1. BYST SigSys - WS2003: Fourier Rep. 120 CPE200 Signals and Systems Chapter 3: Fourier Representations for Signals (Part I)

2. BYST SigSys - WS2003: Fourier Rep. 121 Introduction 3. 1 engineeringanalysis technique linear express as a number of simpler problems A common engineering analysis technique is the partitioning of complex problems into a linear combination of the simpler ones. The simpler problems are then solved, and the total solution becomes the sum of the simpler solutions. Two requirements must be satisfied for a solution as described earlier to be both valid and useful. First, the problem must be linear, such that the solution for the complex problem is equal to the sum of the solutions considering only one simpler problem at a time. Next, the complex problem must be able to express as a number of simpler problems. Analysis techniques in signal and system theory follow the same manner mentioned above. A linear system will allow us to

3. BYST SigSys - WS2003: Fourier Rep. 122 determine the response of each individual input when the input is a combination of more than one signals. The only required procedure is how to decompose the complicated signal into a combination of simpler signals. Fourier series transform Any signals can also be represented as a weighted superposition of complex sinusoids. From Chapter 2, we discovered that any signals can be represented as linear combinations of shifted impulses. When such a signal is applied to a linear time-invariant system, then the system response is based on the convolution sum. However, there is an alternative representation for signals and systems which will be discussed in this chapter and the following two chapters. The alternative representation for signals is known as the c- t and d-t Fourier series and transform.

4. BYST SigSys - WS2003: Fourier Rep. 123 Fourier series periodic aperiodic finite energy Fourier transform frequency domain The Fourier series is named after the French physicist Jean Baptiste Fourier (1768-1830), who was the first one to propose that periodic waveforms could be represented by a sum of sinusoids (or complex exponential signals). When the signals are aperiodic and have finite energy the Fourier series is known as the Fourier transform. A signal with such decomposition is said to be represented in the frequency domain. By representing signals in terms of sinusoids, we will obtain an alternative expression for the input-output behavior of an LTI system which is a weighted sum of the system response to each complex sinusoidal. This type of representation not only leads to a useful expression for the system response but also provides a very

5. BYST SigSys - WS2003: Fourier Rep. 124 perceptive characterization of signals and systems. In this chapter, we focus on the Fourier representations of signals and their properties. We will begin our discussion on studying the response of LTI systems to complex sinusoids. Then, we will consider the development of a representation of signals as linear combinations of complex sinusoids. Properties of c-t and d-t Fourier representations will also be discussed. Finally, the basic concept of filtering will be introduced. Complex Sinusoids and 3. 2 LTI Systems Sinusoidal input signals are often used to characterized the response of a system. In Chapter 2, we discussed how to

6. BYST SigSys - WS2003: Fourier Rep. 125 characterize an LTI system using the impulse and unit step signals. We discovered that when the impulse response is known, we can determine the response of an LTI system to any arbitrary input by a weighted sum (integral) of time-shifted impulse responses. In this section, we will examine the relationship between the impulse response and the steady-state response of an LTI system to a complex sinusoidal input. complex sinusoidal x[n] = e j  n Let consider the d-t complex sinusoidal input expressed as x[n] = e j  n. Hence the response y[n] is given by

7. BYST SigSys - WS2003: Fourier Rep. 126 (3.1) H(e j  )frequency response Let H(e j  ) be the “frequency response” of an LTI system and define as (3.2) Substitute H(e j  ) in Eq. 3.1, we obtain (3.3) Frequency Response Input x[n]

8. BYST SigSys - WS2003: Fourier Rep. 127 e j  n the same frequency Eq. 3.3 states that the response of an LTI system to a complex sinusoid is also, e j  n, a complex sinusoid of the same frequency as the input multiplied by the frequency response H(e j  ). Fig. 3.1 illustrates the response of an LTI system to a complex sinusoidal input. h[n] e j  n H(e j  ) e j  n Figure 3.1 A complex sinusoidal d-t input to an LTI system results in a complex sinusoidal output of the same frequency multiplied by the frequency response of the system. The analogous relationship holds in the c-t case as shown in Fig. 3.2.

