GaussianWaves

Tips & Tricks - Indexing in Matlab

Consider a sample vector in Matlab. >>x=[9 21 6 8 7 3 18 -1 0 4] ans= 9 21 6 8 7 3 18 -1 0 4 Index with single value >>x(2) ans = 21 Index with range of values >> x([1 4 6]) ans = 9 8 3 Select a range of elements using ':' operator. Example: Select elements with index ranging from 1 to 5. >> x(1:5) ans = 9 21 6 8 7 Making a new vector by swapping two halves of the vector x >> x([6:10 1:5]) ans = 3 18 -1 0 4 9 21 6 8 7 To refer the last element of the matrix x use 'end' operator >> x(end) ans = 4 You can do arithmetic on the 'end' operator. Selecting all elements except the last element >> x(1:end-1) ans = 9 21 6 8 7 3 18 -1 0 Reversing the vector >> x(end:-1:1) ans = 4 0 -1 18 3 7 8 6 21 9 Using the end operator in a range. Selecting from fifth element to the end >> x(5:end) ans = 7 3 18 -1 0 4 Replacing specific elements in the array by placing the new values on the right side of the expression. Replacing the values of 'x' at positions [2,5,8] with new values >> x([2 5 8])=[11 14 -6] x = 9 11 6 8 14 3 18 -6 0 4 Replacing specific elements with a same value. Replacing the values of 'x' at position [1 and 10] with 40 >> x([1 10]) = 40 x = 40 11 6 8 14 3 18 -6 0 40 Two dimensional Arrays: >> x=magic(4) %Create a magic array with 4x4 elements x = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 The two dimensional array elements are accessed with two subscript indices like x(i,j) - where read more

Cholesky Factorization and Matlab code

Any $latex n \times n$ positive definite matrix $latex A $ can be factored as $latex A=LL^T &s=2$ where $latex L $ is nxn lower triangular matrix. The lower triangular matrix $latex L$ is often called “Cholesky Factor of $latex A$”. The matrix $latex L$ can be interpreted as square root of the positive definite matrix A. Basic Algorithm to find Cholesky Factorization: Note: In the following text, the variables represented in Greek letters represent scalar values, the variables represented in small Latin letters are column vectors and the variables represented in capital Latin letters are Matrices. Given a positive definite matrix $latex A$, it is partitioned as follows. $latex \alpha_{11}, \lambda_{11} $ = first element of $latex A$ and $latex L$ respectively $latex a_{21} , l_{21}$ = column vector at first column starting from second row of $latex A$ and $latex L$ respectively $latex A_{22} , L_{22}$ = remaining lower part of the matrix of $latex A$ and $latex L$ respectively of size $latex (n-1) \times (n-1)$ Having partitioned the matrix $latex A$ as shown above, the Cholesky factorization can be computed by the following iterative procedure. Steps in computing the Cholesky factorization: Step 1: Compute the scalar: $latex \lambda_{11}=\sqrt{\alpha_{11}}$ Step 2: Compute the column vector: $latex l_{21}= a_{21}/\lambda_{11}$ Step 3: Compute the matrix : $latex A_{22}=l_{21} l^T_{21}+L_{22}L^T_{22}$ Step 4: Replace $latex A$ with $latex A_{22}$, i.e, $latex A=A_{22}$ Step 5: Repeat from step 1 till the matrix size at Step 4 becomes 1x1. Matlab Program (implementing the above algorithm): Function 1: [F]=cholesky(A,option) function [F]=cholesky(A,option) %Function to find the Cholesky factor of a Positive Definite matrix A %Author Mathuranathan for http://www.gaussianwaves.com %Licensed under Creative Commons: CC-NC-BY-SA 3.0 %A = positive definite matrix %Option can be one of the following 'Lower','Upper' %L = Cholesky factorizaton of A such that A=LL^T %If option ='Lower', then it returns the Cholesky factored matrix L in %lower triangular form %If option ='Upper', then it returns the Cholesky factored matrix L in %upper triangular form %Test for positive definiteness if isPositiveDefinite(A), [m,n]=size(A); L=zeros(m,m);%Initialize to all zeros row=1; col=1; j=1; for i=1:m, a11=sqrt(A(1,1)); L(row,col)=a11; read more

