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Fourier Analysis and Power Spectral Density

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Fourier Analysis and Power Spectral Density Chapter 4 Fourier Analysis and Power Spectral Density 4.1 Fourier Series and Transforms Recall Fourier series for periodic functions 1 1 X 2πnt 2πnt x(t) = a + a cos + b sin (4.1) 2 0 n T n T n=1 for x(t + T ) = x(t), where Z T 2 a0 a0 = x(t) dt =x ¯ T 0 2 2 Z T 2π an = x(t) cos n!t dt ! = (4.2) T 0 T 2 Z T bn = x(t) sin n!t dt : T 0 Dirichlet Theorem: For x(t) periodic on 0 t < T , if x(t) is bounded, has a finite number ≤ of maxima, minima, and discontinuities, then the Fourier Series Eq. (4.1) converges t to 8 1 + − 2 [x(t ) + x(t )]. Complex form of Eq. (4.1) is better for experimental applications. Using Euler's (or de Moivre's) formulas we get: 1 cos !t = ei!t + e−i!t 2 (4.3) 1 sin !t = ei!t e−i!t : 2i − Using above Eq. (4.1) can be rewritten as: 1 X in!t x(t) = Xn e ; (4.4) −∞ 31 32 CHAPTER 4. FOURIER ANALYSIS AND POWER SPECTRAL DENSITY where a0 X0 = 2 (4.5) 1 X± = (a ib ) : n 2 n ∓ n ∗ Please also note that Xn = X−n. Therefore: Z T Z T=2 1 −in!t 1 −in!t Xn = x(t) e dt = x(t) e dt : (4.6) T 0 T −T=2 n If \signal" x(t) is not periodic, we let fn = T (i.e., in Eq. (4.6) n! = 2πfn). Now, we define a function X(f) by X(fn) = TXn (i.e., Xn = X(fn)=T ) to get the following: 1 1 1 X X 1 X x(t) = X ein!t = X(f ) ei2πfnt = X(f ) ei2πfnt∆f ; (4.7) n T n n n −∞ n=−∞ n=−∞ where we used the fact that ∆f = f f = n+1 n = 1 . Therefore, in the limit as T n n+1 − n T − T T ! 1 and ∆f 0 in Eq. (4.9), we get our signal in time domain as n ! Z 1 x(t) = X(f) ei2πft df : (4.8) −∞ Now, assuming that everything converges and using Eq. (4.6) we get the corresponding frequency domain expression Z 1 X(f) = x(t) e−i2πft dt : (4.9) −∞ Therefore, x(t) X(f) are Fourier Transform pair, where x(t) is in time domain and X(f) is in $ frequency domain. 4.1.1 Several Important Properties of Fourier Transforms We denote a Fourier transform (FT) as X(f) = (x(t)) and x(t) = −1(X(f)). Now, we can write F F several of the properties of FT: 1. Linearity: [αx(t) + βy(t)] = αX(f) + βY (f) F 2. Duality: x(t) X(f) X(t) x( f) $ ) $ − 3. Conjugation: x(t) X(f) x∗(t) X∗( f). $ ) $ − Therefore, for real signal x(t), X(f) = X∗( f). This instead gives: − X(f) 2 = X(f)X∗(f) = X ( f)X( f) = X( f) 2 ; (4.10) j j ∗ − − j − j i.e., for real x(t), X(f) is symmetric. j j 4. Convolution: Z 1 x(τ)y(t τ) dτ = X(f)Y (f) , [x y] ; F −∞ − F ∗ 4.1. FOURIER SERIES AND TRANSFORMS 33 where x y indicates time convolution between x(t) and y(t). In addition, ∗ Z 1 [xy] = X(φ)Y (f φ) dφ , X Y; F −∞ − ∗ where X Y indicates frequency convolution between X(f) and Y (f) (also, X Y = Y X). ∗ ∗ ∗ 5. Differentiation: dkx = (i2πf)kX(f) ; F dtk 6. Time Scaling and Shifting: 2πif b e a f x(at + b) X : $ a a j j 1 Theorem: Provided x(t) 1 (i.e., R x(t) dt < and x(t) has a finite number of 2 L −∞ j j 1 maxima, minima, and discontinuities) X(f) exists, and 8 < −1 [X(f)] for x continuous at t ; x(t) = F 1 + − : 2 [x(t ) + x(t )] for x discontinuous at t : . There is a problem with the above theorem if we consider the following: Z 1 sin t dt = ; −∞ j j 1 which can be fixed using theory of generalized functions (or distributions), duality and other basic properties. 