a weighted sum of periodic square waves: z(t) = (3/2)x(t) + 1/2y(t).

48 Example : Linearity in the FS. Given z(t) which is the periodic signal. Use the linearity property to determine the FS coefficients Z[k]. Solution: From Example 3.13(Text) we have The linearity property implies that . Cont’d…

Symmetry Properties: Real and Imaginary Signal. Table 3.3: Symmetry Properties for Fourier Representation of Real-and Imaginary–Valued Time Signals. RepresentationReal-Valued Time Signal Imaginary-Valued Time Signal FT X * (j  ) = X(-j  )X * (j  ) = -X(-j  ) FSX * [k] = X[-k]X * [k] = -X[-k] DTFTX*(e j  ) = X(e -j  )X*(e j  ) = -X(e -j  ) DTFSX * [k] = X[-k]X * [k] = -X[-k]

50 Suppose that x(t) is real valued and has even symmetry. Then x*(t)=x(t) and x(-t)=x(t). Substitute x*(t)=x(-t) Change of variable  = -t.  The only way that the condition X * (j  ) =X(j  ) holds is for the imaginary part of X(j  ) to be zero Symmetry Properties: Even and Odd Signal.

51  If time signal is real and even, then the frequency-domain representation of it also real.  If time signal is real and odd, then the frequency-domain representation is imaginary. Cont’d…

52   The convolution of the signals in the time domain transforms to multiplication of their respective Fourier representations in the frequency domain.   The convolution property is a consequences of complex sinusoids being eigen functions of LTI system Convolution of Non periodic Signals Filtering Convolution of Periodic Signals Convolution Property.

Convolution of Nonperiodic Signals.  Convolution of two non periodic continuous-time signals x(t) and h(t) is defines as express x(t-  ) in term of FT:

54 Substitute into convolution integral yields, Cont’d…

55  Convolution of h(t) and x(t) in the time domain corresponds to multiplication of Fourier transforms, H(j  ) and X(j  ) in the frequency domain that is;  Discrete-time nonperiodic signals, if Cont’d…

56 Example : Convolution in Frequency Domain. Let x(t)= (1/(  t))sin(  t) be the input of the system with impulse response h(t)= (1/(  t))sin(2  t). Find the output y(t)=x(t)*h(t) Solution: From Example 3.26 (text, Simon) we have, Since We conclude that . Cont’d…

Filtering.  Filtering implies that some frequency components of the input are eliminated while others are passed by the system unchanged. For example,   low pass filter attenuates high frequencies components of the input and passes the lower frequency components.   high pass filter attenuates low frequencies components of the input and passes the high frequencies.   band pass filter passing signals within a certain frequency band an attenuates signal outside that band.

58  Passband is the band of frequencies that are passed by the system, while the stopband refers to the range of frequencies that are attenuated by the system.  Realistic filters always have transition band; gradual transition from the passedband to the stopband.  Magnitude response (dB);  Magnitude response can be displayed on the logarithmic scale.  Unity gain corresponds to zero dB.  The edge of passed band is usually defined by the frequency which the response is -3dB; magnitude response of (1/sqrt  2). Cont’d…

59  Energy spectrum of filter is given by  -3dB point;   Frequencies at which the filter passed only half of the input power.   Termed as the cutoff frequency of the filter. Cont’d…

60 Figure 3.7: Frequency response of ideal continuous- (left panel) and discrete-time (right panel) filters. (a) Low-pass characteristic. (b) High-pass characteristic. (c) Band-pass characteristic. Cont’d…

Convolution of Periodic Signal.  The convolution of periodic signals often occur at the context of signal analysis and manipulation.  By definition the periodic convolution of two continuous signal x(t) and z(t) each having the period T as,  The symbol * denotes integration is performed over a single period of the signal involve.  Convolution in time transforms to multiplication of the frequency-domain representation.

62 Example : Convolution of Two Periodic Signals. Evaluate the periodic convolution of the sinusoidal signal z(t)= 2cos(2  t)+ sin(4  t), with the periodic square wave shown in Figure 3.56 below. Solution: Both x(t) and y(t) have fundamental period T=1. y(t) = x(t) * z(t). The FS representation has coefficients, . Cont’d…

63 The FS representation of the square wave is (from Example 3.13), The FS coefficient for y(t) are, . Cont’d…Example

Differentiating and Integration Properties.  Differentiation and integration are operations that apply to continuous function.  In this section we will consider the effect of differentiation and integration with respect to time for a continuous-time signal or with respect to frequency in the FT and DTFT Differentiation in Time Differentiation in Frequency Integration.

65  Differentiation and integration are operations that apply to continuous function. - relationship of x(t) and X(j  ) - differentiate both side with respect to t,  Differentiating a signal in a time domain corresponds to multiplying its FT by j  in frequency domain.  Differentiation accentuates the high frequency components of the signal Differentiation in Time.

66 Example: Verifying the Differentiation Property. Use the frequency-differentiation property to find the FT of Verify this result by differentiating and taking the FT of the result. Solution: Step 1: Using the product rule for differentiation we have, Step 2: Taking the FT of each term and using linearity, we have, . Cont’d…

Differentiation in Frequency.  The Fourier transform, - differentiate both side of equation with respect to   Differentiating of a FT in the frequency domain corresponds to multiplication of the signal by –jt in the time domain.

68 Example: Differentiation in Frequency. The differentiation property implies that From Appendix C. Solution: Since, . Cont’d…

Integration.  The operation of integration applies only to continuous dependent variable.  The integration property,

70 Table 3.4: Commonly Used Differentiation and Integration Properties.  Table 3.4 summarize the differentiation and integration properties of Fourier representation. Cont’d…

71 Example: Integration Property. Use the integration property to find the inverse FT of Solution: . Cont’d…

Time and Frequency-Shift Properties.  In this chapter we consider the effect of time and frequency shifts on the Fourier representation Time-Shift Property Frequency-Shift Property.

Time-Shift Properties.  Derivation,  Time shifting the signal x(t) by t 0 is to multiply the FT X(j  ) by e -jωt 0.

74  Table 3.5 is the other three Fourier representations. Table 3.5: Time Shift Properties of Fourier Representation. Cont’d…

75 Example: Finding an FT Using the Time-Shift Property. Use the FT of the rectangular pulse x(t) in Figure 3.8(a) to determine the FT of the time-shifted rectangular pulse z(t) in Figure 3.8(b). Solution: From Table 3.5 Figure 3.8: Time Shift Property. . Cont’d…

Frequency-Shift Properties.  The frequency shift corresponds to multiplication in the time domain by a complex sinusoid whose frequency is equal to shift.  Derivation, Substitute the variable,

77 Table 3.6 Time Shift Properties of Fourier Representation. Cont’d…

78 Example: Finding an FT Using the Frequency-Shift Property. Use the Frequency Shift property to determine the FT of the complex sinusoidal pulse. Solution: Express z(t) as the product of a complex sinusoid e j10t and a rectangular pulse Using the result of Example 3.25 (text), . Cont’d…

79 Employing frequency shift property, Obtain, . Cont’d…Example