# Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

Download "Purpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I"

## Transcription

1 Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes of the statistical analysis of time series are: To understand the variability of the time series. To identify the regular and irregular oscillations of the time series. To describe the characteristics of these oscillations. To understand the physical processes that give rise to each of these oscillations. To achieve the above, we need to: Identify the regular cycle (autocovariance; harmonic analysis) Estimate the importance of these cycles (power spectral analysis) Isolate or remove these cycles (filtering) Autocovariance Function Originally, autocorrelation/autocovariance function is used to estimate the dominant periods in the time series. The autocovariance is the covariance of a variable with itself at some other time, measured by a time lag (or lead) τ. The autocovariance as a function of the time lag (τ and L): Autocorrelation Function The Autocorrelation function is the normalized autocovariance function: Symmetric property of the autocovarinace/autocorrelation function: φ(-τ)=φ(τ) and r(-τ)=r(τ). 1

2 Typical Autocorrelation Function (Figure from Panofsky and Brier 1968) Degree of Freedom The typical autocorrelation function tells us that data points in a time series are not independent from each other. The degree of freedom is less than the number of data points (N). Can we estimate the degree of freedom from the autocorrelation function? If the lag is small, the autocorrelation is still positive for many geophysical variables. This means there is some persistence in the variables. Therefore, if there are N observations in sequence, they can not be considered independent from each other. This means the degree of freedom is less than N. For a time series of red noise, it has been suggested that the degree of freedom can be determined as following: N* = N t/ (2T e ). Here T e is the e-folding decay time of autocorrelation (where autocorrelation drops to 1/e). t is the time interval between data. The number of degrees is only half of the number of e-folding times of the data. Example for Periodic Time Series Example Red Noise Time Series Autocorrelation Function The mathematic form of red noise is as following: a: the degree of memory from previous states (0 < a < 1) ε: random number t: time interval between data points x: standardized variable (mean =0; stand deviation = 1) It can be shown that the autocorrelation function of the red noise is: T e is the e-folding decay time. 2

3 Example White Noise Example Noise + Periodic Time Series If a = o in the red noise, then we have a white noise: x(t) = ε(t) a series of random numbers The autocorrelation function of white noise is: r(τ)=δ(0) non-zero only at τ=0 Autocorrelation Function White noise has no prediction value. Red noise is useful for persistence forecasts. Harmonic Analysis Harmonic analysis is used to identify the periodic (regular) variations in geophysical time series. If we have N observations of (x i, y i ), the time series y(t) can be approximated by cosine and sine functions : t: Time T: Period of observation = N t A k, B k : Coefficients of kth harmonic How many harmonics (cosine/sine functions) do we need? In general, if the number of observations is N, the number of harmonic equal to N/2 (pairs of cosine and sine functions). What Does Each Harmonic Mean? As an example, if the time series is the monthly-mean temperature from January to December: N=12, t =1 month, and T=12 t = 12 month = one year 1 s harmonic (k=1) annual harmonic (one cycle through the period) Period = N t = 12 months 2 nd harmonic (k=2) semi-annual harmonic (2 cycles through the period) Period = 0.5N t = 6 months Last harmonic (k=n/2) the smallest period to be included. Period = 2 t = 2 months Nyquist frequency 3

4 Aliasing Orthogonality The variances at frequency higher than the Nyquist frequency (k>k*) will be aliased into lower frequency (k<k*) in the power spectrum. This is the so-called aliasing problem. In Vector Form: In Continuous Function Form (1) The inner product of orthogonal vectors or functions is zero. (2) The inner product of an orthogonal vector or function with itself is one. True Period = 2/3 t Aliased Period = 2 t This is a problem if there are large variances in the data that have frequencies smaller than k*. A Set of Orthogonal Functions f n (x) Multiple Regression (shown before) If we want to regress y with more than one variables (x 1, x 2, x 3,..x n ): After perform the least-square fit and remove means from all variables: Solve the following matrix to obtain the regression coefficients: a 1, a 2, a 3, a 4,.., a n : Fourier Coefficients Because of the orthogonal property of cosine and sine function, all the coefficients A and B can be computed independently (you don t need to know other A i=2,3, N/2 or B i=1, 2, 3 N/2 in order to get A 1, for example). This is a multiple regression case. Using least-square fit, we can show that: For k=1,n/2 (no B N/2 component) 4

