DOCSLIB.ORG

Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation electronics Article Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation Khaled Gharaibeh Department of Civil and Architectural Engineering, Applied Science University-Bahrain, Al Ekir 3201, Bahrain; [email protected] Abstract: Passive Intermodulation (PIM) distortion is a major problem in wireless communications which limits cell coverage and data rates. Passive nonlinearities result in weak intermodulation (IMD) products that are very difficult to diagnose, troubleshoot and model. To predict PIM, behavioral models are used where spectral components of PIM are estimated from the power spectral density at the output of the model. The primary goal of this paper is to study the effect of window functions on the capability of the power spectral density computed from the periodogram of signal realizations to predict low power PIM components in a wideband multichannel communication system. Different window functions are analyzed and it is shown that windows with high side-lobe level fail to predict PIM components which are close to the main channels due to their high spectral leakage; while window functions with low roll-off rate of the side lobes fail to predict low power higher order PIM components especially when the frequency separation between the main carriers is high. These results are supported by simulations of the power spectral density computed using signal realizations of an LTE carrier aggregated system. Keywords: passive intermodulation; piecewise linear model; LTE; behavioral models; threshold decomposition Citation: Gharaibeh, K. Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation. Electronics 2021, 10, 1. Introduction 1034. https://doi.org/10.3390/ Passive intermodulation (PIM) distortion has been a significant problem in modern electronics10091034 mobile communication systems as it limits system performance and capacity. Unlike active nonlinear distortion which is usually manifested as intermodulation (IMD) products with Academic Editor: Yasir Al-Yasir significant power levels, passive nonlinearities result in very weak IMD products which make it very difficult to measure and model [1]. In carrier aggregated systems such as 4G Received: 21 March 2021 systems, PIM is manifested as IMD components which result in interference with the main Accepted: 18 April 2021 Published: 27 April 2021 channels in the receive band. A simple illustration of PIM in wireless systems is shown in Figure1a where the input f Publisher’s Note: MDPI stays neutral to the passive nonlinearity consists of two band pass signals centered at frequencies 1 with regard to jurisdictional claims in and f2. The power spectral density (PSD) of the output of the nonlinearity is shown in published maps and institutional affil- Figure1b which consists of the spectra at the fundamental frequencies in addition to the iations. IMD components at frequencies 2 f1 − f2 and 2 f2 − f1 which may lie within the receive band 2. In this scenario, the IMD components produced by the passive nonlinearity can result in significant interference in the receive band which limits system performance and capacity. To predict PIM in in such scenarios, behavioral models that relate the output PIM to Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. the input signal level and frequency are used [2–10]. Behavioral models are developed This article is an open access article using measured characteristics of the nonlinear device and then they can be used to predict distributed under the terms and PIM from the computation of the power spectrum at the output of the model. conditions of the Creative Commons Estimating PIM from the power spectral density at the output of the behavioral model Attribution (CC BY) license (https:// using direct application of the Fast Fourier Transform (FFT) to a signal realization can be creativecommons.org/licenses/by/ a simple task when the power level of the IMD components is high as the case of active 4.0/). nonlinearities (above −60 dBc). However, with passive nonlinearity, the IMD components Electronics 2021, 10, 1034. https://doi.org/10.3390/electronics10091034 https://www.mdpi.com/journal/electronics Electronics 2021, 10, 1034 2 of 12 can have power levels of less than −140 dBc which makes it difficult to predict PIM [7,8]. The reason for this is that the finite length of signal realizations results in high levels of spectral leakage in the computed power spectrum which tends to obscure the IMD Electronics 2021, 10, x FOR PEER REVIEWcomponents as spectral leakage level are usually much higher than the intermodulation2 of 12 power level. (a) 2f1 − f2 f1 f2 2f2 − f1 (b) FigureFigure 1. 