Sinusoids. Lecture #2 Chapter 2. BME 310 Biomedical Computing - J.Schesser

Size: px
Start display at page:

Download "Sinusoids. Lecture #2 Chapter 2. BME 310 Biomedical Computing - J.Schesser"

Transcription

1 Sinusoids Lecture # Chapter BME 30 Biomedical Computing - 8

2 What Is this Course All About? To Gain an Appreciation of the Various Types of Signals and Systems To Analyze The Various Types of Systems To Learn the Skills and Tools needed to Perform These Analyses. To Understand How Computers Process Signals and Systems BME 30 Biomedical Computing - 9

3 Sinusoidal Signal Sinusoidal Signals are periodic functions which are based on the sine or cosine function from trigonometry. The general form of a Sinusoidal Signal x(t)=a cos(ω o t+ϕ) Or x(t)=a cos(πf o t +ϕ) where cos ( ) represent the cosine function We can also use sin( ), the sine function ω o t+ϕ or πf o t +ϕ is angle (in radians) of the cosine function Since the angle depends on time, it makes x(t) a signal ω o is the radian frequency of the sinusoidal signal f o is called the cyclical frequency of the sinusoidal signal ϕ is the phase shift or phase angle A is the amplitude of the signal BME 30 Biomedical Computing - 0

4 Example x(t)=0 cos(π(440)t -0.4π) One cycle takes /440 =.007 seconds This is called the period, T, of the sinusoid and is equal to the inverse of the frequency, f BME 30 Biomedical Computing -

5 Sine and Cosine Functions Definition of sine and cosine y θ r x Depending upon the quadrant of θ the sine and cosine function changes As the θ increases from 0 to π/, the cosine decreases from to 0 and the sine increases from 0 to As the θ increases beyond π/ to π, the cosine decreases from 0 to - and the sine decreases from to 0 As the θ increases beyond π to 3π/, the cosine increases from - to 0 and the sine decreases from 0 to - As the θ increases beyond 3π/ to π, the cosine increases from 0 to and the sine increases from - to 0 BME 30 Biomedical Computing - y sin r y r sin x cos r x r cos

6 Properties of Sinusoids cosine sine sine cosine Equivalence Property Equation sin θ = cos (θ π / ) or cos θ = sin (θ + π/) Periodicity cos (θ + πk)=cos θ or sin (θ +πk)=sin θ where k is an integer Evenness of cosine cos θ = cos (-θ ) Oddness of sine sin θ = -sin (-θ ) Zeros of sine sin πk = 0, when k is an integer Zeros of cosine cos [π(k+)/] = 0, when k is an even integer; odd multiples of π/ Ones of the cosine cos πk =, when k is an integer; even multiples of π Ones of the sine sin [π(k+/)] =, when k is an even integer; alternate odd multiples of π/ Negative ones of the cosine cos [π(k +)/]= -, when k is an integer; odd multiples of π Negative ones of the sine sin [π(k +/)]= -, when k is an odd integer; alternate odd multiples of 3π/ BME 30 Biomedical Computing - 3

7 Properties of Sinusoids K (K+)/ X pi() cosine K K+/ X pi() sine BME 30 Biomedical Computing - 4

8 Identities and Derivatives Number Equation sin θ +cos θ = cos θ = cos θ sin θ 3 sin θ = sin θ cos θ 4 sin (a ± b) = sin a cos b ± cos a sin b cos (a ± b) = cos a cos b sin a sin b 6 cos a cos b = [cos (a + b) + cos (a - b)]/ 7 sin a sin b = [cos (a - b) - cos (a + b)]/ 8 cos θ = [ + cos θ]/ 9 sin θ = [ - cos θ]/ 0 d sin θ / dθ = cos θ d cos θ / dθ = -sin θ BME 30 Biomedical Computing -

9 Sinusoidal Signal The general form of a Sinusoidal Signal x(t)=a cos(ω o t+θ) = A cos(πf o t+θ) ω o = πf o is the radian frequency of the sinusoidal signal Since ω o t has units of radians which is dimensionless, ω o has units of rad/sec f o is called the cyclical frequency of the sinusoidal signal Has units of sec - or Hz (formerly, cycles per second) θ is the phase shift or phase angle Has units of radians A is the amplitude of the signal and is the scaling factor that determines how large the signal will be. Since the cosine function varies from - to + then our signal will vary from A to +A. A is sometimes called the peak of the signal and A is called the peak-to-peak value BME 30 Biomedical Computing - 6

10 Sinusoid Signals x(t)=0 cos(π(40)t -0.4π) A = 0, ω o = π(40), f o = 40, θ = - 0.4π Maxima at π(40)t-0.4π =πk or when t =, -0.0, 0.00, 0.03, Minima at π(40)t-0.4π =π(k+.) or when t = , 0.07 Time Period (/f o ) between = (-0.0) =0.0 sec BME 30 Biomedical Computing - 7

11 Relation of Period to Frequency Periodof a sinusoid, T o, is the length of one cycle and T o = /f o The following relationship must be true for all Signals which are periodic (not ust sinusoids) x(t + T o ) = x(t) So A cos(ω o (t + T o ) + θ) = A cos(ω o t+θ) A cos(ω o t + ω o T o + θ) = A cos(ω o t+θ) BME 30 Biomedical Computing - 8

12 Relation of Period to Frequency Continued Since a sinusoid is periodic in π, this means: since T o = /f o Then ω o T o = πf o T o ω o T o = π therefore, T o = π/ω o The period is in units of seconds The frequency is in units of sec - or Hz (formerly, cycles per second) BME 30 Biomedical Computing - 9

