Category: Random Process

Bayes’ theorem

Key focus: Bayes’ theorem is a method for revising the prior probability for specific event, taking into account the evidence available about the event.

Introduction

In statistics, the process of drawing conclusions from data subject to random variations – is called “statistical inference”. Usually, in any random experiment, the observations are recorded and conclusions have to be drawn based on the recorded data set. Conclusions over the underlying random process are necessary to establish one or many of the following:

* Estimation of a parameter of interest (For example: the carrier frequency estimation in the receiver)
* Confidence and credibility of the estimate
* Rejecting a preconceived hypothesis
* Classification of data set into groups

Several schools of statistical inference have evolved over time. Bayesian inference is one of them.

Bayes’ theorem

Bayes’ theorem is central to scientific discovery and a core tool in machine learning/AI. It has numerous applications including but not limited to areas such as: mathematics, medicine, finance, marketing and engineering.

The Bayes’ theorem is used in Bayesian inference, usually dealing with a sequence of events, as new information becomes available about a subsequent event, that new information is used to update the probability of the initial event. In this context, we encounter two flavors of probabilities: prior probability and posterior probability.

Prior probability : This is the initial probability about an event before any information is available about the event. In other words, this is the initial belief about a particular hypothesis before any evidence is available about the hypothesis.

Posterior probability: This is the probability value that has been revised by using new information that is later obtained from a subsequent event. In other words, this is the updated belief about the hypothesis as new evident becomes available.

The formula for Bayes’ theorem is

Figure 1: Formula for Bayes’ theorem

A very simple thought experiment

You are asked to conduct a random experiment with a given coin. You are told that the coin is unbiased (probability of obtaining head or tail is equal and is exactly 50%). You believe (before conducting the experiment) that the coin is unbiased and that the chance of getting head or tail is equal to be 0.5.

Assume that you have not looked at both sides of the coin and simply you start to conduct the experiment. You start to toss the coin repeatedly and record the events (This is the observed new information/evidences). On the first toss you observe the coin lands on the ground with head faced up. On the second toss, again the head shows up. On subsequent tosses, the coin always shows up head. You have tossed 100 times and all these tosses you observe only head. Now what will you think about the coin? You will really start to think that both sides of the coin are engraved with “head” (no tail etched on the coin). Now, based on the new evidences, your belief about the “unbiasedness” of the coin is altered.

This is what Bayes’ theorem or Bayesian inference is all about. It is a general principle about learning from experience. It connects beliefs (called prior probabilities) and evidences (observed data). Based on the evidence, the degree of belief is refined. The degree of belief after conducting the experiment is called posterior probability.

Figure 2: Bayes’ theorem – the process

Real world example

Suppose, a person X falls sick and goes to the doctor for diagnosis. The doctor runs a series of tests and the test result came positive for a rare disease that affects 0.1% of the population. The accuracy of the test is 99%. That is, the test can correctly identify 99% of people that have the disease and will incorrectly report disease in only 1% of the people that do not have the disease. Now, how certain is that the person X actually have the disease ?

In this scenario, we can apply the extended form of Bayes’ theorem

Figure 3: Bayes’ theorem – extended form

Extended form of Bayes’ theorem is applied in special scenarios where P(H) is a binary variable, which implies it can take only two possible states. In the given problem above, the hypothesis can take only two states – H – “having the disease” and H̅ – “not having the disease”.

For the given problem, we can come up with the following numbers for the various quantities in the extended form of Bayes’ theorem.

P(H) = prior probability of having the disease before the availability of test results. This is often guess work, but luckily we have the probability that affects the population (0.1% = 0.001) to replace this.
P(E/H) = probability to test positive for the disease if person X has the disease (99% = 0.99)
P(H̅) = probability of NOT having the disease (1-0.001 = 0.999)
P(E/H̅) = probability of NOT having the disease and falsely identified positive by the test (1% = 0.01).
P(H/E) = probability of person X actually have the disease given the test result is positive.

Plugging-in these numbers in the extended form of Bayes’ theorem, we get the probability that X actually have the disease is just 9%.

Figure 4: Calculation using extended form of Bayes’ theorem

Person X doubts the result and goes for a second opinion to another doctor and gets tested from an independent laboratory. The second test result came back positive this time too. Now what is the probability that person X actually have the disease ?

