The Fourier Transform and Applications
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Frequency Response = K − Ml
Frequency Response 1. Introduction We will examine the response of a second order linear constant coefficient system to a sinusoidal input. We will pay special attention to the way the output changes as the frequency of the input changes. This is what we mean by the frequency response of the system. In particular, we will look at the amplitude response and the phase response; that is, the amplitude and phase lag of the system’s output considered as functions of the input frequency. In O.4 the Exponential Input Theorem was used to find a particular solution in the case of exponential or sinusoidal input. Here we will work out in detail the formulas for a second order system. We will then interpret these formulas as the frequency response of a mechanical system. In particular, we will look at damped-spring-mass systems. We will study carefully two cases: first, when the mass is driven by pushing on the spring and second, when the mass is driven by pushing on the dashpot. Both these systems have the same form p(D)x = q(t), but their amplitude responses are very different. This is because, as we will see, it can make physical sense to designate something other than q(t) as the input. For example, in the system mx0 + bx0 + kx = by0 we will consider y to be the input. (Of course, y is related to the expression on the right- hand-side of the equation, but it is not exactly the same.) 2. Sinusoidally Driven Systems: Second Order Constant Coefficient DE’s We start with the second order linear constant coefficient (CC) DE, which as we’ve seen can be interpreted as modeling a damped forced harmonic oscillator. -
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: [email protected] e-mail: [email protected] tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions. -
Music Synthesis
MUSIC SYNTHESIS Sound synthesis is the art of using electronic devices to create & modify signals that are then turned into sound waves by a speaker. Making Waves: WGRL - 2015 Oscillators An oscillator generates a consistent, repeating signal. Signals from oscillators and other sources are used to control the movement of the cones in our speakers, which make real sound waves which travel to our ears. An oscillator wiggles an audio signal. DEMONSTRATE: If you tie one end of a rope to a doorknob, stand back a few feet, and wiggle the other end of the rope up and down really fast, you're doing roughly the same thing as an oscillator. REVIEW: Frequency and pitch Frequency, measured in cycles/second AKA Hertz, is the rate at which a sound wave moves in and out. The length of a signal cycle of a waveform is the span of time it takes for that waveform to repeat. People generally hear an increase in the frequency of a sound wave as an increase in pitch. F DEMONSTRATE: an oscillator generating a signal that repeats at the rate of 440 cycles per second will have the same pitch as middle A on a piano. An oscillator generating a signal that repeats at 880 cycles per second will have the same pitch as the A an octave above middle A. Types of Waveforms: SINE The SINE wave is the most basic, pure waveform. These simple waves have only one frequency. Any other waveform can be created by adding up a series of sine waves. In this picture, the first two sine waves In this picture, a sine wave is added to its are added together to produce a third. -
To Learn the Basic Properties of Traveling Waves. Slide 20-2 Chapter 20 Preview
Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide 20-2 Chapter 20 Preview Slide 20-3 Chapter 20 Preview Slide 20-5 • result from periodic disturbance • same period (frequency) as source 1 f • Longitudinal or Transverse Waves • Characterized by – amplitude (how far do the “bits” move from their equilibrium positions? Amplitude of MEDIUM) – period or frequency (how long does it take for each “bit” to go through one cycle?) – wavelength (over what distance does the cycle repeat in a freeze frame?) – wave speed (how fast is the energy transferred?) vf v Wavelength and Frequency are Inversely related: f The shorter the wavelength, the higher the frequency. The longer the wavelength, the lower the frequency. 3Hz 5Hz Spherical Waves Wave speed: Depends on Properties of the Medium: Temperature, Density, Elasticity, Tension, Relative Motion vf Transverse Wave • A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave Longitudinal Wave A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave: Pulse Tuning Fork Guitar String Types of Waves Sound String Wave PULSE: • traveling disturbance • transfers energy and momentum • no bulk motion of the medium • comes in two flavors • LONGitudinal • TRANSverse Traveling Pulse • For a pulse traveling to the right – y (x, t) = f (x – vt) • For a pulse traveling to -
Resonance Beyond Frequency-Matching