Check Positive Definite Matrix in Matlab

It is often required to check if a given matrix is positive definite or not. Three methods to check the positive definiteness of a matrix were discussed in a previous article . I will utilize the test method 2 to implement a small matlab code to check if a matrix is positive definite.The test method 2 relies on the fact that for a positive definite matrix, the determinants of all upper-left sub-matrices are positive.The following Matlab code uses an inbuilt Matlab function -'det' - which gives the determinant of an input matrix. Furthermore, the successive upper kxk sub-matrices are got by using the following notation subA=A(1:i,1:i) I will explain how this notation works to give the required sub-matrices. Consider a sample matrix given below $latex \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}$ The sub-matrices for the various combinations for row and column values for the above mentioned code snippet is given below NotationSubmatrixComment subA=A(1:1,1:1)$latex \begin{bmatrix} 1 \end{bmatrix}$Select elements from 1st row-1st column to 1st row-1st column subA=A(1:2,1:2)$latex \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$Select elements from 1st row-1st column to 2nd row-2nd column subA=A(1:3,1:3)$latex \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$Select elements from 1st row-1st column to 3rd row-3rd column subA=A(1:3,1:2)$latex \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 7 & 8 \end{bmatrix}$Select elements from 1st row-1st column to 3rd row-2nd column Matlab Code to test if a matrix is positive definite: function x=isPositiveDefinite(A) %Function to check whether a given matrix A is positive definite %Author Mathuranathan for http://www.gaussianwaves.com %Returns x=1, if the input matrix is positive definite %Returns x=0, if the input matrix is not positive definite %Throws error if the input matrix is not symmetric %Check if the matrix is symmetric [m,n]=size(A); if m~=n, error('A is not Symmetric'); end %Test for positive definiteness x=1; %Flag to check for positiveness for i=1:m subA=A(1:i,1:i); %Extract upper left kxk submatrix if(det(subA)> A=[1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 read more

Solving a Triangular Matrix using Forward & Backward Substitution

Forward Substitution: Consider a set of equations in a matrix form $latex Ax=b $, where A is a lower triangular matrix with non-zero diagonal elements. The equation is re-written in full matrix form as It can be solved using the following algorithm From the DSP implementation point of view, computation of $latex x_1 $ requires one FLoating Point Operation per Second (FLOPS) – only one division. Computing $latex x_2 $ will require 3 FLOPS – 1 multiplication, 1 division and 1 subtraction, $latex x_3 $ will require 5 FLOPS – 2 multiplications, 1 division and two subtractions. Thus the computation of $latex x_{mm} $ will require $latex (2n-1) $ FLOPS. Thus the overall FLOPS required for forward substitution is $latex 1+3+5+\cdots+(2m-1)$ $latex = m^2 $ FLOPS Backward substitution: Consider a set of equations in a matrix form $latex Ax=b $, where A is a upper triangular matrix with non-zero diagonal elements. The equation is re-written in full matrix form as Solved using the following algorithm This one also requires $latex m^2 $ FLOPS. More on Estimation Theory: Recommended Books on Estimation read more

A tutorial on Fourier Analysis - Fourier Series

Fourier analysis and Fourier Synthesis: Fourier analysis – a term named after the French mathematician Joseph Fourier, is the process of breaking down a complex function and expressing it as a combination of simpler functions. The opposite process of combining simpler functions to reconstruct the complex function is termed as Fourier Synthesis. Mostly, the simpler functions are chosen to be sine and cosine functions. Thus, the term “Fourier analysis” expresses a complex function in terms of sine and cosine terms and the term “Fourier Analysis” reconstructs the complex function from the sine and cosine terms. Frequency is the measure of number of repetitive occurrences of a particular event. By definition, a sine wave is a smooth curve that repeats at a certain frequency. Thus, the term “frequency” and sine are almost synonymous. A cosine wave is also a sine wave but with 90* phase shift. Therefore, when you talk about sine and cosine functions, you are taking in terms of “frequencies”. That is why in signal processing, the Fourier analysis is applied in frequency (or spectrum) analysis. Fourier series, Continuous Fourier Transform, Discrete Fourier Transform, and Discrete Time Fourier Transform are some of the variants of Fourier analysis. Fourier series: Applied on functions that are periodic. A periodic function is broken down and expressed in terms of sine and cosine terms. In mathematics, the term “series” represents a sum of sequence of numbers. For example we can make a series with a sequence of numbers that follows Geometric Progression (common ratio between the numbers) Common ratio =3 : 1+3+9+27+… An infinite series is a series that has infinite number of terms. If the elements of the infinite series has a common ratio less than 1, then there is a possibility of the sum converging at a particular value. Fourier series falls under the category of trigonometric infinite series, where the individual elements of the series are expressed trigonometrically. The construct of the Fourier series is given by Here f(x) is the complex periodic function we wish to break down in terms of sine and cosine basis functions. The coefficients a0,a1,… and b1,b2,… can be found by Functions and Symmetry: It is necessary to classify the functions according to its symmetry properties. Doing so will save computation time and effort. Functions either fall into “odd symmetry” or “even symmetry” or “no symmetry”. Symmetry can be ascertained by plotting the function in a graph paper and folding it along the y axis. Symmetry read more

