EE2006_1_FourierTransform

1.8 Fourier Transform & Properties So far, we have derived Fourier series for periodic functions. What about non-periodic functions? Solution: Extend T to infinity! (T=2L) See for example: f T ( x ) ; T = 4  2   /2  /2 2 T x     f T ( x ) ; T = 8  4    4 x            f ( x )  /2 x Recall that for periodic function with period T and Separation between adjacent harmonics is As T ,  d Furthermore, the frequency moves from being discrete to continuous, i.e. n0  as T 1 f T ( x ) xTransition of the amplitude spectrum (i.e. cn) as fT(x) goes from periodic to non-periodic: 'time' domain n  0 0.2 c n 0 'frequency' domain 0.1 n  0 c n 0  F (  ) 0 'time' domain 'frequency' domain c 0 c 1 c 2   0 2  0 0 discrete frequency c n x f T (x)    T   periodic function Fourier Series f ( x )   x   non-periodic function Fourier Transform T T  F (  ) F (  )  continuous frequency Thus from (F3.2) as T Hence we define --- (1) - Fourier Transform of f(x) Similarly Hence --- (2) - Inverse Fourier Transform Equation (1) and (2) defines the Fourier transform pair. 1.9 Convergence of Fourier Transform · f(x) has a Fourier transform if converges. Generally, if f(x) is single-valued and encloses a finite area over the range of integration (i.e. well-behaved), then there is no problem with convergence. · a single-value function that is non-zero over an infinite interval will have a Fourier transform if exists and if any discontinuities of f(x) are finite. Example 1: a > 0 3 Recall , hence, ( e-jt = cos t – j sin t , sin 2t =2 sin t cos t, cos 2t =2 cos2t-1) Example 4: f(x) = A , a constant and 0 It diverges! For certain functions, their Fourier transforms have to be defined through a limit process. Some special functions: unit step function f ( x ) x 0 1) shifted unit step function f ( x ) 1 x 0 T 2) Signum function f ( x ) 1 x 0  1 3) or u(x). Unit impulse function f ( x ) 1 x 0 1.10 Fourier Transform of Elementary Functions f(x) F() (x) (impulse) 1 sgn(x) (signum) u(x) (step) e-ax u(x) eax u(x) e-a|x| (positive- & negative- time exponential) As mentioned earlier, the Fourier transform of some practical functions must be defined through a limit process. Example: 1.0 t  1.0 1) or 1.0 t  1.0 u ( t ) e t u ( t ) e t u ( t ) 1.11 Some Observations about Fourier Transform i.e. where or Some observations: i) A() is an even function of also known as Fourier cosine transform. ii) B() is an odd function of Fourier sine transform. iii) is an even function iv) is an odd function v) F*() = F() (conjugate) From the above observations, we have a) If f(x) is even, the F() is real (and even), i.e. B (  ) = 0 b) If f(x) is odd, the F() is imaginary (and odd), i.e. A (  ) = 0 , 1.12 Properties of Fourier Transform Let F()=F{f(x)}, F1()=F{f1(x)}, F2()=F{f2(x)}, 1. Multiplication by a constant F{k f(x)} = k F{f(x)} 2. Linearity F{a f1(x) b f2(x)} = a F1() b F2() 3. Differentiation , NB: valid provided f(x) = 0 at x = Proof. 4. Integration: If , then NB: valid provided 5. Scalar change - Time and frequency are reciprocal of each other, when time is stretched out, frequency will be compressed and vice versa. Proof. Let at=x, , Let , 6. Translation (or shifting) in time/space domain F{ f (t-a) } = e-ja F() - Translation in time domain alters the phase spectrum. Proof. Let t-a=x, then t=a+x, dt=dx 7. Translation (shifting) in frequency domain Let , Proof. 8. Modulation – a process of varying the amplitude of a sinusoidal carrier. Modulated carrier f(t) cos (ct), then Proof. Using the frequency shifting property: 9. Convolution (in the time/space domain) Let F{ f1(x)} = F1() and F{ f2(x)} = F2() Then F{ f1(x) * f2(x) } = F1() F2() - Convolution in time domain corresponds to multiplication in the frequency domain. Example: Show that given that F{f(t)}=F(); F() = 0 at  = Solution: In fact Prove it! 1.13 Discrete Fourier transform The sequence of N complex numbers x0, ..., xN−1 is transformed into the sequence of N complex numbers X0, ..., XN−1 by the DFT according to the formula: where i is the imaginary unit and is a primitive N'th root of unity. (This expression can also be written in terms of a DFT matrix; when scaled appropriately it becomes a unitary matrix and the Xk can thus be viewed as coefficients of x in an orthonormal basis.) (, normalized, hence called orthonormal) The inverse discrete Fourier transform (IDFT) is given by A simple description of these equations is that the complex numbers Xk represent the amplitude and phase of the different sinusoidal components of the input "signal" xn. The DFT computes the Xk from the xn, while the IDFT shows how to compute the xn as a sum of sinusoidal components with frequency k / N cycles per sample. By writing the equations in this form, we are making extensive use of Euler's formula to express sinusoids in terms of complex exponentials, which are much easier to manipulate. Application of DFT in object recognition and classification: (In MATLAB, DFT is called FFT, Fast Fourier Transform) 1.14 Fourier cosine transform (no complex!) Note the integral is not . The inverse Fourier cosine transform 1.15 Fourier sine transform Example 1: Find the Fourier cosine and sine transforms of Example 2: Find the Fourier cosine and sine transforms of Example 3: Find f(x) satisfying integral equation Solution: The Fourier cosine transform of Fc() is given by, The Fourier cosine transform of Fc() is given by, 1-40