Key focus: Learn how to generate color noise using auto regressive (AR) model. Apply Yule Walker equations for generating power law noises: pink noise, Brownian noise.
Auto-Regressive (AR) model
An uncorrelated Gaussian random sequence can be transformed into a correlated Gaussian random sequence using an AR time-series model. If a time series random sequence is assumed to be following an auto-regressive model of form,
where is the uncorrelated Gaussian sequence of zero mean and variance , the natural tendency is to estimate the model parameters . Least Squares method can be applied here to find the model parameters, but the computations become cumbersome as the order increases. Fortunately, the AR model coefficients can be solved for using Yule Walker equations.
Yule Walker equations relate auto-regressive model parameters to the auto-correlation of random process . Finding the model parameters using Yule-Walker equations, is a two step process:
1. Given , estimate auto-correlation of the process . If is already specified as a function, utilize it as it is (see auto-correlation equations for Jakes spectrum or Doppler spectrum in section 11.3.2 in the book).
2. Solve Yule Walker equations to find the model parameters and the noise sigma .
Yule-Walker equations
Yule-Walker equations can be compactly written as
Equation (2) Yule Walker equation
Written in matrix form, the Yule-Walker equations that comprises of a set of linear equations and unknown parameters.
Representing equation (3) in a compact form,
The AR model parameters can be found by solving
After solving for the model parameters , the noise variance can be found by applying the estimated values of in equation (2) by setting . The aryule command (in Matlab and Python’s spectrum package) efficiently solves the Yule-Walker equations using Levinson Algorithm [1][2]. Once the model parameters are obtained, the AR model can be implemented as an \emph{infinte impulse response (IIR)} filter of form
Example: power law noise generation
The power law in the power spectrum characterizes the fluctuating observables in many natural systems. Many natural systems exhibit some noise which is a stochastic process with a power spectral density having a power exponent that can take values . Simply put, noise is a colored noise with a power spectral density of over its entire frequency range.
The noise can be classified into different types based on the value of .
Violet noise – = -2, the power spectral density is proportional to . Blue noise – = -1, the power spectral density is proportional to . White noise – = 0, the power spectral density is flat across the whole spectrum. Pink noise – = 1, the power spectral density is proportional to , i.e, it decreases by per octave with increase in frequency. Brownian noise – = 2, the power spectral density is proportional to , therefore it decreases by per octave with increase in frequency.
The power law noise can be generated by sequencing a zero-mean white noise through an auto-regressive (AR) filter of order :
where, is a zero-mean white noise process. Referring the AR generation method described in [3], the coefficients of the AR filter can be generated as
which can be implemented as an infinite impulse response (IIR) filter using the filter transfer function described in equation (6).
The following script implements this method and the sample results are plotted in the next Figure.
The aim of this article is to demonstrate the application of spectral factorization method in generating colored noise having Jakes power spectral density. Before continuing, I urge the reader to go through this post: Introduction to generating correlated Gaussian sequences.
In spectral factorization method, a filter is designed using the desired frequency domain characteristics (like PSD) to transform an uncorrelated Gaussian sequence into a correlated sequence . In the model shown in Figure 1, the input to the LTI system is a white noise whose amplitude follows Gaussian distribution with zero mean and variance and the power spectral density (PSD) of the white noise is a constant across all frequencies.
The white noise sequence drives the LTI system with frequency response producing the signal of interest . The PSD of the output process is therefore
Figure 1: Relationship among various power spectral densities in a filtering process
If the desired power spectral density of the colored noise sequence is given, assuming , the impulse response of the LTI filter can be found by taking the inverse Fourier transform of the frequency response
Once, the impulse response of the filter is obtained, the colored noise sequence can be produced by driving the filter with a zero-mean white noise sequence of unit variance.
Example: Generating colored noise with Jakes PSD
For example, we wish to generate a Gaussian noise sequence whose power spectral density follows the normalized Jakes power spectral density (see section 11.3.2 in the book) given by
Applying spectral factorization method, the frequency response of the desired filter is
The impulse response of the filter is [1]
where, is the fractional Bessel function of the first kind, is the sampling interval for implementing the digital filter and is a constant. The impulse response of the filter can be normalized by dividing by .