Data Acquisition through Wireless Sensor Networks – Part 2

Authors: Akhilesh Agrawal & Rishav Rej - Birla Institute of Technology and Science (BITS), Pilani Hyderabad Campus Continuation of Part 1: Data Acquisition through Wireless Sensor Networks - Part 1 III. Background Literature survey about the Temperature Sensor This section gives a brief information about the temperature sensor LM35, its connection diagram and its features. A. General Description The LM35 series IC’s are precision integrated circuit temperature sensors whose output voltage is linearly proportional to the Celsius temperature. The LM35 thus has an advantage over the linear temperature sensors calibrated in Kelvin as the as the user is not required to subtract a large constant voltage from its output to obtain convent centigrade scaling. The LM35 does not require any external calibration to provide typical accuracy of + or – ¼qC at room temperature and + or – ¾qC over a full –55 to +150qC temperature range. The LM35’s low output impedance linear output and Precise inherent calibration make interfacing to readout or control circulatory especially easy. It can be used with single power supplies, or with plus and minus supply; it has very low self-heating less than 0.1 degrees Celsius in still air. The LM35 is rated to operate over a –55q to +150qC temperature range, while the LM35 is rated for a –40q to +110qC range. The LM35 series is available packaged in hermetic TO-46 transistor packages. The LM35D is also available in an 8 lead surface mount small outline package and a plastic TO-220 package. B. Features * Calibrated directly in Celsius (centigrade). Linear + 10.0 m V / qC scale factor. * 0.5q C accuracy guarantee-able (at +25q C). Rated for full -55q to +150q C range. Operates from 4 to 30 volts. * Low self-heating, 0.08qC in still air. * Low impedance output, 0.1: for 1 mA load. C. Connection Diagram The connection diagram for LM35 packages is shown in Fig. 3. Here we are using TO-220 plastic package temperature sensor. It has three leads namely +Vs, ground and Vout Fig 3. Connection diagram for LM35 packages IV. Universal Synchronous Asynchronous Receiver and Transmitter This section gives a brief description of the USART and the registers used for its operation at the transmitter and receiver section. The universal synchronous asynchronous receiver and transmitter (USART) module is one of the two serial input or output modules. USART is also known as a serial communications interface or SCI. The USART can be configured as a full duplex asynchronous system that can read more

Data Acquisition through Wireless Sensor Networks - Part 1

Authors: Akhilesh Agrawal & Rishav Rej - Birla Institute of Technology and Science (BITS), Pilani Hyderabad Campus Abstract: This report deals with the design, development and implementation of a temperature Sensor using ZigBee module interfaced with the PIC microcontroller. The main aim of the work undertaken in this report is to sense the temperature and to display the result on the CENTRAL SERVER using the ZigBee technology. ZigBee operates in the industrial, scientific and medical (ISM) radio bands; 868 MHz in Europe,915MHz in the USA and 2.4 GHz in most jurisdictions worldwide. The technology is intended to be simpler and cheaper than other WPANs such as Bluetooth. The most capable ZigBee node type is said to require only about 10 % of the software of a typical Bluetooth or Wireless Internet node, while the simplest nodes are about 2%. However, actual code sizes are much higher, more like 50 % of the Bluetooth code size. In this work undertaken in the design & development of the temperature sensor, it senses the temperature and after amplification is then fed to the micro controller, this is then connected to the ZigBee module, which transmits the data and at the other end the ZigBee reads the data and displays on to the CENTRAL SERVER. The software developed is highly accurate and works at a very high speed. The method developed shows the effectiveness of the scheme employed. Keywords—Zigbee, Microcontroller, PIC, Transmitter, Receiver, Synchronous, Blue tooth, Communication. I. Introduction This section gives a brief introduction about the work, which describes all the components namely Zigbee, temperature sensor, PIC18F4550. Zigbee is a wireless network protocol specifically designed for low data rate sensors and control networks. Also, a brief literature survey of the work related to the research topic is also presented in the following paragraphs Zigbee is a consortium of software, hardware and services companies that have developed a common standard for wireless, networking of sensors and controllers. While other wireless standards are concerned with exchanging large amounts of data, Zigbee is for devices that have smaller throughout needs. The other driving factors are low cost, high reliability, high security, low battery usage, simplicity and inter-operability with other Zigbee devices. Compared to other wireless protocols, Zigbee wireless protocol offers low complexity. It also offers three frequency bands of operation along with a number of network configurations and optional security capability. It requires a supply voltage in the range of 2.8V to 3.3V. read more