P(H) = Replace this with the posterior probability from first test (we are refining the belief about the result of the first test) = 9.016% = 0.09016
P(E/H) = probability to test positive for the disease if person X has the disease (99% = 0.99)
P(H̅) = probability of NOT having the disease from first test (1-0.09016 = 0.90984)
P(E/H̅) = probability of NOT having the disease and falsely identified positive by the second test (1% = 0.01).
P(H/E) = probability of person X actually have the disease given the second test result is also positive.

Figure 5: Refining the belief about the first test using results from second test

Therefore, the updated probability based on two positive tests is 90.75%. This implies that there is a 90.75% chance that person X has the disease.

I hope the reader got a better understanding of what Bayes’ theorem is, various parameters in the equation for Bayes’ theorem and how to apply it.

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References

[1] Jeremy Orloff and Jonathan Bloom, “Conditional Probability, Independence and Bayes’ Theorem”, MIT OCW, Class 3, 18.05 Introduction to Probability and Statistics ↗.
[2] Veritasium, “The Bayesian Trap”, YouTube

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Statistical measures for stochastic signals

Key focus: Discuss statistical measures for stochastic signals : mean, variance, skewness, kurtosis, histogram, scatterplot, cross-correlation and auto-correlation.

Deterministic and stochastic signals

A deterministic signal is exactly predictable for the given time span of interest. It could be expressed using analytic form (example: x(t) = sin (2 π fc t) ).

Many of the signals that are encountered in real world signal processing applications cannot be expressed or predicted using analytic equations, because they have an element of uncertainty associated with them. As a consequence, such signals are characterized using statistical concepts. Therefore, these signals are outside the realm of deterministic signals and are classified as stochastic signals.

For example, we look at an electrical circuit and monitor the voltage across a resistor for a period of time. Under an applied electric field, atomic particles inside resister tend to randomly move and it manifests as heat. This random thermal motion causes random fluctuation in the voltage measured across the resistor. Therefore, the measured voltage can be characterized by a probability distribution and can be analyzed using statistical methods, but it cannot be predicted with precision. This is an example of signal that is stochastic function of time. Such functions usually evolve according to certain probabilistic laws and are assumed to be generated by an underlying stochastic process (thermal motion in the example above).

Figure 1: Examples for deterministic and stochastic signals

Given the amplitude of the stochastic voltage signal at time , now we know, we cannot predict the value at t’. However, if we observed the signal for a sufficient amount of time, we can empirically determine its probability distribution based on which we should be able to answer questions like

  • Given the amplitude of the voltage at time t, what is the average (expected or mean) of the voltage at time t’?
  • How much can we expect the voltage at time t’, to fluctuate from the mean ? In other words, we are interested in the variance of the voltage at time t’.
  • What is the probability that the voltage at t’ exceeds or falls below a certain threshold value ?
  • How the voltages at times t and t’ are related ? In other words, we are interested in correlation.

Summary of descriptive statistical measures

Different statistical measures are available to gather, review, analyze and draw conclusions from stochastic signals. Some of the statistical measures are summarized below:

Quantitative measures of shape:

In many statistical analysis, the fundamental task is to get a general idea about the shape of a distribution and it is done by using moments.

Measure of central tendency – mean – the first moment:

Measures of central tendency attempt to identify the central position in the distribution of samples that make up the stochastic signal. Mean, mode and median are different measures of central tendency. In signal processing, we are mostly interested in computing mean which is the average of values of the given samples. Given a discrete random variable X with probability mass function pX(x) the mean is denoted by

\[\mu = E \left[ X\right] = \sum_{x: p_X(x) > 0} x p_X(x) \]

Measure of dispersion – variance – the second moment:

Measures of dispersion describe how the values in the signal samples are spread out around a central location. Variance and standard deviation are some of the measures of dispersion. If the values are widely dispersed, the central location is said to be less representative of the values as a whole. If the values are tightly dispersed, the central location is considered more reliable.

