2 The Fourier Transform

Transcription

1 2 The Fourier Transform The Fourier transform decomposes a signal defined on an infinite time interval into a λ-frequency component, where λ can be any real(or even complex) number. 2.1 Informal Development of the Fourier Transform The Fourier Inversion Theorem Theorem 2.1 If f is a continuously differentiable function with f(t) dt <, then f(x) = 1 ˆf(λ)e iλx dλ where ˆf(λ) (the Fourier transform of f) is given by ˆf(λ) = 1 f(λ)e iλt dt. (p. 93) 2.2 Properties of the Fourier Transform Basic properties The Fourier operator F should be thought of as a mapping whose domein and range are the space of complex-valued functions defined on the real line. The input of F is a function, say f, and returns another function, F[f] = ˆf, as its output. We define the inverse Fourier transform operator as F 1 [f](x) = 1 ˆf(λ)e iλx dλ. Theorem 2.1 implies that F 1 [F[f]] = f. Theorem 2.6 Let f and g be differentiable functions defined on the real line with f(t)=0 for large t. Refer to the text p.100 for some properties of the Fourier transform and its inverse. (p. 99) Fourier Transform of a Convolution Definition 2.9 Suppose f and g are two square-integrable functions. The convolution of f and g, denoted f g, is defined by (f g)(t) = f(t x)g(x)dx = f(x)g(t x)dx. (p. 105) 9

2 Theorem 2.10 Suppose f and g are two integrable functions. Then F[f g] = ˆf ĝ F 1 [ ˆf ĝ] = 1 f g. (p. 106) Adjoint of the Fourier Transform Recall from Section that the adjoint of a linear operator T : V W between two inner product spaces is an operator T : W V. The following theorem shows that the adjoint of the Fourier transform is the inverse of the Fourier transform. Theorem 2.11 Suppose f and g are square integrable. Then < F[f], g > L 2 = < f, F 1 [g] > L 2. (p. 107) Plancherel Formula The Fourier transform preserves the L 2 inner product. Theorem 2.12 Suppose f and g are square integrable. Then < F[f], F[g] > L 2 = < f, g > L 2. In particular, < F 1 [f], F 1 [g] > L 2 = < f, g > L 2. F[f] L 2 = f L 2. (p. 107) Plancherel s formula states that the energy of a signal in the time domain, f 2 L 2, is the same as the energy in the frequency domain ˆf 2 L Linear Filters Time Invariant Filters A filter is a transformation L that maps a signal, f, into another signal f. This transformation must satisfy the following two properties in order to be a linear filter : Additivity : L[f + g] = L[f] + L[g] Homogeneity : L[cf] = cl[f], where c is constant 10

3 Definition 2.13 A transformation L (mapping signals to signals) is said to be time invariant if for any signal f and any real number a, L[f a ](t) = (Lf)(t a) for all t (orl[f a ] = (Lf) a ). In words, L is time invariant if the time-shifted input signal f(t a) is transformed by L into the time-shifted output signal (Lf)(t a). (p. 109) Lemma 2.16 Let L be a linear, time-invariant transformation and let λ be any fixed real number. Then there is a function h with L(e iλt ) = ĥ(λ)eiλt. (t is the variable) (p. 110) Theorem 2.17 Let L be a linear, time invariant transformation on the space of signals that are piecewise continuous functions. Then there exists an integrable function, h, such that L(f) = f h for all signals f (p. 112) Physical Interpretation. Assume that h(t) is continuous and that δ is a small positive number. We apply L to the following impulse signal : { 1/(2δ), δ t δ f δ (t) = 0, o.w. If δ > 0 is small, then f δ represents a signal that is very strong but only lasts a short period of time. From several steps, h(t) is the approximate response form applying L to an input signal that is an impulse. For that reason h(t) is called the impulse response function, and ĥ(λ) is called the system function Causality and the Design of Filters Designing a time invariant filter is equivalent to construction the impulse function, h. We consider filters that reduce high frequencies. Such filters are called low-pass filters. Theorem 2.19 Let L be a time invariant filter with response function h (i.e., Lf = f h). L is a causal filter if and only if h(t) = 0 for all t < 0. Theorem 2.20 Suppose L is a causal filter with response function h. Then the system function associated with L is ĥ(λ) = L[h](iλ) where L is the Laplace transform. (p. 116) 11

4 2.4 The Sampling Theorem In this section, we examine a class of signals (i.e., functions) whose Fourier transform is zero outside a finite interval [ Ω,Ω]; these are (frequency) band-limited functions. Definition 2.22 A function f is said to be frequency band-limited if there exists a constant Ω > 0 such that ˆf(λ) = 0 for λ > Ω. When Ω is the smallest frequency for which the preceding equation is true, the natural frequency ν := Ω is called the Nyquist frequency, and 2ν = Ω π is the Nyquist rate. (p. 118) Theorem 2.23 (Shannon-Whittaker Sampling Theorem) Suppose that ˆf(λ) is piecewise smooth, continuous, and that ˆf(λ) = 0 for λ > Ω, where Ω is some fixed, positive frequency. Then f = F 1 [ ˆf] is completely determined by its values at the points t j = jπ/ω, j = 0, ±1, ±2,. More precisely, f has the following series expansion : f(t) = j= sin(ωt jπ) f(jπ/ω) Ωt jπ where the series on the right converges uniformly. The convergence rate can be increased by a technique called oversampling. At the opposite extreme, if a signal is sampled below the Nyquist rate, then the signal reconstructed via the above equation will not only be missing high frequency components, but it will also have the energy in those components transferred to low frequencies that may not have been in the signal at all. This is a phenomenon called aliasing. 2.5 The Uncertainty Principle A function cannot simultaneously have restricted support in time as well as in frequency. Definition 2.25 Suppose f is a function in L 2 (R). The dispersion of f about the point a R is the quantity a f = (t a)2 f(t) 2 dt. f(t) 2 dt (p. 121) 12

5 Applying the preceding definition of dispersion to the Fourier transform of f gives α ˆf = (λ α)2 ˆf(λ) 2 dλ ˆf(λ). 2 dλ By the Plancherel formula, the denominators in the dispersions of f and ˆf are the same. Theorem 2.26 (Uncertainty Principle) Suppose f is a function in L 2 (R) that vanishes at + and. Then for all points a R α R. a f α ˆf