Appendix A Mathematical appendix A.1 The fourier transform The Fourier transform permits us to decompose a complicated field structure into elemental components. This can simplify the computation of fields and provide physical insight into their spatiotemporal behavior. In this section we review the properties of the transform and demonstrate its usefulness in solving field equation One-dimensional case Let f be a function of a single variable x. The Fourier transform of f(r)is the function F(k) defined by the integral Flf())=F()=f(r)e-jkx dx Note that x and the corresponding transform variable k must have reciprocal units: if x is time in seconds, then k is a temporal frequency in radians per second; if x is a length in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k) as the frequency spectrum of f(x) Not every function has a outer tl ansform. The existence of(A 1)can be guaranteed by a set of sufficient conditions such as the following: 1. f is absolutely integrable: /_oo If(x)ldx <oo 2. f has no infinite discontinuities 3. f has at most finitely many discontinuities and finitely many extrema in any finite interval(a, b) While such rigor is certainly of mathematical value, it may be of less ultimate use to the engineer than the following heuristic observation offered by Bracewell 22 mathematical model of a physical process should be Fourier transformable. That is, if the Fourier transform of a mathematical model does not exist, the model cannot precisely describe a physical process. The usefulness of the transform hinges on our ability to recover f through the inverse F{F(k)}=f(x)= (A.2) 0 2001 by CRC Press LLC
Appendix A Mathematical appendix A.1 The Fourier transform The Fourier transform permits us to decompose a complicated field structure into elemental components. This can simplify the computation of fields and provide physical insight into their spatiotemporal behavior. In this section we review the properties of the transform and demonstrate its usefulness in solving field equations. One-dimensional case Let f be a function of a single variable x. The Fourier transform of f (x) is the function F(k) defined by the integral F{ f (x)} = F(k) = ∞ −∞ f (x)e− jkx dx. (A.1) Note that x and the corresponding transform variable k must have reciprocal units: if x is time in seconds, then k is a temporal frequency in radians per second; if x is a length in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k) as the frequency spectrum of f (x). Not every function has a Fourier transform. The existence of (A.1) can be guaranteed by a set of sufficient conditions such as the following: 1. f is absolutely integrable: ∞ −∞ | f (x)| dx < ∞; 2. f has no infinite discontinuities; 3. f has at most finitely many discontinuities and finitely many extrema in any finite interval (a, b). While such rigor is certainly of mathematical value, it may be of less ultimate use to the engineer than the following heuristic observation offered by Bracewell [22]: a good mathematical model of a physical process should be Fourier transformable. That is, if the Fourier transform of a mathematical model does not exist, the model cannot precisely describe a physical process. The usefulness of the transform hinges on our ability to recover f through the inverse transform: F−1 {F(k)} = f (x) = 1 2π ∞ −∞ F(k) e jkx dk. (A.2)
When this is possible we write f(x)分F(k) and say that f(x) and F(k) form a Fourier transform pair. The Fourier integral theorem states that FFIf(x))=F- FIf(x)= f(r). except at points of discontinuity of f. At a jump discontinuity the inversion formula returns the average value of the one-sided limits f(x+) and f(x")of f(x). At points of continuity the forward and inverse transforms are unique Transform theorems and properties. We now review some basic facts pertaining to the Fourier transform. Let f(x)+ F()=R()+jX(), and g(x)+ G() 1. Linearity. af(x)+ Bg(x)* aF(k)+ BG(k) if a and B are arbitrary constants This follows directly from the linearity of the transform integral, and makes the transform useful for solving linear differential equations(e. g, Maxwells equations) 