Linear Time-Invariant Systems and Their Frequency Response
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J-DSP Lab 2: the Z-Transform and Frequency Responses
J-DSP Lab 2: The Z-Transform and Frequency Responses Introduction This lab exercise will cover the Z transform and the frequency response of digital filters. The goal of this exercise is to familiarize you with the utility of the Z transform in digital signal processing. The Z transform has a similar role in DSP as the Laplace transform has in circuit analysis: a) It provides intuition in certain cases, e.g., pole location and filter stability, b) It facilitates compact signal representations, e.g., certain deterministic infinite-length sequences can be represented by compact rational z-domain functions, c) It allows us to compute signal outputs in source-system configurations in closed form, e.g., using partial functions to compute transient and steady state responses. d) It associates intuitively with frequency domain representations and the Fourier transform In this lab we use the Filter block of J-DSP to invert the Z transform of various signals. As we have seen in the previous lab, the Filter block in J-DSP can implement a filter transfer function of the following form 10 −i ∑bi z i=0 H (z) = 10 − j 1+ ∑ a j z j=1 This is essentially realized as an I/P-O/P difference equation of the form L M y(n) = ∑∑bi x(n − i) − ai y(n − i) i==01i The transfer function is associated with the impulse response and hence the output can also be written as y(n) = x(n) * h(n) Here, * denotes convolution; x(n) and y(n) are the input signal and output signal respectively. -
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 Fall Term 2008 1 Lecture 1 Reading: • Class handout: The Dirac Delta and Unit-Step Functions 1 Introduction to Signal Processing In this class we will primarily deal with processing time-based functions, but the methods will also be applicable to spatial functions, for example image processing. We will deal with (a) Signal processing of continuous waveforms f(t), using continuous LTI systems (filters). a LTI dy nam ical system input ou t pu t f(t) Continuous Signal y(t) Processor and (b) Discrete-time (digital) signal processing of data sequences {fn} that might be samples of real continuous experimental data, such as recorded through an analog-digital converter (ADC), or implicitly discrete in nature. a LTI dis crete algorithm inp u t se q u e n c e ou t pu t seq u e n c e f Di screte Si gnal y { n } { n } Pr ocessor Some typical applications that we look at will include (a) Data analysis, for example estimation of spectral characteristics, delay estimation in echolocation systems, extraction of signal statistics. (b) Signal enhancement. Suppose a waveform has been contaminated by additive “noise”, for example 60Hz interference from the ac supply in the laboratory. 1copyright c D.Rowell 2008 1–1 inp u t ou t p u t ft + ( ) Fi lte r y( t) + n( t) ad d it ive no is e The task is to design a filter that will minimize the effect of the interference while not destroying information from the experiment. -
RANDOM VIBRATION—AN OVERVIEW by Barry Controls, Hopkinton, MA
RANDOM VIBRATION—AN OVERVIEW by Barry Controls, Hopkinton, MA ABSTRACT Random vibration is becoming increasingly recognized as the most realistic method of simulating the dynamic environment of military applications. Whereas the use of random vibration specifications was previously limited to particular missile applications, its use has been extended to areas in which sinusoidal vibration has historically predominated, including propeller driven aircraft and even moderate shipboard environments. These changes have evolved from the growing awareness that random motion is the rule, rather than the exception, and from advances in electronics which improve our ability to measure and duplicate complex dynamic environments. The purpose of this article is to present some fundamental concepts of random vibration which should be understood when designing a structure or an isolation system. INTRODUCTION Random vibration is somewhat of a misnomer. If the generally accepted meaning of the term "random" were applicable, it would not be possible to analyze a system subjected to "random" vibration. Furthermore, if this term were considered in the context of having no specific pattern (i.e., haphazard), it would not be possible to define a vibration environment, for the environment would vary in a totally unpredictable manner. Fortunately, this is not the case. The majority of random processes fall in a special category termed stationary. This means that the parameters by which random vibration is characterized do not change significantly when analyzed statistically over a given period of time - the RMS amplitude is constant with time. For instance, the vibration generated by a particular event, say, a missile launch, will be statistically similar whether the event is measured today or six months from today. -
Lecture 2 LSI Systems and Convolution in 1D