9. BYST SigSys - WS2003: Fourier Rep. 128 h(t) e j  t H(j  ) e j  t Figure 3.2 A complex sinusoidal c-t input to an LTI system results in a complex sinusoidal output of the same frequency multiplied by the frequency response of the system. eigenfunction eigenvalue The complex sinusoidal input in Eq. 3.3 is called an eigenfunction of the system since it produces an output that differs from the input by a constant multiplicative factor H(e j  ). This multiplicative factor is called an eigenvalue of the system. a complex value We can notice that the frequency response H(e j  ) is a complex value. That is, if we represent H(e j  ) in complex form as shown in Eq. 3.4, then we may express H(e j  ) in polar form as shown in Eq. 3.5. H(e j  ) = H R (e j  ) + H I (e j  )(3.4)

10. BYST SigSys - WS2003: Fourier Rep. 129 (3.5) Magnitude Response Phase Response where H R (e j  ) = the real part of H(e j  ), H I (e j  ) = the imaginary part of H(e j  ), the magnitude response |H(e j  )| = the magnitude response (the magnitude of H(e j  )) the phase response arg[H(e j  )] = the phase response (the phase of H(e j  ))

11. BYST SigSys - WS2003: Fourier Rep. 130 For the analysis LTI system with the complicated signal, let consider x[n] represented as weighted sum of N complex sinusoids: (3.6) If is an eigenfunction of the system with eigenvalue H(e j  ), then each term in the input as indicated by Eq. 3.6, produces an output term. Therefore, the output to x[n] defined by Eq. 3.6 can be evaluated as follows: (3.7)

12. BYST SigSys - WS2003: Fourier Rep. 131 That is, when the input x[n] is expressed as a sum of eigenfunctions, the output is also a weighted sum of N complex sinusoids, with the weights a k modified by the frequency response H(e j  ). This property describes the signal behavior as a function of frequency which will be discussed later in this chapter. 3.2.1 Response to an Exponentially Damped Sinusoidal e st s = j  pure imaginary s =  +j  Previously, we only considered a complex exponential signal of the form e st with s = j . Such signal is a pure imaginary complex exponential signal. In this section, we will observe the response of an LTI system to a generalization of the complex exponential signal where s =  +j .

13. BYST SigSys - WS2003: Fourier Rep. 132 exponentially damped sinusoidalx(t) = e st Let consider the c-t exponentially damped sinusoidal input expressed as x(t) = e st. Hence, from Eq. 2.9, the response of the LTI system can be evaluated as follows: (3.8) (3.9) TransferFunction Input x(t) Eq. 3.8 can be rewritten in a simply form as:

14. BYST SigSys - WS2003: Fourier Rep. 133 (3.10) H(s)transfer function where H(s) is called “transfer function” of an LTI system which is related to the system impulse response, h(t), by e st the same complex frequency H(s) seigenvalue eigenfunction Eq. 3.9 states that the response of an LTI system to an exponentially damped sinusoid is also, e st, an exponentially damped sinusoid of the same complex frequency as the input multiplied by the transfer function H(s). That is, the constant H(s) for a specific value of s is then the eigenvalue associated with the eigenfunction e st. x[n]=z n Similarly, the analogous relationship holds in the d-t case when the input sequence of a d-t LTI system is of the form: x[n]=z n. The

15. BYST SigSys - WS2003: Fourier Rep. 134 response of the system can be determined from the following equation: (3.11) SystemFunction Input x(t) H(z)system function where H(z) is called “system function” of a d-t LTI system which is related to the system unit-sample response, h[n], by (3.12) H(z) zeigenvalue eigenfunction Similarly, the constant H(z) for a specific value of z is then the eigenvalue associated with the eigenfunction z n.