2. Symmetry. The property F(x)<>2f(k) is helpful when interpreting transform ables in which transforms are listed only in the forward direction 3. Conjugate function. We have f(x)+F(k) 4. Real function. If f is real, then F(k)=F*(k). Also, R(k)=/f(r)coskxdx, X()=-/f(x)sinkxdx f()=Re/ F(k)/ dk A real function is completely determined by its positive frequency spectrum. It obviously advantageous to know this when planning to collect spectral data 5. Real function with reflection symmetry. If f is real and even, then X()=0 and R(k)=2 f(x)cos kx dx, f(x) R(k)cos kx dk If f is real and odd, then R()=0 an X(k)=-2/f(x) X()sin kx dk (Recall that f is even if f(x)=f(x)for all x. Similarly f is odd if f(x)=-f(x) 6. Causal function. Recall that f is causal if f(x)=0 for x <0 (a)If f is real and causal, then x(=-2C 0 2001 by CRC Press LLC
When this is possible we write f (x) ↔ F(k) and say that f (x) and F(k) form a Fourier transform pair. The Fourier integral theorem states that F F−1 { f (x)} = F−1 F{ f (x)} = f (x), except at points of discontinuity of f . At a jump discontinuity the inversion formula returns the average value of the one-sided limits f (x+) and f (x−) of f (x). At points of continuity the forward and inverse transforms are unique. Transform theorems and properties. We now review some basic facts pertaining to the Fourier transform. Let f (x) ↔ F(k) = R(k) + j X(k), and g(x) ↔ G(k). 1. Linearity. αf (x) + βg(x) ↔ αF(k) + βG(k) if α and β are arbitrary constants. This follows directly from the linearity of the transform integral, and makes the transform useful for solving linear differential equations (e.g., Maxwell’s equations). 2. Symmetry. The property F(x) ↔ 2π f (−k) is helpful when interpreting transform tables in which transforms are listed only in the forward direction. 3. Conjugate function. We have f ∗(x) ↔ F∗(−k). 4. Real function. If f is real, then F(−k) = F∗(k). Also, R(k) = ∞ −∞ f (x) cos kx dx, X(k) = − ∞ −∞ f (x)sin kx dx, and f (x) = 1 π Re ∞ 0 F(k)e jkx dk. A real function is completely determined by its positive frequency spectrum. It is obviously advantageous to know this when planning to collect spectral data. 5. Real function with reflection symmetry. If f is real and even, then X(k) ≡ 0 and R(k) = 2 ∞ 0 f (x) cos kx dx, f (x) = 1 π ∞ 0 R(k) cos kx dk. If f is real and odd, then R(k) ≡ 0 and X(k) = −2 ∞ 0 f (x)sin kx dx, f (x) = − 1 π ∞ 0 X(k)sin kx dk. (Recall that f is even if f (−x) = f (x) for all x. Similarly f is odd if f (−x) = − f (x) for all x.) 6. Causal function. Recall that f is causal if f (x) = 0 for x < 0. (a) If f is real and causal, then X(k) = − 2 π ∞ 0 ∞ 0 R(k ) cos k x sin kx dk dx, R(k) = − 2 π ∞ 0 ∞ 0 X(k )sin k x cos kx dk dx.
(b)If f is real and causal, and f(O) is finite, then R(k) and X(k) are related by the Hilbert transforms X() R(k) dk, R(=-Pv X(k) (c)If f is causal and has finite energy, it is not possible to have F(k)=0 for k I < ks k?. That is. the transform of a causal function cannot vanish over an interval A causal function is completely determined by the real or imaginary part of its spectrum. As with item 4, this is helpful when performing calculations or mea- surements in the frequency domain. If the function is not band-limited however truncation of integrals will give erroneous results 7. Time-limited us. band-limited functions. Assume t2>t1. If f()=0 for both t < tr and t>I2, then it is not possible to have F()=0 for both k kI and k > k2 where k2>k1. That is, a time-limited signal cannot be band-limited. Similarly, a band-limited signal cannot be time-limited 8. Null function. If the forward or inverse transform of a function is identically zero, then the function is identically zero. This important consequence of the Fourier tegral theorem is useful when solving homogeneous partial differential equations n the frequency domain. 9. Space or time shift. For any fixed xo f(x-xo)+ F(k)- Jxta. a temporal or spatial shift affects only the phase of the transform, not the magni- tude 10. Frequency shift. For any fixed ko f(r)e Note that if f < F where f is real, then frequency-shifting F causes f to be- come complex -again, this is important if F has been obtained experimentally through computation in the fre 11. Similarity. We have where a is any real constant. " Reciprocal spreading" is exhibited by the Fourier transform pair; dilation in space or time results in compression in frequency, and 12. Convolution. We have fi(rf2(x-x)+ Fi(k) F2(k) f1(x)f2(x)分 Fc)F2(k-k)di 0 2001 by CRC Press LLC
