# SIGNAL PROCESSING & SIMULATION NEWSLETTER

Size: px
Start display at page:

Transcription

1 1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty with this issue, you may safely stop reading after page 5 without feeling guilty. Hilbert Transform, Analytic Signal and the Complex Envelope In Digital Signal Processing we often need to look at relationships between real and imaginary parts of a complex signal. These relationships are generally described by Hilbert transforms. Hilbert transform not only helps us relate the I and Q components but it is also used to create a special class of causal signals called analytic which are especially important in simulation. The analytic signals help us to represent bandpass signals as complex signals which have specially attractive properties for signal processing. Hilbert Transform is not a particularly complex concept and can be much better understood if we take an intuitive approach first before delving into its formula which is related to convolution and is hard to grasp. The following diagram that is often seen in text books describing modulation gives us a clue as to what a Hilbert Transform does. Figure 1 - Role of Hilbert Transform in modulation The role of Hilbert transform as we can guess here is to take the carrier which is a cosine wave and create a sine wave out of it. So let s take a closer look at a cosine wave to see how this is done by the Hilbert transformer. Figure 2a shows the amplitude and the phase spectrum of a cosine wave. Now recall that the Fourier Series is written as where and and A n and B n are the spectral amplitudes of cosine and sine waves. Now take a look at the phase spectrum. The phase spectrum is computed by

2 2 of 10 1/25/2008 3:38 AM Cosine wave has no sine spectral content, so B n is zero. The phase calculated is 90 for both positive and negative frequency from above formula. The wave has two spectral components each of magnitude 1/2A, both positive and lying in the real plane. (the real plane is described as that passing vertically (R-V plane) and the Imaginary plane as one horizontally (R-I plane) through the Imaginary axis) Figure 2b shows the same two spectrums for a sine wave. The sine wave phase is not symmetric because the amplitude spectrum is not symmetric. The quantity A n is zero and B n has either a positive or negative value. The phase is +90 for the positive frequency and -90 for the negative frequency. Now we wish to convert the cosine wave to a sine wave. There are two ways of doing that, one in time domain and the other in frequency domain. Hilbert Transform in Frequency Domain Now compare Figure 2a and 2b, in particular the spectral amplitudes. The cosine spectral amplitudes are both positive and lie in the real plane. The sine wave has spectral components that lie in the Imaginary plane and are of opposite sign. To turn cosine into sine, as shown in Figure 3 below, we need to rotate the negative frequency component of the cosine by +90 and the positive frequency component by -90. We will need to rotate the +Q phasor by -90 or in other words multiply it by -j. We also need to rotate the -Q phasor by +90 or multiply it by j.

3 3 of 10 1/25/2008 3:38 AM Figure 3 - Rotating phasors to create a sine wave out of a cosine We can describe this transformation process called the Hilbert Transform as follows: All negative frequencies of a signal get a +90 phase shift and all positive frequencies get a -90 phase shift. If we put a cosine wave through this transformer, we get a sine wave. This phase rotation process is true for all signals put through the Hilbert transform and not just the cosine. For any signal g(t), its Hilbert Transform has the following property (Putting a little hat over the capital letter representing the time domain signal is the typical way a Hilbert Transform is written.) A sine wave through a Hilbert Transformer will come out as a negative cosine. A negative cosine will come out a negative sine wave and one more transformation will return it to the original cosine wave, each time its phase being changed by 90. For this reason Hilbert transform is also called a quadrature filter. We can draw this filter as shown below in Figure 4. Figure 4 - Hilbert Transform shifts the phase of positive frequencies by -90 and negative frequencies by +90. So here are two things we can say about the Hilbert Transform. 1. It is a peculiar sort of filter that changes the phase of the spectral components depending on the sign of their frequency. 2. It only effects the phase of the signal. It has no effect on the amplitude at all.

