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Digital Signal Processing

Digital Signal Processing. Prof. Nizamettin AYDIN naydin @ yildiz .edu.tr http:// www . yildiz .edu.tr/~naydin. Digital Signal Processing. Lecture 5 Periodic Signals, Harmonics & Time-Varying Sinusoids. READING ASSIGNMENTS. This Lecture: Chapter 3, Sections 3-2 and 3-3

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Digital Signal Processing

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  1. Digital Signal Processing Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin

  2. Digital Signal Processing Lecture 5 Periodic Signals, Harmonics & Time-Varying Sinusoids

  3. 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

  4. 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

  5. LECTURE OBJECTIVES • FREQUENCY can change vs. TIME • Chirps: • Introduce Spectrogram Visualization (specgram.m) (plotspec.m) • Signals with HARMONIC Frequencies • Add Sinusoids with fk = kf0

  6. SPECTRUM DIAGRAM –250 –100 0 100 250 f (in Hz) • Recall Complex Amplitude vs. Freq

  7. SPECTRUM for PERIODIC ? • Nearly Periodic in the Vowel Region • Period is (Approximately) T = 0.0065 sec

  8. PERIODIC SIGNALS • Repeat every T secs • Definition • Example: • Speech can be “quasi-periodic”

  9. Period of Complex Exponential Definition: Period is T k = integer

  10. Harmonic Signal Spectrum

  11. Define FUNDAMENTAL FREQ

  12. Harmonic Signal (3 Freqs) 3rd 5th What is the fundamental frequency? 10 Hz

  13. POP QUIZ: FUNDAMENTAL f (in Hz) –250 –100 0 100 250 • Here’s another spectrum: What is the fundamental frequency? 100 Hz ? 50 Hz ?

  14. IRRATIONAL SPECTRUM SPECIAL RELATIONSHIP to get a PERIODIC SIGNAL

  15. Harmonic Signal (3 Freqs) T=0.1

  16. NON-Harmonic Signal NOT PERIODIC

  17. 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

  18. Time-Varying FREQUENCIES Diagram 26Ekim2k11 A-440 Frequency is the vertical axis Time is the horizontal axis

  19. SIMPLE TEST SIGNAL • C-major SCALE: stepped frequencies • Frequency is constant for each note IDEAL

  20. SPECTROGRAM • 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)

  21. SPECTROGRAM EXAMPLE • Two Constant Frequencies: Beats

  22. AM Radio Signal • Same as BEAT Notes

  23. SPECTRUM of AM (Beat) • 4 complex exponentials in AM: –672 0 648 –648 672 f (in Hz) What is the fundamental frequency? 648 Hz ? 24 Hz ?

  24. STEPPED FREQUENCIES • C-major SCALE: successive sinusoids • Frequency is constant for each note IDEAL

  25. SPECTROGRAM of C-Scale Sinusoids ONLY From SPECGRAM ANALYSIS PROGRAM ARTIFACTS at Transitions

  26. Spectrogram of LAB SONG Sinusoids ONLY Analysis Frame = 40ms ARTIFACTS at Transitions

  27. Time-Varying Frequency • Frequency can change vs. time • Continuously, not stepped • FREQUENCY MODULATION (FM) • CHIRP SIGNALS • Linear Frequency Modulation (LFM) VOICE

  28. New Signal: Linear FM QUADRATIC • Called Chirp Signals (LFM) • Quadratic phase • Freq will change LINEARLY vs. time • Example of Frequency Modulation (FM) • Define “instantaneous frequency”

  29. INSTANTANEOUS FREQ Derivative of the “Angle” Makes sense • Definition • For Sinusoid:

  30. INSTANTANEOUS FREQof the Chirp • Chirp Signals have Quadratic phase • Freq will change LINEARLY vs. time

  31. CHIRP SPECTROGRAM

  32. CHIRP WAVEFORM

  33. OTHER CHIRPS y(t) can be anything: y(t) could be speech or music: • FM radio broadcast

  34. SINE-WAVE FREQUENCY MODULATION (FM)

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