(b) If f is real and causal, and f (0) is finite, then R(k) and X(k) are related by the Hilbert transforms X(k) = − 1 π P.V. ∞ −∞ R(k) k − k dk , R(k) = 1 π P.V. ∞ −∞ X(k) k − k dk . (c) If f is causal and has finite energy, it is not possible to have F(k) = 0 for k1 < k < k2. That is, the transform of a causal function cannot vanish over an interval. A causal function is completely determined by the real or imaginary part of its spectrum. As with item 4, this is helpful when performing calculations or measurements in the frequency domain. If the function is not band-limited however, truncation of integrals will give erroneous results. 7. Time-limited vs. band-limited functions. Assume t2 > t1. If f (t) = 0 for both t < t1 and t > t2, then it is not possible to have F(k) = 0 for both k < k1 and k > k2 where k2 > k1. That is, a time-limited signal cannot be band-limited. Similarly, a band-limited signal cannot be time-limited. 8. Null function. If the forward or inverse transform of a function is identically zero, then the function is identically zero. This important consequence of the Fourier integral theorem is useful when solving homogeneous partial differential equations in the frequency domain. 9. Space or time shift. For any fixed x0, f (x − x0) ↔ F(k)e− jkx0 . (A.3) A temporal or spatial shift affects only the phase of the transform, not the magnitude. 10. Frequency shift. For any fixed k0, f (x)e jk0 x ↔ F(k − k0). Note that if f ↔ F where f is real, then frequency-shifting F causes f to become complex — again, this is important if F has been obtained experimentally or through computation in the frequency domain. 11. Similarity. We have f (αx) ↔ 1 |α| F k α , where α is any real constant. “Reciprocal spreading” is exhibited by the Fourier transform pair; dilation in space or time results in compression in frequency, and vice versa. 12. Convolution. We have ∞ −∞ f1(x ) f2(x − x ) dx ↔ F1(k)F2(k) and f1(x) f2(x) ↔ 1 2π ∞ −∞ F1(k )F2(k − k ) dk .
The first of these is particularly useful when a problem has been solved in the frequency domain and the solution is found to be a product of two or more functions of k 13. Parseval's identity. We have If(x)- dx IF()Idk Computations of energy in the time and frequency domains always give the same 14. Differentiation. We have df(x) *(k)"F() and (-jx)"f(x)<> d"F(k) dxn The Fourier transform can convert a differential equation in the x domain into an algebraic equation in the k domain, and vice versa 15. Integration. We have f(u)dn分丌F(k)(k)+ F(k) where &()is the Dirac delta or unit impulse Generalized Fourier transforms and distributions. It is worth noting that many useful functions are not Fourier transformable in the sense given above. An example the signum function 1,x<0, Although this function lacks a Fourier transform in the usual sense, for practical purposes it may still be safely associated with what is known as a generalized Fourier transform. a treatment of this notion would be out of place here; however, the reader should certainly be prepared to encounter an entry such as sgn(x)分2/jk in a standard Fourier transform table. Other functions can be regarded as possessing transforms when generalized functions are permitted into the discussion. An important example of a generalized function is the Dirac delta &(x), which has enormous value in describing distributions that are very thin, such as the charge layers often found on conductor surfaces. We shall not delve into the intricacies of distribution theory However, we can hardly avoid dealing with generalized functions: to see this we need look no further than the simple function cos kox with its transform pair cos kox + S(k+ko)+&(k- ko) The reader of this book must therefore know the standard facts about S(x): that it cquires meaning only as part of an integrand, and that it satisfies the sifting property 8(x-xo)f(x)dx= f(xo) 0 2001 by CRC Press LLC
