Poularikas A. D. Fourier Transform The Handbook of Formulas and Tables for Signal Processing. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 999
3 Fourier Transform 3. One-Dimensional Fourier Transform Definitions Properties Tables 3.2 Two-Dimensional Continuous Fourier Transform Definitions References Appendix Examples 3. One-Dimensional Fourier Transform 3.. Definitions 3... Fourier Transform or 2π F( f) = f( t) e dt = F( f) e j2π ft f() t = F( f) e df j ft jφω ( ) If f(t) is piecewise continuous and absolutel integrable, then its Fourier transform is a bounded continuous function, bounded b F( ω ) dω. If F(ω) is absolutel integrable F( ω) then its inverse is f(t). jω t jφω ( ) F( ω) = f ( t) e dt = F( ω) e = R( ω) + jx( ω) jω t f() t = F( ω) e dω 2π 999 b CRC Press LLC
3...2 f(t) = f r (t) + j f i (t) is complex, f r (t) and f i (t) are real functions f()= t real F( ω) = [ f ( t)cos ωt + f ( t)sin ωt] dt j [ f ( t)sin ωt f ( t)cos ωt] dt R( ω) = [ f ( t)cos ωt + f ( t)sin ωt] dt X( ω) = [ f ( t)sin ωt f ( t)cos ωt] dt fr () t = 2π (f r (t) = f(t), f i (t) = 0) r i r i r r fi( t) = [ R( ω)sin ωt + X( ω)cos ωt] dω 2π i i [ R( ω)cos ωt X( ω)sin ωt] dω R( ω) = f ( t)cos ωt dt X( ω) = f ( t)sinωt dt R( ω) = R( ω) X( ω) = X( ω) F( ω) = F ( ω) jω t f() t = Re F( ω) e dω π f() t = j f () t is purel imaginar i (f r (t) = 0) R( ω) = f ( t)sin ωt dt X( ω) = f ( t)cosωt dt f() t is even [ f() t = f( t)] i R( ω) = R( ω) X( ω) = X( ω) F( ω) = F ( ω) R( ω) = 2 f ( t)cos ωt dt X( ω) = 0 o f() t is odd [ f() t = f( t)] f( t) = R( ω)cosωtdω π o R( ω) = 0 X( ω) = 2 f( t)sinωtdt f( t) = X( ω)sinωtdω π o o i 999 b CRC Press LLC
3..2 Properties 3..2. Properties of Fourier Transform TABLE 3. Properties of Fourier Transform Operation. Transform-direct f() t F( ω) f() t f() t e j ω t dt 2. Inverse transform jω t F( ω) e dω 2π F( ω) 3. Linearit af (t) + bf 2 (t) af (ω) + bf 2 (ω) 4. Smmetr Ft () 2πf( ω) 5. Time shifting f( t± t o ) ±ω j t e o F( ω) 6. Scaling 7. Frequenc shifting 8. Modulation 9. Time differentiation 0. Time convolution f( at) a F ω a ±ω j e o t f() t F ( ωm ωo ) f( t)cosω ot 2 [ F( ω+ ωo) + F( ω ωo)] f( t)sin ω ot 2 j [ F( ω ωo) F( ω+ ωo)] n d n dt f() t ( jω) F( ω) f() t h() t = f( τ) h( t τ) dτ F( ω) H( ω). Frequenc convolution f() t h() t F( ω) H( ω) = F( τ) H( ω τ) dτ 2π 2π 2. Autocorrelation f() t f *( t) = f( τ) f *( τ t) dτ F( ω) F*( ω) = F( ω) 2 3. Parseval's formula E = f() t dt 2 2 E = F( ω) dω 2π ( n) n F ( 0) 4. Moments formula mn = t f() t dt = n where ( j) 5. Frequenc differentiation 6. Time reversal 7. Conjugate function n ( n d F F ) ( ω) ( 0) =, n = 02,, L n dω ω= 0 df( ω) ( jt) f ( t) dω n n ( jt) f ( t) d F( ω) n dω f( t) F( ω) f *( t) F *( ω) 8. Integral ( F( 0) = 0) f() t dt t jω F( ω) 9. Integral ( F( 0) 0) t f() t dt jω F( ω) + πf( 0) δ( ω) 999 b CRC Press LLC
3..3 Tables 3..3. Graphical Representations of Some Fourier Transforms TABLE 3.2 Table of Fourier Transforms (x = t; = w) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) 999 b CRC Press LLC
f(x) F() 3.53 3.54 A -A s-a<x<s+a -s-a<x<-s+a -2Aj sin a sin s 3.55 A -A 0<x<2a -2a<x<0-4jA sin a sin a 3.56 999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) f(x) F() J 0 (x) 2-2 3.57 J (x) 2x - 2 3.58 j πx sqn = >0 0 =0 - <0 3.59 999 b CRC Press LLC
f(x) F() xe -πx2 -j 2 2π e - 4π 3.60 δ(x+a) - δ(x-a) 2j sin 2 3.6 tanhπx -j cosech 2 3.62 999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) f(x) F() e - x sinx x tan - 2 2 3.63 p(x) * p(x) * p(x) sin 3 3.64 sin ax πx p a (x) 3.65 999 b CRC Press LLC
