V(t) = Total Power = Calculating the Power Spectral Density (PSD) in IDL. Thomas Ferree, Ph.D. August 23, 1999
|
|
- Megan Andrews
- 6 years ago
- Views:
Transcription
1 Calculating the Power Spectral Density (PSD) in IDL Thomas Ferree, Ph.D. August 23, 1999 This note outlines the calculation of power spectra via the fast Fourier transform (FFT) algorithm. There are several variants of the definition of the power of a signal V(t), distinguished mainly by their normalizations. Emphasis is placed on correct inclusion of negative frequency components within the FFT calculational framework, and on defining the PSD to be intensive so that it is independent of frequency bin size. Other topics such as windowing, averaging and padding to increase resolution are discussed briefly. IDL conventions are used throughout for array indexing and in defining the FFT. Continuous signals Time domain definition of total power To understand the definition of signal power, it is helpful to first consider the ideal situation in which we have an infinite amount of continuously sampled data. Given a continous time series V(t), the total power in the signal is usually defined in the time domain as The first thing to note about this definition is that it has units of [V] 2 sec, where [V] represents the units of the signal V(t). This is different than the familiar physics definition of power, which is mechanical, and has units of energy per unit time. Continuous Fourier transform Power By introducing the continuous Fourier transform (FT) V(t) 2 dt and its inverse V ( f ) = V(t) = V(t)e V ( f )e 2πi f t +2πi f t dt df it is straightforward to show that Total Power = V ( f ) 2 df 1
2 Together with the definition of total power given above, this is called Parseval s theorem, and it shows that the total power can equivalently be computed as an integral over all frequency components. This suggests that we define a power spectral density (PSD) function The term density is used in this context to mean that the PSD is intensive with respect to frequency, i.e., it does not scale linearly with the frequency bin size df, since the factor of df appearing in the total power has been factored from the definition of PSD. The superscript is meant to indicate that this is a two-sided PSD, which must be integrated over both positive and negative frequencies to retrieve the total power. The presence of both positive and negative frequencies in Parseval s theorem is a crucial issue. To begin with, the negative frequencies are not distinguished physically from positive frrequencies, but have their origin in the definition of the inverse FT. Negative frequencies are essential in an exponential representation of the FT, in the sense that they are necessary for the set of Fourier basis functions to comprise a complete set capable of characterizing an arbitrary time series. The two-sided PSD is a fine definition, provided that one knows how to interpret it. When one speaks of power at a particular frequency, however, one usually does not intend to distinguish between positive and negative frequencies. This suggests an alternative definition of PSD which is defined over positive frequencies f> only. The superscript indicates that this is a one-sided PSD, which gives the total signal power when integrated over positive frequencies only. As a further simplification, it can be seen from the definition of the FT that for real (not complex) signals Thus for continuously sampled real signals, the negative frequency components can be accounted for simply by including a factor of two in the definition of the one-sided PSD. This one-sided PSD obeys a different normalization condition to retrieve the total power. Discrete signals PSD (±) ( f ) = V ( f ) 2 PSD (+) ( f ) = V ( f ) 2 + V ( f ) 2 V ( f ) = Total Power = [ V ( f )] * PSD (+ ) ( f )df Now consider the realistic situation in which V(t) is sampled at only N time points t j = j t, j=,..., N 1 2
3 We assume that the sampling rate is given. For example, EGI s data acquisition software typically collects 25 or 5 samples per second, depending on the experimental protocol. The total time elapsed between the first and last data points is T = (N 1) t although this is never actually used in what follows. In what follows, we assume that N is even. In practice, N should be an integer power of 2 for the FFT algorithm to be most efficient. The corresponding discrete positive and negative frequencies are f k = k f, k = N 2,...,,..., + N 2 where the frequency bin size is f = 1 N t The maximum meaningful frequency is called the Nyquist frequency and occurs when k=+n/2 or k=-n/2. The standard tool for signal processing of discretely sampled data, however, is the fast Fourier transform (FFT). IDL defines the FFT as and the corresponding inverse FFT as V (k) 1 N f c = 1 2 t N 1 Note from the inverse transform that the discrete frequency index k=,,n-1, i.e., runs over nonnegative values only, and that those values extend beyond the Nyquist frequency. This is merely a matter of convenience in how the frequency information is stored in the FFT, because in most computer languages it is inconvenient to declare arrays with negative indices. Clearly the numbers stored in bins above the Nyquist bin (k=n/2) do not reflect spectral content of the data above the Nyquist frequency, since the FFT only samples the signal up to the Nyquist frequency. Rather, it is straightforward to show from the definition of the FFT that for a real signal V(t) V (N k) = N 1 j= V( j) e 2πi j k / N V( j) V (k) e +2πi j k / N k = [ V (k)] * = 3 V ( k)
