Generation of bandlimited sync transitions for sine waveforms
|
|
- Chloe Evans
- 6 years ago
- Views:
Transcription
1 Generation of bandlimited sync transitions for sine waveforms Vadim Zavalishin May 4, 9 Abstract A common way to generate bandlimited sync transitions for the classical analog waveforms is the BLEP method. The method works well for the sawtooth, pulse, and triangle waveforms, however fails in case of the sine. A number of methods to overcome that problem is proposed in the article. Introduction The BLEP method including its minimum-phase variant [] provides a way for generating bandlimited zero-order discontinuities discontinuities of the signal, which can be applied for generation of bandlimited saw and pulse waveforms and also sync transitions in such waveforms. A triangle waveform contains first-order discontinuities discontinuities of the first derivative plus zero-order discontinuities which may appear during syncing, therefore in addition to the zero-order BLEPs one also needs to use the first-order BLEPs. A sine waveform on its own doesn t require any bandlimiting provided the sine frequency lies below the Nyquist limit, however a sync transition generates discontinuities of all orders up to infinity, therefore an infinite number of BLEPs of all possible orders is theoretically required. In the present article we will discuss a number of ways to deal with that problem. Native Instruments GmbH Problem specification We are going to consider only the constant-frequency sine waveforms, in which case the sync transition is simply a phase discontinuity. Let x t = cost + ϕ = X t + X t x t = cost + ϕ = X t + X t be the signals before and after the transition and let X t = e jt+ϕ X t = e jt+ϕ be their analytic counterparts the asterisk in X t denotes the complex conjugation. Let ht if t < sgn t = if t = if t > be the step function and let the sync transition occur at t =. Then the output signal can be represented as yt = ht x t + + ht x t where = x t + x t + ht x t x t = xt + ht xt xt = x t + x t
2 is the average of the two signals and xt = x t x t is their difference. Respectively, Xt = X t X t is the difference of the analytic signals. We want to bandlimit the output signal yt to the frequency band from to. Apparently, we can assume the, band without loss of generality, as this means simply a choice of the frequency units. The upper bound of the frequency band would typically correspond to the Nyquist frequency, however doesn t have to, e.g. one could specify a bound slightly below or slightly above the Nyquist limit whatever slightly means. We will also assume that the signals x and x are already bandlimited <. The bandlimited versions of the originally nonbandlimited signals will be denoted by a bar on top, e.g. ȳt. Note that some signals will be bandlimited to other bands than,, in which case we will explicitly mention so. 3 Notation issues In the text of this article we will need to deal with the integral exponents, sines, and cosines. A number of different notations are apparently in use here []. In order to simplify the formulas in the article we will use the following somewhat non-standard notation. Restricting z to taking only purely imaginary values z = jϕ we let Ein z = z e t t dt Ei z = γ + ln z + Ein z E n z = e zt dt n tn ei jϕ = Ei jϕ j sgn ϕ = E jϕ where the absolute value of z in ln z and the term j/ sgn ϕ ensure the conjugate symmetries Ei jϕ = Ei jϕ and ei jϕ = ei jϕ respectively, and γ denotes the Euler s constant. Thus: Therefore Cin ϕ = ϕ cos t t dt Ci ϕ = γ + ln ϕ + Cin ϕ Si ϕ = ϕ sin t t dt si ϕ = Si ϕ sgn ϕ Ein jϕ = Cin ϕ + j Si ϕ Ei jϕ = Ci ϕ + j Si ϕ ei jϕ = Ci ϕ + j si ϕ Ci ϕ = Ci ϕ Si ϕ = Si ϕ si ϕ = si ϕ Ein jϕ = Ein jϕ Ei jϕ = Ei jϕ ei jϕ = ei jϕ E n jϕ = E njϕ ne n+ z = ze n z + e z d dz E nz = E n z The limit values are Ci± = Si± = ± ei± j = E n ± j = E n+ n A robust numerical method to compute Ci ϕ and Si t can be found in [3]. Essentially, it is using the series expansion if ϕ < and the continued fraction for E z otherwise, both of which can be also found in []. The remaining functions can be built from Ci ϕ and Si ϕ. Note that in order to compute Cin ϕ and Ein jϕ it is reasonable to use the series expansion for Cin ϕ instead of the one for Ci ϕ. Also, keep in mind the notation specifics of the present article!