13 Frequencies A cos(πf o t+θ) for 00 Hz, 00 Hz, 0 Hz 00 Hz Hz 00 Hz BME 30 Biomedical Computing - 30

14 Phase Shift and Time Shift The phase shift parameter θ (with frequency) determines the time locations of the maxima and minima of the sinusoid. When θ = 0, then for positive peak at t = 0. When θ 0, then the phase shift determines how much the maximum is shifted from t = 0. However, delaying a signal by t seconds, also shifts its waveform. BME 30 Biomedical Computing - 3

15 Time Shifting Look at the following waveform: s( t) t (4 t) t t elsewhere BME 30 Biomedical Computing - 3

16 Time Shifting Continued Now let s time shifted it by seconds (delay), x(t)=s(t -) ( t - ) t - 4 x( t) (4 ( t )) 3 0 (8 3 t) 0 ( t ) t ( t ) t 4 elsewhere BME 30 Biomedical Computing - 33

17 Time Shifting Continued Now let s time shifted it by - seconds (advance), ( t ) x( t) (4 ( t 3 0 t )) ( 3 x(t)=s(t + ) t) 0 ( t ) t ( t ) t elsewhere BME 30 Biomedical Computing - 34

18 Time Shifting BME 30 Biomedical Computing - 3

19 Phase shift and Time Shift Time (seconds) xt () cos( 40 t ) f 40 Hz; Time shift = s T 0.0 sec 40 phase shift: Phase shift = π / radians time shift: ts sec xt ( ) cos( 40( t0.006)) -. Radians -. BME 30 Biomedical Computing - 36

20 Phase and Time Shift x(t-t ) = A cos(ω o (t-t )) = A cos(ω o t+θ) A cos(ω o t-ω o t ) = A cos(ω o t+θ) ω o t-ω o t = ω o t+ϕ -ω o t = θ t = - θ / ω o = -ϕ /πf o θ= - πf o t =-π(t / T o ) Note that a positive (negative) value of t equates to a delay (advance) And a a positive (negative) value of θ equates to an advance (delay) BME 30 Biomedical Computing - 37

21 Phase and Time Shift Note that a positive (negative) value of t equates to a delay (advance) And a a positive (negative) value of θ equates to an advance (delay) x(t) = cos(π 0t + θ) θ = π / ; -π / t = - π / / (π 0) = -.00 sec; +.00 sec BME 30 Biomedical Computing - 38

22 Time shifting Shift = seconds BME 30 Biomedical Computing - 39

23 Periodicity of Sinusoids What happens when θ = π? When θ = π, then the sinusoidal waveform does not change since sinusoids are periodic in π Therefore, adding or subtracting multiple of π does not change the waveform This is called modulo π BME 30 Biomedical Computing - 40

24 Plotting Sinusoid Signals Case #: Delay x( t) 0 cos( (40) t 0.4 ) A 0, f 40Hz, 0.4 Basic Calculations o )Amplitude is ) Frequency 40Hz Calculation of T o f o sec 0 the Period msec 3) Phase angle/shift radians Calculation of the time shift ft 0 (40) t t s s sec msec delay (40) t msec s Note that 0. T msec o phase angle 0.4 which also equals 0. s BME 30 Biomedical Computing - 4

25 Plotting Sinusoid Signals t ft + 0 cos ( ft+ ) Time to the first (in positive time - delay) maximum is Time Shift = 0.00 sec => Negative Phase shift θ = -0.00*πf = -0.4π BME 30 Biomedical Computing cycle is the Period = = 0.0 sec Since the period is 0.0, chose to plot the graph with 0 points per cycle or a Δ t of 0.00 seconds. 4

26 )Amplitude is ) Frequency 40Hz Calculation of T o f o sec Plotting Sinusoid Signals Case #: No Delay x( t) 0 cos( (40) t) A 0, f 40Hz, 0 0 the Period msec Basic Calculations o 3) Phase angle/shift 0 radians Calculation of the time shift ft s 0 (40)t s 0 0 t s 0 (40) 0sec BME 30 Biomedical Computing - 43

27 Plotting Sinusoid Signals t ft + 0 cos ( ft+ ) cycle is the Period = 0.0 sec Time to the first maximum is Time Shift = 0 sec => Phase shift of θ = 0 Since the period is 0.0, chose to plot the graph with 0 points per cycle or a Δ t of 0.00 seconds. BME 30 Biomedical Computing - 44

28 )Amplitude is Calculation of f 0.0sec Plotting Sinusoid Signals Case #: Advance xt ( ) 0 cos( (40) t0.6 ) A0, fo 40 Hz, 0.6 Basic Calculations ) Frequency 40Hz T o o 40 0 the Period msec 3) Phase angle/shift radians Calculation of the time shift fts 0 (40) ts ts 0.007sec 7.msec advance (40) t 7. msec s Note that 0.3 T msec which also equals o phase angle BME 30 Biomedical Computing - 4

29 Plotting Sinusoid Signals t ft + 0 cos ( ft+ ) Time to the first maximum (in negative time - advance) is Time Shift = sec => Positive Phase Shift θ = 0.007*πf = +0.6π BME 30 Biomedical Computing cycle is the Period = = 0.0 sec Time to the first maximum (in positive time - delay) is Time Shift = sec => Negative Phase Shift θ = 0.007*πf = -.4 Since the period is 0.0, chose to plot the graph with 0 points per cycle or a Δ t of 0.00 seconds. 46

30 Matlab Program to Plot Sinusoids function cosineplotphase(frequency,amplitude,phase,points,timestart,timeend); omega=*pi*frequency; Period=/Frequency; phase=phase; Timedelta=Period/Points; time = (Timestart:Timedelta:Timeend); y=amplitude*cos(omega*time+phase); plot(time,y,'r'); title('sinusoidal Plot'); xlabel('seconds'); axis([ Timestart Timeend -.*Amplitude +.*Amplitude]); BME 30 Biomedical Computing - 47