Top books on basics of Communication Systems

Communication Systems - Simon Haykins The standard text book recommended by most of the engineering schools around the world. You may find it little tough to understand it at first. If you have a fair understanding of circuit theory, probability and signals & systems, understanding the concepts in this book will be a piece of cake for you. Now, the newest edition includes Matlab experiments to demonstrate communication theory concepts. It also covers real world applications like Digital Subscriber Lines (DSL). Most of the mathematical derivations are concise and it is your responsibility to understand the steps involved in those derivations - Of course !!! You cannot expect the authors to spoon feed the steps involved in the derivations. Recommended for beginners with foundation on circuit theory, probability and signals & systems Digital Communications: Fundamentals and Applications - Bernard Sklar One of the most referred books on digital communications. Illustrative examples and lucid explanations will nurture understanding of digital communications. Especially, the chapters on Reed Solomon coding ,Trellis modulation & Turbo codes are one of the best I have read. If you love satellite communications, then this book is a must for you. Sklar - being experienced in satellite communication, throws numerous examples in this area throughout the book. Key topics include: Basics of transmission and reception, performance trade-offs (SNR Vs BER Vs Bandwidth), fading, spread spectrum, synchronization, digital modulations and coding theory. Recommended for beginners.  Modern Digital and Analog Communication Systems - B.P Lathi & Zhi Ding Excellent book to begin with. Though, mathematically not rigorous, provides a clear understanding of the subject. Recommended for undergraduate students beginning to understand  the fundamentals of communication theory. Apart from basics, the newest edition covers concepts like OFDM, equalization, soft-decision-decoding, LDPC coding etc.It also includes MATLAB examples throughout the book. Recommended for Beginners Fundamentals of Communication Systems - John G. Proakis and Masoud Salehi Good for advanced learners. This book provides in-depth treatment of concepts and involves mathematics at higher level. I would place this book at the same page as "Digital communication" by John G. Proakis. You need to have a better understanding of communication theory basics to understand these books. Required for serious learners who want to explore further in the area of communication systems. Sometimes, you may find the rigorous mathematical treatments boring, but you will find them amusing once you develop real interest in the subject. Recommended for advanced learners Analog and Digital Communications (Schaum's Outlines) - Hwei P. Hsu Excellent well read more