3 4 Multiple BLEPs method If the sine frequency lies well below the Nyquist limit, then we would intuitively expect higher-order BLEPs to have rather little effect on the waveform, in which case we may simply discard them. Let s analyse this question in more detail. 4. Fundamentals Recalling that ht sgn t = δτ dτ we can refer to ht as to the zero-order step function, and to δt as to the st-order step function, using notations h t = ht and h t = δt, whereas the higher-order step functions are defined as h n t = h n τ dτ = tn n! sgn t We define the aliasing residual of n-th order as h n t = h n t h n t Then the bandlimited output signal is where ȳt = yt ȳt = yt y n h n t n= y n = y n t + y n t 4. Analytic BLEPs Writing the Fourier integral for h t we have h t j d ε lim ε + j d + ε j lim E jεt E jεt ε + j d Thus where H t + H t h t = lim ε + H t j E jεt j ε j d Therefore, H t can be considered as an analytic counterpart of h t except that it s parametrized with ε. Bandlimiting the signal H t: H t j ε j d j E jt ε = we obtain the expression for the aliasing residual H t = H t H t j j E jt = H t ε= Thus, H t doesn t depend on ε and h t = H t + H t Now consider H t = H t + H t which can be alternatively written as H τ dτ = H τ dτ + Evaluating the last term we get H t = H τ dτ j = E jτ t τ= = E jt j d H τ dτ E jτ dτ + The above is not the most desirable scenario, as we would rather have H t for t. Notice however, that the term / is simply a DC offset, therefore it is bandlimited and can be simply included into H : H t = H t = E jt H τ dτ + 3
4 Continuing in the same manner, we obtain H n t = E n+jt j n+ which is a general expression for an n-th order bandlimited analytic aliasing residue. 4.3 Real-domain BLEPs The real domain expression for the aliasing residuals can be obtained as h n t = H n t + H nt = E n+jt + n+ E n+jt j n+ Notably, for n = the above turns into h t = E jt E jt j sgn t Si t 3 which is the familiar expression for the zero-order aliasing residual. The problem with however, is that the function E jt, which is required to recursively evaluate E n jt, is getting infinitely large around t =. Therefore, for small t one should use 3 at n = and take special care in the evaluation of the term jte jt in the expression for E jt at n >. A better option is to notice that n h n t = n E n+jt + n+ E n+jt j n+ = jt E njt + n+ jte n+jt j n+ + ejt + n+ e jt j n+ = t h n t + sin n t where sin n t is the n-th antiderivative of sin t: { sin n n+/ cos t for odd n t = n/ sin t for even n which allows us to compute h n t from h t. Note that it is also not difficult to obtain the explicit form of ht: h n t Si n t where Si n t is the n-th antiderivative of Si t, which is computed as n Si n t = t Si n+ t sin n t 4.4 Error estimation Now we will estimate the bounds for h n t. First, we notice that for n = we have h t = h t Si t = h For higher orders, using we have hn t E n+ jt + n+ E = n+jt j n+ + n+ e jt n+ d d n+ n = h n Now assume, that instead of the infinite series we use an approximation for ȳ: ȳ N y n h n t n= The error of the approximation is given by the remainder of the series: ε = y n h n t y n h n t n=n+ n=n+ Noticing that y n n, and recalling the bound estimations for h n t we have ε n=n+ n h n = Therefore if then ε. n=n+ n n 4
5 4.5 Windowing In choosing N one should take into account the fact that the BLEP method doesn t provide the full bandlimiting anyway, as each of the aliasing residuals is infinitely long, and some windowing would be used in practice. Therefore the number of BLEPs and the window size have to be chosen together, as to provide comparable levels of imperfectness. The windowed BLEPs can be constructed in a number of ways. The most obvious way is to window the aliasing residuals directly: hn TL t = W t h n t where W t is the window function, and TL denotes the timelimited version of a function. Note that an important side effect of windowing is that the argument of ht becomes restricted to the window size, and thus the functions h n t can be simply TL tabulated. Another approach can be to window the thorder BLEP: h TL t = W t h t = W t sinc t and compute the rest by the recursive integration of the aliasing residuals: hn TL t = a n hn TL τ dτ + c n where the constants a n and c n should be chosen so as to satisfy the requirement h n ± =. TL The latter approach might not seem very reasonable, because of a large amount of numerical integration involved, which should lead to increasing precision losses. However, there is a special case particularly useful at small window sizes where the integration can be done analytically. Recall that the continuous-time impulse responses of symmetric interpolators are typically approximations of the sinc function [4] [5], and thus one may take such impulse response as h t. Then one can construct higher-order BLEPs from h t as de- W W scribed above. 