Fourier Transform: Definition, Properties, and Applications

317 4. FourierAnalysis Section 4.1 Definition of the Fourier Transform Section 4.2 Dirac Delta Function Section 4.3 Properties Section 4.4 Uncertainty Principle Section 4.5 Convolution and Correlation Section 4.6 2D Fourier Transforms Section 4.7 The Operations Related to Fourier Transforms ( ) [1] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., McGraw Hill, Boston, 2000. [2] D. G. Zill and Michael R. Cullen, Differential Equations-with Boundary-Value Problem (metric version), 9th edition, Cengage Learning, 2017. [3] D. G. Zill, W. S. Wright, and J. J. Ding, Engineering Mathematics, Metric Edition, Cengage Learning, Taipei, Taiwan, 2019, Chapter 15.

318 Fourier Transform Definition (Sec. 4-1) Properties (Sec. 4-3) Uncertainty (Sec. 4-4) Basic Transform Pair (Sec. 4-1) Sinc (Sec. 4-1) Special Functions Fourier Transform Dirac Delta (Sec. 4-2) Hermite-Gaussian (Sec. 4-4) Jinc (Sec. 4-6) Convolution (Sec. 4-5) Correlation (Sec. 4-5) 2D FT (Sec. 4-6) Hankel Transform (Sec. 4-6) Related Transforms Hilbert Transform (Sec. 4-7) Laplace Transform (Sec. 4-7) Mellin Transform (Sec. 4-7)

319 4.1 Definition of the Fourier Transform Fourier transform ( ) ( ) ( ) = = 2 j fx g x g x e dx G f Fourier transform inverse Fourier transform ( ) ( ) ( ) = = 1 2 j fx G f G f e df g x [1] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., McGraw Hill, Boston, 2000. [2] D. G. Zill, W. S. Wright, and J. J. Ding, Engineering Mathematics, Metric Edition, Cengage Learning, Taipei, Taiwan, 2019, Sections 15-2.

320 Review: Fourier Series of the Complex Form ( ) 2 T ( ) nx = exp j nx n = , -1, 0, 1, 2, 3, . T T form a complete and orthogonal set within , x 2 2 = 0 T if m if m n n /2 T ( ) x ( ) x dx = n m /2 T

321 Review: Fourier Series of the Complex Form (Compared to pages 273, 276) If g(x) = g(x+T), then ( ) 2 T = ( ) exp g x c j nx n = n where ) ( ( ) /2 T 2 T ( ) exp g x conj j nx dx /2 T = c ) ( ( ) ( ) n /2 T 2 T 2 T exp exp j nx conj j nx dx /2 T ( ) /2 T 2 T ( ) = exp g x j nx dx /2 T c T n

322 4.1.1 Derivation and Physical Meaning Fourier transform can be viewed as the Fourier series where T Note that, if we set gn= cnT ( ) ( ) g T 2 T = /2 T 2 T ( ) ( ) = exp g x j nx n exp g g x j nx dx n /2 T = n = 1/ T Then we set f ( ) ( ) ( ) /2 T ( ) = exp 2 g x g j n x = exp 2 g g x j n x dx n f f n f /2 T = n /2 T ( ) ( ) ( ) ( ) ( ) ( ) = = fx dx exp 2 g x G f j fx exp 2 G f g x j f /2 T = n ( ) = = , f n G f g where f n If T , 0 f ( ) ( ) ( ) = ( ) ( ) ( ) fx df exp 2 g x G f j = fx dx exp 2 G f g x j

323 Physical Meaning of the Fourier Transform: ( ) exp 2 j fx expanding a signal as a combination of ( ) exp 2 j fx period: 1/f, frequency: f ( ) exp 2 j fx G(f): the expansion coefficient for ( ) ( ) = 2 j fx g x G f e df

324 When will the Fourier transform exist? Sufficient Conditions: ( ) (1) < g x dx (2) g(x) is of bounded variations (It means that g(x) can be represented by a curve of finite length in any finite interval of x).