Tests for Positive Definiteness of a Matrix

In order to perform Cholesky Decomposition of a matrix, the matrix has to be a positive definite matrix. I have listed down a few simple methods to test the positive definiteness of a matrix. Methods to test Positive Definiteness: Remember that the term positive definiteness is valid only for symmetric matrices. Test method 1: Existence of all Positive Pivots For a matrix to be positive definite, all the pivots of the matrix should be positive. Hmm.. What is a pivot ? Pivots: Pivots are the first non-zero element in each row of a matrix that is in Row-Echelon form. Row-Echelon form of a matrix is the final resultant matrix of Gaussian Elimination technique. In the following matrices, pivots are encircled. A positive definite matrix will have all positive pivots. Only the second matrix shown above is a positive definite matrix. Also, it is the only symmetric matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all $latex k \times k $ upper-left sub-matrices must be positive. Break the matrix in to several sub matrices, by progressively taking $latex k \times k $ upper-left elements. If the determinants of all the sub-matrices are positive, then the original matrix is positive definite. Is the following matrix Positive Definite? $latex \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \\ \end{bmatrix} &s=2$ Find the determinants of all possible $latex k \times k $ upper sub-matrices. $latex \begin{vmatrix} 2 \end{vmatrix} = 2 &s=2$ $latex \begin{vmatrix} 2 & -1 \\-1 & 2 \\\end{vmatrix} = 3 &s=2$ $latex \begin{vmatrix} 2 & -1 & 0 \\-1 & 2 & -1 \\0 & -1 & 2 \\ \end{vmatrix} = 4 &s=2$ $latex \mbox{All determinants are Positive -> Positive Definite} $ Test method 3: All Positive Eigen Values. If all the Eigen values of the symmetric matrix are positive, then it is a positive definite matrix. Is if following matrix Positive definite ? $latex \begin{bmatrix} 9 & 0 & -8 \\ 6 & -5 & -2 \\ -9 & 3 & 3 \end{bmatrix} &s=2$ $latex \mbox{Eigen Values are : -5.90, -1.57, 14.48} &s=2$ Since, not all the Eigen Values are positive, the above matrix is NOT a positive definite matrix. There exist several methods to determine positive definiteness of a matrix. The method listed here are simple and can be done manually for smaller matrices. Following online tools will be of help: 1) Online tool to generate Eigen Values and Eigen Vectors 2) Transforming a matrix to reduced-row-Echelon form See also: Books on Estimation read more

Why Cholesky Decomposition ? A sample case:

Matrix inversion is seen ubiquitously in signal processing applications. For example, matrix inversion is an important step in channel estimation and equalization. For instance, in GSM normal burst, 26 bits of training sequence are put in place with 114 bits of information bits. When the burst travels over the air interface (channel), it is subject to distortions due to channel effect like Inter Symbol Interference (ISI). It becomes necessary to estimate the channel impulse response (H) and equalize the effects of the channel, before attempting to decode/demodulate the information bits. The training bits are used to estimate the channel impulse response. If the transmitted signal “x” travels over a multipath fading channel (H) with AWGN noise “w”, the received signal is modeled as $latex y = Hx + w &s=2$ A Minimum Mean Square Error (MMSE) linear equalizer employed to cancel out the effects of ISI, attempts to minimize the error between equalizer output – "$latex \hat{x} $" and the transmitted signal "$latex x $". If the AWGN noise power is $latex \sigma^2 $, then the equalizer is represented by the following equation[1]. Note that the expression involves the computation of matrix inversion - . Matrix inversion is a tricky subject. Not all matrices are invertible. Furthermore, ordinary matrix inversion technique of finding the adjoint of a matrix and using it to invert the matrix will consume lots of memory and computation time. Physical layer algorithm (PHY) designers typically use Cholesky decomposition to invert the matrix. This helps to reduce the computational complexity of matrix inversion. Reference: [1] Marius Vollmer et al, ”Comparative Study of Joint Detection Technique for DS-CDMA Based Mobile Radio System”,IEEE Journal on selected areas in communications,Vol. 19, NO. 8, August See Also: read more

Essential Preliminary Matrix Algebra for Signal Processing

I thought of making a post on Cholesky Decomposition, which is a very essential technique in digital signal processing applications like generating correlated random variables, solving linear equations, channel estimation etc.., But jumping straight to the topic of Cholesky Decomposition will leave many flummoxed/confused. So I decided to touch on some essentials in basic matrix algebra before taking up advanced topics. Prerequisite: Basic knowledge on topics like formation of matrices, determinants, rank of a matrix, inverse & transpose of matrix, solving a system of simultaneous equation using matrix algebra. The prerequisites listed above being fulfilled, you will learn different types of matrices in this post. 1. Vector A matrix with only one row or one column. $latex \text{Row Vector with 5 elements: } A = \begin{bmatrix}a & b & c & d & e \end{bmatrix} &s=2$ $latex \text{Column Vector with 5 elements: } B = \begin{bmatrix}a \\ b \\ c \\ d \\ e \end{bmatrix} &s=2$ 2. Transpose of a Matrix Transpose of a matrix is formed by interchanging the elements from row to columns. For example, the first row of the matrix becomes first column in the transpose matrix, second row of the matrix becomes second column in the transpose matrix and so on. $latex A = \begin{bmatrix} a & b & c \\ d & e & f\\ g & h & i \end{bmatrix} \;\;\; A^T = \begin{bmatrix} a & d & g \\ b & e & h\\ c & f & i \end{bmatrix} &s=2$ $latex (A^T)_{ij} = A_{ji} &s=2$ 3. Square Matrix: Matrix with equal number of rows and columns. (Note: The sample matrices shown below are of 3x3 dimension. They can be readily extended to nxn dimension) $latex \begin{bmatrix} a & b & c \\ d & e & f\\ g & h & i \end{bmatrix} &s=2$ Square Matrix is further classified into 4. Symmetric Matrix : If a square matrix and its transpose are equal, then it is called a symmetric square matrix or simply symmetric matrix. $latex A = \begin{bmatrix} a & b & c \\ b & d & e\\ c & e & f \end{bmatrix} \;\;\;\;\;A^T = \begin{bmatrix} a & b & c \\ b & d & e\\ c & e & f \end{bmatrix} \;\;\;\;\;\Rightarrow A = A^T &s=2$ 5. Diagonal Matrix A special kind of symmetric matrix, with zeros in off-diagonal locations. $latex A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0\\ 0 & 0 & c \end{bmatrix} &s=2$ 6. Scalar matrix A special kind of diagonal matrix, with read more

