DOCSLIB.ORG

Fourier Integrals and the Sampling Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fourier Integrals and the Sampling Theorem Fourier integrals and the sampling theorem Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Fourier Integrals Read: Strang, Section 4.3. Fourier series are convenient to describe periodic functions (or functions with support on a finite interval). What to do if the function is not periodic? - Consider f : R C.TheFourier transform fˆ = (f ) is given by ! F 1 ikx fˆ(k)= f (x)e− dx, k R. 2 Z1 Here, f should be in L1(R). - Inverse transformation: 1 1 f (x)= fˆ(k)eikx dk 2⇡ Z1 Note: Often, you will se a more symmetric version by using a di↵erent scaling. - Theorem of Plancherel: f L2(R) fˆ L2(R)and 2 , 2 1 1 1 f (x) 2dx = fˆ(k) 2dk. | | 2⇡ | | Z1 Z1 Fourier Integrals: The Key Rules x\ df\/dx = ikfˆ(k) f (⇠)d⇠ = fˆ(k)/(ik) Z1 ikd icx f (\x d)=e− fˆ(k) e f = fˆ(k c) − − Examples of Fourier transforms: d f (x)=sin(!x) fˆ(k)= i⇡(δ(k !) δ(k + !)) − − − f (x)=cos(!x) fˆ(k)=⇡(δ(k !)+δ(k + !)) − f (x)=δ(x) fˆ(k)=1forallk R. 2 1, if L x L, sin kL f (x)= − fˆ(k)=2 =2Lsinc(kL), 0, if x > L. k ( | | where sinc(t)=sin(t)/t is the Sinus cardinalis function. Sampling of functions - The Nyquist sampling rate How many samples do we need to identify sin !t or cos !t? Assume equidistant sampling with step size T . Definition: Nyquist sampling rate: T = ⇡/!. The Nyquist sampling rate, is exactly 2 equidistant samples over a full period of the function. (period is 2⇡/!). For a given sampling rate T , frequencies higher than the Nyquist frequency !N = ⇡/T cannot be detected. A higher frequency harmonic is mapped to a lower frequency one. This e↵ect is called aliasing. Figure from Strang. Band-limited functions – When the continuous signal is a combination of many frequencies !, the largest frequency !max sets the Nyquist sampling rate at T = ⇡/!max . – No aliasing at any faster rate. (Do not set sampling rate equal to the Nyquist sampling rate. Consider sin !T to see why.) – A function f (x)iscalledband-limitedifthereisafiniteW such that its Fourier integral transform fˆ(k)=0forall k W . | | ≥ The Shannon-Nyquist sampling theorem states that such a function f (x) can be recovered from the discrete samples with sampling frequency T = ⇡/W . (Note that relating to above, W = !max + ", " > 0. ) The Sampling Theorem Theorem: (Shannon-Nyquist) Assume that f is band-limited by W ,i.e., fˆ(k)=0forall k W .LetT = ⇡/W be the Nyquist rate. Then it holds | | ≥ 1 f (x)= f (nT )sinc(⇡(x/T n)) − n= X1 where sinc(t)=sin(t)/t is the Sinus cardinalis function. Note: The sinc function is band-limited: 1, if ⇡ k ⇡, sinc(k)= − (0, elsewhere. d Also note: This interpolation procedure is not used in practice. Slow decay, summing an infinite number of terms. Approximations to the sinc function are used, introducing interpolation errors. The Sampling Theorem: Proof Assume for simplicity W = ⇡. By the inverse Fourier transform, ⇡ 1 1 1 f (x)= fˆ(k)eikx dk = fˆ(k)eikx dk. 2⇡ 2⇡ ⇡ Z1 Z− Define fˆ(k), if ⇡ < k < ⇡, f˜(k)= − periodic continuation, if k ⇡. ( | | ≥ f˜ can be represented as a Fourier series in k-space, 1 ˆ ink f˜(k)= f˜ne , n= X1 where ⇡ ⇡ ˆ 1 ink 1 ink f˜n = f˜(k)e− dk = fˆ(k)e− dk = f ( n). 2⇡ ⇡ 2⇡ ⇡ − Z− Z− Hence, for ⇡ < k < ⇡, − 1 fˆ(k)=f˜(k)= f ( n)eink . − n= X1 The Sampling Theorem: Proof, contd. Therefore, 1 ⇡ f (x)= fˆ(k)eikx dk 2⇡ ⇡ Z− ⇡ 1 1 = f ( n)eink eikx dk 2⇡ − ⇡ n= ! Z− X1 ⇡ 1 1 = f ( n) eik(x+n)dk − 2⇡ n= ⇡ X1 Z− 1 sin ⇡(x + n) = f ( n) − ⇡(x + n) n= X1 1 sin ⇡(x n) = f (n) − ⇡(x n) n= X1 − If W di↵erent from ⇡,rescalex by ⇡/W and k by W /⇡ to complete the proof..