4.1.2 Basic Fourier Transform Pairs 1. Delta (δ) \function": This actually is a generalized function or distribution defined as: Z 1 δ(t)dt = 1 : (4.11) −∞ 1 Now, by definition of δ(t), ∆(f) = R e−2πiftδ(t t )dt = e−2πift0 , also called \sifting −∞ − 0 property." Note that ∆ is a complex constant with ∆ = 1. Therefore: j j δ(t t ) e−2πift0 ; (4.12) − 0 $ and in particular, δ(t) 1. Therefore, by duality property $ e−2πif0t δ(f f ) ; (4.13) $ − 0 and in particular, 1 δ(f). $ 34 CHAPTER 4. FOURIER ANALYSIS AND POWER SPECTRAL DENSITY Figure 4.1: Signal modulation in the frequency domain 2. Trigonometric functions: 1 cos(2πf t) = e2πif0t + e−2πif0t ; (4.14) 0 2 so δ(f f ) + δ(f + f ) [cos(2πf t)] = − 0 0 : (4.15) F 0 2 Similarly, δ(f f ) δ(f + f ) [sin(2πf t)] = − 0 − 0 : (4.16) F 0 2i 3. Modulated trigonometric functions: As an example consider x(t) cos(2πf t) X(f) c $ ∗ C(f), where fc is called carrier frequency and C(f) is given by Eq. (4.15). Then, Z 1 δ(f fc s) + δ(f + fc s) x(t) cos(2πfct) X(s) − − − ds ; (4.17) $ −∞ 2 where on the right hand side we have a convolution integral, which gives: X(f f ) + X(f + f ) x(t) cos(2πf t) − c c : (4.18) c $ 2 Therefore, if we already know X(f), modulation scales and shifts it to f as shown in Fig. 4.1. ± c 4.2 Power Spectral Density The autocorrelation of a real, stationary signal x(t) is defined to by Rx(τ) = E[x(t)x(t + τ)]. The Fourier transform of Rx(τ) is called the Power Spectral Density (PSD) Sx(f). Thus: Z 1 −i2πft Sx(f) = Rx(τ) e dτ : (4.19) −∞ The question is: what is the PSD? What does it mean? What is a \spectral density," and why is Sx called a power spectral density? To answer this question, recall that Z 1 X(f) = x(t) e−i2πft dt : (4.20) −∞ 4.2. POWER SPECTRAL DENSITY 35 To avoid convergence problems, we consider only a version of the signal observed over a finite-time 1 T , xT = x(t)wT (t), where 8 < 1 for 0 t T=2 ; wT = ≤ j j ≤ (4.21) : 0 for t > T=2 : ≤ j j Then xT has the Fourier transform Z 1 −i2πft XT (f) = xT (t) e dt ; (4.22) −∞ Z T=2 = x(t) e−i2πft dt ; (4.23) −T=2 and so "Z T=2 #"Z T=2 # ∗ −i2πft ∗ i2πfs XT XT = x(t) e dt x (s) e ds ; (4.24) −T=2 −T=2 Z T=2 Z T=2 = x(t)x(s) e−i2πf(t−s) dtds ; (4.25) −T=2 −T=2 where the star denotes complex conjugation and for compactness the frequency argument of XT has been suppressed. Taking the expectation of both sides of Eq. (4.26)2 Z T=2 Z T=2 ∗ −i2πf(t−s) E [XT XT ] = E [x(t)x(s)] e dtds : (4.26) −T=2 −T=2 Letting s = t + τ , one sees that E[x(t)x(s)] , E[x(t)x(t + τ)] = Rx(τ), and thusb Z T=2 Z T=2 ∗ −i2πf(t−s) E [XT XT ] = Rx(τ) e dtds : (4.27) −T=2 −T=2 To actually evaluate the above integral, the both variables of integration must be changed. Let τ = f(t; s) = s t (as already defined for Eq: (4:30)) (4.28) − η = g(t; s) = s + t : (4.29) Then, the integral of Eq. (4.30) is transformed (except for the limits of integration) using the change of variables formula:3 Z T=2 Z T=2 Z T Z T −i2πf(t−s) −i2πfτ −1 Rx(τ) e dtds = Rx(τ) e J dτdη ; (4.30) −T=2 −T=2 −T −T j j 1This restriction is necessary because not all of our signals will be square integrable. However, they will be mean square integrable, which is what we will take advantage of here. 2To understand what this means, remember that Eq. (5) holds for any x(t). So imagine computing Eq. (6) for different x(t) obtained from different experiments on the same system (each one of these is called a sample function). The expectation is over all possible sample functions. Since the exponential kernel inside the integral of Eq. (6) is the same for each sample function, it can be pulled outside of the expectation. 3This is a basic result from multivariable calculus. See, for example, I.S. Sokolnikoff and R.M. Redheffer, Mathe- matics of Physics and Modern Engineering, 2nd edition, McGraw- Hill, New York, 1966. 36 CHAPTER 4. FOURIER ANALYSIS AND POWER SPECTRAL DENSITY Figure 4.2: The domain of integration (gray regions) for the Fourier transform of the autocorrelation Eq. (7): (left) for the original variables, t and s; (right) for the transformed variables, η and τ, obtained by the change of variables Eq. (4.28). Notice that the square region on the left is not only rotated (and flipped about the t axis), but its area is increased by a factor of J = 2.
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