2. 1 Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin Digital Signal Processing

3. 2 Lecture 5 Periodic Signals, Harmonics & Time-Varying Sinusoids Digital Signal Processing

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5. 4 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

6. 5 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

7. 6 LECTURE OBJECTIVES HARMONICSignals with HARMONIC Frequencies –Add Sinusoids with f k = kf 0 FREQUENCY can change vs. TIME Chirps: Introduce Spectrogram Visualization (specgram.m) (plotspec.m)

8. 7 SPECTRUM DIAGRAM Recall Complex Amplitude vs. Freq 0100250–100–250 f (in Hz)

9. 8 SPECTRUM for PERIODIC ? Nearly Periodic in the Vowel Region –Period is (Approximately) T = 0.0065 sec

10. 9 PERIODIC SIGNALS Repeat every T secs –Definition –Example: –Speech can be “quasi-periodic”

11. 10 Period of Complex Exponential Definition: Period is T k = integer

12. 11 Harmonic Signal Spectrum

13. 12 Define FUNDAMENTAL FREQ

14. 13 What is the fundamental frequency? Harmonic Signal (3 Freqs) 3rd 5th 10 Hz

15. 14 POP QUIZ: FUNDAMENTAL Here’s another spectrum: What is the fundamental frequency? 100 Hz ?50 Hz ? 0100250–100–250 f (in Hz)

16. 15 SPECIAL RELATIONSHIP to get a PERIODIC SIGNAL IRRATIONAL SPECTRUM

17. 16 Harmonic Signal (3 Freqs) T=0.1

18. 17 NON-Harmonic Signal NOT PERIODIC

19. 18 FREQUENCY ANALYSIS Now, a much HARDER problemNow, 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

20. 19 Time-Varying FREQUENCIES Diagram Frequency is the vertical axis Time is the horizontal axis A-440 26Ekim2k11

21. 20 SIMPLE TEST SIGNAL C-major SCALE: stepped frequencies –Frequency is constant for each note IDEAL

22. 21 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 X k –Breaks x(t) into SHORT TIME SEGMENTS Then uses the FFT (Fast Fourier Transform)

23. 22 SPECTROGRAM EXAMPLE Two Constant Frequencies: Beats

24. 23 AM Radio Signal Same as BEAT Notes

25. 24 SPECTRUM of AM (Beat) 4 complex exponentials in AM: What is the fundamental frequency? 648 Hz ?24 Hz ? 0648 672 f (in Hz) –672 –648

26. 25 STEPPED FREQUENCIES C-major SCALE: successive sinusoids –Frequency is constant for each note IDEAL

27. 26 SPECTROGRAM of C-Scale ARTIFACTS at Transitions Sinusoids ONLY From SPECGRAM ANALYSIS PROGRAM

28. 27 Spectrogram of LAB SONG ARTIFACTS at Transitions Sinusoids ONLY Analysis Frame = 40ms

29. 28 Time-Varying Frequency Frequency can change vs. time –Continuously, not stepped FREQUENCY MODULATION (FM)FREQUENCY MODULATION (FM) CHIRP SIGNALS –Linear Frequency Modulation (LFM) VOICE

30. 29 New Signal: Linear FM Called Chirp Signals (LFM) –Quadratic phase Freq will change LINEARLY vs. time –Example of Frequency Modulation (FM) –Define “instantaneous frequency” QUADRATIC

31. 30 INSTANTANEOUS FREQ Definition For Sinusoid: Derivative of the “Angle” Makes sense

32. 31 INSTANTANEOUS FREQ of the Chirp Chirp Signals have Quadratic phase Freq will change LINEARLY vs. time

33. 32 CHIRP SPECTROGRAM

34. 33 CHIRP WAVEFORM

35. 34 OTHER CHIRPS  (t) can be anything:  (t) could be speech or music: –FM radio broadcast

36. 35 SINE-WAVE FREQUENCY MODULATION (FM)