Load more

Recommended publications
  • Convolution! (CDT-14) Luciano Da Fontoura Costa
    Convolution! (CDT-14) Luciano da Fontoura Costa To cite this version: Luciano da Fontoura Costa. Convolution! (CDT-14). 2019. hal-02334910 HAL Id: hal-02334910 https://hal.archives-ouvertes.fr/hal-02334910 Preprint submitted on 27 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Convolution! (CDT-14) Luciano da Fontoura Costa [email protected] S~aoCarlos Institute of Physics { DFCM/USP October 22, 2019 Abstract The convolution between two functions, yielding a third function, is a particularly important concept in several areas including physics, engineering, statistics, and mathematics, to name but a few. Yet, it is not often so easy to be conceptually understood, as a consequence of its seemingly intricate definition. In this text, we develop a conceptual framework aimed at hopefully providing a more complete and integrated conceptual understanding of this important operation. In particular, we adopt an alternative graphical interpretation in the time domain, allowing the shift implied in the convolution to proceed over free variable instead of the additive inverse of this variable. In addition, we discuss two possible conceptual interpretations of the convolution of two functions as: (i) the `blending' of these functions, and (ii) as a quantification of `matching' between those functions.
    [Show full text]
  • Eliminating Aliasing Caused by Discontinuities Using Integrals of the Sinc Function
    Sound production – Sound synthesis: Paper ISMR2016-48 Eliminating aliasing caused by discontinuities using integrals of the sinc function Fabián Esqueda(a), Stefan Bilbao(b), Vesa Välimäki(c) (a)Aalto University, Dept. of Signal Processing and Acoustics, Espoo, Finland, fabian.esqueda@aalto.fi (b)Acoustics and Audio Group, University of Edinburgh, United Kingdom, [email protected] (c)Aalto University, Dept. of Signal Processing and Acoustics, Espoo, Finland, vesa.valimaki@aalto.fi Abstract A study on the limits of bandlimited correction functions used to eliminate aliasing in audio signals with discontinuities is presented. Trivial sampling of signals with discontinuities in their waveform or their derivatives causes high levels of aliasing distortion due to the infinite bandwidth of these discontinuities. Geometrical oscillator waveforms used in subtractive synthesis are a common example of audio signals with these characteristics. However, discontinuities may also be in- troduced in arbitrary signals during operations such as signal clipping and rectification. Several existing techniques aim to increase the perceived quality of oscillators by attenuating aliasing suf- ficiently to be inaudible. One family of these techniques consists on using the bandlimited step (BLEP) and ramp (BLAMP) functions to quasi-bandlimit discontinuities. Recent work on antialias- ing clipped audio signals has demonstrated the suitability of the BLAMP method in this context. This work evaluates the performance of the BLEP, BLAMP, and integrated BLAMP functions by testing whether they can be used to fully bandlimit aliased signals. Of particular interest are cases where discontinuities appear past the first derivative of a signal, like in hard clipping. These cases require more than one correction function to be applied at every discontinuity.