Resonance Beyond Frequency-Matching Zhenyu Wang (王振宇)1, Mingzhe Li (李明哲)1,2, & Ruifang Wang (王瑞方)1,2* 1 Department of Physics, Xiamen University, Xiamen 361005, China. 2 Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China. *Corresponding author. [email protected] Resonance, defined as the oscillation of a system when the temporal frequency of an external stimulus matches a natural frequency of the system, is important in both fundamental physics and applied disciplines. However, the spatial character of oscillation is not considered in the definition of resonance. In this work, we reveal the creation of spatial resonance when the stimulus matches the space pattern of a normal mode in an oscillating system. The complete resonance, which we call multidimensional resonance, is a combination of both the spatial and the conventionally defined (temporal) resonance and can be several orders of magnitude stronger than the temporal resonance alone. We further elucidate that the spin wave produced by multidimensional resonance drives considerably faster reversal of the vortex core in a magnetic nanodisk. Our findings provide insight into the nature of wave dynamics and open the door to novel applications. I. INTRODUCTION Resonance is a universal property of oscillation in both classical and quantum physics[1,2]. Resonance occurs at a wide range of scales, from subatomic particles[2,3] to astronomical objects[4]. A thorough understanding of resonance is therefore crucial for both fundamental research[4-8] and numerous related applications[9-12]. The simplest resonance system is composed of one oscillating element, for instance, a pendulum. Such a simple system features a single inherent resonance frequency. -
Interference: Two Spherical Sources Superposition
Interference: Two Spherical Sources Superposition Interference Waves ADD: Constructive Interference. Waves SUBTRACT: Destructive Interference. In Phase Out of Phase Superposition Traveling waves move through each other, interfere, and keep on moving! Pulsed Interference Superposition Waves ADD in space. Any complex wave can be built from simple sine waves. Simply add them point by point. Simple Sine Wave Simple Sine Wave Complex Wave Fourier Synthesis of a Square Wave Any periodic function can be represented as a series of sine and cosine terms in a Fourier series: y() t ( An sin2ƒ n t B n cos2ƒ) n t n Superposition of Sinusoidal Waves • Case 1: Identical, same direction, with phase difference (Interference) Both 1-D and 2-D waves. • Case 2: Identical, opposite direction (standing waves) • Case 3: Slightly different frequencies (Beats) Superposition of Sinusoidal Waves • Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude • The waves differ in phase • y1 = A sin (kx - wt) • y2 = A sin (kx - wt + f) • y = y1+y2 = 2A cos (f/2) sin (kx - wt + f/2) Resultant Amplitude Depends on phase: Spatial Interference Term Sinusoidal Waves with Constructive Interference y = y1+y2 = 2A cos (f/2) sin (kx - wt + f /2) • When f = 0, then cos (f/2) = 1 • The amplitude of the resultant wave is 2A – The crests of one wave coincide with the crests of the other wave • The waves are everywhere in phase • The waves interfere constructively Sinusoidal Waves with Destructive Interference y = y1+y2 = 2A cos (f/2) -
Understanding What Really Happens at Resonance
feature article Resonance Revealed: Understanding What Really Happens at Resonance Chris White Wood RESONANCE focus on some underlying principles and use these to construct The word has various meanings in acoustics, chemistry, vector diagrams to explain the resonance phenomenon. It thus electronics, mechanics, even astronomy. But for vibration aspires to provide a more intuitive understanding. professionals, it is the definition from the field of mechanics that is of interest, and it is usually stated thus: SYSTEM BEHAVIOR Before we move on to the why and how, let us review the what— “The condition where a system or body is subjected to an that is, what happens when a cyclic force, gradually increasing oscillating force close to its natural frequency.” from zero frequency, is applied to a vibrating system. Let us consider the shaft of some rotating machine. Rotor Yet this definition seems incomplete. It really only states the balancing is always performed to within a tolerance; there condition necessary for resonance to occur—telling us nothing will always be some degree of residual unbalance, which will of the condition itself. How does a system behave at resonance, give rise to a rotating centrifugal force. Although the residual and why? Why does the behavior change as it passes through unbalance is due to a nonsymmetrical distribution of mass resonance? Why does a system even have a natural frequency? around the center of rotation, we can think of it as an equivalent Of course, we can diagnose machinery vibration resonance “heavy spot” at some point on the rotor. problems without complete answers to these questions. -
Chapter 1 Waves in Two and Three Dimensions
Chapter 1 Waves in Two and Three Dimensions In this chapter we extend the ideas of the previous chapter to the case of waves in more than one dimension. The extension of the sine wave to higher dimensions is the plane wave. Wave packets in two and three dimensions arise when plane waves moving in different directions are superimposed. Diffraction results from the disruption of a wave which is impingent upon an object. Those parts of the wave front hitting the object are scattered, modified, or destroyed. The resulting diffraction pattern comes from the subsequent interference of the various pieces of the modified wave. A knowl- edge of diffraction is necessary to understand the behavior and limitations of optical instruments such as telescopes. Diffraction and interference in two and three dimensions can be manipu- lated to produce useful devices such as the diffraction grating. 1.1 Math Tutorial — Vectors Before we can proceed further we need to explore the idea of a vector. A vector is a quantity which expresses both magnitude and direction. Graph- ically we represent a vector as an arrow. In typeset notation a vector is represented by a boldface character, while in handwriting an arrow is drawn over the character representing the vector. Figure 1.1 shows some examples of displacement vectors, i. e., vectors which represent the displacement of one object from another, and introduces 1 CHAPTER 1. WAVES IN TWO AND THREE DIMENSIONS 2 y Paul B y B C C y George A A y Mary x A x B x C x Figure 1.1: Displacement vectors in a plane. -