The filter can be implemented as a finite impulse response (FIR) filter structure. However, the FIR implementation requires that the impulse response be truncated to a reasonable length. Such truncation leads to ringing effects due to Gibbs phenomenon. To avoid distortions due to truncation, the filter impulse response is usually windowed using a window function such as Hamming window.
where, the Hamming window is defined as
The function given in the book in section 2.6.1 implements a windowed Jakes filterusing the aforementioned equations. The impulse response and the spectral characteristics of the filter are plotted in Figure 2.
Figure 2: Impulse response & spectrum of windowed Jakes filter ( fmax = 10Hz; Ts = 0:01s; N = 512)
A white noise can be transformed into colored noise sequence with Jakes PSD, by processing the white noise through the implemented filter. The script (given in the book in section 2.6.1) illustrates this concept by transforming a white noise sequence into a colored noise sequence. The simulated noise samples and its PSD are plotted in Figure 3.
Key focus: Colored noise sequence (a.k.a correlated Gaussian sequence), is a non-white random sequence, with non-constant power spectral density across frequencies.
Introduction
Speaking of Gaussian random sequences such as Gaussian noise, we generally think that the power spectral density (PSD) of such Gaussian sequences is flat.We should understand that the PSD of a Gausssian sequence need not be flat. This bring out the difference between white and colored random sequences, as captured in Figure 1.
A white noise sequence is defined as any random sequence whose PSD is constant across all frequencies. Gaussian white noise is a Gaussian random sequence, whose amplitude is gaussian distributed and its PSD is a constant. Viewed in another way, a constant PSD in frequency domain implies that the average auto-correlation function in time-domain is an impulse function (Dirac-delta function). That is, the amplitude of noise at any given time instant is correlated only with itself. Therefore, such sequences are also referred as uncorrelated random sequences. White Gaussian noise processes are completely characterized by its mean and variance.
Figure 1: Power spectral densities of white noise and colored noise
A colored noise sequence is simply a non-white random sequence, whose PSD varies with frequency. For a colored noise, the amplitude of noise at any given time instant is correlated with the amplitude of noise occurring at other instants of time. Hence, colored noise sequences will have an auto-correlation function other than the impulse function. Such sequences are also referred as correlated random sequences. Colored Gaussian noise processes are completely characterized by its mean and the shaped of power spectral density (or the shape of auto-correlation function).
In mobile channel model simulations, it is often required to generate correlated Gaussian random sequences with specified mean and power spectral density (like Jakes PSD or Gaussian PSD given in section 11.3.2 in the book). An uncorrelated Gaussian sequence can be transformed into a correlated sequence through filtering or linear transformation, that preserves the Gaussian distribution property of amplitudes, but alters only the correlation property (equivalently the power spectral density). We shall see two methods to generate colored Gaussian noise for given mean and PSD shape
Let’s say we observe a real world signal that has an arbitrary spectrum . We would like to describe the long sequence of using very few parameters, as in applications like linear predictive coding (LPC). The modeling approach, described here, tries to answer the following two questions:
• Is it possible to model the first order (mean/variance) and second order (correlations, spectrum) statistics of the signal just by shaping a white noise spectrum using a transfer function ? (see Figure 1). • Does this produce the same statistics (spectrum, correlations, mean and variance) for a white noise input ?
If the answer is yes to the above two questions, we can simply set the modeled parameters of the system and excite the system with white noise, to produce the desired real world signal. This reduces the amount to data we wish to transmit in a communication system application. This approach can be used to transform an uncorrelated white Gaussian noise sequence to a colored Gaussian noise sequence with desired spectral properties.
Key focus: With examples, let’s estimate and plot the probability density function of a random variable using Matlab histogram function.
Generation of random variables with required probability distribution characteristic is of paramount importance in simulating a communication system. Let’s see how we can generate a simple random variable, estimate and plot the probability density function (PDF) from the generated data and then match it with the intended theoretical PDF. Normal random variable is considered here for illustration. Other types of random variables like uniform, Bernoulli, binomial, Chi-squared, Nakagami-m are illustrated in the next section.