31 What s so important about Sinusoids and periodic signals? Are signals is nature periodic? Name some: Vibrations Voice Electromagnetic Biomedical So are signals is nature periodic? Not always but we may be able to model them with period signals Biomedical Research Summer Institute Examples: Measure blood viscosity Measure cell mass BME 30 Biomedical Computing - 48

32 Complex Numbers Complex numbers: What are they? What is the solution to this equation? ax +bx+c=0 This is a second order equation whose solution is: b b 4ac x, a BME 30 Biomedical Computing - 49

33 What is the solution to?. x +4x+3=0 x, , 3 BME 30 Biomedical Computing - 0

34 What is the solution to?. x +4x+=0 x, ????? BME 30 Biomedical Computing -

35 What is the Square Root of a Negative Number? We define the square root of a negative number as an imaginary number We define x for engineers ( i Then our solution becomes:, BME 30 Biomedical Computing - for mathematicans) 6 0 4,

36 The Complex Plane z= x+yis a complex number where: x= Re{z} is the real part of z y= Im{z} is the imaginary part of z We can define the complex plane and we can define representations for a complex number: Im{z} y z = x+y (x,y) x Re{z} BME 30 Biomedical Computing - 3

37 Rectangular Form Rectangular (or cartesian) form of a complex number is given as z= x+y x= Re{z} is the real part of z y= Im{z} is the imaginary part of z Im{z} y z = x+y (x,y) x Re{z} Rectangular or Cartesian BME 30 Biomedical Computing - 4

38 z= re θ = r Polar Form θ is a complex number where: r is the magnitude of z θ is the angle or argument of z (arg z) Im{z} y z= re θ Polar (r,) r x Re{z} BME 30 Biomedical Computing -

39 Relationships between the Polar and Rectangular Forms z= x+ y= re θ Relationship of Polar to the Rectangular Form: x= Re{z} = r cos θ y= Im{z} = r sin θ Relationship of Rectangular to Polar Form: r x y and arctan( y x ) BME 30 Biomedical Computing - 6

40 Addition of complex numbers When two complex numbers are added, it is best to use the rectangular form. The real part of the sum is the sum of the real parts and imaginary part of the sum is the sum of the imaginary parts. y y Im Example: z 3 = z + z z z 3 x ( x x z x x y z ) ; z y x ( y x y y y ) y x y BME 30 Biomedical Computing - z y y z x x x z x Re 7

41 Multiplication of complex numbers When two complex numbers are multiplied, it is best to use the polar form: ( ) ( ) Example: z 3 = z x z z re ; z r e 3 ( ) ( ) We multiply the magnitudes and add the phase angles θ 3 = θ + θ Im z z rr e ( ) r z e re ( ) rr e r e ( ) r 3 = r r r θ θ BME 30 Biomedical Computing - Re 8

42 e Some examples e cos( ) sin( ) 0 e cos( ) sin( ) 0 e cos( ) sin( ) cos( ) sin( ) 0 tan ( ) e ( ) ( ) e 3 4 e Im Re BME 30 Biomedical Computing - 9

43 BME 30 Biomedical Computing - 60 Some examples 4 e e ) sin( ) cos( ) 4 sin( ) 4 cos( 4 e e tan (.4) tan ) ( tan e e e e.707

44 BME 30 Biomedical Computing - 6 Some examples ) sin( ) cos( ) 4 sin( ) 4 cos( 4 e e (.4) tan ) ( tan e e e e e OR

45 Some examples e 4 e 9. 4 Im e.8 Re BME 30 Biomedical Computing - 6

46 Euler s Formula e θ = cos θ + sin θ Im{z} θ Re{z} We can use Euler s Formula to define complex numbers z = r e θ = r cos θ + r sin θ = x + y BME 30 Biomedical Computing - 63

47 Complex Exponential Signals A complex exponential signal is define as: ( t ) zt () Ae o Note that it is defined in polar form where the magnitude of z(t) is z(t) = A the angle (or argument, arg z(t) ) of z(t) = (ω o t + θ) Where ω o is called the radian frequency and θ is the phase angle (phase shift) BME 30 Biomedical Computing - 64

48 Complex Exponential Signals Note that by using Euler s formula, we can rewrite the complex exponential signal in rectangular form as: zt () Ae ( t) o Acos( t ) Asin( t ) o o Therefore real part is the cosine signal and imaginary part is a sine signal both of radial frequency ω o and phase angle of θ BME 30 Biomedical Computing - 6

49 Plotting the waveform of a complex exponential signal For an complex signal, we plot the real part and the imaginary part separately. Example: z(t) = 0e (π(40)t-0.4π) = 0e (80πt-0.4π) = 0 cos(80πt-0.4π) + 0 sin(80πt-0.4π) real part imaginary part BME 30 Biomedical Computing - 66

50 NOTE!!!! The reason why we prefer the complex exponential representation of the real cosine signal: ( t) xt () ezt {()} eae { o } Acos( t ) In solving equations and making other calculations, it easier to use the complex exponential form and then take the Real Part. o BME 30 Biomedical Computing - 67

51 Homework Exercises:.. Problems: -. Instead plot x(t) cos( t ) for t 0 0 Plot x(t) using Matlab; as part of your answer provide your code a,.3b =0.4, Plot these functions using Matlab; as part of your answer provide your code -.4 BME 30 Biomedical Computing - 68

Lecture 3 Complex Exponential Signals

Lecture 3 Complex Exponential Signals Lecture 3 Complex Exponential Signals Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/3/1 1 Review of Complex Numbers Using Euler s famous formula for the complex exponential The

More information

DSP First. Laboratory Exercise #2. Introduction to Complex Exponentials

DSP First. Laboratory Exercise #2. Introduction to Complex Exponentials DSP First Laboratory Exercise #2 Introduction to Complex Exponentials The goal of this laboratory is gain familiarity with complex numbers and their use in representing sinusoidal signals as complex exponentials.