For a discrete random variable X with probability mass function pX(x) , the variance (σX2) is given by the second central moment. The term central moment implies that this is measured relative to mean. For an electrical signal, the second moment is proportional to the average power.

\[\sigma_X^2 = E \left[ \left(X – \mu \right)^2\right] = \sum_{x: p_X(x) > 0} \left(x – \mu \right)^2 p_X(x) \]

The square root of variance is standard deviation

\[\sigma_X = \sqrt{E \left[ \left(X – \mu \right)^2\right]}\]

Figure 2, demonstrates the use of histogram for getting a visual perception of shape of the distribution of samples from two different stochastic signals. Though the signals look similar in nature in time domain, their histogram reveals different picture altogether. The central location (mean) of the first signal is around zero (no DC shift in the time domain view) and for the second signal the mean is at 0.75. The average power of first signal varies widely (histogram is widely spread out) compared to that of the second signal.

Figure 2: Histogram is a visual method that provides a general idea about the shape of a distribution.

Higher order moments – skewness and kurtosis:

Further characterization of the shape of a distribution includes higher order moments : skewness and kurtosis. They identify anomalies and outliers in many signal processing applications.

Skewness provides a measure to quantify the presence of asymmetry in the shape of the distribution. Actually, skewness measures the relative size of the tails in a distribution. Presence of asymmetry manifests as non-zero value for the standardized third central moment. The term “standardized” stands for the normalization of the third central moment by σ3.

\[Skewness = \frac{E \left[ \left( X – \mu \right)^3 \right]}{\sigma^3}\]
Figure 3: A random sequence showing positive skewness (asymmetry in shape) in its distribution

Kurtosis measures the amount of probability in the two tails of a distribution, relative to a normal distribution of same variance. Kurtosis is 3 for normal distribution (perfect bell shaped curve). If the kurtosis of a distribution is greater than 3, it implies that the tails are heavier compared to that of normal distribution. A value less than 3 for kurtosis implies lighter tails compared to that of the normal distribution. Essentially, kurtosis is a measure of outliers. Kurtosis is calculated as the standardized fourth central moment.

\[Kurtosis = \frac{E \left[ \left( X – \mu \right)^4 \right]}{\sigma^4}\]
Figure 4: Histogram showing kurtosis of a random sequence vs. kurtosis of normal distribution

Measures of association

Statistics, such as measures of central tendency, dispersion and higher order moments, describe about a single distribution are called univariate statistics. If we are interested in the relationship between two or more variables, we have to move to at least the realm of bivariate statistics.

Measures of association, summarize the size of association between two variables. Most measures of associates are scaled to a range of values. For example, a measure of association can be construed to have a range 0 to 1, where the value 0 implies no relationship and a value of 1 implies perfect relationship between the two variables under study. In another case, the measure can range from -1 to 1, which can help us determine if the two variables have negative or positive relationship between each other.

Scatter plot

Scatter plot is a great way to visually assess the nature of relationship between two variables. Following figure contains the scatter plots between different stochastic signals, exhibiting different strengths of relationships between the signals.

Figure 5: Scatter plot of stochastic signals exhibiting different strengths of relationship

Correlation

Correlation functions are commonly used in signal processing, for measuring the similarity of two signals. They are especially used in signal detection and pattern recognition.

Cross-correlation

Cross-correlation is commonly used for measuring the similarity between two different signals. In signal processing, cross-correlation measures the similarity of two waveforms as a function of time lags applied to one of the waveform.

For discrete-time waveforms – x[n] and y[n], cross correlation is defined as

\[Corr_{xy}[l] = \sum_{m=-\infty}^{\infty} x[n]^{\ast} y[n+l] = \sum_{m=-\infty}^{\infty} x[n-l]^{\ast} y[n]\]

where, * denotes complex conjugate operation and l is the discrete time lags applied to one of the waveforms. That is, the cross-correlation can be calculated as the dot product of a sequence with each shifted version of another sequence.

Auto-correlation

Auto-correlation is the cross-correlation of a waveform/sequence with itself.

For discrete-time waveform – x[n], auto-correlation is defined as

\[Corr_{xx}[l] = \sum_{m=-\infty}^{\infty} x[n]^{\ast} x[n+l] = \sum_{m=-\infty}^{\infty} x[n-l]^{\ast} x[n] \]

where, * denotes complex conjugate operation and l is the discrete time lags applied to the copy of the same waveform.

Correlation properties are useful for identifying/distinguishing a known bit sequence from a set of other possible known sequences. For example, in GPS technology, each satellite is assigned a unique 10-bit Gold code sequence (210 = 1023 possible combinations). Cross-correlation between different Gold code sequence is very low, since the Gold codes are orthogonal to each other. However, the auto-correlation of a Gold code is maximum at zero lag. The satellites are identified using these correlation properties of Gold codes. (I have described the hardware implementation of Gold codes, it can be accessed here).