4 4 of 10 1/25/2008 3:38 AM Hilbert transform in Time Domain Now look at the signal in time domain. Given a signal g(t), Hilbert Transform of this signal is defined as (1) Another way to write this definition is to recognize that Hilbert Transform is also the convolution of function 1/ pt with the signal g(t). So we can write the above equation as Achieving a Hilbert Transform in time domain means convolving the signal with the function 1/ pt. Why the function 1/ pt, what is its significance? Let s look at the Fourier Transform of this function. What does that tell us? Given in Eq 3, the transform looks a lot like the Hilbert transform we talked about before. (2) The term sgn in Eq 3 above, called signum is simpler than it seems. Here is the way we could have written it which would have been more understandable. (3) In Figure 5 we show the signum function and its decomposition into two familiar functions. (4) Figure 5 - Signum Function decomposed into a unit function and a constant For shortcut, writing sgn is useful but it is better if it is understood as a sum of the above two much simpler functions. (We will use this relationship later.) We see in 6 figure that although 1/pt is a real function, is has a Fourier transform that lies strictly in the imaginary plane. Do you recall what this means in terms of Fourier Series coefficients? What does it tell us about a function if it has no real components in its Fourier transform? It says that this function can be represented completely by a sum of sine waves. It has no cosine component at all. (5)

5 5 of 10 1/25/2008 3:38 AM In Figure 7, we see a function composed of a sum of 50 sine waves. We see the similarity of this function with that of 1/pt. Now you can see that although the function 1/pt looks nothing at all a sinusoid, we can still approximate it with a sum of sinusoids. The function f(t) = 1/pt gives us a spectrum that explains the Hilbert Transform in time domain, albeit this way of looking at the Hilbert Transform is indeed very hard to grasp. We limit our discussion of Hilbert transform to Frequency domain due to this difficulty. Figure 7 - Approximating function f(t) = 1/p t with a sum of 50 sine wave We can add the following to our list of observations about the Hilbert Transform. 3. The signal and its Hilbert Transform are orthogonal. This is because by rotating the signal 90 we have now made it orthogonal to the original signal, that being the definition of orthogonality. 4. The signal and its Hilbert Transform have identical energy because phase shift do not change the energy of the signal only amplitude changes can do that. Analytic Signal Hilbert Transform has other interesting properties. One of these comes in handy in the formulation of an Analytic signal. Analytic signals are used in Double and Single side-band processing (about SSB and DSB later) as well as in creating the I and Q components of a real signal. An analytic signal is defined as follows. An analytic signal is a complex signal created by taking a signal and then adding in quadrature its Hilbert Transform. It is also called the pre-envelope of the real signal. So what is the analytic signal of a cosine? Substitute cos wt for g(t) in Eq 6, knowing that its Hilbert transform is a sine, we get (6)

6 6 of 10 1/25/2008 3:38 AM The analytic function of a cosine is the now familiar phasor or the complex exponential, e jwt. What is the analytic signal of a sine? Now substitute sin wt for g(t) in Eq 6, knowing that its Hilbert transform is a -cos, we get once again a complex exponential. Do you remember what the spectrum of a complex exponential looks like? To remind you, I repeat here the figure from Tutorial 6. f Figure 8 - Fourier transform of a complex exponential We can see from the figure above, that whereas the spectrum of a sine and cosine spans both the negative and positive frequencies, the spectrum of the analytic signal, in this case the complex exponential, is in fact present only in the positive domain. This is true for both sine and cosine and in fact for all real signals. Restating the results: the Analytic signal for both and sine and cosine is the complex exponential. Even though both sine and cosine have a two sided spectrum as we see in figures above, the complex exponential which is the analytic signal of a sinusoid has a one-sided spectrum. We can generalize from this: An analytic signal (composed of a real signal and its Hilbert transform) has a spectrum that exists only in the positive frequency domain. Let s take at a look at the analytic signal again. The conjugate of this signal is also a useful quantity. (7) This signal has components only in the negative frequencies and can be used to separate out the lower side-bands. Now back to the analytic signal. Let s extend our understanding by taking Fourier Transform of both sides of Eq 7. We get (8) (9)