1.Spectra Spectra in in a widebanda wideband receiver: receiver: (a ()a Input) Input spectrum spectrum of of two two RF RF signals signals to to a a passive passive nonlin- non- earitylinearity and (andb) the (b Output) the Output spectrum; spectrum; dashed dashed line indicates line indicates PSD computed PSD computed using FFTusing with FFT rectangular with rec- tangular window indicating spectral leakage. window indicating spectral leakage. FigureTo predict1b shows PIM an in illustration in such scenarios, of this phenomenon behavioral models where spectral that relate leakage the output that results PIM to fromthe theinput Fast signal Fourier level Transform and frequency (FFT) makesare used it impossible[2–10]. Behavioral to predict models IMD componentsare developed fromusing weak measured passive characteristics nonlinearities. of In the principle, nonlinear the device frequency and then of intermodulation they can be used spectra to pre- anddict their PIM power from the level computation depend on of the the frequency power spectrum separation at the between output theof the main model. channels and theEstimating input power PIM and, from thus, the it power is important spectral to usedensity an appropriate at the output spectral of the estimation behavioral algorithmmodel using so that direct these application IMD components of the Fast can Fourier be predicted. Transform (FFT) to a signal realization canSpectral be a simple leakage task canwhen be the minimized power level through of the windowing IMD components where signalis high samples as the case are of weightedactive nonlinearities by a given time (above window −60 which dBc). enhances However, the with accuracy passive of spectral nonlinearity, estimation the [ 11IMD]. Thecomponents type of the can window have haspower a significant levels of less impact than on −140 spectral dBc which leakage makes where it low-peakdifficult to side pre- lobedict of PIM the [7,8]. frequency The reason response for ofthis the is windowthat the finite function length indicates of signal small realizations spectral leakage.results in Severalhigh levels window of spectral functions leakage have in been the usedcomputed in the power literature spectrum for suppression which tends of to spectral obscure leakagethe IMD [12 –components14]. However, as the spectral choice leakage of the window level are function usually is much a compromise higher than between the inter- the amountmodulation of spectral power leakage level. and the frequency resolution and, hence, the computational complexityFigure of 1b computing shows an the illustration windowed of FFTthis [phenomenon3]. where spectral leakage that re- sultsThis from problem the Fast has Fourier been studied Transform in the (FFT) context makes of prediction it impossible and reduction to predict of IMD harmonics compo- innents power from systems weak where passive it was nonlinearities. shown that In the principle, direct application the frequency of the of FFT intermodulation has inherent performancespectra and limitations, their power such level as spectral depend leakage on the and frequency the picket separation fence effect between [11]. However, the main limedchannels research and hasthe beeninput done power on and, the effect thus, ofit selectionis important of window to use an functions appropriate on spectral spectral identificationestimation algorithm of PIM. This so that paper these aims IMD to components provide an assessmentcan be predicted. of the performance of variousSpectral window leakage functions can onbe inminimized the ability through of the power windowing spectral where density signal to predict samples the are IMDweighted components by a given in a wideband time window system. which The significanceenhances the of thisaccuracy research of spectral in that the estimation results presented[11]. The cantype be of used the window in the design has a ofsignificant simulation impact models on spectral and measurements leakage where setups low-peak for accurateside lobe prediction of the frequency or measurement response of PIM of the in wireless window systems. function indicates small spectral leakage.This paper Several provides window an analysisfunctions of have passive been nonlinearity used in the in aliterature carrier aggregated for suppression system of usingspectral a polynomial leakage [12–14]. based behavioralHowever, modelthe choice and of develops the window the output function of the is nonlinearitya compromise forbetween two carrier the aggregatedamount of spectral signals whichleakage represents and the thefrequency practical resolution scenario byand, which hence, PIM the computational complexity of computing the windowed FFT [3]. This problem has been studied in the context of prediction and reduction of har- monics in power systems where
Load more

Recommended publications
  • Spectrum Estimation Using Periodogram, Bartlett and Welch