4 for k=1,,n/2-1. In other words, the information stored in the upper half of the FFT array is redundant with that stored in the lower half, and corresponds exactly to the negative frequency components already discussed in the context of continuously sampled signals. Note also that the units of the FFT are different than those of the FT, since the FFT contains no factor of dt. This will be relevant for making a discrete definition of the PSD. Note also the factor of 1/N which appears in the FFT, but not in the FT. This is the IDL normalization convention for the FFT, and is different from that adopted in MATLAB (Stearns and David, 1996) and Numerical Recipes in C (Press et al., 1992). Mathematically, the 1/N can appear in either the FFT or the inverse FFT, since the only requirement is that the original signal V(t) be retrieved when the transform and inverse transform are applied sequentially. Its presence is therefore not a major issue, but it must be accounted for when defining the PSD via the FFT. We are now in a position to approach the definition of power spectral density for discrete signals. To maintain consistency with continuously sampled signals, and to arrive at an intensive definition of discree PSD, the total power of a discrete signal can be defined as To motivate a discrete definition of PSD, we again invoke Parseval s theorem. It is straightforward to show from the IDL definition of FFT that the discrete signal V(j) and its FFT obey a discrete form of Parseval s theorem N 1 V( j) 2 = N V (k) 2 j= N 1 Total Power V( j) 2 t j = where the sum on the right hand side runs over all frequency bins of the FFT. This is a crucial point: to compute the total signal power one must sum over all components of the FFT, where the bins above the Nyquist bin correspond to negative frequencies. Multiplying both sides by t yields a frequency-space definition of the total power: Total Power = N t N 1 In the second equality, we have used the definition of f, and introduced another factor of f in the numerator and denominator to make the right hand side look like that for continuously sampled data. This suggests that we define a two-sided PSD for discrete signals as N 1 k= V (k) 2 = k= N 1 k = V (k) 2 ( f ) 2 f PSD (±) (k) = 4 V (k) 2 ( f ) 2
5 To portray the total power in the frequency domain, one can plot this quantity symmetrically about zero frequency. Recognizing the redundancy of information in the FFT for real signals, and in an an attempt to simplify the plot of this quantity, many researchers plot this function only over values of k=,,n/2, i.e., up to the Nyquist frequency. This definition is awkward, however, for the same reasons as for continuous signals. Moreover, and most importantly, this awkwardness can not be remedied simply by introducing an overall factor of two, as was the case for continuous signals. This is because the bin (k=) centered on zero frequency, and the bin (k=n/2) centered on the Nyquist frequency, are not included in the redundant information. To define a properly normalized one-sided PSD, therefore, it is common to write PSD (+) () = PSD (+) (k) = PSD (+) (N / 2) = V () 2 ( f ) 2 Press et al. (1992). This definition has the desired property that summing over k=,,n/2 gives the total power, i.e., Total Power = V (k) 2 + V (N k) 2 ( f ) 2, k =1,..., N / 2 1 V (N / 2) 2 N /2 k = ( f ) 2 PSD (+) (k) f as required for straightfoward interpretation of the power at various frequencies. For real signals this definition results in a factor of two for all bins except those at k= and k=n/2. The distinction between the k= and k=n/2 bins versus all the other bins is essential if one wishes to quantify absolute power in a particular bin, or to compare power in neighboring bins where one of those bins corresponds to the lowest sampled frequency, i.e., k=. (Comparing the power in the Nyquist bin with its neighbors is seldom useful, since one normally assumes that the input signal has been low-pass filtered before sampling to eliminate aliasing, and that no meaningful information is present near the Nyquist bin.) Figure 1 shows the effect of computing the one-sided PSD correctly for discrete signals. Figure 1a shows a typical EEG time series consisting of 256 points, collected at 1/ t=25 samples/sec. Figure 1b shows the two-sided PSD amplitude (defined as the square root of the two-sided PSD for improved visualization), plotted over values k=,,n/2. Figure 1c shows the one-sided PSD amplitude, plotted over the same range. Note that in Figure 1b, the power in the second frequency bin (ranging from.5 to 1.5 Hz, but plotted as a single point at f=1 Hz) is slightly lower than the first frequency bin (ranging from.5 to.5 Hz, but plotted as a single point at f=). Calculation of the one-sided PSD in Figure 1c doubles the power in the second bin, without affecting the power in the first bin. (The 5