325 4.1.2 Transform Pair [Example 1] Find the Fourier transform of ( ) g x ( ) = exp 3 x (Solution): 0 ( ) g x 3 x = = + 2 3 2 3 2 j fx x j fx x j fx F e e dx e e dx e e dx 0 0 3 2 6 3 2 x j fx f x j fx f e e e e 1 j 1 j = + = 3 2 3 2 3 2 3 2 j j f f 0 = 9 (2 + 2 ) f

326 [Example 2] Find the Fourier transform of the rectangular function (x) where for -1/2 < x < 1/2, otherwise. 1 0 ( ) x = 1/2 -1/2 (Solution): ( ) 1/2 sin f 2 j fx e 1/2 ( ) g x 1/2 = = = 2 j fx F e dx 2 j f f 1/2 ( ) g x = F sinc f

327 [Example 3] Find the Fourier transform of the Dirac delta function (x) (Solution): From the sifting property of (x): ( ) ( ) y x dx ( ) y x (see page 342) = x x 0 0 we have ( ) x ( ) x e 0 = = = 2 2 j fx j f F 1 dx e ( ) x x 0 height = 1/2b b 0 area = 1 x0+b x0 b x-axis

328 Note: More generally, ( ) ( ) = = 2 j x f 2 j fx F x x x x e dx e 0 0 0

329 Linearity Property of the Fourier Transform If ( ) ( ) f ( ) x ( ) f = = g x G g G 1 1 2 2 then ( ) ( ) x ( ) f ( ) f g x + = + g G G 1 2 1 2

330 Duality Property of the Fourier Transform ( ) ( ) g x G f = If ( ) ( ) = G x g f then (Proof): Since ( ) ( ) ( ) = = 1 2 j fx G f G f e df g x ( ) ( ) g f = 2 j xf G x e dx ( ) ( ) ( ) ( ) = 2 j fx G x e dx g f = G x g f Q: How do we compute the Fourier transforms of sinc(x) and 1?

331 [Example 4] Find the Fourier transform of sinc(x) where (Solution): Since ( ) x ( ) f = sinc from the duality property, we have ( ) ( ) ( ) f = = sinc x f

332 [Example 5] Find the Fourier transform of exp(j2 k x) (Solution): Since ( ) ( ) = + = 2 2 j k f j k f F F , x k e x k e from the duality property, we have ( ) ( ) = + = 2 j kx F e f k f k (Here we apply the fact that (x) = (-x)). Specially, 1 ( ) f = (Note): Although 1 does not satisfy the sufficient condition on page 324, its Fourier transform exists.

333 [Example 6] Find the Fourier transform of cos(2 k x) (Solution):

334 Some basic Fourier transform pairs ( ) g(x) = F ( ) g x G f (x) 1 (1) (f) (2) 1 (x k) exp( j2 kf) (3) exp(j2 kx) (f k) (4) 1 2 j 1 2 j ( ) ( ) cos(2 kx) (5) + + f k f k ( ) ( ) sin(2 kx) (6) + + f k f k 2 2 (x) sinc(f) (7) (f) (8) sinc(x) 1 j 2 4 (9) exp(-kx)U(x) (k > 0) + 2 k k f (10) exp(-k|x|) (k > 0) + 2 2 2 k f

Some basic Fourier transform pairs 335 F (x) 1 F (f) 1 ( ) 1 / 2 + + 1 / 2 f F cos(2 x) ( ) f ( ( ) ) + + 1 / 2 j f sin(2 x) F 1 / 2 j f F (x) sinc(f) (f) sinc(t) F

336 Summary of Popular Special Functions sin( sinc x ) x = (1) Sinc Function x = = = sinc0 1, sinc 0 if is a nonzero integer, n n sinc sinc( ) x x Applications: sampling theorem; ideal filters

337 (2) Rectangular Function 1 0 | | 1/ 2 x otherwise for ( ) x = rect (3) Triangular Function 1 | | 0 | | 1 x otherwise x for ( ) x = tri

338 (4) Step Function 1 0 0 0 for x for x ( ) = U x

339 4.2 Dirac Delta Functions The Dirac delta function does not have a fixed definition. It is in fact the limitation of a distribution. ( ) br x (1) height = 1/2b area = 1 b b x-axis ( ) ( ) x = limb b r x 0

340 ( ) height = 1/b (2) tri x b area = 1 b b x-axis ( ) x ( ) x = lim b tri b 0 ( ) b x 1sinc b height = 1/b ( ) x = (3) sc b ( ) x dx = = 1 area sc b b b ( ) x ( ) x = lim b sc b 0