16. BYST SigSys - WS2003: Fourier Rep. 135 3.2.2 Response to a Sinusoidal Signal e j  n e j  t e st z n H(e j  )H(s)H(z) Sinusoidal signals are the basic building blocks for generating more complicated signals. That is, knowing the response of LTI systems to a sinusoidal signal is so important. So far, we discovered that the response of an LTI system to the complex exponential signal e j  n (e j  t ) or e st or z n is also the complex exponential signal multiplied by H(e j  ) or H(s) or H(z), respectively. Thus, if the input to an LTI system is a real-valued sinusoidal signal, the response of the system should result in the same manner since, by Euler's formula, any sinusoidal signals can be represented as a combination of complex exponential signals. Let x[n] be a sinusoidal signal;

17. BYST SigSys - WS2003: Fourier Rep. 136 Using Euler’s relation, the real-valued sinusoidal signal can be rewritten as: (3.14) From Eq. 3.14, x[n] is now represented as a linear combination of two complex exponential signals which are complex conjugate to each other. Hence, from Eq. 3.7, the response of an LTI system to x[n] defined by Eq. 3.14 can be evaluated as follows: (3.15) (3.13)

18. BYST SigSys - WS2003: Fourier Rep. 137 From Eq. 3.15, the second term is the complex conjugate of the first term. Therefore, only the real part of either term will be remained. That is, (3.16) same frequency different magnitude phase |H(e j  )| From Eq. 3.16, the response of an LTI system to the sinusoidal signal is another sinusoid of the same frequency but with different magnitude and phase. The magnitude of the input sinusoid is modified by the magnitude response, |H(e j  )| and the

19. BYST SigSys - WS2003: Fourier Rep. 138 arg{H(e j  )} phase of the input sinusoid is modified by the phase response, arg{H(e j  )}. The frequency response characterize the steady-state response of the system to sinusoidal inputs as a function of the sinusoid’s frequency. It is said to be a steady-state response since the input sinusoid is assumed to exist for all time and thus the system is in an equilibrium condition. Fourier Series Representation 3.3 of C-T Periodic Signals Fourier series (FS) are series of sinusoidal or complex exponential signals. It was first introduced by Jean Baptiste Joseph, Baron de Fourier who discovered that a complicate periodic signal (with a pattern

20. BYST SigSys - WS2003: Fourier Rep. 139 a sum of sinusoidal signal with different frequencies Fourier series Fourier expansion that repeats itself) consists of the sum of many simple waves. He suggested that most signals can be represented by a sum of sinusoidal signal with different frequencies. Hence, he has purposed a method to decompose a given periodic signal into a linear combination of sinusoidal signals having different frequencies. The resulting sum is called the Fourier series or the Fourier expansion. spectrum frequency content A fundamental concept in the study of Fourier series is the spectrum which is the notation of the frequency content of a signal. For a large class of signals, the frequency content can be evaluated by decomposing the signal into frequency components given 3.3.1 Representation of Signals in Terms of Frequency Components

21. BYST SigSys - WS2003: Fourier Rep. 140 by sinusoidal signals. In this section, the most general and powerful method for generating new signals from sinusoidal signals will be introduced. This method will create any signals by adding together a constant and N sinusoids, each with a different frequency, amplitude and phase as shown in Eq. 3.17. Mathematically, this signal may be represented by the equation (3.17) Using Euler’s relation, Eq. 3.17 can be rewritten as: (3.18)

22. BYST SigSys - WS2003: Fourier Rep. 141 X k phasor Let X k represent the phasor of the individual sinusoidal components. That is, (3.19) Substitute Eq. 3.19 into Eq. 3.18, the alternative form of x(t) can be represented as follows: (3.20) where X 0 = A 0. each sinusoid two rotating phasors F k -F k We see then that each sinusoid in the sum decomposes into two rotating phasors, one with positive frequency, F k, and the other with negative frequency, -F k. This form

23. BYST SigSys - WS2003: Fourier Rep. 142 follows the fact that the real part of a complex number is equal to one-half the sum of that number and its complex conjugate. frequency domain representation (0.5X k, F k ) (frequency) spectrum Eq. 3.20 indicates the frequency domain representation of the signal x(t) in terms of a pair (0.5X k, F k ) which is usually called the (frequency) spectrum. That is, the spectrum is just the set of pairs: The spectrum of signal x(t) simply provides the information required to synthesize it. The graphical plot of the spectrum can be illustrated in Fig. 3.3. {(X 0, 0), (0.5X 1, F 1 ), (0.5X * 1, -F 1 ), (0.5X 2, F 2 ), (0.5X * 2, -F 2 ), … }.