6 increase seen in the plot is actually only a factor of 1.41, due to the square root in the definition of PSD amplitude.) Thus the one-sided PSD shows a peak at nonzero frequencies, which would not have been identified properly in the two-sided PSD without deliberate mental calculation on the part of the reader. We assert that the one-sided PSD is much easier to interpret. The one-sided PSD in implemented in EGI s spectral analysis tools. 5 Raw EEG Data 15 Two-sided PSD Amplitude One-sided PSD Amplitude t (sec) Figure 1. (a) Raw EEG data, (b) two-sided PSD, and (c) one-sided PSD. Windowing As discussed in Press et al. (1992), having finite width frequency bins f, a direct consequence of having a finite recording interval T, results in artifacts in PSD estimation. Given the sampling interval t and the number of data points N, the discrete frequencies and their bin size f are determined. If the input signal has a single frequency which falls in the center of a bin, then the power of the signal will be represented entirely within that bin. In contrast, if the input signal has a single frequency which is not centered in a bin, which is the general case for EEG data, the power of the signal will be spread to many neighboring bins (Press et al., 1992). This effect can be minimized by windowing. Windowing refers to multiplying the input signal V(t) by a window function w(t), which is zero at the ends of the recording interval, and unity in the center. A common choice is the Hanning window 6
7 which is implemented in EGI s spectral analysis tools. Averaging As another means of improving the approximation of the PSD from discretely sampled data, we can invoke averaging. The original data set of length N is first segmented into M smaller data segments each of length L. After computing the FFT and discrete PSD for each segment, an average PSD is computed by averaging the power in each frequency bin over the M segments. Averaging capability is implemented in EGI s spectral analysis tools. Padding w( j) = 1 2π j 1 cos 2 N, j=,..., N 1 As descussed above, if N data points sampled at a rate 1/ t are fed to an FFT algorithm, one obtains values which correspond to the DFT at the discrete frequencies f=1/(n t). This seems to imply that the frequency resolution of the FFT is limited by the length of the time segment of data that one wishes to analyze, which could be a restriction if one is interested in studying changes in the PSD over time. A little-known fact is that this situation can be improved by padding the original data set with zeros. Since it seems at first glance that this would violate some conservation principle, a brief discussion of is in order. First, it is possible in principle to compute the discrete Fourier transform (DFT) for a finite data set at any frequency. The disadvantage is that the sum over the original data must then be computed explicitly, rather than via the highly efficient FFT algorithm. Second, padding with zeros does not change the value of the FFT at a particular frequency, since the additional terms included in the sum are all equal to zero. (A exception is due to the factor 1/N in the IDL defintion of FFT, but this is taken care of in our definition of PSD.) What padding does do, however, is force the FFT to evaluate the DFT at twice as many frequencies. These new frequencies all still lie between zero and the Nyquist, but there are twice as many of them, hence twice the frequency resolution. In order for the PSD to remain unchanged, however, it must be defined as an intensive quantity, as above. To demonstrate the effect of padding, Figure 2 shows the PSD for the same data set shown in Figure 1, but now plotted only to 1 Hz. Figure 2a shows the PSD computed for N=256 and no padding, resulting in a frequency resolution of f=1 Hz. Figure 2b shows the PSD computed for the same data set padded with 256 zeros, resulting in a frequency resolution of f=.5 Hz. Notice the value of the PSD at integer frequencies, e.g., and 1 Hz, are unchanged, however, as a consequence of making an intensive definition of the PSD. Figure 2c shows the PSD computed for the same data set padded with 234 zeros, resulting in a frequency resolution of f=.1 Hz. This clearly shows structure in the PSD which could not have been inferred at lower resolutions obtained without padding. For example, the niche between.5 and 1 Hz seen in Figure 2c was completely missed at lower resolutions. At the highest resolution, in Figure 2c, the point at zero frequency appears to be discontinuous with the rest of the spectrum at low frequencies. Indeed, the ratio of the first point to the zeroth point is 1.41 to a good approximation, and clearly has its origin in our definition of a one-sided PSD. While this appears 7
8 to be a disadvantage to the one-sided definition of PSD, this definition has the advantage that the plot accurately portrays the total signal power. 15 One-Sided PSDA 15 One-Sided PSDA 15 One-Sided PSDA Padding and Windowing Figure 2. (a) no padding, (b) single padding, and (c) nine-fold padding. If combining methods, windowing should be performed before padding to have the desired effect. This is because the effect of windowing in the time domain is to force both ends of the data to zero in a smooth way. If windowing were applied to the augmented data set after padding, then the last nonzero data point would align with the window in such a way that it would not be forced smoothly to zero as desired. References Press, W. H., S.A. Teukolsky, W. T. Vetterling and B. P. Flannery (1992). Numerical recipes in C: The art of scientific computing. Cambridge University Press. 8
9 Stearns, S. D. and R. A. David (1996). Signal processing algorithms in MATLAB. Prentice-Hall, New Jersey. 9
Vibration Testing. an excitation source a device to measure the response a digital signal processor to analyze the system response
Vibration Testing For vibration testing, you need an excitation source a device to measure the response a digital signal processor to analyze the system response i) Excitation sources Typically either
More informationVibration Testing. Typically either instrumented hammers or shakers are used.