The first of these is particularly useful when a problem has been solved in the frequency domain and the solution is found to be a product of two or more functions of k. 13. Parseval’s identity. We have ∞ −∞ | f (x)| 2 dx = 1 2π ∞ −∞ |F(k)| 2 dk. Computations of energy in the time and frequency domains always give the same result. 14. Differentiation. We have dn f (x) dx n ↔ (jk) nF(k) and (− j x) n f (x) ↔ dnF(k) dkn . The Fourier transform can convert a differential equation in the x domain into an algebraic equation in the k domain, and vice versa. 15. Integration. We have x −∞ f (u) du ↔ π F(k)δ(k) + F(k) jk where δ(k) is the Dirac delta or unit impulse. Generalized Fourier transforms and distributions. It is worth noting that many useful functions are not Fourier transformable in the sense given above. An example is the signum function sgn(x) = −1, x < 0, 1, x > 0. Although this function lacks a Fourier transform in the usual sense, for practical purposes it may still be safely associated with what is known as a generalized Fourier transform. A treatment of this notion would be out of place here; however, the reader should certainly be prepared to encounter an entry such as sgn(x) ↔ 2/jk in a standard Fourier transform table. Other functions can be regarded as possessing transforms when generalized functions are permitted into the discussion. An important example of a generalized function is the Dirac delta δ(x), which has enormous value in describing distributions that are very thin, such as the charge layers often found on conductor surfaces. We shall not delve into the intricacies of distribution theory. However, we can hardly avoid dealing with generalized functions; to see this we need look no further than the simple function cos k0x with its transform pair cos k0x ↔ π[δ(k + k0) + δ(k − k0)]. The reader of this book must therefore know the standard facts about δ(x): that it acquires meaning only as part of an integrand, and that it satisfies the sifting property ∞ −∞ δ(x − x0) f (x) dx = f (x0)
for any continuous function f. With f(x)=l we obtain the familiar relation 8(x)dx=1. With f(x)=e-jkr we obtain 8(x)e y thus 8(x)+1 It follows that Useful transform pairs. Some of the more common Fourier transforms that arise in the study of electromagnetics are given in Appendix C. These often involve the simple functions defined here 1. Unit step function <0 2. Signum function 0, 0. 3. Rectangular pulse function 0.|x>1 4. Triangular pulse function △(x)J1-1x,xl<1 x|>1 5. Sinc function Transforms of multi-variable functions Fourier transformations can be performed over multiple variables by successive appli- cations of(A 1). For example, the two-dimensional Fourier transform over xI and x] of the function f (x1, x2, x3, .. xN) is the quantity F(kx, kx, x3,..., xN) given by f(x1, x2,x xn)e-jka dxi e-jkz2 2 dx2 0 2001 by CRC Press LLC
for any continuous function f . With f (x) = 1 we obtain the familiar relation ∞ −∞ δ(x) dx = 1. With f (x) = e− jkx we obtain ∞ −∞ δ(x)e− jkx dx = 1, thus δ(x) ↔ 1. It follows that 1 2π ∞ −∞ e jkx dk = δ(x). (A.4) Useful transform pairs. Some of the more common Fourier transforms that arise in the study of electromagnetics are given in Appendix C. These often involve the simple functions defined here: 1. Unit step function U(x) = 1, x < 0, 0, x > 0. (A.5) 2. Signum function sgn(x) = −1, x < 0, 1, x > 0. (A.6) 3. Rectangular pulse function rect(x) = 1, |x| < 1, 0, |x| > 1. (A.7) 4. Triangular pulse function (x) = 1 − |x|, |x| < 1, 0, |x| > 1. (A.8) 5. Sinc function sinc(x) = sin x x . (A.9) Transforms of multi-variable functions Fourier transformations can be performed over multiple variables by successive applications of (A.1). For example, the two-dimensional Fourier transform over x1 and x2 of the function f (x1, x2, x3,..., xN ) is the quantity F(kx1 , kx2 , x3,..., xN ) given by ∞ −∞ ∞ −∞ f (x1, x2, x3,..., xN ) e− jkx1 x1 dx1 e− jkx2 x2 dx2