    [Show full text]
  • Discrete - Time Signals and Systems
    Discrete - Time Signals and Systems Sampling – II Sampling theorem & Reconstruction Yogananda Isukapalli Sampling at diffe- -rent rates From these figures, it can be concluded that it is very important to sample the signal adequately to avoid problems in reconstruction, which leads us to Shannon’s sampling theorem 2 Fig:7.1 Claude Shannon: The man who started the digital revolution Shannon arrived at the revolutionary idea of digital representation by sampling the information source at an appropriate rate, and converting the samples to a bit stream Before Shannon, it was commonly believed that the only way of achieving arbitrarily small probability of error in a communication channel was to 1916-2001 reduce the transmission rate to zero. All this changed in 1948 with the publication of “A Mathematical Theory of Communication”—Shannon’s landmark work Shannon’s Sampling theorem A continuous signal xt( ) with frequencies no higher than fmax can be reconstructed exactly from its samples xn[ ]= xn [Ts ], if the samples are taken at a rate ffs ³ 2,max where fTss= 1 This simple theorem is one of the theoretical Pillars of digital communications, control and signal processing Shannon’s Sampling theorem, • States that reconstruction from the samples is possible, but it doesn’t specify any algorithm for reconstruction • It gives a minimum sampling rate that is dependent only on the frequency content of the continuous signal x(t) • The minimum sampling rate of 2fmax is called the “Nyquist rate” Example1: Sampling theorem-Nyquist rate x( t )= 2cos(20p t ), find the Nyquist frequency ? xt( )= 2cos(2p (10) t ) The only frequency in the continuous- time signal is 10 Hz \ fHzmax =10 Nyquist sampling rate Sampling rate, ffsnyq ==2max 20 Hz Continuous-time sinusoid of frequency 10Hz Fig:7.2 Sampled at Nyquist rate, so, the theorem states that 2 samples are enough per period.
    [Show full text]
  • The Best Approximation of the Sinc Function by a Polynomial of Degree N with the Square Norm
    Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 307892, 12 pages doi:10.1155/2010/307892 Research Article The Best Approximation of the Sinc Function by a Polynomial of Degree n with the Square Norm Yuyang Qiu and Ling Zhu College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China Correspondence should be addressed to Yuyang Qiu, [email protected] Received 9 April 2010; Accepted 31 August 2010 Academic Editor: Wing-Sum Cheung Copyright q 2010 Y. Qiu and L. Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The polynomial of degree n which is the best approximation of the sinc function on the interval 0, π/2 with the square norm is considered. By using Lagrange’s method of multipliers, we construct the polynomial explicitly. This method is also generalized to the continuous function on the closed interval a, b. Numerical examples are given to show the effectiveness. 1. Introduction Let sin cxsin x/x be the sinc function; the following result is known as Jordan inequality 1: 2 π ≤ sin cx < 1, 0 <x≤ , 1.1 π 2 where the left-handed equality holds if and only if x π/2. This inequality has been further refined by many scholars in the past few years 2–30. Ozban¨ 12 presented a new lower bound for the sinc function and obtained the following inequality: 2 1 4π − 3 π 2 π2 − 4x2 x − ≤ sin cx.
    [Show full text]
  • Oversampling of Stochastic Processes
    OVERSAMPLING OF STOCHASTIC PROCESSES by D.S.G. POLLOCK University of Leicester In the theory of stochastic differential equations, it is commonly assumed that the forcing function is a Wiener process. Such a process has an infinite bandwidth in the frequency domain. In practice, however, all stochastic processes have a limited bandwidth. A theory of band-limited linear stochastic processes is described that reflects this reality, and it is shown how the corresponding ARMA models can be estimated. By ignoring the limitation on the frequencies of the forcing function, in the process of fitting a conventional ARMA model, one is liable to derive estimates that are severely biased. If the data are sampled too rapidly, then maximum frequency in the sampled data will be less than the Nyquist value. However, the underlying continuous function can be reconstituted by sinc function or Fourier interpolation; and it can be resampled at a lesser rate corresponding to the maximum frequency of the forcing function. Then, there will be a direct correspondence between the parameters of the band-limited ARMA model and those of an equivalent continuous-time process; and the estimation biases can be avoided. 1 POLLOCK: Band-Limited Processes 1. Time-Limited versus Band-Limited Processes Stochastic processes in continuous time are usually modelled by filtered versions of Wiener processes which have infinite bandwidth. This seems inappropriate for modelling the slowly evolving trajectories of macroeconomic data. Therefore, we shall model these as processes that are limited in frequency. A function cannot be simultaneously limited in frequency and limited in time. One must choose either a band-limited function, which extends infinitely in time, or a time-limited function, which extents over an infinite range of frequencies.
    [Show full text]
  • Shannon Wavelets Theory
    Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2008, Article ID 164808, 24 pages doi:10.1155/2008/164808 Research Article Shannon Wavelets Theory Carlo Cattani Department of Pharmaceutical Sciences (DiFarma), University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy Correspondence should be addressed to Carlo Cattani, [email protected] Received 30 May 2008; Accepted 13 June 2008 Recommended by Cristian Toma Shannon wavelets are studied together with their differential properties known as connection coefficients. It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization R ff coincides with the Shannon wavelet reconstruction of L2 functions. The di erential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon ∞ wavelets are C -functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the C -functions. Copyright q 2008 Carlo Cattani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Wavelets 1 are localized functions which are a very useful tool in many different applications: signal analysis, data compression, operator analysis, and PDE solving see, e.g., 2 and references therein. The main feature of wavelets is their natural splitting of objects into different scale components 1, 3 according to the multiscale resolution analysis.