More information

Sinusoids and Phasors (Chapter 9 - Lecture #1) Dr. Shahrel A. Suandi Room 2.20, PPKEE

Sinusoids and Phasors (Chapter 9 - Lecture #1) Dr. Shahrel A. Suandi Room 2.20, PPKEE Sinusoids and Phasors (Chapter 9 - Lecture #1) Dr. Shahrel A. Suandi Room 2.20, PPKEE Email:shahrel@eng.usm.my 1 Outline of Chapter 9 Introduction Sinusoids Phasors Phasor Relationships for Circuit Elements

More information

Fall Music 320A Homework #2 Sinusoids, Complex Sinusoids 145 points Theory and Lab Problems Due Thursday 10/11/2018 before class

Fall Music 320A Homework #2 Sinusoids, Complex Sinusoids 145 points Theory and Lab Problems Due Thursday 10/11/2018 before class Fall 2018 2019 Music 320A Homework #2 Sinusoids, Complex Sinusoids 145 points Theory and Lab Problems Due Thursday 10/11/2018 before class Theory Problems 1. 15 pts) [Sinusoids] Define xt) as xt) = 2sin

More information

ECE 201: Introduction to Signal Analysis

ECE 201: Introduction to Signal Analysis ECE 201: Introduction to Signal Analysis Prof. Paris Last updated: October 9, 2007 Part I Spectrum Representation of Signals Lecture: Sums of Sinusoids (of different frequency) Introduction Sum of Sinusoidal

More information

Digital Signal Processing Lecture 1 - Introduction

Digital Signal Processing Lecture 1 - Introduction Digital Signal Processing - Electrical Engineering and Computer Science University of Tennessee, Knoxville August 20, 2015 Overview 1 2 3 4 Basic building blocks in DSP Frequency analysis Sampling Filtering

More information

ECE 201: Introduction to Signal Analysis. Dr. B.-P. Paris Dept. Electrical and Comp. Engineering George Mason University

ECE 201: Introduction to Signal Analysis. Dr. B.-P. Paris Dept. Electrical and Comp. Engineering George Mason University ECE 201: Introduction to Signal Analysis Dr. B.-P. Paris Dept. Electrical and Comp. Engineering George Mason University Last updated: November 29, 2016 2016, B.-P. Paris ECE 201: Intro to Signal Analysis

More information

ECE 201: Introduction to Signal Analysis

ECE 201: Introduction to Signal Analysis ECE 201: Introduction to Signal Analysis Dr. B.-P. Paris Dept. Electrical and Comp. Engineering George Mason University Last updated: November 29, 2016 2016, B.-P. Paris ECE 201: Intro to Signal Analysis

More information

Alternating voltages and currents

Alternating voltages and currents Alternating voltages and currents Introduction - Electricity is produced by generators at power stations and then distributed by a vast network of transmission lines (called the National Grid system) to

More information

Signals and Systems EE235. Leo Lam

Signals and Systems EE235. Leo Lam Signals and Systems EE235 Leo Lam Today s menu Lab detailed arrangements Homework vacation week From yesterday (Intro: Signals) Intro: Systems More: Describing Common Signals Taking a signal apart Offset

More information

Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3

Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan. Review Problems for Test #3 Arkansas Tech University MATH 1203: Trigonometry Dr. Marcel B. Finan Review Problems for Test #3 Exercise 1 The following is one cycle of a trigonometric function. Find an equation of this graph. Exercise

More information

CMPT 368: Lecture 4 Amplitude Modulation (AM) Synthesis

CMPT 368: Lecture 4 Amplitude Modulation (AM) Synthesis CMPT 368: Lecture 4 Amplitude Modulation (AM) Synthesis Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 8, 008 Beat Notes What happens when we add two frequencies

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Trigonometry Final Exam Study Guide Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a polar equation is given. Select the polar

More information

THE SINUSOIDAL WAVEFORM

THE SINUSOIDAL WAVEFORM Chapter 11 THE SINUSOIDAL WAVEFORM The sinusoidal waveform or sine wave is the fundamental type of alternating current (ac) and alternating voltage. It is also referred to as a sinusoidal wave or, simply,

More information

6.02 Fall 2012 Lecture #12

6.02 Fall 2012 Lecture #12 6.02 Fall 2012 Lecture #12 Bounded-input, bounded-output stability Frequency response 6.02 Fall 2012 Lecture 12, Slide #1 Bounded-Input Bounded-Output (BIBO) Stability What ensures that the infinite sum

More information

Spectrum. Additive Synthesis. Additive Synthesis Caveat. Music 270a: Modulation

Spectrum. Additive Synthesis. Additive Synthesis Caveat. Music 270a: Modulation Spectrum Music 7a: Modulation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) October 3, 7 When sinusoids of different frequencies are added together, the

More information

10.3 Polar Coordinates

10.3 Polar Coordinates .3 Polar Coordinates Plot the points whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > and one with r

More information

Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1

Copyright 2009 Pearson Education, Inc. Slide Section 8.2 and 8.3-1 8.3-1 Transformation of sine and cosine functions Sections 8.2 and 8.3 Revisit: Page 142; chapter 4 Section 8.2 and 8.3 Graphs of Transformed Sine and Cosine Functions Graph transformations of y = sin

More information

2. Be able to evaluate a trig function at a particular degree measure. Example: cos. again, just use the unit circle!

2. Be able to evaluate a trig function at a particular degree measure. Example: cos. again, just use the unit circle! Study Guide for PART II of the Fall 18 MAT187 Final Exam NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will be

More information

Frequency Division Multiplexing Spring 2011 Lecture #14. Sinusoids and LTI Systems. Periodic Sequences. x[n] = x[n + N]

Frequency Division Multiplexing Spring 2011 Lecture #14. Sinusoids and LTI Systems. Periodic Sequences. x[n] = x[n + N] Frequency Division Multiplexing 6.02 Spring 20 Lecture #4 complex exponentials discrete-time Fourier series spectral coefficients band-limited signals To engineer the sharing of a channel through frequency