A survey of commonly used fundamental methods to generate a given random variable is given in [1]. For this demonstration, we will consider the normal random variable with the following parameters : – mean and – standard deviation. First generate a vector of randomly distributed random numbers of sufficient length (say 100000) with some valid values for and . There are more than one way to generate this. Some of them are given below.
● Method 1: Using the in-built random function (requires statistics toolbox)
mu=0;sigma=1;%mean=0,deviation=1
L=100000; %length of the random vector
R = random('Normal',mu,sigma,L,1);%method 1
● Method 2: Using randn function that generates normally distributed random numbers having and = 1
mu=0;sigma=1;%mean=0,deviation=1
L=100000; %length of the random vector
R = randn(L,1)*sigma + mu; %method 2
● Method 3: Box-Muller transformation [2] method using rand function that generates uniformly distributed random numbers
mu=0;sigma=1;%mean=0,deviation=1
L=100000; %length of the random vector
U1 = rand(L,1); %uniformly distributed random numbers U(0,1)
U2 = rand(L,1); %uniformly distributed random numbers U(0,1)
Z = sqrt(-2log(U1)).cos(2piU2);%Standard Normal distribution
R = Z*sigma+mu;%Normal distribution with mean and sigma
Step 2: Plot the estimated histogram
Typically, if we have a vector of random numbers that is drawn from a distribution, we can estimate the PDF using the histogram tool. Matlab supports two in-built functions to compute and plot histograms:
● hist – introduced before R2006a ● histogram – introduced in R2014b
Which one to use ? Matlab’s help page points that the histfunction is notrecommended for several reasons and the issue of inconsistency is one among them. The histogram function is the recommended function to use.
Estimate and plot the normalized histogram using the recommended ‘histogram’ function. And for verification, overlay the theoretical PDF for the intended distribution. When using the histogram function to plot the estimated PDF from the generated random data, use ‘pdf’ option for ‘Normalization’ option. Do not use the ‘probability’ option for ‘Normalization’ option, as it will not match the theoretical PDF curve.
histogram(R,'Normalization','pdf'); %plot estimated pdf from the generated data
X = -4:0.1:4; %range of x to compute the theoretical pdf
fx_theory = pdf('Normal',X,mu,sigma); %theoretical normal probability density
hold on; plot(X,fx_theory,'r'); %plot computed theoretical PDF
title('Probability Density Function'); xlabel('values - x'); ylabel('pdf - f(x)'); axis tight;
legend('simulated','theory');
Estimated PDF (using histogram function) and the theoretical PDF
However, if you do not have Matlab version that was released before R2014b, use the ‘hist’ function and get the histogram frequency counts () and the bin-centers (). Using these data, normalize the frequency counts using the overall area under the histogram. Plot this normalized histogram and overlay the theoretical PDF for the chosen parameters.
%For those who don't have access to 'histogram' function
%get un-normalized values from hist function with same number of bins as histogram function
numBins=50; %choose appropriately
[f,x]=hist(R,numBins); %use hist function and get unnormalized values
figure; plot(x,f/trapz(x,f),'b-*');%plot normalized histogram from the generated data
X = -4:0.1:4; %range of x to compute the theoretical pdf
fx_theory = pdf('Normal',X,mu,sigma); %theoretical normal probability density
hold on; plot(X,fx_theory,'r'); %plot computed theoretical PDF
title('Probability Density Function'); xlabel('values - x'); ylabel('pdf - f(x)'); axis tight;
legend('simulated','theory');
The given code snippets above, already include the command to plot the theoretical PDF by using the ‘pdf’ function in Matlab. It you do not have access to this function, you could use the following equation for computing the theoretical PDF
The code snippet for that purpose is given next.
X = -4:0.1:4; %range of x to compute the theoretical pdf
fx_theory = 1/sqrt(2*pi*sigma^2)*exp(-0.5*(X-mu).^2./sigma^2);
plot(X,fx_theory,'k'); %plot computed theoretical PDF
Note: The functions – ‘random’ and ‘pdf’ , requires statistics toolbox.
Rate this article: Note: There is a rating embedded within this post, please visit this post to rate it.