24. BYST SigSys - WS2003: Fourier Rep. 143 Figure 3.3 Spectrum of the signal x(t) represented by Eq. 3.20 Fourier analysis Fourier seriesFourier transforms A general procedure for computing and plotting the spectrum of any signal is called Fourier analysis. This procedure involves simply decomposing a signal x(t) into the complex exponential form as shown in Eq. 3.20, and selecting off the amplitude, phase, and frequency of each of its rotating phasor components. The mathematical tools for performing this analysis are called "Fourier series" and "Fourier transforms".

25. BYST SigSys - WS2003: Fourier Rep. 144 3.3.2 Linear Combinations of Harmonically Related Complex Exponential We have seen how a signal can be created by a sum of a constant and N sinusoids having difference frequency, amplitude and phase. An interesting situation occurs when a signal x(t) is a periodic signal defined as follows: x(t) = x(t+T) for all t T fundamental period  0 = 2  /T fundamental angular frequency When T is the smallest positive, nonzero value, it is called the "fundamental period" and the value  0 = 2  /T is referred to as the "fundamental angular frequency". To maintain the fundamental period, T, of x(t), the frequency F k in Eq. 3.20 must be the k th harmonic frequency of the

26. BYST SigSys - WS2003: Fourier Rep. 145 harmonically related sinusoids F k kF 0 fundamental frequency F 0 of x(t). That is, the signal x(t) will be a linear combinations of harmonically related sinusoids. Hence, if we substitute F k in Eq. 3.20 with kF 0 and allow the sum to be infinite (i.e., N = ∞), any periodic signal x(t) can be expressed as follows: (3.21) or Fourier Series

27. BYST SigSys - WS2003: Fourier Rep. 146 k  0 = the k th harmonic. k =1 F 0 (  0 ) = the first harmonic F 0 (  0 ) = the first harmonic (the fundamental (angular) frequency) where a k the Fourier series coefficients a k = the Fourier series coefficients, Fourier series (FS) a k discrete integer (k) multiples of the fundamental (angular) frequency  0 The harmonic series defined by Eq. 3.21 is called the Fourier series (FS) of the periodic signal x(t). Fourier series represents any periodic signals in the form of the weights (a k ) on a "discrete" set of the frequencies that are integer (k) multiples of the fundamental (angular) frequency  0. Note that the Fourier series is a special case of Eq. 3.20 in that it is a linear combination of infinite (N = ∞) harmonically complex exponentials.

28. BYST SigSys - WS2003: Fourier Rep. 147 FS coefficients a k spectral coefficients These FS coefficients a k are often referred to as the spectral coefficients of x(t). In general, FS coefficients a k are complex coefficients which can be represented in a polar form of: magnitude spectrum phase spectrum The magnitude of a k (A k ) is known as the magnitude spectrum of x(t). Similarly, the phase of a k (  k ) is known as the phase spectrum of x(t). real If x(t) is real periodic signal, the Fourier series coefficient a k and a -k are a pair of complex conjugate with respect to each other. That is,

29. BYST SigSys - WS2003: Fourier Rep. 148 For a real periodic signal, (3.22) The Fourier series coefficients a k of the complex exponentials in Eq. 3.21 are evaluated by the following equation: (3.23) The coefficient a 0 is the constant or de component of x(t) given by (3.23) which is simply the average of x(t) over one period.