5 Amplitude of Harmonics Using the following relation, we can combine the sine and cosine components of the harmonic to determine the amplitude of each harmonic. C 2 =A 2 +B 2 (amplitude) 2 of the harmonic θ 0 the time (phase) when this harmonic has its largest amplitude With this combined form, the harmonic analysis of y(t) can be rewritten as: Fraction of Variance Explained by Harmonics What is the fraction of the variance (of y) explained by a single harmonic? Remember we have shown in the regression analysis that the fraction is equal to the square of the correlation coefficient between this harmonic and y: It can be shown that this fraction is r 2 How Many Harmonics Do We Need? Time Domain.vs. Frequency Domain Since the harmonics are all uncorrelated, no two harmonics can explain the same part of the variance of y. In other words, the variances explained by the different harmonics can be added. time f(t) Fourier Transform Inverse Fourier Transform frequency F(w) We can add up the fractions to determine how many harmonics we need to explain most of the variations in the time series of y. Time Series Power Spectrum 5

6 Power Spectrum Problems with Line Spectrum smooth spectrum (Figure from Panofsky and Brier 1968) line spectrum By plotting the amplitude of the harmonics as a function of k, we produce the power spectrum of the time series y. The meaning of the spectrum is that it shows the contribution of each harmonic to the total variance. If t is time, then we get the frequency spectrum. If t is distance, then we get the wavenumber spectrum. The C k2 is a line spectrum at a specific frequency and wavenumber (k). We are not interested in these line spectra. Here are the reasons: Integer values of k have no specific meaning. They are determined based on the length of the observation period T (=N t): k = (0, 1, 2, 3,..N/2) cycles during the period T. Since we use N observations to determine a mean and N/2 line spectra, each line spectrum has only about 2 degrees of freedom. With such small dof, the line spectrum is not likely to be reproduced from one sampling interval to the other. Also, most geophysical signals that we are interested in and wish to study are not truly periodic. A lot of them are just quasi-periodic, for example ENSO. So we are more interested in the spectrum over a band of frequencies, not at a specific frequency. Continuous Spectrum Aliasing Φ(k) k 1 k 2 Nyquist frequency k* k So we need a continuous spectrum that tells us the variance of y(t) per unit frequency (wavenumber) interval: k* is called the Nyquist frequency, which has a frequency of one cycle per 2 t. (This is the k=n/2 harmonics). The Nyquist frequency is the highest frequency can be resolved by the given spacing of the data point. True Period = 2/3 t Aliased Period = 2 t The variances at frequency higher than the Nyquist frequency (k>k*) will be aliased into lower frequency (k<k*) in the power spectrum. This is the so-called aliasing problem. This is a problem if there are large variances in the data that have frequencies smaller than k*. 6

7 How to Calculate Continuous Spectrum There are two ways to calculate the continuous spectrum: (1)(1) Direct Method (use Fourier transform) (2)(2) Time-Lag Correlation Method (use autocorrelation function) (1) Direct Method (a more popular method) Step 1: Perform Fourier transform of y(t) to get C 2 (k) Step 2: smooth C 2 (k) by averaging a few adjacent frequencies together. or by averaging the spectra of a few time series together. both ways smooth a line spectrum to a continuous spectrum and increase the degrees of freedom of the spectrum. Examples Example 1 smooth over frequency bands A time series has 900 days of record. If we do a Fourier analysis then the bandwidth will be 1/900 day -1, and each of the 450 spectral estimates will have 2 degrees of freedom. If we averaged each 10 adjacent estimates together, then the bandwidth will be 1/90 day -1 and each estimate will have 20 d.o.f. Example 2 smooth over spectra of several time series Suppose we have 10 time series of 900 days. If we compute spectra for each of these and then average the individual spectral estimates for each frequency over the sample of 10 spectra, then we can derive a spectrum with a bandwidth of 1/900 days -1 where each spectral estimate has 20 degrees of freedom. Time-Lag Correlation Method Resolution of Spectrum - Bandwidth (2) Time-Lag Correlation Method It can be shown that the autocorrelation function and power spectrum are Fourier transform of each other. So we can obtain the continuous spectrum by by performing harmonic analysis on the lag correlation function on the interval -T L τ T L. Φ(k): Power Spectrum in frequency (k) r(τ): Autocorrelation in time lag (τ) Bandwidth ( f)= width of the frequency band resolution of spectrum f = 1/N (cycle per time interval) For example, if a time series has 900 monthly-mean data: bandwidth = 1/900 (cycle per month). Nyquist frequency = ½ (cycle per month) Total number of frequency bands = (0-Nyquist frequency)/bandwidth = (0.5)/(1/900) = 450 = N/2 Each frequency band has about 2 degree of freedom. If we average several bands together, we increase the degrees of freedom but reduce the resolution (larger bandwidth). 7

