Of michigan fall 2012 september 7 2012 91 174 lecture

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Unformatted text preview: igan) Fall 2012 September 7, 2012 89 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise Mean 100 n(t) 50 0 −50 −100 0 10 20 30 40 50 time 60 70 80 90 100 0 10 20 30 40 50 time 60 70 80 90 100 100 E[n(t)] 50 0 −50 −100 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 90 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise Variance 12000 10000 n2(t) 8000 6000 4000 2000 0 0 10 20 30 40 50 time 60 70 80 90 100 0 10 20 30 40 50 time 60 70 80 90 100 10000 E[n2(t)] 8000 6000 4000 2000 0 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 91 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Power Spectral Density The power spectral density function of a random signal is the amount of power in the signal as a function of frequency. The autocorrelation measures the correlation between the noise at different points in time. For noise like signals the autocorrelation does not depend on the time but just the time difference between two samples. In this case (and assuming zero mean) the process is called wide-sense stationary. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 92 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Sample Path and Frequency Content of Noise -4 x 10 2 x(t) 1 0 -1 -2 -2 -1 0 1 time 2 3 4 0 |X(f)|^2 -100 -200 -300 -400 -10 -8 EECS 455 (Univ. of Michigan) -6 -4 -2 0 frequency Fall 2012 2 4 6 8 10 September 7, 2012 93 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Random Process Definitions The autocorrelation of a random process X (t ) is RX (τ ) = E [X (t )X (t + τ )] The power of a (wide-sense stationary) random process X (t ) is PX = E [X 2 (t )] = RX (0). To see the meaning of this definition we start with the time average power of a particular realization of a random process over a finite interval. Then take the (probablistic) average to get the average power. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 94 / 174 Lecture Notes 2 PX =E = 1 T →∞ 2T EECS 455 (Univ. of Michigan) T lim 1 T →∞ 2T T lim 1 T →∞ 2T = RX (0) = Random Variables, Random Processes, Noise −T T −T |X (t )|2 dt E [|X (t )|2 ]dt RX (0)dt lim −T Fall 2012 September 7, 2012 95 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Power Spectral Density Define the power spectral density as the Fourier transform of the autocorrelation function. SX (f ) = RX (τ ) = ∞ RX (τ )e−j 2πf τ d τ ∞ ∞ SX (f )ej 2πf τ df ∞ Then the average power of a random process is ∞ PX = RX (0) = SX (f )df . −∞ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 96 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Power Spectral Density Thus the power spectral density is a measure of how the power is distributed over frequency. Another way to see that the Fourier transform of autocorrelation is power spectral density is to put a random process through a linear time-invariant filter which has a very narrow bandpass transfer function. Then the power at the output of the filter is the power spectral density of the input evaluated at the frequency that the filter passes. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 97 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example: White Noise RN (τ ) = N0 δ(τ ), 2 RN (τ ) SN (f ) = N0 2 SN (f ) τ EECS 455 (Univ. of Michigan) f Fall 2012 September 7, 2012 98 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise White Gaussian Noise White Gaussian Noise (WGN) sometimes also called thermal noise or Johnson noise has a flat power spectral density (same noise power at all frequencies). The power spectral density is N0 /2 where N0 = kTo , k = 1.38 × 10−23 Joules/◦ K (Boltzman’s constant) and To is the temperature in Kelvin. For room temperatures To = 290K which makes N0 = 4 × 10−21 Watts/Hz. In dBs this is N0 = −174 dBm/Hz or N0 = −204 dBW/Hz. The distribution of the noise (at any time) is Gaussian. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 99 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise White Gaussian Noise When using power spectral density the noise has equal power at positive and negative frequencies. “One-sided” power spectral density is the sum of the power at the positive and negative frequencies. The one-sided power spectral density of white noise is N0 . Notice that the power (not the power spectral density) of white noise is infinite. Since there is the same power at all frequencies the total power must be infinite. White noise is only a model for what happens in practice. When operating at frequencies below the optical band this is a very accurate model. Since we always filter the signal and noise, the noise at the output of the filter becomes finite. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 100 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise Power Example What is the noise power in the frequency band from 2400 MHz to 2483 MHz at room tem...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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