Vibration Testing Vibration Testing Equipment For vibration testing, you need an excitation source a device to measure the response a digital signal processor to analyze the system response Excitation
More informationAdjustments for the Display of Quantized Ion Channel Dwell Times in Histograms with Logarithmic Bins
662 Biophysical Journal Volume 78 February 2000 662 667 Adjustments for the Display of Quantized Ion Channel Dwell Times in Histograms with Logarithmic Bins J. Alex Stark and Stephen B. Hladky National
More informationFig. 1: Fourier series Examples
FOURIER SERIES AND ITS SOME APPLICATIONS P. Sathyabama Assistant Professor, Department of Mathematics, Bharath collage of Science and Management, Thanjavur, Tamilnadu Abstract: The Fourier series, the
More informationSNR Calculation and Spectral Estimation [S&T Appendix A]
SR Calculation and Spectral Estimation [S&T Appendix A] or, How not to make a mess of an FFT Make sure the input is located in an FFT bin 1 Window the data! A Hann window works well. Compute the FFT 3
More informationCorrelation, discrete Fourier transforms and the power spectral density
Correlation, discrete Fourier transforms and the power spectral density visuals to accompany lectures, notes and m-files by Tak Igusa tigusa@jhu.edu Department of Civil Engineering Johns Hopkins University
More informationsinc function T=1 sec T=2 sec angle(f(w)) angle(f(w))
T=1 sec sinc function 3 angle(f(w)) T=2 sec angle(f(w)) 1 A quick script to plot mag & phase in MATLAB w=0:0.2:50; Real exponential func b=5; Fourier transform (filter) F=1.0./(b+j*w); subplot(211), plot(w,
More informationSignal Processing COS 323
Signal Processing COS 323 Digital Signals D: functions of space or time e.g., sound 2D: often functions of 2 spatial dimensions e.g. images 3D: functions of 3 spatial dimensions CAT, MRI scans or 2 space,
More informationUCSD SIO221c: (Gille) 1
UCSD SIO221c: (Gille) 1 Edge effects in spectra: detrending, windowing and pre-whitening One of the big challenges in computing spectra is to figure out how to deal with edge effects. Edge effects matter
More information12.1 Fourier Transform of Discretely Sampled Data
494 Chapter 12. Fast Fourier Transform now is: The PSD-per-unit-time converges to finite values at all frequencies except those where h(t) has a discrete sine-wave (or cosine-wave) component of finite
More informationANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8
ANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8 Fm n N fnt ( ) e j2mn N X() X() 2 X() X() 3 W Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 W W 3 W W x () x () x () 2 x ()
More informationSHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 7A. Power Spectral Density Function
SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 7A. Power Spectral Density Function By Tom Irvine Introduction A Fourier transform by itself is a poor format for representing random vibration because
More informationChapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum
Chapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum CEN352, DR. Nassim Ammour, King Saud University 1 Fourier Transform History Born 21 March 1768 ( Auxerre ). Died 16 May 1830 ( Paris ) French
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationComputational Data Analysis!