8 Bandwidth in Time-Lag Correlation Method With the time-lag correlation method, the bandwidth of the power spectrum is determined by the maximum time lag (L) used in the calculation: f = 1 cycle/(2l t). Number of frequency band = (Nyquist frequency 0) / f = (2 t) -1 / (2L t) -1 = L Number of degrees of freedom = N/(number of bands) = N/L An Example For red noise, we know: r(τ)=exp(-τ/t e ) T e = - τ / ln(r(τ)) If we know the autocorrelation at τ= t, then we can find out that For example: Example Spectrum of Red Noise Let s use the Parseval s theory to calculate the power spectrum of red noise. Power Spectrum of Red Noise We already showed that the autocorrelation function of the red noise is: By performing the Fourier transform of the autocorrelation function, we obtain the power spectrum of the red noise: 8

9 Significance Test of Spectral Peak Calculate the Red Noise Spectrum for Test The red noise power spectrum can be calculated using the following formula: Power of the Tested Spectrum Power of the Red Noise Null Hypothesis : the time series is not periodic in the region of interest, but simply noise. We thus compare amplitude of a spectral peak to a background value determined by a red noise fit to the spectrum. Use F-Test: P(h, ρ, M) is the power spectrum at frequency h h = 0, 1, 2, 3,., M ρ = autocorrelation coefficient at one time lag We would normally obtain the parameter ρ from the original time series as the average of the one-lag autocorrelation and the square root of the two-lag autocorrelation. We then make the total power (variance) of this red noise spectrum equal to the total power (variance) of the power spectrum we want to test. How To Plot Power Spectrum? Data Window Φ(ω) Linear Scale Logarithmic Scale ωφ(k) stretch low freq contract high freq The Fourier transform obtains the true power spectrum from a time series with a infinite time domain. In real cases, the time series has a finite length. It is like that we obtain the finite time series from the infinite time domain through a data window: finite sample ω 1 ω 2 ω* k lnω 1 lnω 2 lnω* Data Window 0 T How does the data window affect the power spectrum? Infinite time series 9

10 Parseval s Theorem This theory is important for power spectrum analysis and for time filtering to be discussed later. The theory states that the square of the time series integrated over time is equal to the square (inner product) of the Fourier transform integrated over frequency: Power Spectrum of Finite Sample If the infinite time series is f(t) and the sample time series is g(t), then the power spectrum calculated from the sample is related to the true spectrum in the following way: Based on the Convolution Theory Here F 1 (ω)/f 2 (ω) is the Fourier transform of f1(t)/f2(t). The sample spectrum is not equal to the true spectrum but weighted by the spectrum of the data window used: Square Data Windows 1.0 -T/2 0 T/2 Square data window is: w(t) = 1 within the window domain = 0 everywhere else. Bartlett Window The data window has the following weighting effects on the true spectrum: The Weighting Effect of Square Window Response Function of Square Window Spectral Leakage The square window smooth the true spectrum. The degree of the smoothing is determined by the window length (T). The shorter the window length, the stronger the smoothing will be. In addition to the smoothing effect, data window also cause spectral leakage. This leakage will introduce spurious oscillations at higher and lower frequencies and are out of phase with the true oscillation. 10