5 Ring modulation method Consider again. Obviously, xt is already bandlimited, therefore we only need to bandlimit the remainder ht xt remember that ht = h t. Imagine that instead of ht we have used in its bandlimited to, version ht. Then the remainder of turns into ht xt ht Xt + ht X t 4 Recalling that Xt = e jϕ e jϕ and considering the first term of 4 alone, we notice that the multiplication by Xt shifts the spectrum of ht to the right by. Similarly, in the second term the spectrum of ht is shifted to the left by. Therefore, if, then the spectrum of h x will have only minor differences to the ideally bandlimited h x. Already if < /, then all wrong partials that including the aliasing and the missing partials will be located above /. Therefore, if e.g. the sampling rate is 88kHz and the sine frequencies are limited to khz, then all the wrong partials will be located above khz. Considering that the sampling rate of 88kHz or higher may be desirable for a number of other reasons anyway, the above method might be an acceptable option. The time-limited version of ht can be built from the time-limited residual: h t = ht h t TL TL or by direct integration of h TL t: h htl t = h TL TL τ, dτ τ, dτ It is convenient to write the band- and time-limited version of as ȳt = xt + htl t xt = yt h TL t xt because the latter form, by using the residual, allows intuitive handling of time-overlapping transitions. 5
6 6 Frequency shifting method Now we are going to try to bandlimit h x to exactly,. We ll do so by bandlimiting each of the components h X and h X to,. In order to do that we need to bandlimit ht in h X to, +, and bandlimit ht in h X to, Asymmetric BLEP First, we are going to build an analytic expression for the step function ht bandlimited to the frequency range, +, where we will assume < < +. Writing the Fourier integral we get ht j j + j d + j d + + d j + ln + Ein jt = + Ei jt = 5 where the latter transformation is done noticing that ln + ln = ln j + t ln j t 6. Putting the things together Using 5, the bandlimited signal h x is built as the average of signals h X and h X, where in the first case h is limited to a different frequency band than in the second case: Let h x h X + h X = X Ei j + t Ei j t + X Ei j + t Ei j t L H + 6 Note that since <, it means that L H this will be important for estimating the table size for the functions involved in the resulting formula. Using the notation 6 we can write h x = X Ei jl t Ei j H t + X Ei jh t Ei j L t = X Ei jl t Ei j H t X Ei j L t Ei j H t Im Xt Ei j L t Ei j H t Im Xt Ei jt L = H Im Xt ln L H + Ein jt L = H which gives us a formula for bandlimiting yt. Note that if, then L H and h x Im X j Si t = Si t Re X = Si t x which is the ring modulation method s formula. 6.3 Windowing Let so that ht ln j L H + Ein jt L h xt = Re ht Xt = H 6
7 db Then Figure : Non-bandlimited signal. ȳt = xt + h x = xt + ht xt ht xt + h xt = yt Re ht Xt + Re ht Xt = yt Re ht ht Xt = yt Re ht Xt Noticing that h± =, we conclude that we can timelimit the transition by simply timelimiting the signal ht. Note that in the frequency shifting method the function ht depends on the signal s frequency, therefore one can only tabulate Ein jϕ and the window function, whereas the application of the window must be done in real time. 7 Comparing the spectra In order to compare the methods a test case was created, where the master oscillator frequency was set to M =.5, and the slave oscillator frequency was set to S = 7.3 M. The reset phase was ϕ = Fig.. A samples-long Kaiser window with α = 4 was used for the transition time limiting. The resulting spectra generated numerically for a continuous-time case are plotted in Fig. through Fig. d. The dashed lines correspond to the non-bandlimited spectrum. 8 Conclusion We have described three different methods for generating a bandlimited sync transition in the sine waveform. The third method frequency shifting potentially can generate an almost ideally bandlimited db Figure : Non-bandlimited signal s spectrum. db db Figure a: Single-BLEP method. db db Figure b: Multiple BLEPs N = method. db db Figure c: Ring modulation method. 7