Following plots illustrate the application of auto-correlation for audio analysis. The auto-correlation of an audio signal will have a peak at zero lag (i.e, where there is no time shifting when computing the correlation) as shown in Figure 6. Figure 7 contains the same audio file that is synthetically processed to produce reverberation characteristics. By looking at the time series plot in Figure 7, we cannot infer anything. However, the auto-correlation plot reveals the reverberation characteristics embedded in the audio.

Figure 6: Normalized auto-correlation of an original sound
Figure 7: Normalized auto-correlation of sound with simulated reverberation characteristics (synthetically processed through an IIR comb filter)

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For further reading

[1] Steven M. Kay, “Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory”, ISBN: 978-0133457117, Prentice Hall, Edition 1, 1993.↗

Matplotlib histogram and estimated PDF in Python

Key focus: Shown with examples: let’s estimate and plot the probability density function of a random variable using Python’s Matplotlib 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.

Step 1: Generate random samples

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 : μ – mean and σ – standard deviation. First generate a vector of randomly distributed random numbers of sufficient length (say 100000) with some valid values for μ and σ. There are more than one way to generate this. Two of them are given below.

● Method 1: Using the in-built numpy.random.normal() function (requires numpy package to be installed)

import numpy as np

mu=10;sigma=2.5 #mean=10,deviation=2.5
L=100000 #length of the random vector

#Random samples generated using numpy.random.normal()
samples_normal = np.random.normal(loc=mu,scale=sigma,size=(L,1)) #generate normally distributted samples

● Method 2: Box-Muller transformation [2] method produces a pair of normally distributed random numbers (Z1, Z2) by transforming a pair of uniformly distributed independent random samples (U1,U2). The algorithm for transformation is given by

#Samples generated using Box-Muller transformation

U1 = np.random.uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)
U2 = np.random.uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)

a = np.sqrt(-2*np.log(U1))
b = 2*np.pi*U2

Z = a*np.cos(b) #Standard Normal distributed numbers
samples_box_muller= 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. Matplotlib’s hist function can be used to compute and plot histograms. If the density argument is set to ‘True’, the hist function computes the normalized histogram such that the area under the histogram will sum to 1. Estimate and plot the normalized histogram using the hist function.

#For plotting
import matplotlib.pyplot as plt
%matplotlib inline
plt.style.use('ggplot')

fig, ax0 = plt.subplots(ncols=1, nrows=1) #creating plot axes
(values, bins, _) = ax0.hist(samples_normal,bins=100,density=True,label="Histogram of samples") #Compute and plot histogram, return the computed values and bins

Step 3: Theoretical PDF:

And for verification, overlay the theoretical PDF for the intended distribution. The theoretical PDF of normally distributed random samples is given by

Theoretical PDF for normal distribution is readily obtained from stats.norm.pdf() function in the SciPy package.

from scipy import stats
bin_centers = 0.5*(bins[1:] + bins[:-1])
pdf = stats.norm.pdf(x = bin_centers, loc=mu, scale=sigma) #Compute probability density function
ax0.plot(bin_centers, pdf, label="PDF",color='black') #Plot PDF
ax0.legend()#Legend entries
ax0.set_title('PDF of samples from numpy.random.normal()');
Figure 1: Estimated PDF (histogram) and the theoretical PDF for samples generated using numpy.random.normal() function

The histogram and theoretical PDF of random samples generated using Box-Muller transformation, can be plotted in a similar manner.

#Samples generated using Box-Muller transformation
from numpy.random import uniform
U1 = uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)
U2 = uniform(low=0,high=1,size=(L,1)) #uniformly distributed random numbers U(0,1)
a = np.sqrt(-2*np.log(U1))
b = 2*np.pi*U2
Z = a*np.cos(b) #Standard Normal distribution
samples_box_muller = Z*sigma+mu #Normal distribution with mean and sigma

#Plotting
fig, ax1 = plt.subplots(ncols=1, nrows=1) #creating plot axes
(values, bins, _) = ax1.hist(samples_box_muller,bins=100,density=True,label="Histogram of samples") #Plot histogram
bin_centers = 0.5*(bins[1:] + bins[:-1])
pdf = stats.norm.pdf(x = bin_centers, loc=mu, scale=sigma) #Compute probability density function
ax1.plot(bin_centers, pdf, label="PDF",color='black') #Plot PDF
ax1.legend()#Legend entries
ax1.set_title('Box-Muller transformation');
Figure 2: Estimated PDF (histogram) and theoretical PDF for samples generated using Box-Muller transformation for normal distribution