30. BYST SigSys - WS2003: Fourier Rep. 149 Alternative Forms of Fourier Series real periodic signals The Fourier series described by Eq. 3.21 is in the exponential form. In the case of real periodic signals, the other two alternative forms of the Fourier series can be derived using Euler’s relation. Let x(t) be a real periodic signal having the Fourier series coefficients a k. Since a -k = a * k, then  -k = -  k. For a given value of k, the sum of the two terms of the same frequency k  0 in Eq. 3.21 yields (3.24a) or

31. BYST SigSys - WS2003: Fourier Rep. 150 combined trigonometric form real By using Eq. 3.24 and rearranging the summation in Eq. 3.21, we can easily derive the combined trigonometric form for the Fourier series of real periodic signals: (3.25) (3.24b) real A third form for the Fourier series of real periodic signals can be obtained by writing a k in rectangular form as

32. BYST SigSys - WS2003: Fourier Rep. 151 trigonometric form Substituting Eq. 3.26 into Eq. 3.21 yields the trigonometric form of the Fourier series a k = B k +jC k, B k and C k are real where B k and C k are real. Hence, with this expression for a k, Eq. 3.24a can be rewritten as: (3.26)

33. BYST SigSys - WS2003: Fourier Rep. 152 The exponential form (Eq. 3.21) and the combined trigonometric form (Eq. 3.25) of the Fourier series are probably the most useful forms in signal and system theory. (3.27) where

34. BYST SigSys - WS2003: Fourier Rep. 153 (3.28) b k and c k are real where b k and c k are real. Note that the trigonometric form of the Fourier series expressed by Eq. 3.27 is normally rewritten as follows:

35. BYST SigSys - WS2003: Fourier Rep. 154 Dirichlet conditions Fourier suggested that any periodic signal could be expressed as a sum of complex exponentials (or sinusoids). However, in fact, it is partially true in that Fourier series can be used to represent an extremely large class of periodic signals. In particular, a periodic signal x(t) has a Fourier series if it satisfies the Dirichlet conditions given by Convergence of the Fourier Series 1.x(t) is absolutely integrable over any period; that is, 2.x(t) has only a finite number of maxima and minima over any period. (3.29)

36. BYST SigSys - WS2003: Fourier Rep. 155 3.x(t) has only a finite number of discontinuities over any period. Fourier Series Representation 3.4 of D-T Periodic Signals Discrete- Time Fourier Series (DTFS) In this section, we consider the Discrete- Time Fourier Series (DTFS) which is the Fourier series representation of discrete- time (d-t) periodic signals. Although, the principle concept of DTFS is similar to FS. There are some important differences. In particular, DTFS is a finite series while as FS is a infinite series. Being a finite series, DTFS always exists, as opposed to FS which requires some conditions for existence. period N A d-t signal x[n] is periodic with period N if

37. BYST SigSys - WS2003: Fourier Rep. 156 x[n] = x[n+N] for all n. N fundamental period  0 = 2  /N fundamental angular frequency When N is the smallest positive, nonzero value, it is called the "fundamental period" and the value  0 = 2  /N is referred to as the "fundamental angular frequency". (3.30) or The DTFS represents an N periodic d-t signal x[n] is defined as the series of Eq. 3.30:

38. BYST SigSys - WS2003: Fourier Rep. 157 The D-T Fourier series coefficients a k can be evaluated from Eq. 3.31 as follows: (3.31) a k spectral coefficients As in continuous time, DTFS coefficients a k in Eq. 3.31 are also referred to as the spectral coefficients of x[n]. These coefficients specify a decomposition of x[n] into a sum of N harmonically related complex exponentials. Each a k is associated with a complex sinusoid of a different frequency. Since there are only N distinct complex exponentials that are periodic with period N, DTFS representation is a finite series with N terms. Hence, if we consider more than N sequential values of k, the

39. BYST SigSys - WS2003: Fourier Rep. 158 for a real x[n]. real If x[n] is real periodic d-t signal, DTFS coefficient a k and a -k are a pair of complex conjugate with respect to each other. That is, values a k repeat periodically with period N. That is, a k = a k+N (3.32) (3.33) realperiodic sequences There are two alternative forms for DTFS of real periodic sequences which are similar to Eq. 3.25 and Eq. 3.27. If the polar form of

40. BYST SigSys - WS2003: Fourier Rep. 159 (3.34) If we express a k in the Cartesian form of is used, DTFS of real periodic x[n] can be expressed as: a k = B k +jC k if N is odd or as if N is even.