11 We Wish the Data Window Can Produce a narrow central lobe require a larger T (the length of data window) Tapered Data Window How do we reduce the side lobes associated with the data window? A tapered data window. Produce a insignificant side lobes require a smooth data window without sharp corners A rectangular or Bartlett window leaves the time series undistorted, but can seriously distort the frequency spectrum. A tapered window distorts the time series but may yield a more representative frequency spectrum. (from Hartmann 2003) Bartlett Window Bartlett (square or rectangular) window Hanning Window (Cosine Bell) The cosine bell window is perhaps the most frequently used window in meteorological applications. This is the most commonly used window, but we use it without knowing we are using it. The Bartlett window has a serious side lobe problem. Frequencies that are outside the range of frequencies actually resolved can have too strong an influence on the power spectra at the frequencies resolved. The same as Bartlett window Partially cancel out Side lobs, but also Broaden the central lobe 11

12 Filtering of Time Series Time filtering technique is used to remove or to retain variations at particular bands of frequencies from the time series. There are three types of filtering: (1) High-Pass Filtering keep high-frequency parts of the variations and remove lowfrequency parts of the variations. (2) Low-Pass Filtering keep low-frequency and remove high-frequency parts of the variations. (3) Band-Pass Filtering remove both higher and lower frequencies and keep only certain frequency bands of the variations. Response Function Time filters are the same as the data window we have discussed earlier. By performing Fourier transform, we know that: filter or data window The ration between the filtered and original power spectrum is called the response function : power spectrum after filtering original power spectrum If R(ω)=1 the original amplitude at frequency ω is kept. R(ω)=0 the original amplitude at frequency ω is filtered out. An Perfect Filter A Sharp-Cutoff Filter The ideal filter should have a response of 1 over the frequency bands we want to keep and a response of zero over the frequency bands we want to remove: A Perfect Square Response Function 12

### Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### Probability and Random Variables. Generation of random variables (r.v.)

Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly

### 8 Filtering. 8.1 Mathematical operation

8 Filtering The estimated spectrum of a time series gives the distribution of variance as a function of frequency. Depending on the purpose of analysis, some frequencies may be of greater interest than

### Chapter 3 Discrete-Time Fourier Series. by the French mathematician Jean Baptiste Joseph Fourier in the early 1800 s. The

Chapter 3 Discrete-Time Fourier Series 3.1 Introduction The Fourier series and Fourier transforms are mathematical correlations between the time and frequency domains. They are the result of the heat-transfer

### Analysis/resynthesis with the short time Fourier transform

Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis

### Fourier Analysis. Nikki Truss. Abstract:

Fourier Analysis Nikki Truss 09369481 Abstract: The aim of this experiment was to investigate the Fourier transforms of periodic waveforms, and using harmonic analysis of Fourier transforms to gain information

### Design of FIR Filters

Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as

### Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.

Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal

### Chapter 6. Power Spectrum. 6.1 Outline

Chapter 6 Power Spectrum The power spectrum answers the question How much of the signal is at a frequency ω?. We have seen that periodic signals give peaks at a fundamental and its harmonics; quasiperiodic

### Short-time FFT, Multi-taper analysis & Filtering in SPM12

Short-time FFT, Multi-taper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH 20.03.2015 Overview Refresher Short-time Fourier

### A Tutorial on Fourier Analysis

A Tutorial on Fourier Analysis Douglas Eck University of Montreal NYU March 26 1.5 A fundamental and three odd harmonics (3,5,7) fund (freq 1) 3rd harm 5th harm 7th harmm.5 1 2 4 6 8 1 12 14 16 18 2 1.5

### SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 18. Filtering

SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 18. Filtering By Tom Irvine Email: tomirvine@aol.com Introduction Filtering is a tool for resolving signals. Filtering can be performed on either analog

### ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's

### Topic 4: Continuous-Time Fourier Transform (CTFT)

ELEC264: Signals And Systems Topic 4: Continuous-Time Fourier Transform (CTFT) Aishy Amer Concordia University Electrical and Computer Engineering o Introduction to Fourier Transform o Fourier transform