References:

[1] John Mount, ‘Six Fundamental Methods to Generate a Random Variable’, January 20, 2012
[2] Thomas, D. B., Luk. W., Leong, P. H. W., and Villasenor, J. D. 2007. Gaussian random number generators. ACM Comput. Surv. 39, 4, Article 11 (October 2007), 38 pages DOI = 10.1145/1287620.1287622 http://doi.acm.org/10.1145/1287620.1287622

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Articles in this series
[1] Fibonacci series in python
[2] Central Limit Theorem – a demonstration
[3] Moving Average Filter in Python and Matlab
[4] How to plot FFT in Python – FFT of basic signals : Sine and Cosine waves
[5] How to plot audio files as time-series using Scipy python
[6] How to design a simple FIR filter to reject unwanted frequencies
[7] Analytic signal, Hilbert Transform and FFT
[8] Non-central Chi-squared Distribution
[9] Simulation of M-PSK modulation techniques in AWGN channel (in Matlab and Python)
[10] QPSK modulation and Demodulation (with Matlab and Python implementation)

Hidden Markov Models (HMM) – Simplified !!!

Markov chains are useful in computing the probability of events that are observable. However, in many real world applications, the events that we are interested in are usually hidden, that is we don’t observe them directly. These hidden events need to be inferred. For example, given a sentence in a natural language we only observe the words and characters directly. The parts-of-speech from the sentence are hidden, they have to be inferred. This bring us to the following topic– the hidden Markov models.

Hidden Markov models enables us to visualize both observations and the associated hidden events. Let’s consider an example for understanding the concept.

The cheating casino and the gullible gambler

Consider a dishonest casino that deceives it player by using two types of die : a fair die (F) and a loaded die (L). For a fair die, each of the faces has the same probability of landing facing up. For the loaded die, the probabilities of the faces are skewed as given next

When the gambler throws the die, numbers land facing up. These are our observations at a given time t (denoted as Ot = {1,2,3,4,5,6}). At any given time t, whether these number are rolled from a fair die (state St = F) or a loaded die (St = L), is unknown to an observer and therefore they are the hidden events.

Emission probabilities

The probabilities associated with the observations are the observation likelihoods, also called emission probabilities (B).

Initial probabilities

The initial probability of starting (at time = 0) with any of fair die or loaded die (hidden event) is 50%.

Transition probabilities

A gullible gambler switches from the fair die to loaded die with 10% probability. He switches back from loaded die to fair die with 5% probability.

The probabilities of transitioning from one hidden event to another is described by the transition probability matrix (A). The elements of the probability transition matrix, are the transition probabilities (pij) of moving from one hidden state i to another hidden state j.

The transition probabilities from time t-1 to t, for the hidden events are

Therefore, the transition probability matrix is

Based on the given information so far, a probability model is constructed. This is the Hidden Markov Model (HMM) for the given problem.

Figure 1: Hidden Markov Model for the cheating Casino problem

Assumptions

We saw, in previous article, that the Markov models come with assumptions. Similarly, HMMs models also have such assumptions.

1. Assumption on probability of hidden states

In the model given here, the probability of a given hidden state depends only on the previous hidden state. This is a typical first order Markov chain assumption.

2. Assumption on Output

The probability of any observation (output) depends on the hidden state that produce it and not on any other hidden state or output observations.

Problems and Algorithms

Let’s briefly discuss the different problems and the related algorithms for HMMs. The algorithms will be explained in detail in the future articles.

In the dishonest casino, the gambler rolls the following numbers:

Figure 2: Sample Observations

1. Evaluation

Given the model of the dishonest casino, what is the probability of obtaining the above sequence ? This is a typical evaluation problem in HMMs. Forward algorithm is applied for such evaluation problems.

2. Decoding

What is the most likely sequence of die (hidden states) given the above sequence ? Such problems are addressed by Viterbi decoding.

What is the probability of fourth die being loaded, given the above sequence ? Forward-backward algorithm to our rescue.