7 7 of 10 1/25/2008 3:38 AM The first term is the Fourier transform of the signal g(t), and the second term is the inverse Hilbert Transform. We can rewrite by use of property Eq 9 as (10) One more simplification gives us This is a very important result and is applicable to both lowpass and modulated signals. For modulated or bandpass signals, its net effect is to translate the signal down to baseband, double the spectral magnitudes and then chop-off all negative components. Complex Envelope We can now define a new quantity based on the analytic signal, called the Complex Envelope. The Complex Envelope is defined as (11) The part is called the Complex Envelope of the signal g(t). Let s rewrite it and take its Fourier Transform. We now see clearly that the Complex Envelope is just the frequency shifted version of the analytic signal. Recognizing that multiplication with the complex exponential in time domain results in frequency shift in the Frequency domain, using the Fourier Transform results for the analytic signal above, we get (12) So here is what we have been trying to get at all this time. This result says that the Fourier Transform of the analytic signal is just the one-sided spectrum. The carrier signal drops out entirely and the spectrum is no longer symmetrical. This property is very valuable in simulation. We no longer have to do simulation at carrier frequencies but only at the highest frequency of the baseband signal. The process applies equally to other transformation such as filters etc. which are also down shifted. It even works when non-linearities are present in the channel and result in additional frequencies. There are other uses of complex representation which we will discuss as we explore these topics however

8 8 of 10 1/25/2008 3:38 AM its main use is in simulation. Example Let s do an example. Here is a real baseband signal. (I have left out the factor 2p for purposes of simplification) Figure 9 - A Baseband Signal The spectrum of this signal is shown below, both its individual spectral amplitudes and its magnitude spectrum. The magnitude spectrum shows one spectral component of magnitude 2 at f = 2 and -2 and an another one of magnitude 3 at f = 3 and -3. Figure 10a - Spectral amplitudes Figure 10b- The Magnitude Spectrum Now let s multiply it with a carrier signal of cos(100t) to modulate it and to create a bandpass signal, Figure 11 - The modulated signal and its envelope

9 9 of 10 1/25/2008 3:38 AM Let s take the Hilbert Transform of this signal. But before we do that we need to simplify the above so we only have sinusoids and not their products. This step will make it easy to compute the Hilbert Transform. By using these trigonometric relationships, we rewrite the above signal as Now we take the Hilbert Transform of each term and get Now create the analytic signal by adding the original signal and its Hilbert Transform. Let s once again rearrange the terms in the above signal Recognizing that each pair of terms is the Euler s representation of a sinusoid, we can now rewrite the analytic signal as But wait a minute, isn t this the original signal and the carrier written in the complex exponential? So why all the calculations just to get the original signal back? Now let s take the Fourier Transform of the analytic signal and the complex envelope we have computed to show the real advantage of the complex envelope representation of signals.

10 10 of 10 1/25/2008 3:38 AM Spectrum of the Complex Envelope Spectrum of the Analytic Signal Figure 12 - The Magnitude Spectrum of the Complex Envelope vs. The Analytic Signal Although this was a passband signal, we see that its complex envelope spectrum is centered around zero and not the carrier frequency. Also the spectral components are double those in figure 10b and they are only on the positive side. If you think the result looks suspiciously like a one-sided Fourier transform, then you would be right. We do all this because of something Nyquist said. He said that in order to properly reconstruct a signal, any signal, baseband or passband, needs to be sampled at least two times its highest spectral frequency. That requires that we sample at frequency of 200. But we just showed that if we take a modulated signal and go through all this math and create an analytic signal (which by the way does not require any knowledge of the original signal) we can separate the information signal the baseband signal s(t)) from the carrier. We do this by dividing the analytic signal by the carrier. Now all we have left is the baseband signal. All processing can be done at a sampling frequency which is 6 (two times the maximum frequency of 3) instead of 200. The point here is that this mathematical concept help us get around the signal processing requirements by Nyquist for sampling of bandpass systems. The complex envelope is useful primarily for passband signals. In a lowpass signal the complex envelope of the signal is the signal itself. But in passband signal, the complex envelope representation allows us to easily separate out the carrier. Take a look at the complex envelope again for this signal the analytic signal the complex envelope We see the advantage of this form right away. The complex envelope is just the low-pass part of the analytic signal. The analytic signal low-pass signal has been multiplied by the complex exponential at the carrier frequency. The Fourier transform of this representation will lead to the signal translated back down the baseband (and doubled with no negative frequency components) making it possible to get around the Nyquist sampling requirement and reduce computational load. Copyright 1999 Charan Langton, All Rights Reserved