341 Definition of the Dirac delta function: ( ) x dx = 1 (1) ( ) x = if x 0 0 (2) ( ) x ( ) = x (3)

342 Properties of the Dirac delta function: d dx ( ) x ( ) = U x (1) U(x): unit step function d dx ( ) ( ) = x k U x k (2) Sifting property ( ) ( ) k g x dx ( ) g k = x (Proof): ( ) ( ) k g x dx x k b + 1/(2b) 1 ( ) = g(x) lim2 b g x dx b k b 0 k b + 1 ( ) g k ( ) g k = = lim2 b dx b k b 0 k-b k+b

343 (3) Sifting property (without integral) ( ) ( ) k g x ( ) ( ) k g k = x x (4) Scaling property 1 a ( ) ( ) x = ax | | (to balance the integral) dx a dx a 1 a ( ) ( ) x ( ) x dx 1 a = = ax dx for a > 0 ( ) ( ) x ( ) x dx for a < 0 = = ax dx (5) Convolution property ( ) g x ( ) x ( ) = g x ( ) x ( ) x ( ) x = Specially,

344 (6) Integral for exponential functions ( ) x = 2 j fx e df It is directly from the fact that ( ) x 1 ( ) x = = 1 1, (7) Generalization of the integral for exponential functions ( ) ( ) f g x 2 j = ( ) g x e df How do we define it?

345 (8) (g(x)) If g(x) = 0 only at x = x0, then ( ( ) g x ( g x ) x x ) = 0 | ( )| 0 (Proof): ( ) ( )( g x x x ) g x x x when 0 0 0 ( g x ) x x ( ) ( ) ( )( g x = = ( ) g x ) x x 0 | ( )| 0 0 0 In general, if g(x) = 0 only at x = x1, x2, , xN, then ( ( ) | n g x = ) N x x ( ) = g x n ( )| n 1

346 d dx ( ) x ( ) x (9) Derivative of (x) = ( ) x ( ) ( ) g x = g x (Proof): ( ) x ( ) ( ) ( ) = g x g x d d d ( ) ( ) ( ) ( ) = g x g x d ( ) ( ( ) g x ) g x = d (from the sifting property) =

347 (10) Properties related to derivative of (x) ( ) x ( ) = (i) x ( ) ( ) g x dx ( ) g x = x x (ii) 0 0 ( ) ( ) g x ( ) ( ) g x ( ) ( ) g x = x x x x x x (iii) 0 0 0 0 0 (Proof): Since ( ) ( ) g x ( ) ( ) g x ( x = x x x x 0 0 0 d dx d dx ( ) ( ) g x ) ( ) g x = x x x 0 0 0 ( ( ) ( ) g x ) ( ) g x ( ) ( ) g x ) ( ) 0 x g x ( ( ) ( ) g x ) ( ) g x + = x x x ( x x x 0 0 0 0 = x x x x x 0 0 0 0

348 (11) Higher order derivative of (x) ( ) ( ) x g x ( ) x = ( ) n ( ) n g ( ) ( ) g x dx ( ) x ( ) n = ( ) n ( ) n 1 x x g 0 0 ( ) = ( ) n when n > 0 0 x x dx 0 ( ) x ( ) ( x when x 0 = ( ) n 0 ) ( ) n = ( ) n ( ) n 1 x

349 4.3 Properties of the Fourier Transform 4.3.1 List of Properties ( ) G f = ( ) g x ( ) g x ( ) = f x dx F exp 2 j (1) Recovery (inverse Fourier transform) (2) Integration (DC property) ( ) g x ( ) ( ) = f x dx exp 2 G f j ( ) 0 ( ) ( ) 0 ( ) g x dx = = g G f df G ( ) ( ) g x e = 2 j f x F G f f 0 (3) Modulation 0 ( ) ( ) 2 = j f x F g x x G f e 0 (4) Time Shifting 0 f a 1 a ( ) = F g ax G (5) Scaling | | ( ) ( ) = F g x G f (6) Time Reverse