Lecture 2 LSI systems and convolution in 1D 2.1 Learning Objectives Understand the concept of a linear system. • Understand the concept of a shift-invariant system. • Recognize that systems that are both linear and shift-invariant can be described • simply as a convolution with a certain “impulse response” function. 2.2 Linear Systems Figure 2.1: Example system H, which takes in an input signal f(x)andproducesan output g(x). AsystemH takes an input signal x and produces an output signal y,whichwecan express as H : f(x) g(x). This very general definition encompasses a vast array of di↵erent possible systems.! A subset of systems of particular relevance to signal processing are linear systems, where if f (x) g (x)andf (x) g (x), then: 1 ! 1 2 ! 2 a f + a f a g + a g 1 · 1 2 · 2 ! 1 · 1 2 · 2 for any input signals f1 and f2 and scalars a1 and a2. In other words, for a linear system, if we feed it a linear combination of inputs, we get the corresponding linear combination of outputs. Question: Why do we care about linear systems? 13 14 LECTURE 2. LSI SYSTEMS AND CONVOLUTION IN 1D Question: Can you think of examples of linear systems? Question: Can you think of examples of non-linear systems? 2.3 Shift-Invariant Systems Another subset of systems we care about are shift-invariant systems, where if f1 g1 and f (x)=f (x x )(ie:f is a shifted version of f ), then: ! 2 1 − 0 2 1 f (x) g (x)=g (x x ) 2 ! 2 1 − 0 for any input signals f1 and shift x0. -
Introduction to Aircraft Aeroelasticity and Loads
JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads i JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 ii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Copyright C 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. -
Linear System Theory
Chapter 2 Linear System Theory In this course, we will be dealing primarily with linear systems, a special class of sys- tems for which a great deal is known. During the first half of the twentieth century, linear systems were analyzed using frequency domain (e.g., Laplace and z-transform) based approaches in an effort to deal with issues such as noise and bandwidth issues in communication systems. While they provided a great deal of intuition and were suf- ficient to establish some fundamental results, frequency domain approaches presented various drawbacks when control scientists started studying more complicated systems (containing multiple inputs and outputs, nonlinearities, noise, and so forth). Starting in the 1950’s (around the time of the space race), control engineers and scientists started turning to state-space models of control systems in order to address some of these issues. These time-domain approaches are able to effectively represent concepts such as the in- ternal state of the system, and also present a method to introduce optimality conditions into the controller design procedure. We will be using the state-space (or “modern”) approach to control almost exclusively in this course, and the purpose of this chapter is to review some of the essential concepts in this area. 2.1 Discrete-Time Signals Given a field F,asignal is a mapping from a set of numbers to F; in other words, signals are simply functions of the set of numbers that they operate on. More specifically: • A discrete-time signal f is a mapping from the set of integers Z to F, and is denoted by f[k]fork ∈ Z.Eachinstantk is also known as a time-step. -
Amplifier Frequency Response
EE105 – Fall 2015 Microelectronic Devices and Circuits Prof. Ming C. Wu [email protected] 511 Sutardja Dai Hall (SDH) 2-1 Amplifier Gain vO Voltage Gain: Av = vI iO Current Gain: Ai = iI load power vOiO Power Gain: Ap = = input power vIiI Note: Ap = Av Ai Note: Av and Ai can be positive, negative, or even complex numbers. Nagative gain means the output is 180° out of phase with input. However, power gain should always be a positive number. Gain is usually expressed in Decibel (dB): 2 Av (dB) =10log Av = 20log Av 2 Ai (dB) =10log Ai = 20log Ai Ap (dB) =10log Ap 2-2 1 Amplifier Power Supply and Dissipation • Circuit needs dc power supplies (e.g., battery) to function. • Typical power supplies are designated VCC (more positive voltage supply) and -VEE (more negative supply). • Total dc power dissipation of the amplifier Pdc = VCC ICC +VEE IEE • Power balance equation Pdc + PI = PL + Pdissipated PI : power drawn from signal source PL : power delivered to the load (useful power) Pdissipated : power dissipated in the amplifier circuit (not counting load) P • Amplifier power efficiency η = L Pdc Power efficiency is important for "power amplifiers" such as output amplifiers for speakers or wireless transmitters. 2-3 Amplifier Saturation • Amplifier transfer characteristics is linear only over a limited range of input and output voltages • Beyond linear range, the output voltage (or current) waveforms saturates, resulting in distortions – Lose fidelity in stereo – Cause interference in wireless system 2-4 2 Symbol Convention iC (t) = IC +ic (t) iC (t) : total instantaneous current IC : dc current ic (t) : small signal current Usually ic (t) = Ic sinωt Please note case of the symbol: lowercase-uppercase: total current lowercase-lowercase: small signal ac component uppercase-uppercase: dc component uppercase-lowercase: amplitude of ac component Similarly for voltage expressions. -
Control Theory
Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. -
Frequency Response and Bode Plots