3. Learning

Learning problems involve parametrization of the model. In learning problems, we attempt to find the various parameters (transition probabilities, emission probabilities) of the HMM, given the observation. Baum-Welch algorithm helps us to find the unknown parameters of a HMM.

Some real-life examples

Here are some real-life examples of HMM applications:

  1. Speech recognition: HMMs are widely used in speech recognition systems to model the variability of speech sounds. In this application, the observable events are the acoustic features of the speech signal, while the hidden states represent the phonemes or words that generate the speech signal.
  2. Handwriting recognition: HMMs can be used to recognize handwritten characters by modeling the temporal variability of the pen strokes. In this application, the observable events are the coordinates of the pen on the writing surface, while the hidden states represent the letters or symbols that generate the handwriting.
  3. Stock price prediction: HMMs can be used to model the behavior of stock prices and predict future price movements. In this application, the observable events are the daily price movements, while the hidden states represent the different market conditions that generate the price movements.
  4. Gene prediction: HMMs can be used to identify genes in DNA sequences. In this application, the observable events are the nucleotides in the DNA sequence, while the hidden states represent the different regions of the genome that generate the sequence.
  5. Natural language processing: HMMs are used in many natural language processing tasks, such as part-of-speech tagging and named entity recognition. In these applications, the observable events are the words in the text, while the hidden states represent the grammatical structures or semantic categories that generate the text.
  6. Image and video analysis: HMMs can be used to analyze images and videos, such as for object recognition and tracking. In this application, the observable events are the pixel values in the image or video, while the hidden states represent the object or motion that generates the pixel values.
  7. Bio-signal analysis: HMMs can be used to analyze physiological signals, such as electroencephalograms (EEGs) and electrocardiograms (ECGs). In this application, the observable events are the signal measurements, while the hidden states represent the physiological states that generate the signal.
  8. Radar signal processing: HMMs can be used to process radar signals and detect targets in noisy environments. In this application, the observable events are the radar measurements, while the hidden states represent the presence or absence of targets.

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Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

 

Markov Chains – Simplified !!

Key focus: Markov chains are a probabilistic models that describe a sequence of observations whose occurrence are statistically dependent only on the previous ones.

● Time-series data like speech, stock price movements.
● Words in a sentence.
● Base pairs on the rung of a DNA ladder.

States and transitions

Assume that we want to model the behavior of a driver behind the wheel. The possible behaviors are

● accelerate (state 1)
● constant speed (state 2)
● idling (engine running slowly but the vehicle is not moving – (state 3))
● brake (state 4)

Let’s refer each of these behaviors as a state. In the given example, there are N=4 states, refer them as Q = {q1,q2,q3,q4}.

We observe the following pattern in the driver’s behavior (Figure 1). That is, the driver operates the vehicle through a certain sequence of states. In the graph shown in Figure 1, the states are represented as nodes and the transitions as edges.

Figure 1: Driver’s behavior – operating the vehicle through a sequence of states

We see that, sometimes, the driver changes the state of the vehicle from one state to another and sometimes stays in the same state (as indicated by the arrows).

We also note that either the vehicle stays in the same state or changes to the next state. Therefore, from this model, if we want to predict the future state, all that matters is the current state of the vehicle. The past states has no bearing on the future state except through the current state. Take note of this important assumption for now.

Probabilistic model

We know that we cannot be certain about the driver’s behavior at any given point in time. Therefore, to model this uncertainty, the model is turned into a probabilistic model. A probabilistic model allows us to account for the likelihood of the behaviors or change of states.

An example for a probabilistic model for the given problem is given in Figure 2.

Figure 2: Driver’s behavior – a probabilistic model (transition matrix shown)

In this probabilistic model, we have assigned probability values to the transitions.These probabilities are collectively called transition probabilities. For example, considering the state named “idling”, the probability of the car to transition from this state to the next state (accelerate) is 0.8. In probability mathematics this is expressed as a conditional probability conditioned on the previous state.

p(state = “accelerate” | previous state = “idling” ) = 0.8

Usually, the transition probabilities are formulated in the form of matrix called transition probability matrix. The transition probability matrix is shown in Figure 2. In a transition matrix, denoted as A, each element aij represent the probability of transitioning from state i to state j. The elements of the transition matrix satisfy the following property.

That is, the sum of transition probabilities leaving any given state is 1.