### R.E. White (Birkbeck University of London) & E. Zabihi Naeini* (Ikon Science)

Tu ELI2 10 Broad-band Well Tie R.E. White (Birkbeck University of London) & E. Zabihi Naeini* (Ikon Science) SUMMARY By extending the low-frequency content, broad-band seismic data lessens the dependence

### Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Jesus J. Caban Outline! Assignment #! Paper Presentation & Schedule! Frequency Domain! Mathematical Morphology %& Assignment #! Questions?! How s OpenCV?! You

### What is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter

What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency

### FIR Filter Design. FIR Filters and the z-domain. The z-domain model of a general FIR filter is shown in Figure 1. Figure 1

FIR Filters and the -Domain FIR Filter Design The -domain model of a general FIR filter is shown in Figure. Figure Each - box indicates a further delay of one sampling period. For example, the input to

### FILTER CIRCUITS. A filter is a circuit whose transfer function, that is the ratio of its output to its input, depends upon frequency.

FILTER CIRCUITS Introduction Circuits with a response that depends upon the frequency of the input voltage are known as filters. Filter circuits can be used to perform a number of important functions in

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### Aliasing, Image Sampling and Reconstruction

Aliasing, Image Sampling and Reconstruction Recall: a pixel is a point It is NOT a box, disc or teeny wee light It has no dimension It occupies no area It can have a coordinate More than a point, it is

### L9: Cepstral analysis

L9: Cepstral analysis The cepstrum Homomorphic filtering The cepstrum and voicing/pitch detection Linear prediction cepstral coefficients Mel frequency cepstral coefficients This lecture is based on [Taylor,

### Fourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees

Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This

### Univariate and Multivariate Methods PEARSON. Addison Wesley

Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

### Frequency Domain Analysis

Exercise 4. Frequency Domain Analysis Required knowledge Fourier-series and Fourier-transform. Measurement and interpretation of transfer function of linear systems. Calculation of transfer function of

### L6: Short-time Fourier analysis and synthesis

L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude

### Chapter 8 - Power Density Spectrum

EE385 Class Notes 8/8/03 John Stensby Chapter 8 - Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is

### Curve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.

Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections Least-Squares Regression Linear Regression General Linear Least-Squares Nonlinear

### Synthesis Techniques. Juan P Bello

Synthesis Techniques Juan P Bello Synthesis It implies the artificial construction of a complex body by combining its elements. Complex body: acoustic signal (sound) Elements: parameters and/or basic signals

### Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs

Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in

### where T o defines the period of the signal. The period is related to the signal frequency according to

SIGNAL SPECTRA AND EMC One of the critical aspects of sound EMC analysis and design is a basic understanding of signal spectra. A knowledge of the approximate spectral content of common signal types provides

### TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of

### The Fourier Transform

The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2. Eric W. Weisstein.

### Frequency Response and Continuous-time Fourier Transform

Frequency Response and Continuous-time Fourier Transform Goals Signals and Systems in the FD-part II I. (Finite-energy) signals in the Frequency Domain - The Fourier Transform of a signal - Classification

### 3. Regression & Exponential Smoothing

3. Regression & Exponential Smoothing 3.1 Forecasting a Single Time Series Two main approaches are traditionally used to model a single time series z 1, z 2,..., z n 1. Models the observation z t as a

### 7. FOURIER ANALYSIS AND DATA PROCESSING

7. FOURIER ANALYSIS AND DATA PROCESSING Fourier 1 analysis plays a dominant role in the treatment of vibrations of mechanical systems responding to deterministic or stochastic excitation, and, as has already

### EELE445 - Lab 2 Pulse Signals

EELE445 - Lab 2 Pulse Signals PURPOSE The purpose of the lab is to examine the characteristics of some common pulsed waveforms in the time and frequency domain. The repetitive pulsed waveforms used are

### Fourier Transform and Image Filtering. CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010

Fourier Transform and Image Filtering CS/BIOEN 6640 Lecture Marcel Prastawa Fall 2010 The Fourier Transform Fourier Transform Forward, mapping to frequency domain: Backward, inverse mapping to time domain:

### R U S S E L L L. H E R M A N

R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.

