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Howell, K.B. “Fourier Transforms.” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000

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2 Fourier Transforms Kenneth Howell 2.1 Introduction and Basic Deﬁnitions University of Alabama Basic Deﬁnition, Notation, and Terminology • Alternate in Huntsville Deﬁnitions • The Generalized Transforms • Further Generalization of the Generalized Transforms • Use of the Residue Theorem • Cauchy Principal Values 2.2 General Identities and Relations Invertibility • Near-Equivalence (Symmetry of the Transforms) • Conjugation of Transforms • Linearity • Scaling • Translation and Multiplication by Exponentials • Complex Translation and Multiplication by Real Exponentials • Modulation • Products and Convolution • Correlation • Differentiation and Multiplication by Polynomials • Moments • Integration • Parseval’s Equality • Bessel’s Equality • The Bandwidth Theorem 2.3 Transforms of Speciﬁc Classes of Functions Real/Imaginary Valued Even/Odd Functions • Absolutely Integrable Functions • The Bandwidth Theorem for Absolutely Integrable Functions • Square Integrable (“Finite Energy”) Functions • The Bandwidth Theorem of Finite Energy Functions • Functions with Finite Duration • Band-Limited Functions • Finite Power Functions • Periodic Functions • Regular Arrays of Delta Functions • Periodic Arrays of Delta Functions • Powers of Variables and Derivatives of Delta Functions • Negative Powers and Step Functions • Rational Functions • Causal Functions • Functions on the Half-Line • Functions on Finite Intervals • Bessel Functions 2.4 Extensions of the Fourier Transform and Other Closely Related Transforms Multidimensional Fourier Transforms • Multidimensional Transforms of Separable Functions • Transforms of Circularly Symmetric Functions and the Hankel Transform • Half-Line Sine and Cosine Transforms • The Discrete Fourier Transform • Relations Between the Laplace Transform and the Fourier Transform 2.5 Reconstruction of Sampled Signals Sampling Theorem for Band-Limited Functions • Truncated Sampling Reconstructions of Band-Limited Functions • Reconstruction of Sampled Nearly Band-Limited Functions • Sampling Theorem for Finite Duration Functions • Fundamental Sampling Formulas and Poisson’s Formula 2.6 Linear Systems Linear Shift Invariant Systems • Reality and Stability • System Response to Complex Exponentials and Periodic Functions • Causal Systems • Systems Given by Differential Equations • RLC Circuits • Modulation and Demodulation © 2000 by CRC Press LLC