1 Frequency Response and Bode Plots 1.1 Preliminaries The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function Hj( ) . A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. Of course we can easily program the transfer function into a computer to make such plots, and for very complicated transfer functions this may be our only recourse. But in many cases the key features of the plot can be quickly sketched by hand using some simple rules that identify the impact of the poles and zeroes in shaping the frequency response. The advantage of this approach is the insight it provides on how the circuit elements influence the frequency response. This is especially important in the design of frequency-selective circuits. We will first consider how to generate Bode plots for simple poles, and then discuss how to handle the general second-order response. Before doing this, however, it may be helpful to review some properties of transfer functions, the decibel scale, and properties of the log function. Poles, Zeroes, and Stability The s-domain transfer function is always a rational polynomial function of the form Ns() smm as12 a s m asa Hs() K K mm12 10 (1.1) nn12 n Ds() s bsnn12 b s bsb 10 As we have seen already, the polynomials in the numerator and denominator are factored to find the poles and zeroes; these are the values of s that make the numerator or denominator zero. If we write the zeroes as zz123,, zetc., and similarly write the poles as pp123,, p , then Hs( ) can be written in factored form as ()()()s zsz sz Hs() K 12 m (1.2) ()()()s psp12 sp n 1 © Bob York 2009 2 Frequency Response and Bode Plots The pole and zero locations can be real or complex. -
Evaluation of Audio Test Methods and Measurements for End-Of-Line Loudspeaker Quality Control
Evaluation of audio test methods and measurements for end-of-line loudspeaker quality control Steve Temme1 and Viktor Dobos2 (1. Listen, Inc., Boston, MA, 02118, USA. [email protected]; 2. Harman/Becker Automotive Systems Kft., H-8000 Székesfehérvár, Hungary) ABSTRACT In order to minimize costly warranty repairs, loudspeaker OEMS impose tight specifications and a “total quality” requirement on their part suppliers. At the same time, they also require low prices. This makes it important for driver manufacturers and contract manufacturers to work with their OEM customers to define reasonable specifications and tolerances. They must understand both how the loudspeaker OEMS are testing as part of their incoming QC and also how to implement their own end-of-line measurements to ensure correlation between the two. Specifying and testing loudspeakers can be tricky since loudspeakers are inherently nonlinear, time-variant and effected by their working conditions & environment. This paper examines the loudspeaker characteristics that can be measured, and discusses common pitfalls and how to avoid them on a loudspeaker production line. Several different audio test methods and measurements for end-of- the-line speaker quality control are evaluated, and the most relevant ones identified. Speed, statistics, and full traceability are also discussed. Keywords: end of line loudspeaker testing, frequency response, distortion, Rub & Buzz, limits INTRODUCTION In order to guarantee quality while keeping testing fast and accurate, it is important for audio manufacturers to perform only those tests which will easily identify out-of-specification products, and omit those which do not provide additional information that directly pertains to the quality of the product or its likelihood of failure. -
The Scientist and Engineer's Guide to Digital Signal Processing Properties of Convolution
CHAPTER 7 Properties of Convolution A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on. This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Fourth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers. Common Impulse Responses Delta Function The simplest impulse response is nothing more that a delta function, as shown in Fig. 7-1a. That is, an impulse on the input produces an identical impulse on the output. This means that all signals are passed through the system without change. Convolving any signal with a delta function results in exactly the same signal. Mathematically, this is written: EQUATION 7-1 The delta function is the identity for ( ' convolution. Any signal convolved with x[n] *[n] x[n] a delta function is left unchanged. This property makes the delta function the identity for convolution. This is analogous to zero being the identity for addition (a%0 ' a), and one being the identity for multiplication (a×1 ' a). -
Calibration of Radio Receivers to Measure Broadband Interference
NBSIR 73-335 CALIBRATION OF RADIO RECEIVERS TO MEASORE BROADBAND INTERFERENCE Ezra B. Larsen Electromagnetics Division Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302 September 1973 Final Report, Phase I Prepared for Calibration Coordination Group Army /Navy /Air Force NBSIR 73-335 CALIBRATION OF RADIO RECEIVERS TO MEASORE BROADBAND INTERFERENCE Ezra B. Larsen Electromagnetics Division Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302 September 1973 Final Report, Phase I Prepared for Calibration Coordination Group Army/Navy /Air Force u s. DEPARTMENT OF COMMERCE, Frederick B. Dent. Secretary NATIONAL BUREAU OF STANDARDS. Richard W Roberts Director V CONTENTS Page 1. BACKGROUND 2 2. INTRODUCTION 3 3. DEFINITIONS OF RECEIVER BANDWIDTH AND IMPULSE STRENGTH (Or Spectral Intensity) 5 3.1 Random noise bandwidth 5 3.2 Impulse bandwidth in terms of receiver response transient 7 3.3 Voltage-response bandwidth of receiver 8 3.4 Receiver bandwidths related to voltage- response bandwidth 9 3.5 Impulse strength in terms of random-noise bandwidth 10 3.6 Sum-and-dif ference correlation radiometer technique 10 3.7 Impulse strength in terms of Fourier trans- forms 11 3.8 Impulse strength in terms of rectangular DC pulses 12 3.9 Impulse strength and impulse bandwidth in terms of RF pulses 13 4. EXPERIMENTAL PROCEDURE AND DATA 16 4.1 Measurement of receiver input impedance and phase linearity 16 4.2 Calibration of receiver as a tuned RF volt- meter 20 4.3 Stepped- frequency measurements of receiver bandwidth 23 4.4 Calibration of receiver impulse bandwidth with a baseband pulse generator 37 iii CONTENTS [Continued) Page 4.5 Calibration o£ receiver impulse bandwidth with a pulsed-RF source 38 5.