    [Show full text]
  • The Nyquist Sampling Rate for Spiraling Curves 11
    THE NYQUIST SAMPLING RATE FOR SPIRALING CURVES PHILIPPE JAMING, FELIPE NEGREIRA & JOSE´ LUIS ROMERO Abstract. We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of under- sampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis. 1. Introduction 1.1. The mobile sampling problem. In this article, we consider the reconstruction of a compactly supported function from samples of its Fourier transform taken along certain curves, that we call spiraling. This problem is relevant, for example, in magnetic resonance imaging (MRI), where the anatomy and physiology of a person are captured by moving sensors. The Fourier sampling problem is equivalent to the sampling problem for bandlimited functions - that is, functions whose Fourier transform are supported on a given compact set. The most classical setting concerns functions of one real variable with Fourier transform supported on the unit interval [ 1/2, 1/2], and sampled on a grid ηZ, with η > 0. The sampling rate η determines whether− every bandlimited function can be reconstructed from its samples: reconstruction fails if η > 1 and succeeds if η 6 1 [42]. The transition value η = 1 is known as the Nyquist sampling rate, and it is the benchmark for all sampling schemes: modern sampling strategies that exploit the particular structure of a certain class of signals are praised because they achieve sub-Nyquist sampling rates.
    [Show full text]
  • 2.161 Signal Processing: Continuous and Discrete Fall 2008
    MIT OpenCourseWare http://ocw.mit.edu 2.161 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 INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete 1 Sampling and the Discrete Fourier Transform 1 Sampling Consider a continuous function f(t) that is limited in extent, T1 · t < T2. In order to process this function in the computer it must be sampled and represented by a ¯nite set of numbers. The most common sampling scheme is to use a ¯xed sampling interval ¢T and to form a sequence of length N: ffng (n = 0 : : : N ¡ 1), where fn = f(T1 + n¢T ): In subsequent processing the function f(t) is represented by the ¯nite sequence ffng and the sampling interval ¢T . In practice, sampling occurs in the time domain by the use of an analog-digital (A/D) converter. The mathematical operation of sampling (not to be confused with the physics of sampling) is most commonly described as a multiplicative operation, in which f(t) is multiplied by a Dirac comb sampling function s(t; ¢T ), consisting of a set of delayed Dirac delta functions: X1 s(t; ¢T ) = ±(t ¡ n¢T ): (1) n=¡1 ? We denote the sampled waveform f (t) as X1 ? f (t) = s(t; ¢T )f(t) = f(t)±(t ¡ n¢T ) (2) n=¡1 ? as shown in Fig. 1. Note that f (t) is a set of delayed weighted delta functions, and that the waveform must be interpreted in the distribution sense by the strength (or area) of each component impulse.
    [Show full text]
  • Signal Sampling
    FYS3240 PC-based instrumentation and microcontrollers Signal sampling Spring 2017 – Lecture #5 Bekkeng, 30.01.2017 Content – Aliasing – Sampling – Analog to Digital Conversion (ADC) – Filtering – Oversampling – Triggering Analog Signal Information Three types of information: • Level • Shape • Frequency Sampling Considerations – An analog signal is continuous – A sampled signal is a series of discrete samples acquired at a specified sampling rate – The faster we sample the more our sampled signal will look like our actual signal Actual Signal – If not sampled fast enough a problem known as aliasing will occur Sampled Signal Aliasing Adequately Sampled SignalSignal Aliased Signal Bandwidth of a filter • The bandwidth B of a filter is defined to be between the -3 dB points Sampling & Nyquist’s Theorem • Nyquist’s sampling theorem: – The sample frequency should be at least twice the highest frequency contained in the signal Δf • Or, more correctly: The sample frequency fs should be at least twice the bandwidth Δf of your signal 0 f • In mathematical terms: fs ≥ 2 *Δf, where Δf = fhigh – flow • However, to accurately represent the shape of the ECG signal signal, or to determine peak maximum and peak locations, a higher sampling rate is required – Typically a sample rate of 10 times the bandwidth of the signal is required. Illustration from wikipedia Sampling Example Aliased Signal 100Hz Sine Wave Sampled at 100Hz Adequately Sampled for Frequency Only (Same # of cycles) 100Hz Sine Wave Sampled at 200Hz Adequately Sampled for Frequency and Shape 100Hz Sine Wave Sampled at 1kHz Hardware Filtering • Filtering – To remove unwanted signals from the signal that you are trying to measure • Analog anti-aliasing low-pass filtering before the A/D converter – To remove all signal frequencies that are higher than the input bandwidth of the device.