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2.7 Random Variables Basic Probability and Statistics • Multiple Random Processes and Independence • Sums of Random Processes • Random Signals and Stationary Random Signals • Correlation of Stationary Random Signals and Independence • Systems and Random Signals 2.8 Partial Differential Equations The One-Dimensional Heat Equation • The Initial Value Problem for Heat Flow on an Inﬁnite Rod • An Inﬁnite Rod with Heat Sources and Sinks • A Boundary Value Problem for Heat Flow on a Half-Inﬁnite Rod • Tables 2.1 Introduction and Basic Deﬁnitions The Fourier transform is certainly one of the best known of the integral transforms and vies with the Laplace transform as being the most generally useful. Since its introduction by Fourier in the early 1800s, it has found use in innumerable applications and has, itself, led to the development of other transforms. Today the Fourier transform is a fundamental tool in engineering science. Its importance has been enhanced by the development in the twentieth century of generalizations extending the set of functions that can be Fourier transformed and by the development of efﬁcient algorithms for computing the discrete version of the Fourier transform. There are two parts to this article on the Fourier transform. The ﬁrst (Sections 2.1 through 2.4) contains the fundamental theory necessary for the intelligent use of the Fourier transform in practical problems arising in engineering. The second part (Sections 2.5 through 2.8) is devoted to applications in which the Fourier transform plays a signiﬁcant role. This part contains both fairly detailed descriptions of speciﬁc applications and fairly broad overviews of classes of applications. This particular section deals with the basic deﬁnition of the Fourier transform and some of the integrals used to compute Fourier transforms. Two deﬁnitions for the transform are given. First, the classical deﬁnition is given in Subsection 2.1.1. This is the integral formula for directly computing transforms generally found in elementary texts. From this formula many of the basic formulas and identities involving the Fourier transform can be derived. Inherent in the classical deﬁnition, however, are integrability conditions that cannot be satisﬁed by many functions routinely arising in applications. For this reason, more general deﬁnitions of the Fourier transform are brieﬂy discussed in Subsections 2.1.3 and 2.1.4. These general deﬁnitions will also help clarify the role of generalized functions in Fourier analysis. The computation of Fourier transforms often involves the evaluation of integrals, many of which cannot be evaluated by the elementary methods of calculus. For this reason, this section also contains a brief discussion illustrating the use of the residue theorem in computing certain integrals as well as a brief discussion of how to deal with certain integrals containing singularities in the integrand. 2.1.1 Basic Deﬁnition, Notation, and Terminology ∞ If φ(s) is an absolutely integrable function on (–∞, ∞) (i.e., |φ(s)-冨 ds < ∞), then the (direct) Fourier ∫−∞ transform of φ(s), Ᏺ[φ], and the Fourier inverse transform of φ(s), Ᏺ–1 [φ], are the functions given by ∞ − j x s Ᏺ[φ] x = ∫−∞φ(s)e ds (2.1.1.1) and −1 1 ∞ j x s Ᏺ [φ] x = 2π ∫−∞φ(s)e ds . (2.1.1.2) © 2000 by CRC Press LLC

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Example 2.1.1.1 If φ(s) = e – s u(s), then ∞ − s − j x s ∞ −(1+ j x ) s 1 Ᏺ[φ] x = ∫−∞e u(s)e ds = ∫0 e ds = 1+ jx and −1 1 ∞ − s j x s 1 ∞ −(1− j x ) s 1 Ᏺ [φ] x = 2π ∫−∞e u(s)e ds = 2π ∫0 e ds = 2π − j2πx . Example 2.1.1.2 For α > 0, the transform of the corresponding pulse function, 1, if s < α pα (s) = , 0, if α < s is α − j x s e jα x − e − jα x 2 Ᏺ[pα ] x = ∫−αe ds = jx = x sin(α x) . A function, ψ, is said to be “classically transformable” if either 1. ψ is absolutely integrable on the real line, or 2. ψ is the Fourier transform (or Fourier inverse transform) of an absolutely integrable function, or 3. ψ is a linear combination of an absolutely integrable function and a Fourier transform (or Fourier inverse transform) of an absolutely integrable function. If φ is classically transformable but not absolutely integrable, then it can be shown that formulas (2.1.1.1) and (2.1.1.2) can still be used to deﬁne Ᏺ[φ] and Ᏺ–1[φ] provided the limits are taken symmetrically; that is, a − j x s Ᏺ[φ] x = lai→m∞∫−aφ(s)e ds and −1 1 a j x s Ᏺ [φ] x = 2π lai→m∞∫−aφ(s)e ds . In most applications involving Fourier transforms, the functions of time, t, or position, x, are denoted using lower case letters — for example: f and g. The Fourier transforms of these functions are denoted using the corresponding upper case letters — for example: F = Ᏺ[ f ] and G = Ᏺ[g]. The transformed functions can be viewed as functions of angular frequency, ω . Along these same lines it is standard practice to view a signal as a pair of functions, f(t) and F(ω), with f(t) being the “time domain repre- sentation of the signal” and F(ω) being the “frequency domain representation of the signal.” © 2000 by CRC Press LLC