In this article, the relationship between SNR-per-bit (Eb/N0) and SNR-per-symbol (Es/N0) are defined with respect to M-ary signaling schemes. Then the complex baseband model for an AWGN channel is discussed, followed by the theoretical error rates of various modulations over the additive white Gaussian noise (AWGN) channel. Finally, the complex baseband models for digital modulators and detectors developed in previous chapter of this book, are incorporated to build a complete communication system model.
Assuming a channel of bandwidth B, received signal power Pr and the power spectral density (PSD) of noise N0/2, the signal to noise ratio (SNR) is given by
Let a signal’s energy-per-bit is denoted as Eb and the energy-per-symbol as Es, then γb=Eb/N0and γs=Es/N0 are the SNR-per-bit and the SNR-per-symbol respectively.
For uncoded M-ary signaling scheme with k = log2(M) bits per symbol, the signal energy per modulated symbol is given by
The SNR per symbol is given by
AWGN channel model
In order to simulate a specific SNR point in performance simulations, the modulated signal from the transmitter needs to be added with random noise of specific strength. The strength of the generated noise depends on the desired SNR level which usually is an input in such simulations. In practice, SNRs are specified in dB. Given a specific SNR point for simulation, let’s see how we can simulate an AWGN channel that adds correct level of white noise to the transmitted symbols.
Figure 1: Simplified simulation model for awgn channel
Consider the AWGN channel model given in Figure 1. Given a specific SNR point to simulate, we wish to generate a white Gaussian noise vector of appropriate strength and add it to the incoming signal. The method described can be applied for both waveform simulations and the complex baseband simulations. In following text, the term SNR (γ) refers to γb = Eb/N0 when the modulation is of binary type (example: BPSK). For multilevel modulations such as QPSK and MQAM, the term SNR refers to γs = Es/N0.
(1) Assume, s is a vector that represents the transmitted signal. We wish to generate a vector r that represents the signal after passing through the AWGN channel. The amount of noise added by the AWGN channel is controlled by the given SNR – γ
(3) Let N denotes the length of the vector s. The signal power for the vector s can be measured as,
(4) The required power spectral density of the noise vector n is computed as
(5) Assuming complex IQ plane for all the digital modulations, the required noise variance (noise power) for generating Gaussian random noise is given by
(6) Generate the noise vector n drawn from normal distribution with mean set to zero and the standard deviation computed from the equation given above
(7) Finally add the generated noise vector (n) to the signal (s)
Matlab code
The following custom function written in Matlab, can be used for adding AWGN noise to an incoming signal. It can be used in waveform simulation as well as complex baseband simulation models.
%author - Mathuranathan Viswanathan (gaussianwaves.com
%This code is part of the books: Wireless communication systems using Matlab & Digital modulations using Matlab.
function [r,n,N0] = add_awgn_noise(s,SNRdB,L)
%Function to add AWGN to the given signal
%[r,n,N0]= add_awgn_noise(s,SNRdB) adds AWGN noise vector to signal
%'s' to generate a %resulting signal vector 'r' of specified SNR
%in dB. It also returns the noise vector 'n' that is added to the
%signal 's' and the spectral density N0 of noise added
%
%[r,n,N0]= add_awgn_noise(s,SNRdB,L) adds AWGN noise vector to
%signal 's' to generate a resulting signal vector 'r' of specified
%SNR in dB. The parameter 'L' specifies the oversampling ratio used
%in the system (for waveform simulation). It also returns the noise
%vector 'n' that is added to the signal 's' and the spectral
%density N0 of noise added
s_temp=s;
if iscolumn(s), s=s.'; end; %to return the result in same dim as 's'
gamma = 10ˆ(SNRdB/10); %SNR to linear scale
if nargin==2, L=1; end %if third argument is not given, set it to 1
if isvector(s),
P=L*sum(abs(s).ˆ2)/length(s);%Actual power in the vector
else %for multi-dimensional signals like MFSK
P=L*sum(sum(abs(s).ˆ2))/length(s); %if s is a matrix [MxN]
end
N0=P/gamma; %Find the noise spectral density
if(isreal(s)),
n = sqrt(N0/2)*randn(size(s));%computed noise
else
n = sqrt(N0/2)*(randn(size(s))+1i*randn(size(s)));%computed noise
end
r = s + n; %received signal
if iscolumn(s_temp), r=r.'; end;%return r in original format as s
end
Python code
The following custom function written in Python 3, can be used for adding AWGN noise to an incoming signal. It can be used in waveform simulation as well as complex baseband simulation models.
