1Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37EE422GSignals and Systems LaboratoryNoise AnalysisKevin D. DonohueElectrical and Computer EngineeringUniversity of KentuckyNoise as a Random VariableNoise is modeled as a random variable (RV), which is defined as a function that maps an event into a real number.There are 2 main characterizations of noise that are important for designing signal processing and communication systems:1. Distribution of amplitudes is characterized by the probability density function (pdf), or its integral, the cumulative distribution function (cdf)2. The correlation or influence between neighboring noise samples is characterized by the autocorrelation (AC), or its Fourier transform, the power spectral density (PDF).The PDFGiven random signal x[n] with pdf pX(x), its probability of occurring between values a and b is given by: The cdf is the probability the RV X is less than x denoted by:Pr{a X ≤b}=∫abpXxdxPr{X ≤x}=Pxx=∫−∞xpXd PDF and CDF ExampleConsider an exponential noise distribution with pdf:Compute cdf and find values of x such that the probability of being less than x1 is 0.5132 and less than x2 is 0.9544. Find the probability of x being between x1 and x2 pX(x)=0.9exp(−0.9x)forx≥0and 0 elsewhere0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 500 . 20 . 40 . 60 . 81X : 0 . 8Y : 0 . 5 1 3 2xc d fX : 3 . 4 3Y : 0 . 9 5 4 40 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 500 . 20 . 40 . 60 . 81xp d fParametric PDF EstimateIf the form of the distribution is known, then only the parameters of the distribution need to be estimated from the data. For example consider a normal or Gaussian distribution with pdf given by:with mean  and standard deviation 2.So if N data samples (xn) are collected, the sample mean and standard deviation is estimated by: pX(x)=1√2π σ2exp(−(x−μ)22σ2)=1N∑n=0N−1xn=1N−1∑n=0N−1xn−2Threshold DesignAssume that a noise process is exponential with unknown mean. Given 10 noise samples, estimate a threshold to detect a signal of greater power, such that the probability of false alarm is 1 out of 10k tests.pXx=1bexp−xbfor x≥0Model:Sample Data:s=[0.20, 1.17, 0.69, 3.70, 1.3, 5.55, 0.46, 0.70, 0.36, 0.34]Threshold DesignCompute mean as an estimate of the b parameter (sample mean): Use in cdf to show a relationship between a false alarm probability and the threshold value: b=110∑i=110si=1.45Pfa=1−cdf T =exp−TbT =−b ln Pfa≈−b lnPfaT ≈−1.45ln 1/10k=13.35PDF Non-Parametric EstimationIf the form of the distribution is unknown, it can be estimated with few assumptions using a normalized histogram operation, which estimates the PDF over short intervals (bins) based on percentage of sample data occurring in the bin:Pr[ax≤b]=∫abpxdx≈samples∈[a ,b]total samples collected0 . 5 1 1 . 5 2 2 . 5 3 3 . 500 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 8x0 1 2 3 400 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 8x∫abpxdx=b−apxabfor some xab∈[a b]Signal Power and Moments The RMS value of a random signal is equivalent to its second moment.If the signal is zero mean, the standard deviation is equivalent to the RMS value.Find the RMS value of:Find RMS value of a Gaussian random noise process with mean 0.5 and variance of 4. Srms=√1T∫Ts2(t)dt≈√1N∑i=1Nsi2s(t)=3+5sin(2π100t)VSignal to Noise RatioSignal to noise ratio in Decibels (dB) is defined as the following power ratio:where s is the RMS value of the signal and nis the RMS value of the noise.Assume a zero mean signal has an RMS value of 2 and zero mean Gaussian noise has an RMS value of 1. The noise and signal will be added together to simulate a signal in noise. What must the noise signal be multiplied by so that the resulting SNRdB is -2 dB? SNRdB=10 log10(σ2sσ2n)=20 log10(σsσn)CorrelationThe correlation indicates how similar one signal is to another. Related to this is the covariance, which removes the signal mean:Correlation is the same as above without the mean subtraction. For zero mean signals, the covariance and correlation are identical.To estimate correlation from stationary (i.e. statistics do not change over time) random signal segments of length N, the sample correlation is used:xy=E[xy]=∫x∫yx−xy−ypXYx , ydx dyRxy[k ]=1N −k∑n=kN −1x [n−k ] y [n]AutocorrelationA signal can be correlated with a delayed version of itself to determine influence (statistically) as samples get further apart. This is the autocorrelation (AC) function:This is referred to as the biased AC and k is often called the lag, which represents the relative delay or shift between the signals. The Matlab function xcorr() can be used to perform these operations.Note for k=0 the energy or second moment is computed. This will always be the largest value for all AC lags. The AC is often normalized by this value so the zero lag becomes 1.Rxy[k ]=1N∑n=kN −1y [n−k ] y [n]Convolution and CorrelationRecall the convolution operation:The main difference between convolution and correlation is time reversal of one of the signals before the multiply and sum operation. The conv() operation in Matlab can be used to implement correlation (with some minor modifications to the input arguments)w [k ]=∑n=kN −1x [k −n] y [n]R[k ]=1N∑n=kN −1x [n−k ] y [n]Autocorrelation Example- 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0- 4- 2024x ( n )nAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExampleAutocorrelation ExamplePower Spectral DensityThe power spectral density (PSD) of a random signal is its average magnitude spectrum (phase is considered irrelevant and is removed). For stationary random signals the PSD can be estimated from the average of DFT