### Nyquist Sampling Theorem. By: Arnold Evia

Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier

### Department of Electronics and Communication Engineering 1

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING III Year ECE / V Semester EC 6502 PRINCIPLES OF DIGITAL SIGNAL PROCESSING QUESTION BANK Department of

Measurement Lab Fourier Analysis Last Modified 9/5/06 Any time-varying signal can be constructed by adding together sine waves of appropriate frequency, amplitude, and phase. Fourier analysis is a technique

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### USE OF SYNCHRONOUS NOISE AS TEST SIGNAL FOR EVALUATION OF STRONG MOTION DATA PROCESSING SCHEMES

USE OF SYNCHRONOUS NOISE AS TEST SIGNAL FOR EVALUATION OF STRONG MOTION DATA PROCESSING SCHEMES Ashok KUMAR, Susanta BASU And Brijesh CHANDRA 3 SUMMARY A synchronous noise is a broad band time series having

### NRZ Bandwidth - HF Cutoff vs. SNR

Application Note: HFAN-09.0. Rev.2; 04/08 NRZ Bandwidth - HF Cutoff vs. SNR Functional Diagrams Pin Configurations appear at end of data sheet. Functional Diagrams continued at end of data sheet. UCSP

### B3. Short Time Fourier Transform (STFT)

B3. Short Time Fourier Transform (STFT) Objectives: Understand the concept of a time varying frequency spectrum and the spectrogram Understand the effect of different windows on the spectrogram; Understand

### Bandwidth-dependent transformation of noise data from frequency into time domain and vice versa

Topic Bandwidth-dependent transformation of noise data from frequency into time domain and vice versa Authors Peter Bormann (formerly GeoForschungsZentrum Potsdam, Telegrafenberg, D-14473 Potsdam, Germany),

### IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL

IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL Saurabh Singh Rajput, Dr.S.S. Bhadauria Department of Electronics, Madhav Institute of Technology

### Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

### Implementation of FIR Filter using Adjustable Window Function and Its Application in Speech Signal Processing

International Journal of Advances in Electrical and Electronics Engineering 158 Available online at www.ijaeee.com & www.sestindia.org/volume-ijaeee/ ISSN: 2319-1112 Implementation of FIR Filter using

### 568 Subject Index. exponential distribution, 106, 108, 539 extension field, 545, 551 eye diagram, 193

SUBJECT INDEX additive system, 182 additive white Gaussian noise, 99, 103 aliasing, 52, 54 amplifier AM/AM characteristic, 332, 342 AM/PM characteristic, 332, 342 nonlinear, 344 two-box model, 344 amplifiers,

### Image Processing with. ImageJ. Biology. Imaging

Image Processing with ImageJ 1. Spatial filters Outlines background correction image denoising edges detection 2. Fourier domain filtering correction of periodic artefacts 3. Binary operations masks morphological

### Time Series Analysis in WinIDAMS

Time Series Analysis in WinIDAMS P.S. Nagpaul, New Delhi, India April 2005 1 Introduction A time series is a sequence of observations, which are ordered in time (or space). In electrical engineering literature,

### Introduction to Digital Audio

Introduction to Digital Audio Before the development of high-speed, low-cost digital computers and analog-to-digital conversion circuits, all recording and manipulation of sound was done using analog techniques.

### SIGNAL PROCESSING & SIMULATION NEWSLETTER

1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

### Silicon Laboratories, Inc. Rev 1.0 1

Clock Division with Jitter and Phase Noise Measurements Introduction As clock speeds and communication channels run at ever higher frequencies, accurate jitter and phase noise measurements become more

### Analog and Digital Signals, Time and Frequency Representation of Signals

1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals

### Fast Fourier Transforms and Power Spectra in LabVIEW

Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide

### Mechanical Vibrations

Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

### Introduction to Time Series Analysis. Lecture 22.

Introduction to Time Series Analysis. Lecture 22. 1. Review: The smoothed periodogram. 2. Examples. 3. Parametric spectral density estimation. 1 Review: Periodogram The periodogram is defined as I(ν) =

### SIGNAL PROCESSING FOR EFFECTIVE VIBRATION ANALYSIS

SIGNAL PROCESSING FOR EFFECTIVE VIBRATION ANALYSIS Dennis H. Shreve IRD Mechanalysis, Inc Columbus, Ohio November 1995 ABSTRACT Effective vibration analysis first begins with acquiring an accurate time-varying

### There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all).

Data acquisition There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all). Some deal with the detection system.