# author - Mathuranathan Viswanathan (gaussianwaves.com
# This code is part of the book Digital Modulations using Python
from numpy import sum,isrealobj,sqrt
from numpy.random import standard_normal
def awgn(s,SNRdB,L=1):
"""
AWGN channel
Add AWGN noise to input signal. The function adds AWGN noise vector to signal 's' to generate a resulting signal vector 'r' of specified SNR in dB. It also
returns the noise vector 'n' that is added to the signal 's' and the power spectral density N0 of noise added
Parameters:
s : input/transmitted signal vector
SNRdB : desired signal to noise ratio (expressed in dB) for the received signal
L : oversampling factor (applicable for waveform simulation) default L = 1.
Returns:
r : received signal vector (r=s+n)
"""
gamma = 10**(SNRdB/10) #SNR to linear scale
if s.ndim==1:# if s is single dimensional vector
P=L*sum(abs(s)**2)/len(s) #Actual power in the vector
else: # multi-dimensional signals like MFSK
P=L*sum(sum(abs(s)**2))/len(s) # if s is a matrix [MxN]
N0=P/gamma # Find the noise spectral density
if isrealobj(s):# check if input is real/complex object type
n = sqrt(N0/2)*standard_normal(s.shape) # computed noise
else:
n = sqrt(N0/2)*(standard_normal(s.shape)+1j*standard_normal(s.shape))
r = s + n # received signal
return r
Theoretical symbol error rates for digital modulations in AWGN channel
Denoting the symbol error rate (SER) as , SNR-per-bit as and SNR-per-symbol as , the symbol error rates for various modulation schemes over AWGN channel are listed in Table 1 (refer [1]).
Table 1: Theoretical symbol error rate for various modulations in AWGN channel
The theoretical symbol error rates are coded as a reusable function. In this implementation, erfc function is used instead of the Q function shown in the Table 4.1. The following equation describes the relationship between the erfc function and the Q function.
Unified simulation model for performance simulation
In the previous chapter of the books, the code implementation for complex baseband models for various digital modulators and demodulator are given. Using these models, we can create a unified simulation code for simulating the performance of various modulation techniques over AWGN channel.
The complete simulation model for performance simulation over AWGN channel is given in Figure 2. The figure is illustrated for a coherent communication system model (applicable for MPSK/MQAM/MPAM modulations)
Figure 2: Complete simulation model for a communication system with AWGN channel
The Matlab code implementing the aforementioned simulation model is given in the books. Here, an unified approach is employed to simulate the performance of any of the given modulation technique – MPSK, MQAM, MPAM or MFSK (MFSK simulation technique is available in the following books: Digital Modulations using Python and Digital Modulations using Matlab).
The simulation code will automatically choose the selected modulation type, performs Monte Carlo simulation, computes symbol error rates and plots them against the theoretical symbol error rates. The simulated performance results obtained for MQAM and MPSK modulations are shown in the Figure 3 and Figure 4.
Figure 3: Simulated symbol error rate performance of M-QAM modulation over AWGN channel
In a “coin-flipping” experiment, the outcome is not known prior to the experiment, that is we cannot predict it with certainty (non-deterministic/stochastic). But we know the all possible outcomes – Head or Tail. Assign real numbers to the all possible events (this is called “sample space”), say “0” to “Head” and “1” to “Tail”, and associate a variable “X” that could take these two values. This variable “X” is called a random variable, since it can randomly take any value ‘0’ or ‘1’ before performing the actual experiment.
Obviously, we do not want to wait till the coin-flipping experiment is done. Because the outcome will lose its significance, we want to associate some probability to each of the possible event. In the coin-flipping experiment, all outcomes are equally probable (given that the coin is fair and unbiased). This means that we can say that the probability of getting Head ( our random variable X = 0 ) as well that of getting Tail ( X =1 ) is 0.5 (i.e. 50-50 chance for getting Head/Tail).