12.714 Computational Data Analysis! Alan Chave (alan@whoi.edu)! Thomas Herring (tah@mit.edu),! http://geoweb.mit.edu/~tah/12.714! Concentration Problem:! Today s class! Signals that are near time and band
More informationChapter 3. Data Analysis
Chapter 3 Data Analysis The analysis of the measured track data is described in this chapter. First, information regarding source and content of the measured track data is discussed, followed by the evaluation
More informationCentre for Mathematical Sciences HT 2017 Mathematical Statistics
Lund University Stationary stochastic processes Centre for Mathematical Sciences HT 2017 Mathematical Statistics Computer exercise 3 in Stationary stochastic processes, HT 17. The purpose of this exercise
More informationImage Acquisition and Sampling Theory
Image Acquisition and Sampling Theory Electromagnetic Spectrum The wavelength required to see an object must be the same size of smaller than the object 2 Image Sensors 3 Sensor Strips 4 Digital Image
More informationExperimental Fourier Transforms
Chapter 5 Experimental Fourier Transforms 5.1 Sampling and Aliasing Given x(t), we observe only sampled data x s (t) = x(t)s(t; T s ) (Fig. 5.1), where s is called sampling or comb function and can be
More informationDegrees-of-freedom estimation in the Student-t noise model.
LIGO-T0497 Degrees-of-freedom estimation in the Student-t noise model. Christian Röver September, 0 Introduction The Student-t noise model was introduced in [] as a robust alternative to the commonly used
More informationInformation and Communications Security: Encryption and Information Hiding
Short Course on Information and Communications Security: Encryption and Information Hiding Tuesday, 10 March Friday, 13 March, 2015 Lecture 5: Signal Analysis Contents The complex exponential The complex
More informationwhich arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i
MODULE 6 Topics: Gram-Schmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.
More informationData Sampling Techniques for Fourier Analysis
Session 3226 Data Sampling Techniques for Fourier Analysis John Hartin, Kenneth Belanus University of Pittsburgh at Johnstown/Oklahoma State University Abstract Fourier analysis methods and data sampling
More informationFourier transforms of molecular vibrations
Fourier transforms of molecular vibrations Part I: An Introduction to the Harmonic Oscillator and Fourier Transforms W. Tandy Grubbs, Department of Chemistry, Unit 827, Stetson University, DeLand, FL 32720
More informationChapter 2 Fourier Series Phase Object Spectra
PhD University of Edinburgh 99 Phase-Only Optical Information Processing D. J. Potter Index Chapter 3 4 5 6 7 8 9 Chapter Fourier Series Phase Object Spectra In chapter one, it was noted how one may describe
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationIB Paper 6: Signal and Data Analysis
IB Paper 6: Signal and Data Analysis Handout 5: Sampling Theory S Godsill Signal Processing and Communications Group, Engineering Department, Cambridge, UK Lent 2015 1 / 85 Sampling and Aliasing All of
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal rocessing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts
More informationJim Lambers ENERGY 281 Spring Quarter Lecture 5 Notes
Jim ambers ENERGY 28 Spring Quarter 27-8 ecture 5 Notes These notes are based on Rosalind Archer s PE28 lecture notes, with some revisions by Jim ambers. Fourier Series Recall that in ecture 2, when we
More informationReview of Linear System Theory
Review of Linear System Theory The following is a (very) brief review of linear system theory and Fourier analysis. I work primarily with discrete signals. I assume the reader is familiar with linear algebra
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts
More informationchapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS
chapter MORE MATRIX ALGEBRA GOALS In Chapter we studied matrix operations and the algebra of sets and logic. We also made note of the strong resemblance of matrix algebra to elementary algebra. The reader
More informationError Reporting Recommendations: A Report of the Standards and Criteria Committee
Error Reporting Recommendations: A Report of the Standards and Criteria Committee Adopted by the IXS Standards and Criteria Committee July 26, 2000 1. Introduction The development of the field of x-ray
More informationComputer Vision Lecture 3
Computer Vision Lecture 3 Linear Filters 03.11.2015 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de leibe@vision.rwth-aachen.de Demo Haribo Classification Code available on the class website...