As we know, in this example, the driver cannot start car in any state (example, it is impossible to start the car in “constant speed” state). He can only start the car from at rest (i.e, brake state). To model this uncertainty, we introduce πi – the probability that the Markov chain starts in a given state i. The set of starting probabilities for all the N states are called initial probability distribution (π = π1, π2, …, πN). In Figure 3, the starting probabilities are denoted by green arrows.

Figure 3: Markov Chain model for driver’s behavior

Markov Assumption

As noted in the definition, the Markov chain in this example, assumes that the occurrence of each event/observation is statistically dependent only on the previous one. This is a first order Markov chain (or termed as bigram language model in natural language processing application). For the states Q = {q1, …, qn}, predicting the probability of a future state depends only on the current observation, all other previous observations do not matter. In probabilistic terms, this first order Markov chain assumption is denoted as

Extending the assumption for mth order Markov chain, predicting the probability of a future observation depends only on the previous m observations. This is an m-gram model.

Given a set of n arbitrary random variables/observations Q = {q1, …, qn}, their joint probability distribution is usually computed by applying the following chain rule.

However, if the random observations in Q are of sequential in nature and follows the generic mth order Markov chain model, then the computation of joint probability gets simplified.

The Markov assumptions for first and second order of Markov models are summarized in Figure 4.Figure 4: Assumptions for 1st order and 2nd order Markov chains

Hidden Markov Model (HMM)

Markov chains are useful in computing the probability of events that are observable. However, in many real world applications, the events that we are interested in are usually hidden, that is we don’t observe them directly. These hidden events need to be inferred. For example, given a sentence in a natural language we only observe the words and characters directly. The parts-of-speech from the sentence are hidden, they have to be inferred. This brings us to the next topic of discussion – the hidden Markov models.

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Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

Generate color noise using Auto-Regressive (AR) model

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.

Refer the book for the Matlab code

Figure 1: Simulated color noise samples and their PSD estimates: pink noise (α =1) and Brown noise (α =2)

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References

[1] Gene H. Golub, Charles F. Van Loan, Matrix Computations, ISBN-9780801854149, Johns Hopkins University Press, 1996, p. 143.↗
[2] J. Durbin, The fitting of time series in models, Review of the International Statistical Institute, 28:233-243, 1960.↗
[3] Kasdin, N.J. Discrete Simulation of Colored Noise and Stochastic Processes and Power Law Noise Generation, Proceedings of the IEEE, Vol. 83, No. 5, 1995, pp. 802-827.↗

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Generating colored noise with Jakes PSD: Spectral factorization

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.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

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 filter using 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.

Figure 3: Simulating colored noise with Jakes PSD

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Reference

[1] Jeruchim et., al, Simulation of communication systems – modeling, methodology, and techniques, second edition, Kluwer academic publishers, 2002, ISBN: 0306462672.↗

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Generate correlated Gaussian sequence (colored noise)

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.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.
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

• Spectral factorization method
• Auto-regressive (AR) model

Motivation

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.

Linear time invariant (LTI) system model

In the given model, the random signal is observed. Given the observed signal , the goal here is to find a model that best describes the spectral properties of under the following assumptions
• The sequence is WSS (wide sense stationary) and ergodic.
• The input sequence to the LTI system is white noise, whose amplitudes follow Gaussian distribution with zero-mean and variance with flat the power spectral density.
• The LTI system is BIBO (bounded input bounded output) stable.

Read the continuation of this post : Spectral factorization method

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Reference

[1] Jeruchim et., al, Simulation of communication systems – modeling, methodology, and techniques, second edition, Kluwer academic publishers, 2002, ISBN: 0306462672.↗

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Exponential random variable – simulation & application

Introduction

An exponential random variable (RV) is a continuous random variable that has applications in modeling a Poisson process. Poisson processes find extensive applications in tele-traffic modeling and queuing theory. They are used to model random points in time or space, such as the times when call requests arriving at an exchange, the times when a shot noise occurs in the photon counting processing of an optical device, the times when file requests arrive at a serve etc.

Figure 1: Estimated PDF from an exponential random variable

Univariate random variables

This section focuses on some of the most frequently encountered univariate random variables in communication systems design. Basic installation of Matlab provides access to two fundamental random number generators: uniform random number generator (rand) and the standard normal random number generator (randn). They are fundamental in the sense that all other random variables like Bernoulli, Binomial, Chi, Chi-square, Rayleigh, Ricean, Nakagami-m, exponential etc.., can be generated by transforming them.