    Spectrum estimation using Periodogram, Bartlett and Welch Guido Schuster Slides follow closely chapter 8 in the book „Statistical Digital Signal Processing and Modeling“ by Monson H. Hayes and most of the figures and formulas are taken from there 1 Introduction • We want to estimate the power spectral density of a wide- sense stationary random process • Recall that the power spectrum is the Fourier transform of the autocorrelation sequence • For an ergodic process the following holds 2 Introduction • The main problem of power spectrum estimation is – The data x(n) is always finite! • Two basic approaches – Nonparametric (Periodogram, Bartlett and Welch) • These are the most common ones and will be presented in the next pages – Parametric approaches • not discussed here since they are less common 3 Nonparametric methods • These are the most commonly used ones • x(n) is only measured between n=0,..,N-1 • Ensures that the values of x(n) that fall outside the interval [0,N-1] are excluded, where for negative values of k we use conjugate symmetry 4 Periodogram • Taking the Fourier transform of this autocorrelation estimate results in an estimate of the power spectrum, known as the Periodogram • This can also be directly expressed in terms of the data x(n) using the rectangular windowed function xN(n) 5 Periodogram of white noise 32 samples 6 Performance of the Periodogram • If N goes to infinity, does the Periodogram converge towards the power spectrum in the mean squared sense? • Necessary conditions – asymptotically unbiased: – variance
    [Show full text]
  • Configuring Spectrogram Views
    Chapter 5: Spectrographic Analysis Configuring spectrographic views To create a new spectrogram, spectrogram slice, or selection spectrum view, click on the appropriate New View button in the view toolbar (Figure 5.3) or choose a view type from the View > New menu. Figure 5.3. New View buttons Figure 5.3. The New View buttons, in Raven’s view toolbar. 112 Raven Pro 1.4 User’s Manual Chapter 5: Spectrographic Analysis A dialog box appears, containing parameters for configuring the requested type of spectrographic view (Figure 5.4). The dialog boxes for configuring spectrogram and spectrogram slice views are identical, except for their titles. The dialog boxes are identical because both view types calculate a spectrogram of the entire sound; the only difference between spectrogram and spectrogram slice views is in how the data are displayed (see “How the spectrographic views are related” on page 110). The dialog box for configuring a selection spectrum view is the same, except that it lacks the Averaging parameter. The remainder of this section explains each of the parameters in the configuration dialog box. Figure 5.4 Configure Spectrogram dialog Figure 5.4. The Configure New Spectrogram dialog box. Window type Each data record is multiplied by a window function before its spectrum is calculated. Window functions are used to reduce the magnitude of spurious “sidelobe” energy that appears at frequencies flanking each analysis frequency in a spectrum. These sidelobes appear as a result of Raven Pro 1.4 User’s Manual 113 Chapter 5: Spectrographic Analysis analyzing a finite (truncated) portion of a signal.
    [Show full text]
  • An Information Theoretic Algorithm for Finding Periodicities in Stellar Light Curves Pablo Huijse, Student Member, IEEE, Pablo A
    IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 1, NO. 1, JANUARY 2012 1 An Information Theoretic Algorithm for Finding Periodicities in Stellar Light Curves Pablo Huijse, Student Member, IEEE, Pablo A. Estevez*,´ Senior Member, IEEE, Pavlos Protopapas, Pablo Zegers, Senior Member, IEEE, and Jose´ C. Pr´ıncipe, Fellow Member, IEEE Abstract—We propose a new information theoretic metric aligned to Earth. Periodic drops in brightness are observed for finding periodicities in stellar light curves. Light curves due to the mutual eclipses between the components of the are astronomical time series of brightness over time, and are system. Although most stars have at least some variation in characterized as being noisy and unevenly sampled. The proposed metric combines correntropy (generalized correlation) with a luminosity, current ground based survey estimations indicate periodic kernel to measure similarity among samples separated that 3% of the stars varying more than the sensitivity of the by a given period. The new metric provides a periodogram, called instruments and ∼1% are periodic [2]. Correntropy Kernelized Periodogram (CKP), whose peaks are associated with the fundamental frequencies present in the data. Detecting periodicity and estimating the period of stars is The CKP does not require any resampling, slotting or folding of high importance in astronomy. The period is a key feature scheme as it is computed directly from the available samples. for classifying variable stars [3], [4]; and estimating other CKP is the main part of a fully-automated pipeline for periodic light curve discrimination to be used in astronomical survey parameters such as mass and distance to Earth [5].