More information

Basic Trigonometry You Should Know (Not only for this class but also for calculus)

Basic Trigonometry You Should Know (Not only for this class but also for calculus) Angle measurement: degrees and radians. Basic Trigonometry You Should Know (Not only for this class but also for calculus) There are 360 degrees in a full circle. If the circle has radius 1, then the circumference

More information

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2

More information

1 Introduction and Overview

1 Introduction and Overview DSP First, 2e Lab S-0: Complex Exponentials Adding Sinusoids Signal Processing First Pre-Lab: Read the Pre-Lab and do all the exercises in the Pre-Lab section prior to attending lab. Verification: The

More information

What is a Sine Function Graph? U4 L2 Relate Circle to Sine Activity.pdf

What is a Sine Function Graph? U4 L2 Relate Circle to Sine Activity.pdf Math 3 Unit 6, Trigonometry L04: Amplitude and Period of Sine and Cosine AND Translations of Sine and Cosine Functions WIMD: What I must do: I will find the amplitude and period from a graph of the sine

More information

Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan. Figure 50.1

Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan. Figure 50.1 50 Polar Coordinates Arkansas Tech University MATH 94: Calculus II Dr. Marcel B. Finan Up to this point we have dealt exclusively with the Cartesian coordinate system. However, as we will see, this is

More information

Calculus for the Life Sciences

Calculus for the Life Sciences Calculus for the Life Sciences Lecture Notes Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego

More information

5.3-The Graphs of the Sine and Cosine Functions

5.3-The Graphs of the Sine and Cosine Functions 5.3-The Graphs of the Sine and Cosine Functions Objectives: 1. Graph the sine and cosine functions. 2. Determine the amplitude, period and phase shift of the sine and cosine functions. 3. Find equations

More information

Complex Numbers in Electronics

Complex Numbers in Electronics P5 Computing, Extra Practice After Session 1 Complex Numbers in Electronics You would expect the square root of negative numbers, known as complex numbers, to be only of interest to pure mathematicians.

More information

Here are some of Matlab s complex number operators: conj Complex conjugate abs Magnitude. Angle (or phase) in radians

Here are some of Matlab s complex number operators: conj Complex conjugate abs Magnitude. Angle (or phase) in radians Lab #2: Complex Exponentials Adding Sinusoids Warm-Up/Pre-Lab (section 2): You may do these warm-up exercises at the start of the lab period, or you may do them in advance before coming to the lab. You

More information

Signal Processing First Lab 02: Introduction to Complex Exponentials Direction Finding. x(t) = A cos(ωt + φ) = Re{Ae jφ e jωt }

Signal Processing First Lab 02: Introduction to Complex Exponentials Direction Finding. x(t) = A cos(ωt + φ) = Re{Ae jφ e jωt } Signal Processing First Lab 02: Introduction to Complex Exponentials Direction Finding Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over

More information

Introduction to signals and systems

Introduction to signals and systems CHAPTER Introduction to signals and systems Welcome to Introduction to Signals and Systems. This text will focus on the properties of signals and systems, and the relationship between the inputs and outputs

More information

Chapter 6: Periodic Functions

Chapter 6: Periodic Functions Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle. We noticed how the x and y

More information

Phase demodulation using the Hilbert transform in the frequency domain

Phase demodulation using the Hilbert transform in the frequency domain Phase demodulation using the Hilbert transform in the frequency domain Author: Gareth Forbes Created: 3/11/9 Revision: The general idea A phase modulated signal is a type of signal which contains information

More information

CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION

CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION CHAPTER 6 INTRODUCTION TO SYSTEM IDENTIFICATION Broadly speaking, system identification is the art and science of using measurements obtained from a system to characterize the system. The characterization

More information

Math 1205 Trigonometry Review

Math 1205 Trigonometry Review Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of

More information

Signal Processing First Lab 02: Introduction to Complex Exponentials Multipath. x(t) = A cos(ωt + φ) = Re{Ae jφ e jωt }

Signal Processing First Lab 02: Introduction to Complex Exponentials Multipath. x(t) = A cos(ωt + φ) = Re{Ae jφ e jωt } Signal Processing First Lab 02: Introduction to Complex Exponentials Multipath Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over all exercises

More information

Circuit Analysis-II. Circuit Analysis-II Lecture # 2 Wednesday 28 th Mar, 18

Circuit Analysis-II. Circuit Analysis-II Lecture # 2 Wednesday 28 th Mar, 18 Circuit Analysis-II Angular Measurement Angular Measurement of a Sine Wave ü As we already know that a sinusoidal voltage can be produced by an ac generator. ü As the windings on the rotor of the ac generator

More information

Trigonometric Identities

Trigonometric Identities Trigonometric Identities Scott N. Walck September 1, 010 1 Prerequisites You should know the cosine and sine of 0, π/6, π/4, π/, and π/. Memorize these if you do not already know them. cos 0 = 1 sin 0

More information

PREREQUISITE/PRE-CALCULUS REVIEW

PREREQUISITE/PRE-CALCULUS REVIEW PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which

More information

CHAPTER 9. Sinusoidal Steady-State Analysis

CHAPTER 9. Sinusoidal Steady-State Analysis CHAPTER 9 Sinusoidal Steady-State Analysis 9.1 The Sinusoidal Source A sinusoidal voltage source (independent or dependent) produces a voltage that varies sinusoidally with time. A sinusoidal current source

More information

Basic Signals and Systems

Basic Signals and Systems Chapter 2 Basic Signals and Systems A large part of this chapter is taken from: C.S. Burrus, J.H. McClellan, A.V. Oppenheim, T.W. Parks, R.W. Schafer, and H. W. Schüssler: Computer-based exercises for

More information

CMPT 468: Frequency Modulation (FM) Synthesis

CMPT 468: Frequency Modulation (FM) Synthesis CMPT 468: Frequency Modulation (FM) Synthesis Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 6, 23 Linear Frequency Modulation (FM) Till now we ve seen signals

More information

Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh

Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions By Dr. Mohammed Ramidh Trigonometric Functions This section reviews the basic trigonometric functions. Trigonometric functions are important because

More information

10. Introduction and Chapter Objectives

10. Introduction and Chapter Objectives Real Analog - Circuits Chapter 0: Steady-state Sinusoidal Analysis 0. Introduction and Chapter Objectives We will now study dynamic systems which are subjected to sinusoidal forcing functions. Previously,

More information

Outline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK.