This article is part of the book
● Wireless Communication Systems in Matlab (second edition), ISBN: 979-8648350779 available in ebook (PDF) format and Paperback (hardcopy) format.

Exponential RV

An exponential random variable takes value in the interval and has the following continuous distribution function (CDF).

The rate parameter specifies the mean number of occurrences per unit time and is the number of time units until the occurrence of next event that happens in the modeled process. The probability density function of the exponential rv is given by

By applying the inverse transform method [1], an uniform random variable can be transformed into an exponential random variable. This method is coded in the Matlab function that is shown at the end of this article. Using the function, a sequence of exponentially distributed random numbers can be generated, whose estimated pdf is plotted against the theoretical pdf as shown in the Figure 1.

Application to Poisson process

Poisson process is a continuous-time discrete state process that is widely used to model independent events occurring in time or space. It is widely applied to model a counting process in which the events occur at independently random times but appear to happen at certain rate. In practice, Poisson process has been used to model counting processes like

  • photons landing on a photo-diode
  • arrivals of phone calls at a telephone exchange
  • request for file downloads at a web server
  • location of users in a wireless network

Poisson process is closely related to a number of vital random variables (RV) including the uniform RV, binomial RV, the exponential RV and the Poisson RV. For example, the inter-arrival times (duration between the subsequent arrivals of events) in a Poisson process are independent exponential random variables.

For example, the inter-arrival times (duration between the subsequent arrivals of events) in a Poisson process are independent exponential random variables

Refer the book Wireless Communication Systems in Matlab for full Matlab code

function T = expRV(lambda,L)
%Generate random number sequence that is exponentially distributed
%lambda - rate parameter, L - length of the sequence generated
U = rand(1,L); %continuous uniform random numbers in (0,1)
T = -1/lambda*(log(1-U));
end

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References

● L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986.↗

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Binomial random variable using Matlab

Binomial random variable, a discrete random variable, models the number of successes in mutually independent Bernoulli trials, each with success probability . The term Bernoulli trial implies that each trial is a random experiment with exactly two possible outcomes: success and failure. It can be used to model the total number of bit errors in the received data sequence of length that was transmitted over a binary symmetric channel of bit-error probability .

Generating binomial random sequence in Matlab

Let X denotes the total number of successes in mutually independent Bernoulli trials. For ease of understanding, let’s denote success as ‘1’ and failure as ‘0’. Suppose if a particular outcome of the experiment contains ones and zeros (example outcome: 1011101), the probability mass function↗ of is given by

A binomial random variable can be simulated by generating independent Bernoulli trials and summing up the results.

function X = binomialRV(n,p,L)
%Generate Binomial random number sequence
%n - the number of independent Bernoulli trials
%p - probability of success yielded by each trial
%L - length of sequence to generate
X = zeros(1,L);
for i=1:L,
   X(i) = sum(bernoulliRV(n,p));
end
end

Following program demonstrates how to generate a sequence of binomially distributed random numbers, plot the estimated and theoretical probability mass functions for the chosen parameters (Figure 1).

n=30; p=1/6; %number of trails and success probability
X = binomialRV(n,p,10000);%generate 10000 bino rand numbers
X_pdf = pdf('Binomial',0:n,n,p); %theoretical probility density
histogram(X,'Normalization','pdf'); %plot histogram
hold on; plot(0:n,X_pdf,'r'); %plot computed theoreical PDF
Figure 1: PMF generated from binomial random variable for three different cases of n and p

PMF sums to unity

Let’s verify theoretically, the fact that the PMF of the binomial distribution sums to unity. Using the result of Binomial theorem↗,

Mean and variance

The mean number of success in a binomial distribution is

The variance is

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
 □ Uniform random variable
 □ Bernoulli random variable
 □ Binomial random variable
 □ Exponential random variable
 □ Poisson process
 □ Gaussian random variable
 □ Chi-squared random variable
 □ Non-central Chi-Squared random variable
 □ Chi distributed random variable
 □ Rayleigh random variable
 □ Ricean random variable
 □ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
 □ Generating two sequences of correlated random variables
 □ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
 □ Spectral factorization method
 □ Auto-Regressive (AR) model

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
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Digital Modulations using Python
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