More informationLecture 3: Linear Filters
Lecture 3: Linear Filters Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Images as functions Linear systems (filters) Convolution and correlation Discrete Fourier Transform (DFT)
More informationThe Discrete Fourier Transform
In [ ]: cd matlab pwd The Discrete Fourier Transform Scope and Background Reading This session introduces the z-transform which is used in the analysis of discrete time systems. As for the Fourier and
More informationThe discrete Fourier transform
3 The discrete Fourier transform Page 1 3.1 Introduction Let f(t) be a function describing a physical process in the time domain. An equivalent representation of the process, F (ν), is possible in the
More informationINTRODUCTION TO THE DFS AND THE DFT
ITRODUCTIO TO THE DFS AD THE DFT otes: This brief handout contains in very brief outline form the lecture notes used for a video lecture in a previous year introducing the DFS and the DFT. This material
More informationEach of these functions represents a signal in terms of its spectral components in the frequency domain.
N INTRODUCTION TO SPECTRL FUNCTIONS Revision B By Tom Irvine Email: tomirvine@aol.com March 3, 000 INTRODUCTION This tutorial presents the Fourier transform. It also discusses the power spectral density
More informationNotes on FFT-based differentiation
Notes on FFT-based differentiation Steven G Johnson, MIT Applied Mathematics Created April, 2011, updated May 4, 2011 Abstract A common numerical technique is to differentiate some sampled function y(x)
More informationTutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling
2.39 utorial Sheet #2 discrete vs. continuous functions, periodicity, sampling We will encounter two classes of signals in this class, continuous-signals and discrete-signals. he distinct mathematical
More informationThe Hilbert Transform
The Hilbert Transform David Hilbert 1 ABSTRACT: In this presentation, the basic theoretical background of the Hilbert Transform is introduced. Using this transform, normal real-valued time domain functions
More informationReview of Linear Systems Theory
Review of Linear Systems Theory The following is a (very) brief review of linear systems theory, convolution, and Fourier analysis. I work primarily with discrete signals, but each result developed in
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Filtering in the Frequency Domain http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Background
More informationVII. Bandwidth Limited Time Series
VII. Bandwidth Limited Time Series To summarize the discussion up to this point: (1) In the general case of the aperiodic time series, which is infinite in time and frequency, both the time series and
More informationFFT, total energy, and energy spectral density computations in MATLAB Aaron Scher
FFT, total energy, and energy spectral density computations in MATLAB Aaron Scher Everything presented here is specifically focused on non-periodic signals with finite energy (also called energy signals
More informationL6: 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
More informationLecture 3: Linear Filters
Lecture 3: Linear Filters Professor Fei Fei Li Stanford Vision Lab 1 What we will learn today? Images as functions Linear systems (filters) Convolution and correlation Discrete Fourier Transform (DFT)
More informationFourier analysis of discrete-time signals. (Lathi Chapt. 10 and these slides)
Fourier analysis of discrete-time signals (Lathi Chapt. 10 and these slides) Towards the discrete-time Fourier transform How we will get there? Periodic discrete-time signal representation by Discrete-time
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
More informationMath 56 Homework 5 Michael Downs
1. (a) Since f(x) = cos(6x) = ei6x 2 + e i6x 2, due to the orthogonality of each e inx, n Z, the only nonzero (complex) fourier coefficients are ˆf 6 and ˆf 6 and they re both 1 2 (which is also seen from
More informationA523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011
A523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011 Lecture 6 PDFs for Lecture 1-5 are on the web page Problem set 2 is on the web page Article on web page A Guided
More informationω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the
he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus
More informationBernstein polynomials of degree N are defined by
SEC. 5.5 BÉZIER CURVES 309 5.5 Bézier Curves Pierre Bézier at Renault and Paul de Casteljau at Citroën independently developed the Bézier curve for CAD/CAM operations, in the 1970s. These parametrically
More information1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
More informationFourier Analysis of Signals Using the DFT
Fourier Analysis of Signals Using the DFT ECE 535 Lecture April 29, 23 Overview: Motivation Many applications require analyzing the frequency content of signals Speech processing study resonances of vocal
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 12 Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial,
More informationSpectra (2A) Young Won Lim 11/8/12
Spectra (A) /8/ Copyright (c) 0 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 4 Digital Signal Processing Pro. Mark Fowler ote Set # Using the DFT or Spectral Analysis o Signals Reading Assignment: Sect. 7.4 o Proakis & Manolakis Ch. 6 o Porat s Book /9 Goal o Practical Spectral
More informationTransform Representation of Signals
C H A P T E R 3 Transform Representation of Signals and LTI Systems As you have seen in your prior studies of signals and systems, and as emphasized in the review in Chapter 2, transforms play a central
More informationThis pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication.
