Periodic Signals, Harmonics and Time-Varying Sinusoids - Slides | ECE 2025, Study notes of Electrical and Electronics Engineering

Download Periodic Signals, Harmonics and Time-Varying Sinusoids - Slides | ECE 2025 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! EE-2025 Fall-2004 Lecture 5 Periodic Signals, Harmonics & Time-Varying Sinusoids 30-Aug-04 9/3/2004 EE-2025 Spring-2004 jMc 2 General Information Bulletin Board: OFFICIAL ANNOUNCEMENTS Old Quizzes & Problems are linked via WebCT Quiz #1 on 13-Sept-04 (Monday) Allowed one page of notes (Handwritten) Calculators permitted Review Session planned for Sunday 12-Sept @ 6pm HW #3 due NEXT WEEK in Recitation Solutions will be posted Thurs evening 9/3/2004 EE-2025 Spring-2004 jMc 3 Lab Info Prepare for On-line Pre-Post-Lab Questions Practice version is available Take advantage of Help Sessions Lab #2 Report Turn in at beginning your lab time Write-up lab report on Beamforming Discuss lab report standards with your TA Miscellaneous ERRORS ? ALWAYS Check Bulletin Board Complete INSTRUCTOR VERIFICATION in Lab 9/3/2004 EE-2025 Spring-2004 jMc 4 Concept Map for Chapter 3 Lecture 9/3/2004 EE-2025 Spring-2004 jMc 5 READING ASSIGNMENTS This Lecture: Chapter 3, Sections 3-2 and 3-3 Chapter 3, Sections 3-7 and 3-8 Next Lecture: Fourier Series ANALYSIS Sections 3-4, 3-5 and 3-6 9/3/2004 EE-2025 Spring-2004 jMc 6 Problem Solving Skills Math Formula Sum of Cosines Amp, Freq, Phase Recorded Signals Speech Music No simple formula Plot & Sketches S(t) versus t Spectrum MATLAB Numerical Computation Plotting list of numbers 9/3/2004 EE-2025 Spring-2004 jMc 7 LECTURE OBJECTIVES Signals with HARMONIC Frequencies Add Sinusoids with fk = kf0 FREQUENCY can change vs. TIME Chirps: Introduce Spectrogram Visualization (specgram.m) (plotspec.m) x(t) = cos(αt2 ) ∑ = ++= N k kk tkfAAtx 1 00 )2cos()( ϕπ 9/3/2004 EE-2025 Spring-2004 jMc 8 SPECTRUM DIAGRAM Recall Complex Amplitude vs. Freq kk aX =2 1 0 100 250–100–250 f (in Hz) 3/7 πje 3/7 πje− 2/4 πje− 2/4 πje 10 )2/)250(2cos(8 )3/)100(2cos(1410)( ππ ππ ++ −+= t ttx kj kk eAX ϕ= ∗ kX2 1 9/3/2004 EE-2025 Spring-2004 jMc 17 Harmonic Signal (3 Freqs) T=0.1 9/3/2004 EE-2025 Spring-2004 jMc 18 NON-Harmonic Signal NOT PERIODIC 9/3/2004 EE-2025 Spring-2004 jMc 19 FREQUENCY ANALYSIS Now, a much HARDER problem Given a recording of a song, have the computer write the music Can a machine extract frequencies? Yes, if we COMPUTE the spectrum for x(t) During short intervals 9/3/2004 EE-2025 Spring-2004 jMc 20 Time-Varying FREQUENCIES Diagram Fr eq ue nc y is th e ve rt ic al a xi s Time is the horizontal axis A-440 9/3/2004 EE-2025 Spring-2004 jMc 21 SIMPLE TEST SIGNAL C-major SCALE: stepped frequencies Frequency is constant for each note IDEAL 9/3/2004 EE-2025 Spring-2004 jMc 22 R-rated: ADULTS ONLY SPECTROGRAM Tool MATLAB function is specgram.m SP-First has plotspec.m & spectgr.m ANALYSIS program Takes x(t) as input & Produces spectrum values Xk Breaks x(t) into SHORT TIME SEGMENTS Then uses the FFT (Fast Fourier Transform) 9/3/2004 EE-2025 Spring-2004 jMc 23 SPECTROGRAM EXAMPLE Two Constant Frequencies: Beats ))12(2sin())660(2cos( tt ππ 9/3/2004 EE-2025 Spring-2004 jMc 24 ( ) ( )tjtjjtjtj eeee )12(2)12(221)660(2)660(221 ππππ −− −+ AM Radio Signal Same as BEAT Notes ))12(2sin())660(2cos( tt ππ ))648(2cos())672(2cos( 22 1 22 1 ππ ππ ++− tt ( )tjtjtjtjj eeee )648(2)648(2)672(2)672(241 ππππ −− +−− 9/3/2004 EE-2025 Spring-2004 jMc 25 SPECTRUM of AM (Beat) 4 complex exponentials in AM: What is the fundamental frequency? 648 Hz ? 24 Hz ? 0 648 672 f (in Hz) –672 –648 4 1 4 1 4 1 4 1 9/3/2004 EE-2025 Spring-2004 jMc 26 STEPPED FREQUENCIES C-major SCALE: successive sinusoids Frequency is constant for each note IDEAL 9/3/2004 EE-2025 Spring-2004 jMc 27 SPECTROGRAM of C-Scale ARTIFACTS at Transitions Sinusoids ONLY From SPECGRAM ANALYSIS PROGRAM 9/3/2004 EE-2025 Spring-2004 jMc 28 Spectrogram of LAB SONG ARTIFACTS at Transitions Sinusoids ONLY Analysis Frame = 40ms