Outline. EECS 3213 Fall Sebastian Magierowski York University. Review Passband Modulation. Constellations ASK, FSK, PSK. EECS 3213 Fall 2014 L12: Modulation Sebastian Magierowski York University 1 Outline Review Passband Modulation ASK, FSK, PSK Constellations 2 1 Underlying Idea Attempting to send a sequence of digits through

More information

Math 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b

Math 102 Key Ideas. 1 Chapter 1: Triangle Trigonometry. 1. Consider the following right triangle: c b Math 10 Key Ideas 1 Chapter 1: Triangle Trigonometry 1. Consider the following right triangle: A c b B θ C a sin θ = b length of side opposite angle θ = c length of hypotenuse cosθ = a length of side adjacent

More information

HW 6 Due: November 3, 10:39 AM (in class)

HW 6 Due: November 3, 10:39 AM (in class) ECS 332: Principles of Communications 2015/1 HW 6 Due: November 3, 10:39 AM (in class) Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do

More information

Introduction to Mathematical Modeling of Signals and Systems

Introduction to Mathematical Modeling of Signals and Systems Introduction to Mathematical Modeling of Signals and Systems Mathematical Representation of Signals Signals represent or encode information In communications applications the information is almost always

More information

Digital Video and Audio Processing. Winter term 2002/ 2003 Computer-based exercises

Digital Video and Audio Processing. Winter term 2002/ 2003 Computer-based exercises Digital Video and Audio Processing Winter term 2002/ 2003 Computer-based exercises Rudolf Mester Institut für Angewandte Physik Johann Wolfgang Goethe-Universität Frankfurt am Main 6th November 2002 Chapter

More information

In Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function.

In Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function. 0.5 Graphs of the Trigonometric Functions 809 0.5. Eercises In Eercises -, graph one ccle of the given function. State the period, amplitude, phase shift and vertical shift of the function.. = sin. = sin.

More information

Solution to Chapter 4 Problems

Solution to Chapter 4 Problems Solution to Chapter 4 Problems Problem 4.1 1) Since F[sinc(400t)]= 1 modulation index 400 ( f 400 β f = k f max[ m(t) ] W Hence, the modulated signal is ), the bandwidth of the message signal is W = 00

More information

5-5 Multiple-Angle and Product-to-Sum Identities

5-5 Multiple-Angle and Product-to-Sum Identities Find the values of sin 2, cos 2, and tan 2 for the given value and interval. 1. cos =, (270, 360 ) Since on the interval (270, 360 ), one point on the terminal side of θ has x-coordinate 3 and a distance

More information

The period is the time required for one complete oscillation of the function.

The period is the time required for one complete oscillation of the function. Trigonometric Curves with Sines & Cosines + Envelopes Terminology: AMPLITUDE the maximum height of the curve For any periodic function, the amplitude is defined as M m /2 where M is the maximum value and

More information

Graphs of other Trigonometric Functions

Graphs of other Trigonometric Functions Graphs of other Trigonometric Functions Now we will look at other types of graphs: secant. tan x, cot x, csc x, sec x. We will start with the cosecant and y csc x In order to draw this graph we will first

More information

Topic 6. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)

Topic 6. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) Topic 6 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) 10 20 30 40 50 60 70 80 90 100 0-1 -0.8-0.6-0.4-0.2 0 0.2 0.4

More information

CHAPTER 14 ALTERNATING VOLTAGES AND CURRENTS

CHAPTER 14 ALTERNATING VOLTAGES AND CURRENTS CHAPTER 4 ALTERNATING VOLTAGES AND CURRENTS Exercise 77, Page 28. Determine the periodic time for the following frequencies: (a) 2.5 Hz (b) 00 Hz (c) 40 khz (a) Periodic time, T = = 0.4 s f 2.5 (b) Periodic

More information

Laboratory Assignment 5 Amplitude Modulation

Laboratory Assignment 5 Amplitude Modulation Laboratory Assignment 5 Amplitude Modulation PURPOSE In this assignment, you will explore the use of digital computers for the analysis, design, synthesis, and simulation of an amplitude modulation (AM)

More information

Linear Frequency Modulation (FM) Chirp Signal. Chirp Signal cont. CMPT 468: Lecture 7 Frequency Modulation (FM) Synthesis

Linear Frequency Modulation (FM) Chirp Signal. Chirp Signal cont. CMPT 468: Lecture 7 Frequency Modulation (FM) Synthesis Linear Frequency Modulation (FM) CMPT 468: Lecture 7 Frequency Modulation (FM) Synthesis Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 26, 29 Till now we

More information

Graph of the Sine Function

Graph of the Sine Function 1 of 6 8/6/2004 6.3 GRAPHS OF THE SINE AND COSINE 6.3 GRAPHS OF THE SINE AND COSINE Periodic Functions Graph of the Sine Function Graph of the Cosine Function Graphing Techniques, Amplitude, and Period

More information

Practical Application of Wavelet to Power Quality Analysis. Norman Tse

Practical Application of Wavelet to Power Quality Analysis. Norman Tse Paper Title: Practical Application of Wavelet to Power Quality Analysis Author and Presenter: Norman Tse 1 Harmonics Frequency Estimation by Wavelet Transform (WT) Any harmonic signal can be described