### Online Chapter 12 Time and Frequency: A Closer Look at Filtering and Time-Frequency Analysis

Online Chapter 12 and : A Closer Look at Filtering and - Analysis 12.1 A Note About This Chapter This is a revised version of Chapter 5 from the first edition of An Introduction to the Event-Related Potential

### APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell)

1 APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell) (Reference The Fourier Transform and its Application, second edition, R.N. Bracewell, McGraw-Hill Book Co., New

### Lecture 1-6: Noise and Filters

Lecture 1-6: Noise and Filters Overview 1. Periodic and Aperiodic Signals Review: by periodic signals, we mean signals that have a waveform shape that repeats. The time taken for the waveform to repeat

### Computer Graphics. Course SS 2007 Antialiasing. computer graphics & visualization

Computer Graphics Course SS 2007 Antialiasing How to avoid spatial aliasing caused by an undersampling of the signal, i.e. the sampling frequency is not high enough to cover all details Supersampling -

### PHYS 331: Junior Physics Laboratory I Notes on Noise Reduction

PHYS 331: Junior Physics Laboratory I Notes on Noise Reduction When setting out to make a measurement one often finds that the signal, the quantity we want to see, is masked by noise, which is anything

### CHAPTER 2 FOURIER SERIES

CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =

### ADC Familiarization. Dmitry Teytelman and Dan Van Winkle. Student name:

Dmitry Teytelman and Dan Van Winkle Student name: RF and Digital Signal Processing US Particle Accelerator School 15 19 June, 2009 June 16, 2009 Contents 1 Introduction 2 2 Exercises 2 2.1 Measuring 48

### Lecture 9, Multirate Signal Processing, z-domain Effects

Lecture 9, Multirate Signal Processing, z-domain Effects Last time we saw the effects of downsampling and upsampling. Observe that we can perfectly reconstruct the high pass signal in our example if we

### T = 1 f. Phase. Measure of relative position in time within a single period of a signal For a periodic signal f(t), phase is fractional part t p

Data Transmission Concepts and terminology Transmission terminology Transmission from transmitter to receiver goes over some transmission medium using electromagnetic waves Guided media. Waves are guided

### An introduction to generalized vector spaces and Fourier analysis. by M. Croft

1 An introduction to generalized vector spaces and Fourier analysis. by M. Croft FOURIER ANALYSIS : Introduction Reading: Brophy p. 58-63 This lab is u lab on Fourier analysis and consists of VI parts.

### Fourier Series and Sturm-Liouville Eigenvalue Problems

Fourier Series and Sturm-Liouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Half-range Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex

### Convolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005

Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall

### Time Series Analysis

Time Series Analysis Autoregressive, MA and ARMA processes Andrés M. Alonso Carolina García-Martos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 212 Alonso and García-Martos

### Frequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao

Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Time-domain description Waveform representation

### Measuring Line Edge Roughness: Fluctuations in Uncertainty

Tutor6.doc: Version 5/6/08 T h e L i t h o g r a p h y E x p e r t (August 008) Measuring Line Edge Roughness: Fluctuations in Uncertainty Line edge roughness () is the deviation of a feature edge (as

### Chapter 10 Introduction to Time Series Analysis

Chapter 1 Introduction to Time Series Analysis A time series is a collection of observations made sequentially in time. Examples are daily mortality counts, particulate air pollution measurements, and

### Image Enhancement: Frequency domain methods

Image Enhancement: Frequency domain methods The concept of filtering is easier to visualize in the frequency domain. Therefore, enhancement of image f ( m, n) can be done in the frequency domain, based

Performance of an IF sampling ADC in receiver applications David Buchanan Staff Applications Engineer Analog Devices, Inc. Introduction The concept of direct intermediate frequency (IF) sampling is not