    [Show full text]
  • Windowing Techniques, the Welch Method for Improvement of Power Spectrum Estimation
    Computers, Materials & Continua Tech Science Press DOI:10.32604/cmc.2021.014752 Article Windowing Techniques, the Welch Method for Improvement of Power Spectrum Estimation Dah-Jing Jwo1, *, Wei-Yeh Chang1 and I-Hua Wu2 1Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University, Keelung, 202-24, Taiwan 2Innovative Navigation Technology Ltd., Kaohsiung, 801, Taiwan *Corresponding Author: Dah-Jing Jwo. Email: [email protected] Received: 01 October 2020; Accepted: 08 November 2020 Abstract: This paper revisits the characteristics of windowing techniques with various window functions involved, and successively investigates spectral leak- age mitigation utilizing the Welch method. The discrete Fourier transform (DFT) is ubiquitous in digital signal processing (DSP) for the spectrum anal- ysis and can be efciently realized by the fast Fourier transform (FFT). The sampling signal will result in distortion and thus may cause unpredictable spectral leakage in discrete spectrum when the DFT is employed. Windowing is implemented by multiplying the input signal with a window function and windowing amplitude modulates the input signal so that the spectral leakage is evened out. Therefore, windowing processing reduces the amplitude of the samples at the beginning and end of the window. In addition to selecting appropriate window functions, a pretreatment method, such as the Welch method, is effective to mitigate the spectral leakage. Due to the noise caused by imperfect, nite data, the noise reduction from Welch’s method is a desired treatment. The nonparametric Welch method is an improvement on the peri- odogram spectrum estimation method where the signal-to-noise ratio (SNR) is high and mitigates noise in the estimated power spectra in exchange for frequency resolution reduction.
    [Show full text]
  • Windowing Design and Performance Assessment for Mitigation of Spectrum Leakage
    E3S Web of Conferences 94, 03001 (2019) https://doi.org/10.1051/e3sconf/20199403001 ISGNSS 2018 Windowing Design and Performance Assessment for Mitigation of Spectrum Leakage Dah-Jing Jwo 1,*, I-Hua Wu 2 and Yi Chang1 1 Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University 2 Pei-Ning Rd., Keelung 202, Taiwan 2 Bison Electronics, Inc., Neihu District, Taipei 114, Taiwan Abstract. This paper investigates the windowing design and performance assessment for mitigation of spectral leakage. A pretreatment method to reduce the spectral leakage is developed. In addition to selecting appropriate window functions, the Welch method is introduced. Windowing is implemented by multiplying the input signal with a windowing function. The periodogram technique based on Welch method is capable of providing good resolution if data length samples are selected optimally. Windowing amplitude modulates the input signal so that the spectral leakage is evened out. Thus, windowing reduces the amplitude of the samples at the beginning and end of the window, altering leakage. The influence of various window functions on the Fourier transform spectrum of the signals was discussed, and the characteristics and functions of various window functions were explained. In addition, we compared the differences in the influence of different data lengths on spectral resolution and noise levels caused by the traditional power spectrum estimation and various window-function-based Welch power spectrum estimations. 1 Introduction knowledge. Due to computer capacity and limitations on processing time, only limited time samples can actually The spectrum analysis [1-4] is a critical concern in many be processed during spectrum analysis and data military and civilian applications, such as performance processing of signals, i.e., the original signals need to be test of electric power systems and communication cut off, which induces leakage errors.
    [Show full text]
  • Discrete-Time Fourier Transform
    .@.$$-@@4&@$QEG'FC1. CHAPTER7 Discrete-Time Fourier Transform In Chapter 3 and Appendix C, we showed that interesting continuous-time waveforms x(t) can be synthesized by summing sinusoids, or complex exponential signals, having different frequencies fk and complex amplitudes ak. We also introduced the concept of the spectrum of a signal as the collection of information about the frequencies and corresponding complex amplitudes {fk,ak} of the complex exponential signals, and found it convenient to display the spectrum as a plot of spectrum lines versus frequency, each labeled with amplitude and phase. This spectrum plot is a frequency-domain representation that tells us at a glance “how much of each frequency is present in the signal.” In Chapter 4, we extended the spectrum concept from continuous-time signals x(t) to discrete-time signals x[n] obtained by sampling x(t). In the discrete-time case, the line spectrum is plotted as a function of normalized frequency ωˆ . In Chapter 6, we developed the frequency response H(ejωˆ ) which is the frequency-domain representation of an FIR filter. Since an FIR filter can also be characterized in the time domain by its impulse response signal h[n], it is not hard to imagine that the frequency response is the frequency-domain representation,orspectrum, of the sequence h[n]. 236 .@.$$-@@4&@$QEG'FC1. 7-1 DTFT: FOURIER TRANSFORM FOR DISCRETE-TIME SIGNALS 237 In this chapter, we take the next step by developing the discrete-time Fouriertransform (DTFT). The DTFT is a frequency-domain representation for a wide range of both finite- and infinite-length discrete-time signals x[n].
    [Show full text]
  • Understanding the Lomb-Scargle Periodogram
    Understanding the Lomb-Scargle Periodogram Jacob T. VanderPlas1 ABSTRACT The Lomb-Scargle periodogram is a well-known algorithm for detecting and char- acterizing periodic signals in unevenly-sampled data. This paper presents a conceptual introduction to the Lomb-Scargle periodogram and important practical considerations for its use. Rather than a rigorous mathematical treatment, the goal of this paper is to build intuition about what assumptions are implicit in the use of the Lomb-Scargle periodogram and related estimators of periodicity, so as to motivate important practical considerations required in its proper application and interpretation. Subject headings: methods: data analysis | methods: statistical 1. Introduction The Lomb-Scargle periodogram (Lomb 1976; Scargle 1982) is a well-known algorithm for de- tecting and characterizing periodicity in unevenly-sampled time-series, and has seen particularly wide use within the astronomy community. As an example of a typical application of this method, consider the data shown in Figure 1: this is an irregularly-sampled timeseries showing a single ob- ject from the LINEAR survey (Sesar et al. 2011; Palaversa et al. 2013), with un-filtered magnitude measured 280 times over the course of five and a half years. By eye, it is clear that the brightness of the object varies in time with a range spanning approximately 0.8 magnitudes, but what is not immediately clear is that this variation is periodic in time. The Lomb-Scargle periodogram is a method that allows efficient computation of a Fourier-like power spectrum estimator from such unevenly-sampled data, resulting in an intuitive means of determining the period of oscillation.
    [Show full text]
  • Effect of Data Gaps: Comparison of Different Spectral Analysis Methods
    Ann. Geophys., 34, 437–449, 2016 www.ann-geophys.net/34/437/2016/ doi:10.5194/angeo-34-437-2016 © Author(s) 2016. CC Attribution 3.0 License. Effect of data gaps: comparison of different spectral analysis methods Costel Munteanu1,2,3, Catalin Negrea1,4,5, Marius Echim1,6, and Kalevi Mursula2 1Institute of Space Science, Magurele, Romania 2Astronomy and Space Physics Research Unit, University of Oulu, Oulu, Finland 3Department of Physics, University of Bucharest, Magurele, Romania 4Cooperative Institute for Research in Environmental Sciences, Univ. of Colorado, Boulder, Colorado, USA 5Space Weather Prediction Center, NOAA, Boulder, Colorado, USA 6Belgian Institute of Space Aeronomy, Brussels, Belgium Correspondence to: Costel Munteanu ([email protected]) Received: 31 August 2015 – Revised: 14 March 2016 – Accepted: 28 March 2016 – Published: 19 April 2016 Abstract. In this paper we investigate quantitatively the ef- 1 Introduction fect of data gaps for four methods of estimating the ampli- tude spectrum of a time series: fast Fourier transform (FFT), Spectral analysis is a widely used tool in data analysis and discrete Fourier transform (DFT), Z transform (ZTR) and processing in most fields of science. The technique became the Lomb–Scargle algorithm (LST). We devise two tests: the very popular with the introduction of the fast Fourier trans- single-large-gap test, which can probe the effect of a sin- form algorithm which allowed for an extremely rapid com- gle data gap of varying size and the multiple-small-gaps test, putation of the Fourier Transform. In the absence of modern used to study the effect of numerous small gaps of variable supercomputers, this was not just useful, but also the only size distributed within the time series.