### Introduction to Complex Fourier Series

Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function

### Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

### Convolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005

Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall

### Experiment 3: Double Sideband Modulation (DSB)

Experiment 3: Double Sideband Modulation (DSB) This experiment examines the characteristics of the double-sideband (DSB) linear modulation process. The demodulation is performed coherently and its strict

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### 2.2 Magic with complex exponentials

2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or

### Adding Sinusoids of the Same Frequency. Additive Synthesis. Spectrum. Music 270a: Modulation

Adding Sinusoids of the Same Frequency Music 7a: Modulation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) February 9, 5 Recall, that adding sinusoids of

### CIRCUITS LABORATORY EXPERIMENT 3. AC Circuit Analysis

CIRCUITS LABORATORY EXPERIMENT 3 AC Circuit Analysis 3.1 Introduction The steady-state behavior of circuits energized by sinusoidal sources is an important area of study for several reasons. First, the

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### 9 Fourier Transform Properties

9 Fourier Transform Properties The Fourier transform is a major cornerstone in the analysis and representation of signals and linear, time-invariant systems, and its elegance and importance cannot be overemphasized.

### 2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

### 7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )

34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using

### Positive Feedback and Oscillators

Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active

### CHAPTER 6 Frequency Response, Bode Plots, and Resonance

ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal

### Appendix D Digital Modulation and GMSK

D1 Appendix D Digital Modulation and GMSK A brief introduction to digital modulation schemes is given, showing the logical development of GMSK from simpler schemes. GMSK is of interest since it is used

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Convolution. The Delta Function and Impulse Response

CHAPTER 6 Convolution Convolution is a mathematical way of combining two signals to form a third signal. It is the single most important technique in Digital Signal Processing. Using the strategy of impulse

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Trigonometry Hard Problems

Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Short-time FFT, Multi-taper analysis & Filtering in SPM12

Short-time FFT, Multi-taper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH 20.03.2015 Overview Refresher Short-time Fourier

### Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs

Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in

### Introduction to Complex Numbers in Physics/Engineering

Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The

### chapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective

Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### TRIGONOMETRY Compound & Double angle formulae

TRIGONOMETRY Compound & Double angle formulae In order to master this section you must first learn the formulae, even though they will be given to you on the matric formula sheet. We call these formulae

### 6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

### Introduction to IQ-demodulation of RF-data

Introduction to IQ-demodulation of RF-data by Johan Kirkhorn, IFBT, NTNU September 15, 1999 Table of Contents 1 INTRODUCTION...3 1.1 Abstract...3 1.2 Definitions/Abbreviations/Nomenclature...3 1.3 Referenced

### Aliasing, Image Sampling and Reconstruction

Aliasing, Image Sampling and Reconstruction Recall: a pixel is a point It is NOT a box, disc or teeny wee light It has no dimension It occupies no area It can have a coordinate More than a point, it is

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.

Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### How to Graph Trigonometric Functions

How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

### SOLVING TRIGONOMETRIC EQUATIONS

Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### The front end of the receiver performs the frequency translation, channel selection and amplification of the signal.

Many receivers must be capable of handling a very wide range of signal powers at the input while still producing the correct output. This must be done in the presence of noise and interference which occasionally

### Analysis/resynthesis with the short time Fourier transform

Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis

### MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### SGN-1158 Introduction to Signal Processing Test. Solutions

SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### FRICTION, WORK, AND THE INCLINED PLANE

FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle

### L9: Cepstral analysis

L9: Cepstral analysis The cepstrum Homomorphic filtering The cepstrum and voicing/pitch detection Linear prediction cepstral coefficients Mel frequency cepstral coefficients This lecture is based on [Taylor,