41. BYST SigSys - WS2003: Fourier Rep. 160 B k and C k are real where B k and C k are real, the alternative form for DTFS of a real periodic x[n] can be expressed as: (3.35) if N is odd or as if N is even.

42. BYST SigSys - WS2003: Fourier Rep. 161 The Continuous-Time Fourier 3.5 Transform (FT) periodic nonperiodic aperiodic infinitesimally close in frequency harmonically related frequency periodictime discrete frequency aperiodic timecontinuous frequency A key feature of the Fourier series representation of periodic signals is the description of such signals in terms of the frequency content given by sinusoidal components. Now we would like to extend this notation to nonperiodic signals, also called aperiodic signals. As will be seen, the frequency components of aperiodic signals are infinitesimally close in frequency rather than harmonically related frequency as in the case of periodic signals. In other words, signals that are periodic in time have discrete frequency domain representations, while aperiodic time signals have continuous frequency domain representations. Therefore, the representation in terms of a linear combination takes the form of an

43. BYST SigSys - WS2003: Fourier Rep. 162 Fourier transform integral instead of a sum and such representation will be called the Fourier transform. The extension of the Fourier series to aperiodic signals is based on the idea of extending the period to infinity. Let x T (t) denote the pulse train with period T as shown in Fig. 3.4. T 2 TT1T1 T 2 - -T 1 -T... x T (t) Figure 3.4 A continuous-time periodic pulse train. That is, (3.36)

44. BYST SigSys - WS2003: Fourier Rep. 163 To demonstrate the change in frequency component of x T (t) as T → ∞, the Fourier series coefficients a k for this pulse train will be evaluated. Since then, for k = 0 gives For a k where k ≠ 0, we get k = 0 which we can rewrite as (3.37a) (3.37b)

45. BYST SigSys - WS2003: Fourier Rep. 164 k ≠ 0 where  0 T = 2 . To illustrate the change in the amplitude spectrum |a k |, it turns out to be more appropriate to plot the amplitude spectrum |a k | scaled by T; that is, a plot of T|a k | versus  = k  0 will be generated. (3.38) 2sin  T 1 /  envelope That is, with  thought of as a continuous variable, the function “2sin  T 1 /  ” represents the envelope of Ta k, and the (3.39)

46. BYST SigSys - WS2003: Fourier Rep. 165 coefficients ak are simply equally space samples of this envelope. As T 1 is fixed, the envelope of Ta k is independent of T. The plots of Ta k for T=4T 1, T=8T 1, and T=16T 1 are illustrated in Fig. 3.5. Figure 3.5 The Fourier series coefficients and their envelope of the periodic square wave in Fig. 3.4 for several values of T (with T1 fixed): (a) T=4T 1 ; (b) T=8T 1 ; (c) T=16T 1.

47. BYST SigSys - WS2003: Fourier Rep. 166 From Fig. 3.5 it can be seen that as T is increased (with T 1 fixed), the “density” of the frequency components, a k, increase, whereas the envelope of the magnitudes of the scaled spectral components, Ta k, remains the same. In other words, the envelope is sampled with a closer and closer spacing. In the limit as T → ∞, Ta k converge into a continuum of frequency components whose magnitudes have the same shape as the envelope of the discrete spectra shown in Fig. 3.5 and the pulse train x T (t) approaches a rectangular pulse, x(t), as shown in Fig. 3.6. That is, in mathematical terms, (3.40)

48. BYST SigSys - WS2003: Fourier Rep. 167 T1T1 -T 1... Figure 3.6 A pulse signal as limit of a pulse train signal. This example illustrates the effect of extending the period of a periodic signal to infinity which is the basic idea behind the development of Fourier representation for aperiodic signals. Previously, we have seen the effect of increasing the period of a periodic signal to “density” of the frequency components. Let us now examine the effect of this on the Fourier series representation of a periodic signal.