    [Show full text]
  • Chapter 11: Fourier Transform Pairs
    CHAPTER 11 Fourier Transform Pairs For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. For example, a rectangular pulse in the time domain coincides with a sinc function [i.e., sin(x)/x] in the frequency domain. Duality provides that the reverse is also true; a rectangular pulse in the frequency domain matches a sinc function in the time domain. Waveforms that correspond to each other in this manner are called Fourier transform pairs. Several common pairs are presented in this chapter. Delta Function Pairs For discrete signals, the delta function is a simple waveform, and has an equally simple Fourier transform pair. Figure 11-1a shows a delta function in the time domain, with its frequency spectrum in (b) and (c). The magnitude is a constant value, while the phase is entirely zero. As discussed in the last chapter, this can be understood by using the expansion/compression property. When the time domain is compressed until it becomes an impulse, the frequency domain is expanded until it becomes a constant value. In (d) and (g), the time domain waveform is shifted four and eight samples to the right, respectively. As expected from the properties in the last chapter, shifting the time domain waveform does not affect the magnitude, but adds a linear component to the phase. The phase signals in this figure have not been unwrapped, and thus extend only from -B to B. Also notice that the horizontal axes in the frequency domain run from -0.5 to 0.5. That is, they show the negative frequencies in the spectrum, as well as the positive ones.
    [Show full text]
  • Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing
    Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 1 News Homework #5 . Due this week . Submit via canvas Coding Problem #4 . Due this week . Submit via canvas Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 2 Exam 1 Grades The class did exceedingly well . Mean: 89.3 . Median: 91.5 Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 3 Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform Foundations of Digital Signal Processing Outline • Review of Sampling • The Nyquist-Shannon Sampling Theorem • Continuous-time Reconstruction / Interpolation • Aliasing and anti-Aliasing • Deriving Transforms from the Fourier Transform • Discrete-time Fourier Transform, Fourier Series, Discrete-time Fourier Series • The Discrete Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing, and the Discrete Fourier Transform 4 Sampling Discrete-Time Fourier Transform Foundations of Digital Signal Processing Lecture 12: Sampling, Aliasing,
    [Show full text]
  • Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems †
    S S symmetry Article Adaptive Sub-Nyquist Spectrum Sensing for Ultra-Wideband Communication Systems † Yong Lu , Shaohe Lv and Xiaodong Wang * Science and Technology on Parallel and Distributed Processing Laboratory, National University of Defense Technology, Changsha 410073, China; [email protected] (Y.L.); [email protected] (S.L.) * Correspondence: [email protected] † This paper is an extended version of our paper published in 2014 IEEE 17th International Conference on Computational Science and Engineering (CSE), Chengdu, China, 19–21 December 2014. Received: 14 January 2019; Accepted: 3 March 2019; Published: 7 March 2019 Abstract: With the ever-increasing demand for high-speed wireless data transmission, ultra-wideband spectrum sensing is critical to support the cognitive communication over an ultra-wide frequency band for ultra-wideband communication systems. However, it is challenging for the analog-to-digital converter design to fulfill the Nyquist rate for an ultra-wideband frequency band. Therefore, we explore the spectrum sensing mechanism based on the sub-Nyquist sampling and conduct extensive experiments to investigate the influence of sampling rate, bandwidth resolution and the signal-to-noise ratio on the accuracy of sub-Nyquist spectrum sensing. Afterward, an adaptive policy is proposed to determine the optimal sampling rate, and bandwidth resolution when the spectrum occupation or the strength of the existing signals is changed. The performance of the policy is verified by simulations. Keywords: ultra-wideband spectrum sensing; energy detection; cognitive radio; sub-Nyquist sensing; Chinese remainder theorem 1. Introduction After decades of rapid development, wireless technologies have been intensively applied to almost every field from daily communications to mobile medical and battlefield communications.
    [Show full text]