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2.1.2 Alternate Deﬁnitions Pairs of formulas other than formulas (2.1.1.1) and (2.1.1.2) are often used to deﬁne Ᏺ[φ] and Ᏺ–1[φ]. Some of the other formula pairs commonly used are: ∞ ∞ − j2π xs −1 j2π xs Ᏺ[φ] x = ∫−∞φ(s)e ds, Ᏺ [φ] x = ∫−∞φ(s)e ds (2.1.2.1) and 1 ∞ − jxs −1 1 ∞ jxs Ᏺ[φ] x = 2π ∫−∞φ(s)e ds, Ᏺ [φ] x = 2π ∫−∞φ(s)e ds . (2.1.2.2) Equivalent analysis can be performed using the theory arising from any of these pairs; however, the resulting formulas and equations will depend on which pair is used. For this reason care must be taken to ensure that, in any particular application, all the Fourier analysis formulas and equations used are derived from the same deﬁning pair of formulas. Example 2.1.2.1 Let φ(t) = e–t u(t) and let ψ1, ψ2, and ψ3 be the Fourier transforms of φ as deﬁned, respectively, by formulas (2.1.1.1), (2.1.2.1), and (2.1.2.2). Then, ∞ −t − jtω 1 ψ 1(ω) = ∫−∞e u(t)e dt = 1+ jω , ∞ −t − j2π tω 1 ψ 2(ω) = ∫−∞e u(t)e dt = 1+ j2πω , and 1 ∞ −t − jtω 1 1 ψ 3(ω) = 2π ∫−∞e u(t)e dt = 2π ⋅1+ jω . 2.1.3 The Generalized Transforms Many functions and generalized functions* arising in applications are not sufﬁciently integrable to apply the deﬁnitions given in subsection 2.1.1 directly. For such functions it is necessary to employ a generalized deﬁnition of the Fourier transform constructed using the set of “rapidly decreasing test functions” and a version of Parseval’s equation (see subsection 2.2.14). A function, φ, is a “rapidly decreasing test function” if 1. every derivative of φ exists and is a continuous function on (–∞, ∞) and 2. for every pair of nonnegative integers, n and p, 冨φ(n)(s)冨 = O(冨s 冨–p) as 冨s 冨 → ∞. The set of all rapidly decreasing test functions is denoted by and includes the Gaussian functions as well as all test functions that vanish outside of some ﬁnite interval (such as those discussed in the ﬁrst *For a detailed discussion of generalized functions, see the ﬁrst chapter in this handbook. © 2000 by CRC Press LLC

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chapter of this handbook. If φ is a rapidly decreasing test function then it is easily veriﬁed that φ is classically transformable and that both Ᏺ[φ] and Ᏺ–1[φ] are also rapidly decreasing test functions. It can also be shown that Ᏺ–1[Ᏺ[φ]] = φ. Moreover, if f and G are classically transformable, then ∞ ∞ ∫−∞Ᏺ[ f ] x φ(x)dx = ∫−∞f (y)Ᏺ[φ] y dy (2.1.3.1) and ∞ ∞ −1 −1 ∫−∞Ᏺ [G] x φ(x)dx = ∫−∞G(y)Ᏺ [φ] y dy . (2.1.3.2) If f is a function or a generalized function for which the right-hand side of equation (2.1.3.1) is well deﬁned for every rapidly decreasing test function, φ, then the generalized Fourier transform of f, Ᏺ[f], is that generalized function satisfying (2.1.3.1) for every φ in . Likewise, if G is a function or generalized function for which the right-hand side of (2.1.3.2) is well deﬁned for every rapidly decreasing test function, φ, then the generalized inverse Fourier transform of G, Ᏺ–1[G], is that generalized function satisfying equation (2.1.3.2) for every φ in . Example 2.1.3.1 Let α be any real number. Then, for every rapidly decreasing test function φ, ∞ ∞ jαy jαy ∫−∞Ᏺ[e ] x φ(x)dx = ∫−∞e Ᏺ[φ] y dy 1 ∞ jαy = 2π 2π ∫−∞Ᏺ[φ] y e dy −1 = 2πᏲ [Ᏺ[φ]] α = 2πφ(α) ∞ = ∫−∞2πδ(x −α)φ(x)dx where δ(x) is the delta function. This shows that, for every φ in , ∞ ∞ jαy ∫−∞2πδ(x −α)φ(x)dx = ∫−∞e Ᏺ[φ] y dy and thus, Ᏺ[e jα y]冨x = 2πδ (x – α) . Any (generalized) function whose Fourier transform can be computed via the above generalized deﬁnition is called “transformable.” The set of all such functions is sometimes called the set of “tempered generalized functions” or the set of “tempered distributions.” This set includes any piecewise continuous function, f, which is also polynomially bounded, that is, which satisﬁes 冨f(s)冨 = O(冨s 冨p) as 冨s 冨 → ∞ © 2000 by CRC Press LLC