49. BYST SigSys - WS2003: Fourier Rep. 168 Let consider an aperiodic signal x(t) that is of finite duration (band limited). That is, for some number T 1, x(t) = 0 if |t| > T 1, as illustrated in Fig. 3.6 (a pulse signal). Then, we can reconstruct a periodic signal x T (t) for which x(t) is one period, as shown in Fig. 3.4(a pulse train signal). The periodic signal x T (t) has an exponential Fourier series that is given by where  0 T = 2  and (3.41) (3.42)

50. BYST SigSys - WS2003: Fourier Rep. 169 As far as the integration is concerned in Eq. 3.42, the integrand on the integral can be rewritten as: Now, let us generalize the formula for the Fourier series coefficient in Eq. 3.43 by defining the envelope X(j  ) of Ta k as (3.43) (3.44)

51. BYST SigSys - WS2003: Fourier Rep. 170 From Eq. 3.48, the signal x T (t) becomes x(t) when T → ∞. That is, Using Eq. 3.44, we can rewrite Eq. 3.43 as (3.45) Combining Eq. 3.41 and Eq. 3.45, we can express x T (t) in terms of X(j  ) as (3.46)  = 2  /T. Let  = 2  /T. (3.48) Then, we can rewrite Eq. 3.46 as: (3.47)

52. BYST SigSys - WS2003: Fourier Rep. 171 (3.49) From Eq. 3.47, the space between each frequency component, , becomes dense as T → ∞. That is, k  0 → ,  → d , and the summation becomes an integral. Hence in the limit Eq. 3.49 becomes (3.50) Fourier transformFourier integral inverse Fourier transform X(j  ) in Eq. 3.44 is referred to as the Fourier transform or Fourier integral of x(t) and Eq. 3.50 is referred to as the inverse Fourier transform equation.

53. BYST SigSys - WS2003: Fourier Rep. 172 3.5.1 The Definition Let x(t) be a signal such that: (a) x(t), -∞ < t < ∞, and (b) for 0 < M < ∞ Fourier transform Then the Fourier transform of x(t) is defined as Absolutely integrable Analysis equation direct transform (3.51)

54. BYST SigSys - WS2003: Fourier Rep. 173 And X(j  ) ↔ x(t) Fourier transform pair is called a Fourier transform pair. X(j  ) (3.52) Spectrum (of signal) (a graphical representation of the frequency content of signal) inverse Fourier transform The inverse Fourier transform is defined as Synthesis equation inverse transform X(j  ) provides the information needed for describing x(t) as a linear combination of sinusoidal signals at different frequencies.

55. BYST SigSys - WS2003: Fourier Rep. 174 periodic discrete aperiodic continuum We can notice that both Fourier series coefficients, {a k } and Eq. 3.52 represent a signal as a linear combination of complex exponential signals. For periodic signals, these complex exponential signals have amplitudes {a k } and occur at a discrete set of harmonically related frequencies k  0, k =. For aperiodic signals, these complex exponential signals occur at a continuum of frequencies. That is, the essential difference between the Fourier series and the Fourier transform is that the spectrum in the latter case is continuous. Therefore, the synthesis of an aperiodic signal from its spectrum (Eq. 3.52) is accomplished by means of integration instead of summation.