### Time series analysis Matlab tutorial. Joachim Gross

Time series analysis Matlab tutorial Joachim Gross Outline Terminology Sampling theorem Plotting Baseline correction Detrending Smoothing Filtering Decimation Remarks Focus on practical aspects, exercises,

### 5 Signal Design for Bandlimited Channels

225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g

### Moving Average Filters

CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average

### Introduction to time series analysis

Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples

### Determining the Optimal Sampling Rate of a Sonic Anemometer Based on the Shannon-Nyquist Sampling Theorem

Determining the Optimal Sampling Rate of a Sonic Anemometer Based on the Shannon-Nyquist Sampling Theorem Andrew Mahre National Oceanic and Atmospheric Administration Research Experiences for Undergraduates

### MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

### APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (David Sandwell, Copyright, 2004)

1 APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (David Sandwell, Copyright, 2004) (Reference The Fourier Transform and its Application, second edition, R.N. Bracewell, McGraw-Hill Book Co., New York,

### SGN-1158 Introduction to Signal Processing Test. Solutions

SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:

### Introduction to IQ-demodulation of RF-data

Introduction to IQ-demodulation of RF-data by Johan Kirkhorn, IFBT, NTNU September 15, 1999 Table of Contents 1 INTRODUCTION...3 1.1 Abstract...3 1.2 Definitions/Abbreviations/Nomenclature...3 1.3 Referenced

### Ged Ridgway Wellcome Trust Centre for Neuroimaging University College London. [slides from the FIL Methods group] SPM Course Vancouver, August 2010

Ged Ridgway Wellcome Trust Centre for Neuroimaging University College London [slides from the FIL Methods group] SPM Course Vancouver, August 2010 β β y X X e one sample t-test two sample t-test paired

### Modulation methods. S-72. 333 Physical layer methods in wireless communication systems. Sylvain Ranvier / Radio Laboratory / TKK 16 November 2004

Modulation methods S-72. 333 Physical layer methods in wireless communication systems Sylvain Ranvier / Radio Laboratory / TKK 16 November 2004 sylvain.ranvier@hut.fi SMARAD / Radio Laboratory 1 Line out

### Filtering Data using Frequency Domain Filters

Filtering Data using Frequency Domain Filters Wouter J. Den Haan London School of Economics c Wouter J. Den Haan August 27, 2014 Overview Intro lag operator Why frequency domain? Fourier transform Data

### Electronic Communications Committee (ECC) within the European Conference of Postal and Telecommunications Administrations (CEPT)

Page 1 Electronic Communications Committee (ECC) within the European Conference of Postal and Telecommunications Administrations (CEPT) ECC RECOMMENDATION (06)01 Bandwidth measurements using FFT techniques

### Signal to Noise Instrumental Excel Assignment

Signal to Noise Instrumental Excel Assignment Instrumental methods, as all techniques involved in physical measurements, are limited by both the precision and accuracy. The precision and accuracy of a

### SWISS ARMY KNIFE INDICATOR John F. Ehlers

SWISS ARMY KNIFE INDICATOR John F. Ehlers The indicator I describe in this article does all the common functions of the usual indicators, such as smoothing and momentum generation. It also does some unusual

### Applications of the DFT

CHAPTER 9 Applications of the DFT The Discrete Fourier Transform (DFT) is one of the most important tools in Digital Signal Processing. This chapter discusses three common ways it is used. First, the DFT

### Agilent Creating Multi-tone Signals With the N7509A Waveform Generation Toolbox. Application Note

Agilent Creating Multi-tone Signals With the N7509A Waveform Generation Toolbox Application Note Introduction Of all the signal engines in the N7509A, the most complex is the multi-tone engine. This application

### Overview. Speech Signal Analysis. Speech signal analysis for ASR. Speech production model. Vocal Organs & Vocal Tract. Speech Signal Analysis for ASR

Overview Speech Signal Analysis Hiroshi Shimodaira and Steve Renals Automatic Speech Recognition ASR Lectures 2&3 7/2 January 23 Speech Signal Analysis for ASR Reading: Features for ASR Spectral analysis