This can be written as,
Cumulative Distribution Function:
Mathematically, a complete description of a random variable is given be “Cumulative Distribution Function”- FX(x). Here the bold faced “X” is a random variable and “x” is a dummy variable which is a place holder for all possible outcomes ( “0” and “1” in the above mentioned coin flipping experiment). The Cumulative Distribution Function is defined as,
If we plot the CDF for our coin-flipping experiment, it would look like the one shown in the figure on your right. The example provided above is of discrete nature, as the values taken by the random variable are discrete (either “0” or “1”) and therefore the random variable is called Discrete Random Variable.
If the values taken by the random variables are of continuous nature (Example: Measurement of temperature), then the random variable is called Continuous Random Variable and the corresponding cumulative distribution function will be smoother without discontinuities.
Probability Distribution function :
Consider an experiment in which the probability of events are as follows. The probabilities of getting the numbers 1,2,3,4 individually are respectively. It will be more convenient for us if we have an equation for this experiment which will give these values based on the events. For example, the equation for this experiment can be given by where . This equation ( equivalently a function) is called probability distribution function.
Probability Density function (PDF) and Probability Mass Function(PMF):
Its more common deal with Probability Density Function (PDF)/Probability Mass Function (PMF) than CDF.
The PDF (defined for Continuous Random Variables) is given by taking the first derivate of CDF.
For discrete random variable that takes on discrete values, is it common to defined Probability Mass Function.
The previous example was simple. The problem becomes slightly complex if we are asked to find the probability of getting a value less than or equal to 3. Now the straight forward approach will be to add the probabilities of getting the values which comes out to be . This can be easily modeled as a probability density function which will be the integral of probability distribution function with limits 1 to 3.
The mean of a random variable is defined as the weighted average of all possible values the random variable can take. Probability of each outcome is used to weight each value when calculating the mean. Mean is also called expectation (E[X])
For continuos random variable X and probability density function fX(x)
For discrete random variable X, the mean is calculated as weighted average of all possible values (xi) weighted with individual probability (pi)
Variance :
Variance measures the spread of a distribution. For a continuous random variable X, the variance is defined as
For discrete case, the variance is defined as
Standard Deviation () is defined as the square root of variance
Properties of Mean and Variance:
For a constant – “c” following properties will hold true for mean
For a constant – “c” following properties will hold true for variance
PDF and CDF define a random variable completely. For example: If two random variables X and Y have the same PDF, then they will have the same CDF and therefore their mean and variance will be same. On the otherhand, mean and variance describes a random variable only partially. If two random variables X and Y have the same mean and variance, they may or may not have the same PDF or CDF.
Gaussian Distribution :
Gaussian PDF looks like a bell. It is used most widely in communication engineering. For example , all channels are assumed to be Additive White Gaussian Noise channel. What is the reason behind it ? Gaussian noise gives the smallest channel capacity with fixed noise power. This means that it results in the worst channel impairment. So the coding designs done under this most adverse environment will give superior and satisfactory performance in real environments. For more information on “Gaussianity” refer [1]
The PDF of the Gaussian Distribution (also called as Normal Distribution) is completely characterized by its mean () and variance(),
Since PDF is defined as the first derivative of CDF, a reverse engineering tell us that CDF can be obtained by taking an integral of PDF. Thus to get the CDF of the above given function,
Equations for PDF and CDF for certain distributions are consolidated below
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Necessary cookies are absolutely essential for the website to function properly. These cookies ensure basic functionalities and security features of the website, anonymously.
Cookie
Duration
Description
cookielawinfo-checbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-analytics
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Analytics".
cookielawinfo-checbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-functional
11 months
The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional".
cookielawinfo-checbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checbox-others
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Other.
cookielawinfo-checkbox-necessary
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookies is used to store the user consent for the cookies in the category "Necessary".
cookielawinfo-checkbox-performance
11 months
This cookie is set by GDPR Cookie Consent plugin. The cookie is used to store the user consent for the cookies in the category "Performance".
viewed_cookie_policy
11 months
The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. It does not store any personal data.
Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features.
Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors.