### The Fourier Analysis Tool in Microsoft Excel

The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier

### The Calculation of G rms

The Calculation of G rms QualMark Corp. Neill Doertenbach The metric of G rms is typically used to specify and compare the energy in repetitive shock vibration systems. However, the method of arriving

### Elements of a graph. Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

### Jitter Measurements in Serial Data Signals

Jitter Measurements in Serial Data Signals Michael Schnecker, Product Manager LeCroy Corporation Introduction The increasing speed of serial data transmission systems places greater importance on measuring

### Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

### 2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

### ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS

Texas College Mathematics Journal Volume 6, Number 2, Pages 18 24 S applied for(xx)0000-0 Article electronically published on September 23, 2009 ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE

1 25. AM radio receiver The chapter describes the programming of a microcontroller to demodulate a signal from a local radio station. To keep the circuit simple the signal from the local amplitude modulated

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

### ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's

### G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

### Chap#5 (Data communication)

Chap#5 (Data communication) Q#1: Define analog transmission. Normally, analog transmission refers to the transmission of analog signals using a band-pass channel. Baseband digital or analog signals are

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

Reading 30 Ron Bertrand VK2DQ http://www.radioelectronicschool.com AM TRANSMITTERS & RECEIVERS Revision: our definition of amplitude modulation. Amplitude modulation is when the modulating audio is combined

### Finding Equations of Sinusoidal Functions From Real-World Data

Finding Equations of Sinusoidal Functions From Real-World Data **Note: Throughout this handout you will be asked to use your graphing calculator to verify certain results, but be aware that you will NOT

### Lecture 9. Poles, Zeros & Filters (Lathi 4.10) Effects of Poles & Zeros on Frequency Response (1) Effects of Poles & Zeros on Frequency Response (3)

Effects of Poles & Zeros on Frequency Response (1) Consider a general system transfer function: zeros at z1, z2,..., zn Lecture 9 Poles, Zeros & Filters (Lathi 4.10) The value of the transfer function

### Implementation of Digital Signal Processing: Some Background on GFSK Modulation

Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 4 (February 7, 2013)

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Agilent PN 89400-13 Extending Vector Signal Analysis to 26.5 GHz with 20 MHz Information Bandwidth

Agilent PN 89400-13 Extending Vector Signal Analysis to 26.5 GHz with 20 MHz Information Bandwidth Product Note The Agilent Technologies 89400 series vector signal analyzers provide unmatched signal analysis

### Phasors. Phasors. by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department. ^ V cos (wt + θ) ^ V sin (wt + θ)

V cos (wt θ) V sin (wt θ) by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department EE 209 Fundamentals of Electrical and Electronics Engineering, Prof. Dr. O. SEVAİOĞLU, Page 1 Vector

### Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

### Trigonometry for AC circuits

Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

### (Refer Slide Time: 2:03)

Control Engineering Prof. Madan Gopal Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 11 Models of Industrial Control Devices and Systems (Contd.) Last time we were

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Trigonometric functions and sound

Trigonometric functions and sound The sounds we hear are caused by vibrations that send pressure waves through the air. Our ears respond to these pressure waves and signal the brain about their amplitude

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Non-Data Aided Carrier Offset Compensation for SDR Implementation

Non-Data Aided Carrier Offset Compensation for SDR Implementation Anders Riis Jensen 1, Niels Terp Kjeldgaard Jørgensen 1 Kim Laugesen 1, Yannick Le Moullec 1,2 1 Department of Electronic Systems, 2 Center

### Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

### Analog and Digital Signals, Time and Frequency Representation of Signals

1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### 5 Signal Design for Bandlimited Channels

225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g

### How To Understand The Nyquist Sampling Theorem

Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### Mathematics Scope and Sequence, K-8

Standard 1: Number and Operation Goal 1.1: Understands and uses numbers (number sense) Mathematics Scope and Sequence, K-8 Grade Counting Read, Write, Order, Compare Place Value Money Number Theory K Count

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### 2312 test 2 Fall 2010 Form B

2312 test 2 Fall 2010 Form B 1. Write the slope-intercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function

### Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.

INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe