Understanding Ffts and Windowing
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
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 filter is optimal for a common task: reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. However, the moving average is the worst filter for frequency domain encoded signals, with little ability to separate one band of frequencies from another. Relatives of the moving average filter include the Gaussian, Blackman, and multiple- pass moving average. These have slightly better performance in the frequency domain, at the expense of increased computation time. Implementation by Convolution As the name implies, the moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written: EQUATION 15-1 Equation of the moving average filter. In M &1 this equation, x[ ] is the input signal, y[ ] is ' 1 % y[i] j x [i j ] the output signal, and M is the number of M j'0 points used in the moving average. This equation only uses points on one side of the output sample being calculated. Where x[ ] is the input signal, y[ ] is the output signal, and M is the number of points in the average. For example, in a 5 point moving average filter, point 80 in the output signal is given by: x [80] % x [81] % x [82] % x [83] % x [84] y [80] ' 5 277 278 The Scientist and Engineer's Guide to Digital Signal Processing As an alternative, the group of points from the input signal can be chosen symmetrically around the output point: x[78] % x[79] % x[80] % x[81] % x[82] y[80] ' 5 This corresponds to changing the summation in Eq. -
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. -
Spatial Domain Low-Pass Filters
Low Pass Filtering Why use Low Pass filtering? • Remove random noise • Remove periodic noise • Reveal a background pattern 1 Effects on images • Remove banding effects on images • Smooth out Img-Img mis-registration • Blurring of image Types of Low Pass Filters • Moving average filter • Median filter • Adaptive filter 2 Moving Ave Filter Example • A single (very short) scan line of an image • {1,8,3,7,8} • Moving Ave using interval of 3 (must be odd) • First number (1+8+3)/3 =4 • Second number (8+3+7)/3=6 • Third number (3+7+8)/3=6 • First and last value set to 0 Two Dimensional Moving Ave 3 Moving Average of Scan Line 2D Moving Average Filter • Spatial domain filter • Places average in center • Edges are set to 0 usually to maintain size 4 Spatial Domain Filter Moving Average Filter Effects • Reduces overall variability of image • Lowers contrast • Noise components reduced • Blurs the overall appearance of image 5 Moving Average images Median Filter The median utilizes the median instead of the mean. The median is the middle positional value. 6 Median Example • Another very short scan line • Data set {2,8,4,6,27} interval of 5 • Ranked {2,4,6,8,27} • Median is 6, central value 4 -> 6 Median Filter • Usually better for filtering • - Less sensitive to errors or extremes • - Median is always a value of the set • - Preserves edges • - But requires more computation 7 Moving Ave vs. Median Filtering Adaptive Filters • Based on mean and variance • Good at Speckle suppression • Sigma filter best known • - Computes mean and std dev for window • - Values outside of +-2 std dev excluded • - If too few values, (<k) uses value to left • - Later versions use weighting 8 Adaptive Filters • Improvements to Sigma filtering - Chi-square testing - Weighting - Local order histogram statistics - Edge preserving smoothing Adaptive Filters 9 Final PowerPoint Numerical Slide Value (The End) 10. -
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. -
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. -
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]. -
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. -
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. -
Review of Smoothing Methods for Enhancement of Noisy Data from Heavy-Duty LHD Mining Machines
E3S Web of Conferences 29, 00011 (2018) https://doi.org/10.1051/e3sconf/20182900011 XVIIth Conference of PhD Students and Young Scientists Review of smoothing methods for enhancement of noisy data from heavy-duty LHD mining machines Jacek Wodecki1, Anna Michalak2, and Paweł Stefaniak2 1Machinery Systems Division, Wroclaw University of Science and Technology, Wroclaw, Poland 2KGHM Cuprum R&D Ltd., Wroclaw, Poland Abstract. Appropriate analysis of data measured on heavy-duty mining machines is essential for processes monitoring, management and optimization. Some particular classes of machines, for example LHD (load-haul-dump) machines, hauling trucks, drilling/bolting machines etc. are characterized with cyclicity of operations. In those cases, identification of cycles and their segments or in other words – simply data segmen- tation is a key to evaluate their performance, which may be very useful from the man- agement point of view, for example leading to introducing optimization to the process. However, in many cases such raw signals are contaminated with various artifacts, and in general are expected to be very noisy, which makes the segmentation task very difficult or even impossible. To deal with that problem, there is a need for efficient smoothing meth- ods that will allow to retain informative trends in the signals while disregarding noises and other undesired non-deterministic components. In this paper authors present a review of various approaches to diagnostic data smoothing. Described methods can be used in a fast and efficient way, effectively cleaning the signals while preserving informative de- terministic behaviour, that is a crucial to precise segmentation and other approaches to industrial data analysis. -
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. -
Data Mining Algorithms
Data Management and Exploration © for the original version: Prof. Dr. Thomas Seidl Jiawei Han and Micheline Kamber http://www.cs.sfu.ca/~han/dmbook Data Mining Algorithms Lecture Course with Tutorials Wintersemester 2003/04 Chapter 4: Data Preprocessing Chapter 3: Data Preprocessing Why preprocess the data? Data cleaning Data integration and transformation Data reduction Discretization and concept hierarchy generation Summary WS 2003/04 Data Mining Algorithms 4 – 2 Why Data Preprocessing? Data in the real world is dirty incomplete: lacking attribute values, lacking certain attributes of interest, or containing only aggregate data noisy: containing errors or outliers inconsistent: containing discrepancies in codes or names No quality data, no quality mining results! Quality decisions must be based on quality data Data warehouse needs consistent integration of quality data WS 2003/04 Data Mining Algorithms 4 – 3 Multi-Dimensional Measure of Data Quality A well-accepted multidimensional view: Accuracy (range of tolerance) Completeness (fraction of missing values) Consistency (plausibility, presence of contradictions) Timeliness (data is available in time; data is up-to-date) Believability (user’s trust in the data; reliability) Value added (data brings some benefit) Interpretability (there is some explanation for the data) Accessibility (data is actually available) Broad categories: intrinsic, contextual, representational, and accessibility. WS 2003/04 Data Mining Algorithms 4 – 4 Major Tasks in Data Preprocessing -
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.