PHASOR DIAGRAMS II Fault Analysis Ron Alexander – Bonneville Power Administration
PHASOR DIAGRAMS II Fault Analysis Ron Alexander – Bonneville Power Administration For any technician or engineer to understand the characteristics of a power system, the use of phasors and polarity are essential. They aid in the understanding and analysis of how the power system is connected and operates both during normal (balanced) conditions, as well as fault (unbalanced) conditions. Thus, as J. Lewis Blackburn of Westinghouse stated, “a sound theoretical and practical knowledge of phasors and polarity is a fundamental and valuable resource.” C C A A B B Balanced System Unbalanced System With the proper identification of circuits and assumed direction established in a circuit diagram (with the use of polarity), the corresponding phasor diagram can be drawn from either calculated or test data. Fortunately, most relays today along with digital fault recorders supply us with recorded quantities as seen during fault conditions. This in turn allows us to create a phasor diagram in which we can visualize how the power system was affected during a fault condition. FAULTS Faults are unavoidable in the operation of a power system. Faults are caused by: • Lightning • Insulator failure • Equipment failure • Trees • Accidents • Vandalism such as gunshots • Fires • Foreign material Faults are essentially short circuits on the power system and can occur between phases and ground in virtually any combination: • One phase to ground • Two phases to ground • Three phase to ground • Phase to phase As previously instructed by Cliff Harris of Idaho Power Company: Faults come uninvited and seldom leave voluntarily. Faults cause voltage to collapse and current to increase. Fault voltage and current magnitude depend on several factors, including source strength, location of fault, type of fault, system conditions, etc. -
Tektronix Signal Generator
Signal Generator Fundamentals Signal Generator Fundamentals Table of Contents The Complete Measurement System · · · · · · · · · · · · · · · 5 Complex Waves · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15 The Signal Generator · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6 Signal Modulation · · · · · · · · · · · · · · · · · · · · · · · · · · · 15 Analog or Digital? · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 7 Analog Modulation · · · · · · · · · · · · · · · · · · · · · · · · · 15 Basic Signal Generator Applications· · · · · · · · · · · · · · · · 8 Digital Modulation · · · · · · · · · · · · · · · · · · · · · · · · · · 15 Verification · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 8 Frequency Sweep · · · · · · · · · · · · · · · · · · · · · · · · · · · 16 Testing Digital Modulator Transmitters and Receivers · · 8 Quadrature Modulation · · · · · · · · · · · · · · · · · · · · · 16 Characterization · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 8 Digital Patterns and Formats · · · · · · · · · · · · · · · · · · · 16 Testing D/A and A/D Converters · · · · · · · · · · · · · · · · · 8 Bit Streams · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 17 Stress/Margin Testing · · · · · · · · · · · · · · · · · · · · · · · · · · · 9 Types of Signal Generators · · · · · · · · · · · · · · · · · · · · · · 17 Stressing Communication Receivers · · · · · · · · · · · · · · 9 Analog and Mixed Signal Generators · · · · · · · · · · · · · · 18 Signal Generation Techniques -
Fourier Analysis
FOURIER ANALYSIS Lucas Illing 2008 Contents 1 Fourier Series 2 1.1 General Introduction . 2 1.2 Discontinuous Functions . 5 1.3 Complex Fourier Series . 7 2 Fourier Transform 8 2.1 Definition . 8 2.2 The issue of convention . 11 2.3 Convolution Theorem . 12 2.4 Spectral Leakage . 13 3 Discrete Time 17 3.1 Discrete Time Fourier Transform . 17 3.2 Discrete Fourier Transform (and FFT) . 19 4 Executive Summary 20 1 1. Fourier Series 1 Fourier Series 1.1 General Introduction Consider a function f(τ) that is periodic with period T . f(τ + T ) = f(τ) (1) We may always rescale τ to make the function 2π periodic. To do so, define 2π a new independent variable t = T τ, so that f(t + 2π) = f(t) (2) So let us consider the set of all sufficiently nice functions f(t) of a real variable t that are periodic, with period 2π. Since the function is periodic we only need to consider its behavior on one interval of length 2π, e.g. on the interval (−π; π). The idea is to decompose any such function f(t) into an infinite sum, or series, of simpler functions. Following Joseph Fourier (1768-1830) consider the infinite sum of sine and cosine functions 1 a0 X f(t) = + [a cos(nt) + b sin(nt)] (3) 2 n n n=1 where the constant coefficients an and bn are called the Fourier coefficients of f. The first question one would like to answer is how to find those coefficients.