This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication. Copyright Pearson Canada Inc. All rights reserved. Copyright Pearson
More informationChirp Transform for FFT
Chirp Transform for FFT Since the FFT is an implementation of the DFT, it provides a frequency resolution of 2π/N, where N is the length of the input sequence. If this resolution is not sufficient in a
More informationThe Simplex Method: An Example
The Simplex Method: An Example Our first step is to introduce one more new variable, which we denote by z. The variable z is define to be equal to 4x 1 +3x 2. Doing this will allow us to have a unified
More informationSystem Modeling and Identification CHBE 702 Korea University Prof. Dae Ryook Yang
System Modeling and Identification CHBE 702 Korea University Prof. Dae Ryook Yang 1-1 Course Description Emphases Delivering concepts and Practice Programming Identification Methods using Matlab Class
More informationCS711008Z Algorithm Design and Analysis
CS711008Z Algorithm Design and Analysis Lecture 5 FFT and Divide and Conquer Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 56 Outline DFT: evaluate a polynomial
More informationENGIN 100: Music Signal Processing. LAB #5: Spectrograms
ENGIN : Music Signal Processing LAB #5: Spectrograms Professor Andrew E. Yagle Dept. of EECS, The University of Michigan, Ann Arbor, MI 489-222 I. ABSTRACT This lab will teach you how to use another tool
More information1.4 Fourier Analysis. Random Data 17
Random Data 17 1.4 Fourier Analysis In the physical world we sample (and acquire) signals in either the temporal domain or the spatial domain. However, we are frequently interested in thefrequencycontentof
More informationThe Discrete Fourier Transform (DFT) Properties of the DFT DFT-Specic Properties Power spectrum estimate. Alex Sheremet.
4. April 2, 27 -order sequences Measurements produce sequences of numbers Measurement purpose: characterize a stochastic process. Example: Process: water surface elevation as a function of time Parameters:
More informationExamples of linear systems and explanation of the term linear. is also a solution to this equation.
. Linear systems Examples of linear systems and explanation of the term linear. () ax b () a x + a x +... + a x b n n Illustration by another example: The equation x x + 5x 7 has one solution as x 4, x
More informationIn this Lecture. Frequency domain analysis
In this Lecture Frequency domain analysis Introduction In most cases we want to know the frequency content of our signal Why? Most popular analysis in frequency domain is based on work of Joseph Fourier
More informationDHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A
DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete
More informationDiscrete Wavelet Transform
Discrete Wavelet Transform [11] Kartik Mehra July 2017 Math 190s Duke University "1 Introduction Wavelets break signals up and then analyse them separately with a resolution that is matched with scale.
More informationFourier Series. (Com S 477/577 Notes) Yan-Bin Jia. Nov 29, 2016
Fourier Series (Com S 477/577 otes) Yan-Bin Jia ov 9, 016 1 Introduction Many functions in nature are periodic, that is, f(x+τ) = f(x), for some fixed τ, which is called the period of f. Though function
More informationE : Lecture 1 Introduction
E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation
More informationLecture 20: Discrete Fourier Transform and FFT
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept of Electrical Engineering Lecture 20: Discrete Fourier Transform and FFT Dec 10, 2001 Prof: J Bilmes TA:
More informationDOING PHYSICS WITH MATLAB FOURIER ANALYSIS FOURIER TRANSFORMS
DOING PHYSICS WITH MATLAB FOURIER ANALYSIS FOURIER TRANSFORMS Ian Cooper School of Physics, University of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS maths_ft_01.m mscript used
More information/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by
Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,
More informationEngineering Physics 3W4 "Acquisition and Analysis of Experimental Information" Part II: Fourier Transforms
Engineering Physics 3W4 "Acquisition and Analysis of Experimental Information" Part II: Fourier Transforms prepared by: Wm. J. Garland, Professor Department of Engineering Physics McMaster University Hamilton,
More informationNumerical Methods I Orthogonal Polynomials
Numerical Methods I Orthogonal Polynomials Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 Nov. 4th and 11th, 2010 A. Donev (Courant Institute)