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for some p < ∞. Finally, it should also be noted that if f is classically transformable, then it is transformable, and the generalized deﬁnition of Ᏺ[f] yields exactly the same function as the classical deﬁnition. 2.1.4 Further Generalization of the Generalized Transforms Unfortunately, even with the theory discussed in the previous subsection, it is not possible to deﬁne or discuss the Fourier transform of the real exponential, e t. It may be of interest to note, however, that a further generalization that does permit all exponentially bounded functions to be considered “Fourier transformable” is currently being developed using a recently discovered alternate set of test functions. This alternate set, denoted by Ᏻ, is the subset of the rapidly decreasing test functions that satisfy the following two additional properties: 1. Each test function is an analytic test function on the entire complex plane. 2. Each test function, φ(x + jy), satisﬁes φ(x + jy) = O(e–α|x|) as x → ± ∞ for every real value of y and α. The second additional property of these test functions ensures that all exponentially bounded functions are covered by this theory. The very same computations given in example 2.1.3.1 can be used to show that, for any complex value, α + jβ, Ᏺ[ej(α+jβ)t]|ω = 2πδα+jβ(ω), where δα+jβ(t) is “the delta function at α + jβ.” This delta function, δα+jβ(t), is the generalized function satisfying ∞ ∫−∞δα+ jβ (t )φ(t)dt = φ(α + jβ) for every test function φ(t), in Ᏻ. In particular, letting α + jβ = –j, Ᏺ[et]冨ω = 2πδ–j(ω) and Ᏺ[δj(t)]冨ω = eω. In addition to allowing delta functions to be deﬁned at complex points, the analyticity of the test functions allows a generalization of translation. Let α + jβ be any complex number and f(t) any (exponentially bounded) (generalized) function. The “generalized translation of f(t) by α + jβ ,” denoted by Tα+jβ f(t), is that generalized function satisfying ∞ ∞ ∫−∞Tα+ jβ f (t )φ(t)dt = ∫−∞f (t )φ(t + (α + jβ))dt (2.1.4.1) for every test function, φ(t), in Ᏻ. So long as β = 0 or f(t) is, itself, an analytic function on the entire complex plane, then the generalized translation is exactly the same as the classical translation. Tα+jβ f (t) = f(t – (α + jβ )). © 2000 by CRC Press LLC