How to estimate unknown parameters using Ordinary Least Squares (OLS)

As mentioned in the previous post, it is often required to estimate parameters that are unknown to the receiver. For example, if a fading channel is encountered in a communication system, it is desirable to estimate the channel response and cancel out the fading effects during reception. Assumption and model selection: For any estimation algorithm, certain assumptions have to be made. One of the main assumption is choosing an appropriate model for estimation. The chosen model should produce minimum statistical deviation and therefore should provide a “good fit”. A metric is often employed to determine the “goodness of fit”. Tests like “likelihood ratio test”, “Chi-Square test”, “Akaike Information Criterion” etc.., are used to measure the goodness of the assumed statistical model and decisions are made on the validity of the model assumption. To keep things simple, we will consider only polynomial models. Observed data and Least Squares estimation: Suppose, an experiment is performed and the following data points are observed in the following table. The variable “x” is the input to the experiment and “y” is the output obtained. The input variable is often called “regressor”, “independent variable”, “manipulated variable”, etc. Similarly, the regression analysis jargon, the output variable or the observed variable is called “observed variable”, “explanatory variable”, “regressand” , “response variable” etc.  The aim of the experiment is to fit the experiment into a model with appropriate parameters. Once we know the model, we no longer need to perform the experiment to determine the output for any given arbitrary input. All we need to do is to use the model and generate the desired output. Generic outline of an estimation algorithm: We have a very small number of data points and it is four in our case. To illustrate the concept, we will choose linear model for parameter estimation. Higher order models will certainly give better performance. Remember!!! More the polynomial order, more is the number of parameters to be estimated and therefore the computational complexity will be more. It is necessary to strike a balance between the required performance and the model order. We assume that the data points follow a linear trend. So, the linear model is chosen for the estimation problem. $latex y = \alpha_2 x + \alpha_1 &s=2$ Now the estimation problem simplifies to finding the parameters $latex \alpha_2 $ and $latex \alpha_1 $. We can write the relationship between the observed variable and the input variable as Next step is to solve for the above read more

The Mean Square Error - Why do we use it for estimation problems

"Mean Square Error", abbreviated as MSE, is an ubiquitous term found in texts on estimation theory. Have you ever wondered what this term actually means and why is this getting used in estimation theory very often ? Any communication system has a transmitter, a channel or medium to communicate and a receiver. Given the channel impulse response and the channel noise, the goal of a receiver is to decipher what was sent from the transmitter. A simple channel is usually characterized by a channel response - $latex h $ and an additive noise term - $latex n $. In time domain, this can be written as $latex y = h \circledast x + n &s=2$ Here, $latex \circledast $ is the convolution operation. Equivalently, in frequency domain, the convolution operation is equivalent to multiplication in frequency domain and vice-versa. $latex Y = HX + N &s=2$ Remember!!! Capitalized letters indicate frequency domain representation and small caps indicate time domain representation. The frequency domain equation looks simple and the only spoiler is the noise term. The receiver receives the information over the channel corrupted by noise and tries to decipher what was sent from the transmitter - $latex X $. If the noise is cancelled out in the receiver, $latex N =0 $, the observed spectrum at the receiver will look like, $latex Y = HX &s=2 $ Now, to know $latex X $, the receiver has to know $latex H $. Then it can simple divide the observed spectrum $latex Y $ with the channel frequency response $latex H $ to get $latex X $. Unfortunately, things are not that easy. Cancellation of noise from the received samples/spectrum is the hardest part. Complete nullification of noise at the receiver is hardest to achieve and the entire communication system design engineering revolves around reducing this noise to minimum acceptable level to achieve acceptable performance. Given the noise term, how do we know $latex H $ from the observed/received spectrum $latex Y $. This is a classical estimation problem. Usually a known sequence ( pilot sequence in OFDM and training sequence in GSM etc.., ) is transmitted and sent across the channel and from that the channel response $latex H $ or the impulse response $latex h $ is estimated. This estimated channel response is used to decipher the transmitted spectrum/sequence when receiving the actual data. This type of estimation is useful if and only if the channel response remains constant read more