    [Show full text]
  • Optimal and Adaptive Subband Beamforming
    Optimal and Adaptive Subband Beamforming Principles and Applications Nedelko Grbi´c Ronneby, June 2001 Department of Telecommunications and Signal Processing Blekinge Institute of Technology, S-372 25 Ronneby, Sweden c Nedelko Grbi´c ISBN 91-7295-002-1 ISSN 1650-2159 Published 2001 Printed by Kaserntryckeriet AB Karlskrona 2001 Sweden v Preface This doctoral thesis summarizes my work in the field of array signal processing. Frequency domain processing comprise a significant part of the material. The work is mainly aimed at speech enhancement in communication systems such as confer- ence telephony and handsfree mobile telephony. The work has been carried out at Department of Telecommunications and Signal Processing at Blekinge Institute of Technology. Some of the work has been conducted in collaboration with Ericsson Mobile Communications and Australian Telecommunications Research Institute. The thesis consists of seven stand alone parts: Part I Optimal and Adaptive Beamforming for Speech Signals Part II Structures and Performance Limits in Subband Beamforming Part III Blind Signal Separation using Overcomplete Subband Representation Part IV Neural Network based Adaptive Microphone Array System for Speech Enhancement Part V Design of Oversampled Uniform DFT Filter Banks with Delay Specifica- tion using Quadratic Optimization Part VI Design of Oversampled Uniform DFT Filter Banks with Reduced Inband Aliasing and Delay Constraints Part VII A New Pilot-Signal based Space-Time Adaptive Algorithm vii Acknowledgments I owe sincere gratitude to Professor Sven Nordholm with whom most of my work has been conducted. There have been enormous amount of inspiration and valuable discussions with him. Many thanks also goes to Professor Ingvar Claesson who has made every possible effort to make my study complete and rewarding.
    [Show full text]
  • Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation Comparison with Traditional Method
    Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation Comparison with Traditional Method Linus Ekvall Engineering Physics and Electrical Engineering, master's level 2018 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering Airborne Radar Ground Clutter Suppression Using Multitaper Spectrum Estimation & Comparison with Traditional Method Linus C. Ekvall Lule˚aUniversity of Technology Dept. of Computer Science, Electrical and Space Engineering Div. Signals and Systems 20th September 2018 ABSTRACT During processing of data received by an airborne radar one of the issues is that the typical signal echo from the ground produces a large perturbation. Due to this perturbation it can be difficult to detect targets with low velocity or a low signal-to-noise ratio. Therefore, a filtering process is needed to separate the large perturbation from the target signal. The traditional method include a tapered Fourier transform that operates in parallel with a MTI filter to suppress the main spectral peak in order to produce a smoother spectral output. The difference between a typical signal echo produced from an object in the environment and the signal echo from the ground can be of a magnitude corresponding to more than a 60 dB difference. This thesis presents research of how the multitaper approach can be utilized in concurrence with the minimum variance estimation technique, to produce a spectral estimation that strives for a more effective clutter suppression. A simulation model of the ground clutter was constructed and also a number of simulations for the multitaper, minimum variance estimation technique was made. Compared to the traditional method defined in this thesis, there was a slight improve- ment of the improvement factor when using the multitaper approach.