More information

1 Graphs of Sine and Cosine

1 Graphs of Sine and Cosine 1 Graphs of Sine and Cosine Exercise 1 Sketch a graph of y = cos(t). Label the multiples of π 2 and π 4 on your plot, as well as the amplitude and the period of the function. (Feel free to sketch the unit

More information

Amplitude, Reflection, and Period

Amplitude, Reflection, and Period SECTION 4.2 Amplitude, Reflection, and Period Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the amplitude of a sine or cosine function. Find the period of a sine or

More information

Problem Set 1 (Solutions are due Mon )

Problem Set 1 (Solutions are due Mon ) ECEN 242 Wireless Electronics for Communication Spring 212 1-23-12 P. Mathys Problem Set 1 (Solutions are due Mon. 1-3-12) 1 Introduction The goals of this problem set are to use Matlab to generate and

More information

Section 5.2 Graphs of the Sine and Cosine Functions

Section 5.2 Graphs of the Sine and Cosine Functions A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in

More information

Section 7.1 Graphs of Sine and Cosine

Section 7.1 Graphs of Sine and Cosine Section 7.1 Graphs of Sine and Cosine OBJECTIVE 1: Understanding the Graph of the Sine Function and its Properties In Chapter 7, we will use a rectangular coordinate system for a different purpose. We

More information

Math 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas

Math 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas Math 180 Chapter 6 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 6.1 Section 6.1 Verifying Trigonometric Identities Verify the identity. a. sin x + cos x cot x = csc

More information

The Formula for Sinusoidal Signals

The Formula for Sinusoidal Signals The Formula for I The general formula for a sinusoidal signal is x(t) =A cos(2pft + f). I A, f, and f are parameters that characterize the sinusoidal sinal. I A - Amplitude: determines the height of the

More information

Syllabus Cosines Sampled Signals. Lecture 1: Cosines. ECE 401: Signal and Image Analysis. University of Illinois 1/19/2017

Syllabus Cosines Sampled Signals. Lecture 1: Cosines. ECE 401: Signal and Image Analysis. University of Illinois 1/19/2017 Lecture 1: Cosines ECE 401: Signal and Image Analysis University of Illinois 1/19/2017 1 Syllabus 2 Cosines 3 Sampled Signals Outline 1 Syllabus 2 Cosines 3 Sampled Signals Who should take this course?

More information

AC Theory and Electronics

AC Theory and Electronics AC Theory and Electronics An Alternating Current (AC) or Voltage is one whose amplitude is not constant, but varies with time about some mean position (value). Some examples of AC variation are shown below:

More information

1 Introduction and Overview

1 Introduction and Overview GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2026 Summer 2018 Lab #2: Using Complex Exponentials Date: 31 May. 2018 Pre-Lab: You should read the Pre-Lab section of

More information

2. (8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given

2. (8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given Trigonometry Joysheet 1 MAT 145, Spring 2017 D. Ivanšić Name: Covers: 6.1, 6.2 Show all your work! 1. 8pts) If θ is an acute angle, find the values of all the trigonometric functions of θ given that sin

More information

Signals Arthur Holly Compton

Signals Arthur Holly Compton Signals The story is told that young King Solomon was given the choice between wealth and wisdom. When he chose wisdom, God was so pleased that he gave Solomon not only wisdom but wealth also. So it is

More information

Music 270a: Modulation

Music 270a: Modulation Music 7a: Modulation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) October 3, 7 Spectrum When sinusoids of different frequencies are added together, the

More information

DSP First Lab 03: AM and FM Sinusoidal Signals. We have spent a lot of time learning about the properties of sinusoidal waveforms of the form: k=1

DSP First Lab 03: AM and FM Sinusoidal Signals. We have spent a lot of time learning about the properties of sinusoidal waveforms of the form: k=1 DSP First Lab 03: AM and FM Sinusoidal Signals Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over all exercises in the Pre-Lab section before

More information

Alternative View of Frequency Modulation

Alternative View of Frequency Modulation Alternative View of Frequency Modulation dsauersanjose@aol.com 8/16/8 When a spectrum analysis is done on a FM signal, a odd set of side bands show up. This suggests that the Frequency modulation is a

More information

Chapter 7 Repetitive Change: Cyclic Functions

Chapter 7 Repetitive Change: Cyclic Functions Chapter 7 Repetitive Change: Cyclic Functions 7.1 Cycles and Sine Functions Data that is periodic may often be modeled by trigonometric functions. This chapter will help you use Excel to deal with periodic

More information

Lecture 2: SIGNALS. 1 st semester By: Elham Sunbu

Lecture 2: SIGNALS. 1 st semester By: Elham Sunbu Lecture 2: SIGNALS 1 st semester 1439-2017 1 By: Elham Sunbu OUTLINE Signals and the classification of signals Sine wave Time and frequency domains Composite signals Signal bandwidth Digital signal Signal

More information

2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal.

2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 1 2.1 BASIC CONCEPTS 2.1.1 Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 2 Time Scaling. Figure 2.4 Time scaling of a signal. 2.1.2 Classification of Signals

More information

Section 2.4 General Sinusoidal Graphs

Section 2.4 General Sinusoidal Graphs Section. General Graphs Objective: any one of the following sets of information about a sinusoid, find the other two: ) the equation ) the graph 3) the amplitude, period or frequency, phase displacement,

More information

1 Trigonometry. Copyright Cengage Learning. All rights reserved.

1 Trigonometry. Copyright Cengage Learning. All rights reserved. 1 Trigonometry Copyright Cengage Learning. All rights reserved. 1.2 Trigonometric Functions: The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives Identify a unit circle and describe

More information

Multiple-Angle and Product-to-Sum Formulas

Multiple-Angle and Product-to-Sum Formulas Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite

More information

Lecture 17 z-transforms 2

Lecture 17 z-transforms 2 Lecture 17 z-transforms 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/3 1 Factoring z-polynomials We can also factor z-transform polynomials to break down a large system into

More information

Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) Amplitude Amplitude Discrete Fourier Transform (DFT) DFT transforms the time domain signal samples to the frequency domain components. DFT Signal Spectrum Time Frequency DFT is often used to do frequency

More information

Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine

Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine 14A Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine Find these vocabulary words in Lesson 14-1 and the Multilingual Glossary. Vocabulary periodic function cycle period amplitude frequency

More information

Principles of Communications ECS 332

Principles of Communications ECS 332 Principles of Communications ECS 332 Asst. Prof. Dr. Prapun Suksompong prapun@siit.tu.ac.th 5. Angle Modulation Office Hours: BKD, 6th floor of Sirindhralai building Wednesday 4:3-5:3 Friday 4:3-5:3 Example

More information

Trigonometric Equations

Trigonometric Equations Chapter Three Trigonometric Equations Solving Simple Trigonometric Equations Algebraically Solving Complicated Trigonometric Equations Algebraically Graphs of Sine and Cosine Functions Solving Trigonometric

More information

Real Analog Chapter 10: Steady-state Sinusoidal Analysis

Real Analog Chapter 10: Steady-state Sinusoidal Analysis 1300 Henley Court Pullman, WA 99163 509.334.6306 www.store. digilent.com Real Analog Chapter 10: Steadystate Sinusoidal Analysis 10 Introduction and Chapter Objectives We will now study dynamic systems

More information

Real and Complex Modulation

Real and Complex Modulation Real and Complex Modulation TIPL 4708 Presented by Matt Guibord Prepared by Matt Guibord 1 What is modulation? Modulation is the act of changing a carrier signal s properties (amplitude, phase, frequency)

More information

Review of Filter Types

Review of Filter Types ECE 440 FILTERS Review of Filters Filters are systems with amplitude and phase response that depends on frequency. Filters named by amplitude attenuation with relation to a transition or cutoff frequency.

More information

Real Analog - Circuits 1 Chapter 11: Lab Projects

Real Analog - Circuits 1 Chapter 11: Lab Projects Real Analog - Circuits 1 Chapter 11: Lab Projects 11.2.1: Signals with Multiple Frequency Components Overview: In this lab project, we will calculate the magnitude response of an electrical circuit and

More information

Bakiss Hiyana binti Abu Bakar JKE, POLISAS BHAB

Bakiss Hiyana binti Abu Bakar JKE, POLISAS BHAB 1 Bakiss Hiyana binti Abu Bakar JKE, POLISAS 1. Explain AC circuit concept and their analysis using AC circuit law. 2. Apply the knowledge of AC circuit in solving problem related to AC electrical circuit.

More information

of the whole circumference.

of the whole circumference. TRIGONOMETRY WEEK 13 ARC LENGTH AND AREAS OF SECTORS If the complete circumference of a circle can be calculated using C = 2πr then the length of an arc, (a portion of the circumference) can be found by

More information

Chapter 4 Trigonometric Functions

Chapter 4 Trigonometric Functions Chapter 4 Trigonometric Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Section 8 Radian and Degree Measure Trigonometric Functions: The Unit Circle Right Triangle Trigonometry

More information

Solutions to Exercises, Section 5.6

Solutions to Exercises, Section 5.6 Instructor s Solutions Manual, Section 5.6 Exercise 1 Solutions to Exercises, Section 5.6 1. For θ = 7, evaluate each of the following: (a) cos 2 θ (b) cos(θ 2 ) [Exercises 1 and 2 emphasize that cos 2

More information

ω d = driving frequency, F m = amplitude of driving force, b = damping constant and ω = natural frequency of undamped, undriven oscillator.

ω d = driving frequency, F m = amplitude of driving force, b = damping constant and ω = natural frequency of undamped, undriven oscillator. Physics 121H Fall 2015 Homework #14 16-November-2015 Due Date : 23-November-2015 Reading : Chapter 15 Note: Problems 7 & 8 are tutorials dealing with damped and driven oscillations, respectively. It may

More information

Objectives. Presentation Outline. Digital Modulation Lecture 03

Objectives. Presentation Outline. Digital Modulation Lecture 03 Digital Modulation Lecture 03 Inter-Symbol Interference Power Spectral Density Richard Harris Objectives To be able to discuss Inter-Symbol Interference (ISI), its causes and possible remedies. To be able

More information

Massachusetts Institute of Technology Dept. of Electrical Engineering and Computer Science Fall Semester, Introduction to EECS 2

Massachusetts Institute of Technology Dept. of Electrical Engineering and Computer Science Fall Semester, Introduction to EECS 2 Massachusetts Institute of Technology Dept. of Electrical Engineering and Computer Science Fall Semester, 2006 6.082 Introduction to EECS 2 Modulation and Demodulation Introduction A communication system

More information

6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function.

6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function. Math 160 www.timetodare.com Periods of trigonometric functions Definition A function y f ( t) f ( t p) f ( t) 6.4 & 6.5 Graphing Trigonometric Functions = is periodic if there is a positive number p such

More information

Unit 6 Test REVIEW Algebra 2 Honors

Unit 6 Test REVIEW Algebra 2 Honors Unit Test REVIEW Algebra 2 Honors Multiple Choice Portion SHOW ALL WORK! 1. How many radians are in 1800? 10 10π Name: Per: 180 180π 2. On the unit circle shown, which radian measure is located at ( 2,

More information

You found trigonometric values using the unit circle. (Lesson 4-3)

You found trigonometric values using the unit circle. (Lesson 4-3) You found trigonometric values using the unit circle. (Lesson 4-3) LEQ: How do we identify and use basic trigonometric identities to find trigonometric values & use basic trigonometric identities to simplify

More information