HDD servo system using GA based Fixed Structure Robust Loop Shaping Control

Abstract: The capacity of the Hard Disk Drive (HDD) is increasing at very aggressive rate this leads to higher Tracks Per Inch (TPI). The servo system in HDD needs to improve in tracking performance and robustness. H∞ robust controller is one of the most popular techniques for the mentioned problem; however, order of this controller is usually much higher than that of the plant, making it difficult to implement. To overcome this problem, this paper proposes a new technique for designing a robust controller for HDD Voice Coil motor (VCM) actuator. The control design problem, H∞ loop shaping with structured controller, is solved by using GA. Infinity norm of the transfer function from disturbances to states is used as the cost function in our G.A optimization problem. The proposed technique not only solves the problem of complicated and high order controller but also retains the robust performance of conventional H∞ control technique. In addition, structured pre-filter is also designed by using GA to enhance the performance in time-domain tracking. Simulation results in a HDD control system show the effectiveness of the proposed technique. Full Paper: HDD servo system using GA based Fixed Structure Robust Loop Shaping Control About the Authors: *Amar Nath is an expert in Servo Engineering working with Seagate Technologies, Singapore - the world's top hard-disk manufacturer. *S. Kaitwanidvilai is with the Electrical Engg Department and Center of Excellence for Innovative Energy Systems KMITL,Bangkok,Thailand. ***Reproduced with permission from the first author of the read more

Characteristics of Noise Received by Software Defined Radio - Part 2

Abstract To study the characteristics of the noise received by Software defined radio (SDR) and the characteristics of signal transimtted from one SDR to the other SDR using Intermediate Frequency (IF) and Radio Frequency (RF). Click here for the first part of this series Signal and its characteristics: 2.1 Introduction A baseband signal of frequency F Hz is generated and was upconverted to IF and transmitted via channel from one SDR to the other SDR. Sine wave generation In this model, we have generated a samples of sine wave for different frequencies with different sampling rates and was upsampled by 8 in the Matlab and pulse shaped using RRC filter and given to the SDR which upconverts the signal to IF for trasmission of the signal via channel from one SDR to the other SDR. The pots of the generated sine wave are shown in figure. The Figure above shows the upsampled version of the baseband signal. We can see that there are zeros between any two samples this is because it is padded with zeros for upsampling. The figure above shows the pulse shaped version of the previous figure. Pulse shaping was done using Raise Cosine filter with roll of factor 0.9. 2.3 Autocorrelation of received signal The transmitted samples from the SDR was received by the other SDR. The received samples where analyzed using the autocorrelation function as given in equation (1.1) and by the Power Spectral Density (PSD) which is the DFT of the samples received by the SDR. The DFT is performed using Fast Fourier Transform (FFT) in Matlab. The received signal, its Power Spectral Density and its autocorrelation are shown in the following figure. The above figure has three plots. The rst plot is the sine wave received from the channel which was transmitted by the SDR. The second plot is the PSD of the signal. The spikes we see here are because of the sine nature. Since its frequency is 120 kHz we can see two spikes at 120kHz. The last part of plot is the autocorrelated version of the signal received by the SDR. This autocorrelation was done without taking the number of samples (N-k) in the equation (1.1). If that was taken we are expected to get a perfectly sine wave which was actually received by the SDR i.e., we are expected to get the first plot. 2.4 Conclusion From the graphs above we can infer that the received samples were almost with zero noise read more