Analytical cookies are used to understand how visitors interact with the website. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc.
![x[n]](https://s0.wp.com/latex.php?latex=x%5Bn%5D&bg=ffffff&fg=000&s=0&c=20201002)
![y[n]](https://s0.wp.com/latex.php?latex=y%5Bn%5D&bg=ffffff&fg=000&s=0&c=20201002)
![R_{yy}[k]](https://s0.wp.com/latex.php?latex=R_%7Byy%7D%5Bk%5D&bg=ffffff&fg=000&s=0&c=20201002)
![R_{yy}[n]](https://s0.wp.com/latex.php?latex=R_%7Byy%7D%5Bn%5D&bg=ffffff&fg=000&s=0&c=20201002)
![\sum_{k=0}^{k=N} a_k y[n - k] = x[n] \quad \quad \quad \quad (8)](https://s0.wp.com/latex.php?latex=%5Csum_%7Bk%3D0%7D%5E%7Bk%3DN%7D+a_k+y%5Bn+-+k%5D+%3D+x%5Bn%5D+%5Cquad+%5Cquad+%5Cquad+%5Cquad+%288%29+&bg=ffffff&fg=000&s=2&c=20201002)
![h[n]](https://s0.wp.com/latex.php?latex=h%5Bn%5D&bg=ffffff&fg=000&s=0&c=20201002)
![H(f) = \sqrt{S_{yy}(f)} = \sqrt{\pi f_{max}} \left[1- \left(f/f_{max}\right)^2\right]^{-1/4} \quad (5)](https://s0.wp.com/latex.php?latex=H%28f%29+%3D+%5Csqrt%7BS_%7Byy%7D%28f%29%7D+%3D+%5Csqrt%7B%5Cpi+f_%7Bmax%7D%7D+%5Cleft%5B1-+%5Cleft%28f%2Ff_%7Bmax%7D%5Cright%29%5E2%5Cright%5D%5E%7B-1%2F4%7D+%5Cquad+%285%29+&bg=ffffff&fg=000&s=2&c=20201002)
![h[n] = C f_{max} x^{-1/4} J_{1/4} (x) , \quad \quad x = 2 \pi f_{max} |n T_s| \quad (6)](https://s0.wp.com/latex.php?latex=h%5Bn%5D+%3D+C+f_%7Bmax%7D+x%5E%7B-1%2F4%7D+J_%7B1%2F4%7D+%28x%29+%2C+%5Cquad+%5Cquad+x+%3D+2+%5Cpi+f_%7Bmax%7D+%7Cn+T_s%7C+%5Cquad+%286%29+&bg=ffffff&fg=000&s=2&c=20201002)
![h_{norm}[n] = x^{-1/4} J_{1/4} (x), \quad \quad x = 2 \pi f_{max} |n T_s| \quad (7)](https://s0.wp.com/latex.php?latex=h_%7Bnorm%7D%5Bn%5D+%3D+x%5E%7B-1%2F4%7D+J_%7B1%2F4%7D+%28x%29%2C+%5Cquad+%5Cquad+x+%3D+2+%5Cpi+f_%7Bmax%7D+%7Cn+T_s%7C+%5Cquad+%287%29+&bg=ffffff&fg=000&s=2&c=20201002)
![h_{w}[n] = h_{norm}[n] w_{H}[n] \quad \quad (8)](https://s0.wp.com/latex.php?latex=h_%7Bw%7D%5Bn%5D+%3D+h_%7Bnorm%7D%5Bn%5D+w_%7BH%7D%5Bn%5D+%5Cquad+%5Cquad+%288%29+&bg=ffffff&fg=000&s=2&c=20201002)
![w_{H}[n] = 0.54-0.46 \; cos(2 \pi n /N) \quad \quad (9)](https://s0.wp.com/latex.php?latex=w_%7BH%7D%5Bn%5D+%3D+0.54-0.46+%5C%3B+cos%282+%5Cpi+n+%2FN%29+%5Cquad+%5Cquad+%289%29+&bg=ffffff&fg=000&s=2&c=20201002)