8 db References [] Brandt E. Hard Sync without Aliasing eli db Figure d: Frequency shifting method. transition, given a long enough window. However it requires the computation of about 6 additional tabulated functions times the real and imaginary parts of the integral exponent, the window function, and the imaginary counterpart of the sine waveform. At a similar computation complexity we could have used the multiple BLEPs method to the 5th order inclusive instead. Another limitation of the frequency shifting method is that it is not immediately offering a conversion option to the minimum-phase version. The linear phase bandlimiting requires a lookahead, which means that a latency needs to be introduced in the output. However, often the latency introduced by the necessary lookahead can be either tolerated or compensated by introducing an equal latency in the parallel signal paths. Also, personally, the author prefers the linear phase antialiasing, because it preserves the phase relationships in the waveform, particularly, the generated waveforms are practically identical to the sampled analog waveforms. Note that all three methods tend to produce larger errors as the signal s frequency gets closer to the band limit. [] Abramowitz M., Stegun I.A. Handbook of Mathematical Functions National Bureau of Standards Scanned online copies at cbm/aands, and other places [3] William H. Press et al. Numerical Recipes in C nd ed. Cambridge University Press 99. A full preview of the current edition is also available online at [4] Smith J.O. III Digital Audio Resampling Home Page jos [5] Olli Niemitallo Polynomial Interpolators for High-Quality Resampling of Oversampled Audio oniemita The http links are at the time of writing the article Acknowledgements The author would like to thank NI, and Stephan Schmitt in particular, for their support and for helping him to release this article. c Vadim Zavalishin. The right is hereby granted to freely distribute this article further, provided no profit is made from such distribution. 8
Review: 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 information6.003 Homework #10 Solutions
6.3 Homework # Solutions Problems. DT Fourier Series Determine the Fourier Series coefficients for each of the following DT signals, which are periodic in N = 8. x [n] / n x [n] n x 3 [n] n x 4 [n] / n
More informationNotes 07 largely plagiarized by %khc
Notes 07 largely plagiarized by %khc Warning This set of notes covers the Fourier transform. However, i probably won t talk about everything here in section; instead i will highlight important properties
More informationFourier Transform for Continuous Functions
Fourier Transform for Continuous Functions Central goal: representing a signal by a set of orthogonal bases that are corresponding to frequencies or spectrum. Fourier series allows to find the spectrum
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More information4 The Continuous Time Fourier Transform
96 4 The Continuous Time ourier Transform ourier (or frequency domain) analysis turns out to be a tool of even greater usefulness Extension of ourier series representation to aperiodic signals oundation
More informationPreserving the LTI system topology in s- to z-plane transforms
Preserving the LTI system topology in s- to z-plane transforms Vadim Zavalishin May 5, 2008 Abstract A method for preserving the LTI system topology during the application of an s- to z-plane transform
More informationBridge between continuous time and discrete time signals
6 Sampling Bridge between continuous time and discrete time signals Sampling theorem complete representation of a continuous time signal by its samples Samplingandreconstruction implementcontinuous timesystems
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 informationFourier Series. Spectral Analysis of Periodic Signals
Fourier Series. Spectral Analysis of Periodic Signals he response of continuous-time linear invariant systems to the complex exponential with unitary magnitude response of a continuous-time LI system at
More informationLine Spectra and their Applications
In [ ]: cd matlab pwd Line Spectra and their Applications Scope and Background Reading This session concludes our introduction to Fourier Series. Last time (http://nbviewer.jupyter.org/github/cpjobling/eg-47-
More informationENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function
More informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationENGIN 211, Engineering Math. Fourier Series and Transform
ENGIN 11, Engineering Math Fourier Series and ransform 1 Periodic Functions and Harmonics f(t) Period: a a+ t Frequency: f = 1 Angular velocity (or angular frequency): ω = ππ = π Such a periodic function
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationLOPE3202: Communication Systems 10/18/2017 2
By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.
More informationTHE LIMIT PROCESS (AN INTUITIVE INTRODUCTION)
The Limit Process THE LIMIT PROCESS (AN INTUITIVE INTRODUCTION) We could begin by saying that limits are important in calculus, but that would be a major understatement. Without limits, calculus would
More informationPART 1. Review of DSP. f (t)e iωt dt. F(ω) = f (t) = 1 2π. F(ω)e iωt dω. f (t) F (ω) The Fourier Transform. Fourier Transform.