350 If g(x) is real, then G(f) = G*( f); If g(x) is pure imaginary, then G(f) = G*( f) (7) Real / Imaginary Input If g(x) = g( x), then G(f) = G( f); If g(x) = g( x), then G(f) = G( f); (8) Even / Odd Input ( ) x ( ) ( ) ( ) f = = F (9) Conjugation F g G f g x G ( ) ( ) g x = f G f F 2 j (10) Differentiation j ( ) ( ) f = F xg x G (11) Multiplication by x 2 ( ) g x x = f ( ) F 2 (12) Division by x j G d (13) Parseval s Theorem (Energy Preservation) ( ) g x ( ) 2 2 = dx G f df ( ) g x h ( ) x dx ( ) ( ) f df (14) Generalized Parseval s Theorem = G f H

351 ( ) ( ) ( ) ( ) + = + F (15) Linearity ag x bh x aG f bH f ( ) z x ( ) g x ( ) Z f ( ) h x ( ) G f H f ( ) ( ) ( ) = = , g h x d If (16) Convolution = then If , then ( ) ( ) ( ) z x g x h x = ( ) ( ) Z f G f = ( ) Z f = (17) Multiplication ( ) ( ) ( ) = H f G H f d ( ) z x ( ) ( ) ( ) = , g h x d If (18) Correlation ( ) f then G f H (19) Two Times of Fourier Transforms ( ) g x ( ) = F F g x ( ) (20) Four Times of Fourier Transforms ( ) g x ( ) g x = F F F F

352 (Proof of (2) Integration Property) ( ) ( ) ( ) f x g x dx = exp 2 G f j ( ) 0 ( ) ( ) x g x dx ( ) = 2 0 = exp G j g x dx (Proof of (5) Scaling Property) ( ) ( ( ) ( ) ( ) ) = f x g ax dx F exp 2 g ax j x a dx = exp 2 j f g x | | a f a f a 1 a 1 a ( ) x g x dx = = exp 2 j G | | | | Property (6) is a special case of Property (5) where a = -1.

353 (Proof of (7) and (9)) ( ) ( ) ( ) f x g x dx = exp 2 G f j ( ) ( ) x dx ( ) x = f x g = F exp 2 j ((9) is proven) g If g(x) is real, then ( ) ( ) x ( ) g x ( ) = = = F F G f g G f (Proof of (10) Differentiation Property) ( ) g x ( ) ( ) ( ) f x G f df = = 1 F exp 2 G f j dg x dx ( ) ( ) ( ) f x G f df ( ) = = f G f 1 F 2 exp 2 2 j f j j

354 [Example 1] Determine the Fourier transform of the following signal. ( ) 3 g x = for |x| < 1, ( ) 1 g x = for 1< |x| < 3, ( ) g x = for |x| > 3 0 (Solution): Note that ( ) 2 ( ) 6 ( x x x ( ) g x = + 2 ( ) g x ) ( ) 2 / 2 /6 x + = ( ) ( f ) ( f ) 2 2sinc 2 = = + Therefore, 6sinc 6 G f f + f ( ) ( ) 4sinc 2 6sinc 6

355 [Example 2] Determine the Fourier transform of the following signal. ( ) ( exp g x x = ) | | x (Solution): From page 334, we have F 2 ( ) = exp | | x 1 4 + 2 2 f j ( ) ( ) f = F Then, from the differentiation property xg x G 2 j d 2 ( ) = F exp | | x x 2 df + 2 2 1 4 8 + f f = j 2 2 2 ) (1 4 f

356 [Example 3] Determine the Fourier transform of the following signal. ( ) ( exp 3| 1| g x x = ) + 6 j x (Solution): Since 6 ( ) 3| | = F exp x 9 4 + 2 2 f 6 ( ) = 2 j f F exp 3| 1| x e time shifting property 9 4 + 2 2 f 6 ( ( ) 2 ( + = 3) j f F exp 3| 1| 6 x j x e 9 4 + 2 2 3) f modulation property

357 [Example 4] Determine the Fourier transform of the following signal. ( ) ( ) cos 6 g x x = for 0 < x < 8, ( ) 0 g x = otherwise (Solution): Note that ( ) g x ) ) ( 4 x ( = cos 6 x 8 ) ) ) ( ) ( 4 4 x x 1 2 1 2 ( ( = + exp 6 exp 6 j x j x 8 8 Since ( ) x ( ) 8 ( ( ) f = F sinc x ( ) = (scaling) F 8sinc 8 f ) 4 x ( ) (time shifting) = 8 j f F 8 sinc 8 e f 8