![f_X(x) = \frac{1}{ \sqrt{ 2 \pi \sigma^2 }} exp \left[ -\frac{ \left( x-\mu \right) ^2}{2 \sigma^2} \right]](https://s0.wp.com/latex.php?latex=f_X%28x%29+%3D+%5Cfrac%7B1%7D%7B+%5Csqrt%7B+2+%5Cpi+%5Csigma%5E2+%7D%7D+exp+%5Cleft%5B+-%5Cfrac%7B+%5Cleft%28+x-%5Cmu+%5Cright%29+%5E2%7D%7B2+%5Csigma%5E2%7D+%5Cright%5D+&bg=ffffff&fg=000&s=2&c=20201002)
![E\left[X \right] = \int_{-\infty }^{\infty}xf_X(x)dx](https://s0.wp.com/latex.php?latex=E%5Cleft%5BX+%5Cright%5D+%3D+%5Cint_%7B-%5Cinfty+%7D%5E%7B%5Cinfty%7Dxf_X%28x%29dx+&bg=ffffff&fg=000&s=1&c=20201002)
![E\left[X \right] = \mu{_X} = \sum_{-\infty }^{\infty}x_{i}p_{i}](https://s0.wp.com/latex.php?latex=E%5Cleft%5BX+%5Cright%5D+%3D+%5Cmu%7B_X%7D+%3D+%5Csum_%7B-%5Cinfty+%7D%5E%7B%5Cinfty%7Dx_%7Bi%7Dp_%7Bi%7D+&bg=ffffff&fg=000&s=1&c=20201002)
![var \left[X\right] = \int_{-\infty }^{\infty} \left(x - E\left[X \right] \right)^2 f_X(x) dx](https://s0.wp.com/latex.php?latex=var+%5Cleft%5BX%5Cright%5D+%3D+%5Cint_%7B-%5Cinfty+%7D%5E%7B%5Cinfty%7D+%5Cleft%28x+-+E%5Cleft%5BX+%5Cright%5D+%5Cright%29%5E2+f_X%28x%29+dx+&bg=ffffff&fg=000&s=1&c=20201002)
![var \left[X\right] = {\sigma^2}_X = \sum_{-\infty }^{\infty} \left( x_i - \mu_X\right)^2 p_{i}](https://s0.wp.com/latex.php?latex=var+%5Cleft%5BX%5Cright%5D+%3D+%7B%5Csigma%5E2%7D_X+%3D+%5Csum_%7B-%5Cinfty+%7D%5E%7B%5Cinfty%7D+%5Cleft%28+x_i+-+%5Cmu_X%5Cright%29%5E2+p_%7Bi%7D+&bg=ffffff&fg=000&s=1&c=20201002)
![E\left[cX\right] = c E\left[X\right]](https://s0.wp.com/latex.php?latex=E%5Cleft%5BcX%5Cright%5D+%3D+c+E%5Cleft%5BX%5Cright%5D+&bg=ffffff&fg=000&s=1&c=20201002)
![E\left[X+c\right] = E\left[X\right]+c](https://s0.wp.com/latex.php?latex=E%5Cleft%5BX%2Bc%5Cright%5D+%3D+E%5Cleft%5BX%5Cright%5D%2Bc+&bg=ffffff&fg=000&s=1&c=20201002)
![E\left[c\right] = c](https://s0.wp.com/latex.php?latex=E%5Cleft%5Bc%5Cright%5D+%3D+c+&bg=ffffff&fg=000&s=1&c=20201002)
![var\left[cX\right] = c^2 var\left[X\right]](https://s0.wp.com/latex.php?latex=var%5Cleft%5BcX%5Cright%5D+%3D+c%5E2+var%5Cleft%5BX%5Cright%5D+&bg=ffffff&fg=000&s=1&c=20201002)
![var\left[X+c\right] = var\left[X\right]](https://s0.wp.com/latex.php?latex=var%5Cleft%5BX%2Bc%5Cright%5D+%3D+var%5Cleft%5BX%5Cright%5D+&bg=ffffff&fg=000&s=1&c=20201002)
![var\left[c\right] = 0](https://s0.wp.com/latex.php?latex=var%5Cleft%5Bc%5Cright%5D+%3D+0+&bg=ffffff&fg=000&s=1&c=20201002)
