The psdf at frequency f is defined to be the expected power that the voltage N(), bandlimited to an increment band df centered at f would dissipate in a 1-f2 resistance, divided by df. Equations(73.20)are known as the Wiener-Khinchin relations that establish that SxN(w)and RNT)are a Fourier transform pair for ws random processes [Brown, 1983; Gardner, 1990, P. 230; Peebles, 1987 ]. The psdf SNN(w)has units of w/(rad/s), whereas the autocorrelation function RNn()has units of watts. when T=0 in the second integral of Eq(73. 20), the exponential becomes e=l, so that RNNO)(= EIN(02=ON) is the integral of the psdf SNn()over all radian frequencies,-o0 w<oo. The rms(root-mean-square)voltage is Nms =U, (the standard deviation). The power spectrum in W/(rad/s)is a density that is summed up via an integral over the radian frequency band wi to w2 to obtain the total power over that band. P(w1,w2) x(w)· dw watts PNN =ON=EN(: (w)· dw watts The variance on=RNN(O)is the mean instantaneous power PNN over all frequencies at any time t. Effect of linear transformations on autocorrelation and Power Spectral Density Let h(t) be the impulse response function of a time-invariant linear system L and H(w)=F[h(o)] be its transfer function. Let an input noise signal N(n) have autocorrelation function RNN(T)and psdf SNN(w). We denote the output noise signal by Y(r)=LIN(o]. The Fourier transforms Y(w)=FlY(o] and N(w)= FN(o] do not exist, but they are not needed. The output Y(o) of a linear system is ws whenever the input N(r) is ws [see Gardner, 1990, P. 195; or Peebles, 1987, P. 215]. The output psdf S(w)and autocorrelation function Rry(T) are given by, respectively, Syr(w)=Hw)SNOw), Ryr(T)=F-ISrrw)I (73.22) Isee Gardner, 1990, P. 223]. The output noise power is H(w)2 White, Gaussian, and Pink Noise Models White noise[see Brown, 1983; Gardner, 1990, P. 234; or Peebles, 1987] is a theoretical model W(n)of noise that (时 W/(rad/s)-& w< a, The inverse Fourier transform of this is the i to lse te light, p its p (n,)8(T), which is zero for all offsets except T=O. Therefore, white noise W(t) is a process that is uncorrelated over time, ie, EW(t)w(t)]=0 for t, not equal to t,. Figure 73. 4(a) shows the autocorrelation and psdf for white noise where the offset is s=T A Gaussian white noise is white noise such that the probability distribution of each random variable w= W(r)is Gaussian. When two gaussian random variables w, and W2 related, i. e, E[W,W2]=0, they are independent [see Gardner, 1990, P. 37]. We use Gaussian models because of the central limit theorem that states that the sum of a number of random variables is approximately Gaussian. Actual circuits attenuate signals above cut-off frequencies, and also the power must be finite. However, for white noise, Pww= RNO)=oo, so we often truncate the white noise spectral density(psdf)at cu w. The result is known as pink noise, P(t), and is usually taken to be Gaussian because linear filtering of any white noise(through the effect of the central limit theorem)tends to make the noise Gaussian [see Gardner, e 2000 by CRC Press LLC© 2000 by CRC Press LLC The psdf at frequency f is defined to be the expected power that the voltage N(t), bandlimited to an incremental band df centered at f, would dissipate in a 1-W resistance, divided by df. Equations (73.20) are known as the Wiener-Khinchin relations that establish that SNN(w) and RNN(t) are a Fourier transform pair for ws random processes [Brown, 1983; Gardner, 1990, p. 230; Peebles, 1987]. The psdf SNN(w) has units of W/(rad/s), whereas the autocorrelation function RNN(t) has units of watts. When t = 0 in the second integral of Eq. (73.20), the exponential becomes e0 = 1, so that RNN(0) (= E[N(t)2 ] = sN 2 ) is the integral of the psdf SNN(w) over all radian frequencies, –` < w < `. The rms (root-mean-square) voltage is Nrms = sN (the standard deviation). The power spectrum in W/(rad/s) is a density that is summed up via an integral over the radian frequency band w1 to w2 to obtain the total power over that band. (73.21) The variance sN 2 = RNN(0) is the mean instantaneous power PNN over all frequencies at any time t. Effect of Linear Transformations on Autocorrelation and Power Spectral Density Let h(t) be the impulse response function of a time-invariant linear system L and H(w) = F[h(t)] be its transfer function. Let an input noise signal N(t) have autocorrelation function RNN(t) and psdf SNN(w). We denote the output noise signal by Y(t) = L[N(t)]. The Fourier transforms Y(w) [ F[Y(t)] and N(w) [ F[N(t)] do not exist, but they are not needed. The output Y(t) of a linear system is ws whenever the input N(t) is ws [see Gardner, 1990, p. 195; or Peebles, 1987, p. 215]. The output psdf SYY(w) and autocorrelation function RYY (t) are given by, respectively, SYY(w) = *H(w)* 2SNN(w), RYY(t) = F–1[SYY(w)] (73.22) [see Gardner, 1990, p. 223]. The output noise power is (73.23) White, Gaussian, and Pink Noise Models White noise [see Brown, 1983; Gardner, 1990, p. 234; or Peebles, 1987] is a theoretical model W(t) of noise that is ws with zero mean. It has a constant power level no over all frequencies (analogous to white light), so its psdf is SWW(w) = no W/(rad/s), –` < w < `. The inverse Fourier transform of this is the impulse function RWW(t) = (no)d(t), which is zero for all offsets except t = 0. Therefore, white noise W(t) is a process that is uncorrelated over time, i.e., E[W(t1)W(t2)] = 0 for t1 not equal to t2 . Figure 73.4(a) shows the autocorrelation and psdf for white noise where the offset is s = t. A Gaussian white noise is white noise such that the probability distribution of each random variable Wt = W(t) is Gaussian. When two Gaussian random variables W1 and W2 are uncor￾related, i.e., E[W1W2] = 0, they are independent [see Gardner, 1990, p. 37]. We use Gaussian models because of the central limit theorem that states that the sum of a number of random variables is approximately Gaussian. Actual circuits attenuate signals above cut-off frequencies, and also the power must be finite. However, for white noise, PWW = RNN(0) = `, so we often truncate the white noise spectral density (psdf) at cut-offs –wc to wc . The result is known as pink noise, P(t), and is usually taken to be Gaussian because linear filtering of any white noise (through the effect of the central limit theorem) tends to make the noise Gaussian [see Gardner, P w w S w dw P E N t S w dw NN NN w w NN N NN ( , ) ( ) [ ( ) ] ( ) 1 2 1 2 2 2 1 2 1 2 = × = = = × Ú Ú-• • p s p watts watts s p p Y YY YY NN P S w dw H w S w dw 2 2 1 2 1 2 = = = -• • -• • Ú Ú ( ) * * ( ) ( )