56. BYST SigSys - WS2003: Fourier Rep. 175 3.5.2 Convergence of Fourier Transforms Eq. 3.44 indicates that the definition of the Fourier transform relies on the existence of infinite integrals. Hence we should verify that the infinite integrals in the definition exist for a class of signals before we use the Fourier transform. In this section we simply state several convergence conditions on the signal x(t) since an analysis of convergence is beyond the scope of this class. Dirichlet conditions The set of conditions that guarantee the existence of the Fourier transform is the Dirichlet conditions, which may be express as: 1. The signal x(t) is absolutely integrable, that is,

57. BYST SigSys - WS2003: Fourier Rep. 176 2. The signal x(t) has a finite number of finite discontinuities. 3. The signal x(t) has a finite number of maxima and minima. The first condition implies that x(t) also has finite energy: that is, (3.54) However, the converse is not true. That is, a signal may have finite energy but may not be absolutely integrable. Hence, the finite energy condition of a signal is a weaker condition. In any case, nearly all finite energy signals have a Fourier transform, although, they violate the first condition. (3.53)

58. BYST SigSys - WS2003: Fourier Rep. 177 Thus, the Dirichlet conditions are sufficient but not necessary for the existence of the Fourier transform. Fourier transform can be used to represent periodic signals. Previously, we only concentrate on the Fourier transform of an aperiodic signal. In this section, we are going to consider both periodic and aperiodic signals within a unified context. Let consider a signal x(t) with Fourier transform X(j  ) that defined as: X(j  ) =  (  - k  0 ) (3.55) 3.5.3 The Fourier Transform for Periodic Signals

59. BYST SigSys - WS2003: Fourier Rep. 178 The signal x(t) having the spectrum defined by Eq. 3.55 can be obtained by taking the inverse Fourier transform; that is, complex sinusoidal Hence, the inverse Fourier transform of a single impulse defined by Eq. 3.55 is a complex sinusoidal. In fact, it is a periodic signal x(t). a linear combination Now let consider the case when X(j  ) represented as a linear combination of (3.56) single impulse at  = k  0 That is, the spectrum of signal x(t) is a single impulse at  = k  0.

60. BYST SigSys - WS2003: Fourier Rep. 179 the Fourier transform of a periodic signal can be constructed directly from its Fourier series Eq. 3.58 corresponds exactly to the Fourier series representation of a periodic signal. Therefore, the Fourier transform of a periodic signal can be constructed directly from its Fourier series. (3.57) Similarly, the inverse Fourier transform yields (3.58) scaled impulses equally spaced in frequency, that is, impulse train of area 2  a k at  = k  0 That is, the spectrum of signal x(t) is now a impulse train of area 2  a k at  = k  0.

61. BYST SigSys - WS2003: Fourier Rep. 180 3.5.4 Properties of Continuous-Time Fourier Transform Let x(t) ↔ X(j  ) and y(t) ↔ Y(j  ). Linearity where a and b are constants. (3.59) Time Shifting (3.40) phase changed-  t 0 That is, a signal which is shifted in time has only the phase of its Fourier transform changed by the factor of “-  t 0 ”. ax(t) + by(t) ↔ aX(j  ) + bY(j  ).

62. BYST SigSys - WS2003: Fourier Rep. 181 Frequency Shifting (3.41) real If x(t) is real-valued time signal, (3.43) That is, |X(j  )| = |X(-j  )| and arg{X(j  )} = -arg{X(-j  )} (3.44) or, Conjugation and Conjugate Symmetry (3.42)

63. BYST SigSys - WS2003: Fourier Rep. 182 Re{X(j  )} = Re{X(-j  )} and Im{X(j  )} = -Im{X(-j  )} (3/45) (3.47) real Eq. 3.47 implies that X(j  ) is real. realodd Similarly, if x(t) is real valued and has odd symmetry, (3.48) imaginary Eq. 3.48 implies that X(j  ) is imaginary. imaginary If x(t) is imaginary-valued time signal, (3.46) realeven If x(t) is real valued and has even symmetry,

64. BYST SigSys - WS2003: Fourier Rep. 183 Differentiation (3.49) Integration (3.50) Time and Frequency Scaling (3.51)

65. BYST SigSys - WS2003: Fourier Rep. 184 Duality (3.52) Parseval’s Relation (3.53) Convolution (3.54) Time Domain where

66. BYST SigSys - WS2003: Fourier Rep. 185 (4.35) Frequency Domain where