    [Show full text]
  • Chapter 2: the Fourier Transform Chapter 2
    “EEE305”, “EEE801 Part A”: Digital Signal Processing Chapter 2: The Fourier Transform Chapter 2 The Fourier Transform 2.1 Introduction The sampled Fourier transform of a periodic, discrete-time signal is known as the discrete Fourier transform (DFT). The DFT of such a signal allows an interpretation of the frequency domain descriptions of the given signal. 2.2 Derivation of the DFT The Fourier transform pair of a continuous-time signal is given by Equation 2.1 and Equation 2.2. ∞ − j2πft X ( f ) = ∫ x(t) ⋅e dt (2.1) −∞ ∞ j2πft x(t) = ∫ X ( f ) ⋅e df (2.2) −∞ Now consider the signal x(t) to be periodic i.e. repeating. The Fourier transform pair from above can now be represented by Equation 2.3 and Equation 2.4 respectively. T0 1 − π = ⋅ j2 kf0t X (kf 0 ) ∫ x(t) e dt (2.3) T0 0 ∞ π = ⋅ j 2 kf0t x(t) ∑ X (kf0 ) e (2.4) k=−∞ = 1 where f 0 . Now suppose that the signal x(t) is sampled N times per period, as illustrated in Figure 2.1, with a T0 sampling period of T seconds. The discrete signal can now be represented by Equation 2.5, where δ is the dirac delta impulse function and has a unit area of one. ∞ x[nT ] = ∑ x(t) ⋅δ (t − nT ) (2.5) n=−∞ Now replace x(t) in the Fourier integral of Equation 2.3, with x[nT] of Equation 2.5. The Equation can now be re- written as: ∞ T0 1 − π = ⋅δ − ⋅ jk 2 f0t X (kf0 ) ∑ ∫ x(t) (t nT ) e dt (2.6) T0 n=−∞ 0 1 for t = nT Since the dirac delta function δ ()t − nT = , Equation 2.6 can be modified even further to become 0 otherwise Equation 2.7 below: N −1 1 − π = ⋅ jk 2 f0nT X[kf 0 ] ∑ x[nT] e (2.7) N n=0 University of Newcastle upon Tyne Page 2.1 “EEE305”, “EEE801 Part A”: Digital Signal Processing Chapter 2: The Fourier Transform Sampled sequence δ(t-nT) litude p Am nT Time (t) Figure 2.1: Sampled sequence for the DFT.
    [Show full text]
  • The Sliding Window Discrete Fourier Transform Lee F
    The Sliding Window Discrete Fourier Transform Lee F. Richardson Department of Statistics and Data Science, Carnegie Mellon University, Pittsburgh, PA The Sliding Window DFT Sliding Window DFT for Local Periodic Signals Leakage The Sliding Window Discrete Fourier Transform (SW- Leakage occurs when local signal frequency is not one DFT) computes a time-frequency representation of a The Sliding Window DFT is a useful tool for analyzing data with local periodic signals (Richardson and Eddy of the Fourier frequencies (or, the number of complete signal, and is useful for analyzing signals with local (2018)). Since SW-DFT coefficients are complex-numbers, we analyze the squared modulus of these coefficients cycles in a length n window is not an integer). In this periodicities. Shortly described, the SW-DFT takes 2 2 2 case, the largest squared modulus coefficients occur at sequential DFTs of a signal multiplied by a rectangu- |ak,p| = Re(ak,p) + Im(ak,p) frequencies closest to the true frequency: lar sliding window function, where the window func- Since the squared modulus SW-DFT coefficients are large when the window is on a periodic part of the signal: Frequency: 2 Frequency: 2.1 Frequency: 2.2 Frequency: 2.3 Frequency: 2.4 20 Freq 2 20 Freq 2 20 Freq 2 20 Freq 2 20 Freq 2 tion is only nonzero for a short duration. Let x = Freq 3 Freq 3 Freq 3 Freq 3 Freq 3 15 15 15 15 15 [x0, x1, . , xN−1] be a length N signal, then for length 10 10 10 10 10 n ≤ N windows, the SW-DFT of x is: 5 5 5 5 5 0 0 0 0 0 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 1 n−1 X −jk Frequency: 2.5 Frequency: 2.6 Frequency: 2.7 Frequency: 2.8 Frequency: 2.9 ak,p = √ xp−n+1+jω 20 Freq 2 20 Freq 2 20 Freq 2 20 Freq 2 20 Freq 2 n n Freq 3 Freq 3 Freq 3 Freq 3 Freq 3 j=0 15 15 15 15 15 k = 0, 1, .
    [Show full text]