More informationRadar Systems Engineering Lecture 3 Review of Signals, Systems and Digital Signal Processing
Radar Systems Engineering Lecture Review of Signals, Systems and Digital Signal Processing Dr. Robert M. O Donnell Guest Lecturer Radar Systems Course Review Signals, Systems & DSP // Block Diagram of
More informationOSE801 Engineering System Identification. Lecture 05: Fourier Analysis
OSE81 Engineering System Identification Lecture 5: Fourier Analysis What we will study in this lecture: A short introduction of Fourier analysis Sampling the data Applications Example 1 Fourier Analysis
More informationDesigning Information Devices and Systems I Spring 2018 Lecture Notes Note Introduction to Linear Algebra the EECS Way
EECS 16A Designing Information Devices and Systems I Spring 018 Lecture Notes Note 1 1.1 Introduction to Linear Algebra the EECS Way In this note, we will teach the basics of linear algebra and relate
More informationINTRODUCTION TO DELTA-SIGMA ADCS
ECE37 Advanced Analog Circuits INTRODUCTION TO DELTA-SIGMA ADCS Richard Schreier richard.schreier@analog.com NLCOTD: Level Translator VDD > VDD2, e.g. 3-V logic? -V logic VDD < VDD2, e.g. -V logic? 3-V
More informationContinuous Time Signal Analysis: the Fourier Transform. Lathi Chapter 4
Continuous Time Signal Analysis: the Fourier Transform Lathi Chapter 4 Topics Aperiodic signal representation by the Fourier integral (CTFT) Continuous-time Fourier transform Transforms of some useful
More informationSome of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e
Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More information000=0 001=1 010=2 011=3 100=4 101=5 110=6 111=7 000=0 100=4 010=2 110=6 001=1 101=5 011=3 111=7
8 CHAPTER : Fourier Series The second idea behind the FFT concerns how to get the correct h k s into the final DFT s of size. If the subscripts to 7 of the h k s are written in binary form, as shown in
More informationTime and Spatial Series and Transforms
Time and Spatial Series and Transforms Z- and Fourier transforms Gibbs' phenomenon Transforms and linear algebra Wavelet transforms Reading: Sheriff and Geldart, Chapter 15 Z-Transform Consider a digitized
More informationSignals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters
Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (
More information6.003: Signal Processing
6.003: Signal Processing Discrete Fourier Transform Discrete Fourier Transform (DFT) Relations to Discrete-Time Fourier Transform (DTFT) Relations to Discrete-Time Fourier Series (DTFS) October 16, 2018
More informationIn-Structure Response Spectra Development Using Complex Frequency Analysis Method
Transactions, SMiRT-22 In-Structure Response Spectra Development Using Complex Frequency Analysis Method Hadi Razavi 1,2, Ram Srinivasan 1 1 AREVA, Inc., Civil and Layout Department, Mountain View, CA
More informationDepartment of Electrical and Telecommunications Engineering Technology TEL (718) FAX: (718) Courses Description:
NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York 300 Jay Street Brooklyn, NY 11201-2983 Department of Electrical and Telecommunications Engineering Technology TEL (718) 260-5300 - FAX:
More information1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
More informationLecture 7 Random Signal Analysis
Lecture 7 Random Signal Analysis 7. Introduction to Probability 7. Amplitude Distributions 7.3 Uniform, Gaussian, and Other Distributions 7.4 Power and Power Density Spectra 7.5 Properties of the Power
More informationThe discrete and fast Fourier transforms
The discrete and fast Fourier transforms Marcel Oliver Revised April 7, 1 1 Fourier series We begin by recalling the familiar definition of the Fourier series. For a periodic function u: [, π] C, we define
More informationSampling in 1D ( ) Continuous time signal f(t) Discrete time signal. f(t) comb
Sampling in 2D 1 Sampling in 1D Continuous time signal f(t) Discrete time signal t ( ) f [ k] = f( kt ) = f( t) δ t kt s k s f(t) comb k 2 Nyquist theorem (1D) At least 2 sample/period are needed to represent
More informationEE 435. Lecture 32. Spectral Performance Windowing
EE 435 Lecture 32 Spectral Performance Windowing . Review from last lecture. Distortion Analysis T 0 T S THEOREM?: If N P is an integer and x(t) is band limited to f MAX, then 2 Am Χ mnp 1 0 m h N and
More informationDistortion Analysis T
EE 435 Lecture 32 Spectral Performance Windowing Spectral Performance of Data Converters - Time Quantization - Amplitude Quantization Quantization Noise . Review from last lecture. Distortion Analysis
More information