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It may be observed, however, that equation (2.1.4.1) deﬁnes the generalized function Tα+jβ f even when f(z) is not deﬁned for nonreal values of z. 2.1.5 Use of the Residue Theorem Often a Fourier transform or inverse transform can be described as an integral of a function that either is analytic on the entire complex plane, or else has a few isolated poles in the complex plane. Such integrals can often be evaluated through intelligent use of the reside theorem from complex analysis (see Appendix I). Two examples illustrating such use of the residue theorem will be given in this subsection. The ﬁrst example illustrates its use when the function is analytic throughout the complex plane, while the second example illustrates its use when the function has poles off the real axis. The use of the residue theorem to compute transforms when the function has poles on the real axis will be discussed in the next subsection. Example 2.1.5.1 Transform of an Analytic Function Consider computing the Fourier transform of g(t) = e – t 2, ∞ −t2 − jω t G(ω) = Ᏺ[g(t)] ω = ∫−∞e e dt . Because 2 ω 2 ω 2 t + jωt = t + j + , 2 4 it follows that G(ω) = e − 14ω 2 ∫−∞exp −t + j ω2 2dt (2.1.5.1) = e − 41ω 2 ∞+ jω2 e − z2dz . ∫−∞+ jω 2 Consider, now, the integral of e – z 2 over the contour Cγ where, for each γ > 0, Cγ = C1,γ + C2,γ + C3,γ + C4,γ is the contour in Figure 2.1. Because e – z 2 is analytic everywhere on the complex plane, the residue theorem states that − z2 0 = e dz ∫C γ − z2 − z2 − z2 − z2 = e dz + e dz + e dz + e dz . ∫C 1,γ ∫C2,γ ∫C3,γ ∫C4,γ Thus, − z2 − z2 − z2 − z2 − e dz = e dz + e dz + e dz . (2.1.5.2) ∫C 3,γ ∫C1,γ ∫C2,γ ∫C4,γ © 2000 by CRC Press LLC

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FIGURE 2.1 Contour for computing Ᏺ[e–t2]. Now, − z2 ω 2 −(γ + j y)2 lim e dz = lim e dy γ →∞∫C2,γ γ →∞∫y=0 −γ 2 ω 2 y2− j2γ y = lim e e dy γ →∞ ∫y=0 = 0 . Likewise, − z2 lim e dz = 0 , γ →∞∫C4,γ while − z2 −γ + jω2 − z2 ∞+ jω2 − z2 lim e dz = lim e dz = − e dz γ →∞∫C3,γ γ →∞∫γ + jω 2 ∫−∞+ jω2 and − z2 γ − x 2 ∞ − x 2 lim e dz = lim e dz = e dx . γ →∞∫C1,γ γ →∞∫x=−γ ∫−∞ The last integral is well known and equals π . Combining equations (2.1.5.1) and (2.1.5.2) with the above limits yields © 2000 by CRC Press LLC

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G(ω) = e − 14ω 2 ∫−∞+ jω2 e − z2dz 2 = e − 41ω 2 γli→m∞ −∫C3,γe − z2dz = e − 14ω 2 γli→m∞ ∫C1,γe − z2dz + ∫C2,γe − z2dz + ∫C4,γe − z2dz − 1ω 2 = e 4 π . So, Ᏺ[e −t2 ] = G(ω) = π e − 14ω 2 . ω Example 2.1.5.2 Transform of a Function with a Pole Off the Real Axis Consider computing the Fourier inverse transform of F(ω) = (1+ ω 2)−1 , f (t ) = Ᏺ−1[F(ω)] t = 21π ∫−∞ 1e+jtωω 2 dω . (2.1.5.3) For t = 0, 1 ∞ 1 1 ∞ 1 f (0) = 2π ∫−∞ 1+ω 2 dω = 2π arctanω −∞ = 2 . (2.1.5.4) To evaluate f(t) when t ≠ 0, ﬁrst observe that the integrand in formula (2.1.5.3), viewed as a function of the complex variable, jtz e Φ(z) = 1+ z 2 , has simple poles at z = ± j. The residue at z = j is Res j[Φ] = l zi→mj(z − j)Φ(z) = lzi→mj(z − j) (z −ej)j(tzz + j) = 21j e −t , while the residue at z = –j is Res− j [Φ] = zl→im− j(z + j)Φ(z) = − 21j e t . For each γ > 1, let Cγ , C+,γ , and C–,γ be the curves sketched in Figure 2.2. By the residue theorem: © 2000 by CRC Press LLC

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