f(x) F() x e -ax a>0 x 0 a 2-2 2a -j (a 2 + 2 ) 2 (a 2 + 2 ) 2 3.66 a 2 + x 2 π a e -a 3.67 cos bx a 2 + x 2 π 2a e -a -b + e -a +b 3.68 999 b CRC Press LLC
TABLE 3.2 Table of Fourier Transforms (x = t; = w) (continued) f(x) F() sin bx a 2 + x 2 π 2aj e -a -b - e -a +b 3.69 dδ(x) dx j 3.70 x 2πj dδ() d 3.7 999 b CRC Press LLC
f(x) F() sin w 0 x 0 x 0 x < 0 w 0 w 0 2-2 -j π 2 [δ(-w 0 ) - δ(+w 0 )] 3.72 cos w 0 x 0 x 0 x < 0 j w 0 2-2 + π 2 [δ(-w 0 ) - δ(+w 0 )] 3.73 x 0 x 0 x < 0 jπ dδ() d - 2 3.74 999 b CRC Press LLC
3.2 Two-Dimensional Continuous Fourier Transform 3.2. Definitions 3.2.. Two-Dimensional Fourier Transform 3.2..2 Properties of Two-Dimensional Fourier Transform TABLE 3.3 Properties of Two-Dimensional Fourier Transform Rotation f( ± x, ± ) F( ± ω, ± ω2) Linearit af ( x, ) + a2f2( x, ) af ( ω, ω2) + af 2 2( ω, ω2) Conjugation f *( x, ) F *( ω, ω2) Separabilit f( x) f2( ) F( ω) F2( ω2) 2 Scaling f( ax, b) ab F ω ω, a b ± j( ω Shifting f( x± xo, ± o) xo+ ω2o) e F( ω, ω2) ± j( ω Modulation cx+ ωc2) e f( x, ) F( ω mωc, ω2 mωc2) Convolution gx (, ) = hx (, ) f( x, ) G( ω, ω2) = H( ω, ω2) F( ω, ω2) Multiplication gx (, ) = hx (, ) f( x, ) G( ω, ω2) = H( ω, ω ) F( ω, ω ) 2 2 2 ( 2π) Correlation cx (, ) = hx (, ) µµ f( x, ) G( ω, ω ) = H( ω, ω ) F( ω, ω ) Inner Product 2 F{ f( x, )} = F( ω, ω ) = f( x, ) e dxd F 2 j( xω + ω ) + 2 2 2 2 j( ωx ω2) { F( ω, ω )} = F( ω, ω ) e dω dω ( 2π) 2 2 2 I = f( x, ) h*( x, ) dxd I = F( ω, ω ) H*( ω, ω ) dω dω ( 2π) 2 2 2 2 Parseval Formula I = f ( x, ) dx d 2 2 I = F( ω d d 2 2, ω2) ω ω2 ( π) References Bracewell, Ron, The Fourier Transform and Its Applications, McGraw-Hill Book Compan, New York, NY, 965. Campbell, G. A., and R. M. Foster, Fourier Integrals for Practical Applications, Van Nostrand Compan, Princeton, NJ, 948. Champenc, D. C., Fourier Transforms and Their Phsical Applications, Academic Press, New York, 973. Howell, K., Fourier transform, in The Transforms and Application Handbook, Edited b A. D. Poularikas, CRC Press Inc., Boca Raton, FL, 996. Papoulis, Athanasios, The Fourier Integral and Its Applications, McGraw-Hill Book Compan, New York, NY, 962. Walker, James S., Fourier Analsis, Oxford Universit Press, New York, NY, 988. 999 b CRC Press LLC
Appendix Examples. Gibbs Phenomenon Example 3. Let U(ω) be the spectrum of unit step function, and let the truncated spectrum U ( ω) = U(ω) for ω ω o ω o and zero otherwise. We can also write the truncated spectrum as follows: U ( ω) = U(ω) p ( ω), where ( ω) ω o ωo is a unit pulse of 2ω o duration and centered at the origin. The approximate step function pω o ot ua()=f t U p F U F sinω { ( ω) ω ( ω)} = { ( ω)} { p ( )} = u( t) = o ω ω o π t π t sinωτ o dτ τ and is shown in Figure 3.. FIGURE 3..2 Special Functions Example 3.2 Example 3.3 Example 3.4 ε t ε t j t ω F{sgn( t)} = F{lime sgn( t)} = lim e sgn( t) e dt ε 0 ε 0 o ( j ) t ( j ) t = lim ε ω ε+ ω e dt + e dt lim = 2 + = ε 0 ε 0 ε jω ε + jω jω o F{ ut ( )} = F{ + sgn( t)} = 2πδ( ω) + jω = πδ( ω) + j 2 2 2 2 ω 2 F n= jω t δ ( t nt)} = F{ combt ( t) = δ( t nt) e dt n= 999 b CRC Press LLC
But comb T (t) is periodic with the period ω o = 2π/T, and can be expanded in the complex form of Fourier series comb T (t) = jnω t o comb t e Hence T () =. T COMB n= ωo j( ω nωo ) t 2π ( ω) = e dt = δω ( nωo ) T T n= n= 999 b CRC Press LLC