PART 1 Review of DSP Mauricio Sacchi University of Alberta, Edmonton, AB, Canada The Fourier Transform F() = f (t) = 1 2π f (t)e it dt F()e it d Fourier Transform Inverse Transform f (t) F () Part 1 Review
More information3 Fourier Series Representation of Periodic Signals
65 66 3 Fourier Series Representation of Periodic Signals Fourier (or frequency domain) analysis constitutes a tool of great usefulness Accomplishes decomposition of broad classes of signals using complex
More informationDamped harmonic motion
Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of under-damping,
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 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 informationCH.4 Continuous-Time Fourier Series
CH.4 Continuous-Time Fourier Series First step to Fourier analysis. My mathematical model is killing me! The difference between mathematicians and engineers is mathematicians develop mathematical tools
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
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 informationLecture 28 Continuous-Time Fourier Transform 2
Lecture 28 Continuous-Time Fourier Transform 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/14 1 Limit of the Fourier Series Rewrite (11.9) and (11.10) as As, the fundamental
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science. Fall Solutions for Problem Set 2
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Issued: Tuesday, September 5. 6.: Discrete-Time Signal Processing Fall 5 Solutions for Problem Set Problem.
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationAntialiased Soft Clipping using an Integrated Bandlimited Ramp
Budapest, Hungary, 31 August 2016 Antialiased Soft Clipping using an Integrated Bandlimited Ramp Fabián Esqueda*, Vesa Välimäki*, and Stefan Bilbao** *Dept. Signal Processing and Acoustics, Aalto University,
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 informationECE 451 Black Box Modeling
Black Box Modeling Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jose@emlab.uiuc.edu Simulation for Digital Design Nonlinear Network Nonlinear Network 3 Linear N-Port with
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More information3.2 Complex Sinusoids and Frequency Response of LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationIntroduction to Initial Value Problems
Chapter 2 Introduction to Initial Value Problems The purpose of this chapter is to study the simplest numerical methods for approximating the solution to a first order initial value problem (IVP). Because
More informationSolutions to Problems in Chapter 4
Solutions to Problems in Chapter 4 Problems with Solutions Problem 4. Fourier Series of the Output Voltage of an Ideal Full-Wave Diode Bridge Rectifier he nonlinear circuit in Figure 4. is a full-wave
More informationLIMITS AND DERIVATIVES
2 LIMITS AND DERIVATIVES LIMITS AND DERIVATIVES 2.2 The Limit of a Function In this section, we will learn: About limits in general and about numerical and graphical methods for computing them. THE LIMIT
More informationPart 3.3 Differentiation Taylor Polynomials
Part 3.3 Differentiation 3..3.1 Taylor Polynomials Definition 3.3.1 Taylor 1715 and Maclaurin 1742) If a is a fixed number, and f is a function whose first n derivatives exist at a then the Taylor polynomial
More informationChapter 7: Filter Design 7.1 Practical Filter Terminology
hapter 7: Filter Design 7. Practical Filter Terminology Analog and digital filters and their designs constitute one of the major emphasis areas in signal processing and communication systems. This is due
More information2 Fourier Transforms and Sampling
2 Fourier ransforms and Sampling 2.1 he Fourier ransform he Fourier ransform is an integral operator that transforms a continuous function into a continuous function H(ω) =F t ω [h(t)] := h(t)e iωt dt
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 information1 Solutions in cylindrical coordinates: Bessel functions
1 Solutions in cylindrical coordinates: Bessel functions 1.1 Bessel functions Bessel functions arise as solutions of potential problems in cylindrical coordinates. Laplace s equation in cylindrical coordinates
More information1 Understanding Sampling
1 Understanding Sampling Summary. In Part I, we consider the analysis of discrete-time signals. In Chapter 1, we consider how discretizing a signal affects the signal s Fourier transform. We derive the
More informationVer 3808 E1.10 Fourier Series and Transforms (2014) E1.10 Fourier Series and Transforms. Problem Sheet 1 (Lecture 1)
Ver 88 E. Fourier Series and Transforms 4 Key: [A] easy... [E]hard Questions from RBH textbook: 4., 4.8. E. Fourier Series and Transforms Problem Sheet Lecture. [B] Using the geometric progression formula,
More informationTherefore the new Fourier coefficients are. Module 2 : Signals in Frequency Domain Problem Set 2. Problem 1
Module 2 : Signals in Frequency Domain Problem Set 2 Problem 1 Let be a periodic signal with fundamental period T and Fourier series coefficients. Derive the Fourier series coefficients of each of the
More informationInfinite Series. 1 Introduction. 2 General discussion on convergence
Infinite Series 1 Introduction I will only cover a few topics in this lecture, choosing to discuss those which I have used over the years. The text covers substantially more material and is available for
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 informationEC Signals and Systems
UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J
More informationFourier Series Example
Fourier Series Example Let us compute the Fourier series for the function on the interval [ π,π]. f(x) = x f is an odd function, so the a n are zero, and thus the Fourier series will be of the form f(x)
More informationThe Hilbert Transform
The Hilbert Transform Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto October 22, 2006; updated March 0, 205 Definition The Hilbert
More informationELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization
ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)
More informationLecture 10. Applied Econometrics. Time Series Filters. Jozef Barunik ( utia. cas. cz
Applied Econometrics Lecture 10 Time Series Filters Please note that for interactive manipulation you need Mathematica 6 version of this.pdf. Mathematica 6 is available at all Lab's Computers at IE http://staff.utia.cas.cz/baruni
More informationElec4621 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis
Elec461 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis Dr. D. S. Taubman May 3, 011 In this last chapter of your notes, we are interested in the problem of nding the instantaneous
More information1 otherwise. Note that the area of the pulse is one. The Dirac delta function (a.k.a. the impulse) can be defined using the pulse as follows:
The Dirac delta function There is a function called the pulse: { if t > Π(t) = 2 otherwise. Note that the area of the pulse is one. The Dirac delta function (a.k.a. the impulse) can be defined using the
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces
More information4 Uniform convergence
4 Uniform convergence In the last few sections we have seen several functions which have been defined via series or integrals. We now want to develop tools that will allow us to show that these functions
More informationECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are
ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that:
More information13. Power Spectrum. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if.
For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if jt X ( ) = xte ( ) dt, (3-) then X ( ) represents its energy spectrum. his follows from Parseval
More informationZ-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) =
Z-Transform The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = Z : G(z) = It is Used in Digital Signal Processing n= g(t)e st dt g[n]z n Used to Define Frequency
More information3 What You Should Know About Complex Numbers
3 What You Should Know About Complex Numbers Life is complex it has a real part, and an imaginary part Andrew Koenig. Complex numbers are an extension of the more familiar world of real numbers that make
More informationFOURIER ANALYSIS using Python
(version September 5) FOURIER ANALYSIS using Python his practical introduces the following: Fourier analysis of both periodic and non-periodic signals (Fourier series, Fourier transform, discrete Fourier
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 informationFourier transform. Alejandro Ribeiro. February 1, 2018
Fourier transform Alejandro Ribeiro February 1, 2018 The discrete Fourier transform (DFT) is a computational tool to work with signals that are defined on a discrete time support and contain a finite number
More informationLecture 27 Frequency Response 2
Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period
More informationCHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2
CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial
More informationFilter Banks For "Intensity Analysis"
morlet.sdw 4/6/3 /3 Filter Banks For "Intensity Analysis" Introduction In connection with our participation in two conferences in Canada (August 22 we met von Tscharner several times discussing his "intensity
More informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationChapter 17 : Fourier Series Page 1 of 12
Chapter 7 : Fourier Series Page of SECTION C Further Fourier Series By the end of this section you will be able to obtain the Fourier series for more complicated functions visualize graphs of Fourier series
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 informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationAPPENDIX A. The Fourier integral theorem
APPENDIX A The Fourier integral theorem In equation (1.7) of Section 1.3 we gave a description of a signal defined on an infinite range in the form of a double integral, with no explanation as to how that
More informationSummary: ISI. No ISI condition in time. II Nyquist theorem. Ideal low pass filter. Raised cosine filters. TX filters
UORIAL ON DIGIAL MODULAIONS Part 7: Intersymbol interference [last modified: 200--23] Roberto Garello, Politecnico di orino Free download at: www.tlc.polito.it/garello (personal use only) Part 7: Intersymbol
More informationContinuous Fourier transform of a Gaussian Function
Continuous Fourier transform of a Gaussian Function Gaussian function: e t2 /(2σ 2 ) The CFT of a Gaussian function is also a Gaussian function (i.e., time domain is Gaussian, then the frequency domain
More information8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
More informationENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University
ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier
More informationDESIGN OF CMOS ANALOG INTEGRATED CIRCUITS
DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete
More informationDigital Filters Ying Sun
Digital Filters Ying Sun Digital filters Finite impulse response (FIR filter: h[n] has a finite numbers of terms. Infinite impulse response (IIR filter: h[n] has infinite numbers of terms. Causal filter:
More information2. Signal Space Concepts
2. Signal Space Concepts R.G. Gallager The signal-space viewpoint is one of the foundations of modern digital communications. Credit for popularizing this viewpoint is often given to the classic text of
More informationFunctions not limited in time
Functions not limited in time We can segment an arbitrary L 2 function into segments of width T. The mth segment is u m (t) = u(t)rect(t/t m). We then have m 0 u(t) =l.i.m. m0 u m (t) m= m 0 This wors
More information6.003: Signals and Systems. Sampling and Quantization
6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): x[n] = x(nt ) t n Impulse reconstruction: x p (t) =
More informationESS Dirac Comb and Flavors of Fourier Transforms
6. Dirac Comb and Flavors of Fourier ransforms Consider a periodic function that comprises pulses of amplitude A and duration τ spaced a time apart. We can define it over one period as y(t) = A, τ / 2
More informationPredicting the future with Newton s Second Law
Predicting the future with Newton s Second Law To represent the motion of an object (ignoring rotations for now), we need three functions x(t), y(t), and z(t), which describe the spatial coordinates of
More informationSummary of Fourier Transform Properties
Summary of Fourier ransform Properties Frank R. Kschischang he Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of oronto January 7, 207 Definition and Some echnicalities
More information06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1
IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11]
More informationSensors. Chapter Signal Conditioning
Chapter 2 Sensors his chapter, yet to be written, gives an overview of sensor technology with emphasis on how to model sensors. 2. Signal Conditioning Sensors convert physical measurements into data. Invariably,
More informationz = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1)
11 Complex numbers Read: Boas Ch. Represent an arb. complex number z C in one of two ways: z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) Here i is 1, engineers call
More informationLecture 3 - Design of Digital Filters
Lecture 3 - Design of Digital Filters 3.1 Simple filters In the previous lecture we considered the polynomial fit as a case example of designing a smoothing filter. The approximation to an ideal LPF can
More informationContinuous-time Fourier Methods
ELEC 321-001 SIGNALS and SYSTEMS Continuous-time Fourier Methods Chapter 6 1 Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity
More informationChapter 4 The Fourier Series and Fourier Transform
Chapter 4 The Fourier Series and Fourier Transform Fourier Series Representation of Periodic Signals Let x(t) be a CT periodic signal with period T, i.e., xt ( + T) = xt ( ), t R Example: the rectangular
More informationAspects of Continuous- and Discrete-Time Signals and Systems
Aspects of Continuous- and Discrete-Time Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling the
More informationDigital Baseband Systems. Reference: Digital Communications John G. Proakis
Digital Baseband Systems Reference: Digital Communications John G. Proais Baseband Pulse Transmission Baseband digital signals - signals whose spectrum extend down to or near zero frequency. Model of the
More informationExamples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.
s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:
More informationECE Unit 4. Realizable system used to approximate the ideal system is shown below: Figure 4.47 (b) Digital Processing of Analog Signals
ECE 8440 - Unit 4 Digital Processing of Analog Signals- - Non- Ideal Case (See sec8on 4.8) Before considering the non- ideal case, recall the ideal case: 1 Assump8ons involved in ideal case: - no aliasing
More informationSTABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse
SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential
More informationFinding Limits Graphically and Numerically
Finding Limits Graphically and Numerically 1. Welcome to finding limits graphically and numerically. My name is Tuesday Johnson and I m a lecturer at the University of Texas El Paso. 2. With each lecture
More informationCast of Characters. Some Symbols, Functions, and Variables Used in the Book
Page 1 of 6 Cast of Characters Some s, Functions, and Variables Used in the Book Digital Signal Processing and the Microcontroller by Dale Grover and John R. Deller ISBN 0-13-081348-6 Prentice Hall, 1998
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 informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More information2 M. Hasegawa-Johnson. DRAFT COPY.
Lecture Notes in Speech Production Speech Coding and Speech Recognition Mark Hasegawa-Johnson University of Illinois at Urbana-Champaign February 7 2000 2 M. Hasegawa-Johnson. DRAFT COPY. Chapter Basics
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 informationPower Series Solutions We use power series to solve second order differential equations
Objectives Power Series Solutions We use power series to solve second order differential equations We use power series expansions to find solutions to second order, linear, variable coefficient equations
More information