( ) 358 4 x ( ) = 8 j f F 8 sinc 8 e f 8 ) ( ) ( ) ) 4 x ( ( ) 8 ( = 3) j f F exp 6 8 sinc 8( 3) j x e f (modulation) 8 x 4 ( ( ) 8 ( + = + 3) j f F exp 6 8 sinc 8( 3) j x e f 8 Therefore, 1 2 4 ( ) F g x ) ( ( ) + ) ( + ) 4 4 x x 1 2 ( ( = + F F exp 6 exp 6 j x j x 8 3) 8 ) ( ) 8 ( 8 ( + = 3) 3) j f j f sinc 8( 4 sinc 8( 3) e f e f

359 4.3.2 Real, Imaginary, Even, and Odd Parts Moreover, from Properties (7), (8), (9), we can conclude that ( ) 1 2 ( ) ( ) ( ) = + Re g x G f G f (i) ( ) 1 2 ( ) ( ) ( ) (ii) = Im j g x G f G f (Practice to prove them)

360 Also, any function can be decomposed into ( ) ( ) e g x g x = ( ) ( ) ( 2 ( ) ( ) ( 2 ( ) x ( g + g (1) o ) ) 1 ) where = + e g x g x x 1 ( ) = o g x g x g x ( ) ( ) x ( ) x ( ) x ( ) x = + + + (2) g x g g g g , , , , e r e i o r o i ) ) ( ) 1 2 1 2 ( ) x ( ) ( ) where = + Re g g x g x , e r ( ) 1 2 ( ( ) x ( ) x ( ) ( ) g x ( = + Im g j g x g x , e i ) ( ) = Re g g x , o r ( ) 1 2 ( ) x ( ) ( = Im g j g x g x , o i

361 One can prove that e g ( ) x ( ) f ( ) x ( ) , e r ( ) , o r ( ) f = = g G G o o e ( ) f ( ) f = ( ) x ( ) x ( ) f ( ) f g x G = g G , e r , , e i e i = g x G = g G , o i , , o i o r note where ( ) 1 2 ( ) 1 2 ( ) f ( ) ( ) ( ) f ( ) 2 2 ( ) = = + G ) ) ) G f G f G G f G f o e ( ) 1 ( ) f ( ) ( ) = + Re G G f G f , e r ( 1 2 ( ( ) f ( ) f ( ) ( ) G f ( = + Im G j G f G f , e i ) 1 ( ) = Re G G f , o r ( ) 1 2 ( ) f ( ) ( = Im G j G f G f , o i

362 ( ) ( ) x ( ) x ( ) x ( ) x = + + + g x g g g g , , , , e r e i o r o i F F F F ( ) ( ) f ( ) f ( ) f ( ) f = + + + G f G G G G , , , , e r e i o r o i

363 4.3.3 Parseval s Theorem ( ) ( ) 2 2 = g x dx G f df Parseval s theorem is also called the energy preservation property, Rayleigh s Theorem, or Plancheral s Theorem. (Proof): ( ) ( ) f df ( ) ( ) = d df 2 2 j fx j f G f G g x e dx g e ( ) g x g ( ) = 2 ( ) j f x e df dxd ( ) g x g ( ) ( ) = x d dx (from page 344) ( ) g x g ( ) x dx = (from the sifting property) ( ) g x 2 = dx

364 Generalized Parseval s Theorem ( ) ( ) x dx ( ) ( ) f df = g x h G f H It also called the power theorem.

365 [Example 5] Determine the following integral: ( ) sinc 2 x dx [Example 6] Determine the following integral: ( ) ( ) cos 8 sinc 3 x x dx (Solution): Since cos 8 F ( ) 1 2 ( ) ( ) ( ) = + + 4 4 x f f f 1 3 ( ) = F sinc 3 x 3 f ( ) 1 2 1 3 ( ) ( ) ( ) ( ) = + + cos 8 sinc 3 4 4 x x dx f f df 3 ( ) 3/2 1 6 ( ) ( ) 3/2 = + + = 4 4 0 f f df

366 4.4 Uncertainty Principles 4.4.1 Uncertainty Principles from Different Views (1) From the Point of View of the Scaling Property ( g ax F f a 1 a ) = G | | wide in the time domain narrow in the frequency domain narrow in the time domain wide in the frequency domain ( ) ( ) x sinc f F f 1sinc 2 ( ) F 2x 2