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

3. 2 Lecture 4 Spectrum Representation Digital Signal Processing

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5. 4 READING ASSIGNMENTS This Lecture: –Chapter 3, Section 3-1 Other Reading: –Appendix A: Complex Numbers –Next Lecture: Ch 3, Sects 3-2, 3-3, 3-7 & 3-8

6. 5 LECTURE OBJECTIVES Sinusoids with DIFFERENT Frequencies –SYNTHESIZE by Adding Sinusoids SPECTRUM Representation DIFFERENT –Graphical Form shows DIFFERENT Freqs

7. 6 FREQUENCY DIAGRAM Plot Complex Amplitude vs. Freq 0100250–100–250 f (in Hz)

8. 7 Another FREQ. Diagram Frequency is the vertical axis Time is the horizontal axis A-440

9. 8 MOTIVATION Synthesize Complicated Signals –Musical Notes Piano uses 3 strings for many notes Chords: play several notes simultaneously –Human Speech Vowels have dominant frequencies Application: computer generated speech –Can all signals be generated this way? Sum of sinusoids?

10. 9 Fur Elise WAVEFORM Beat Notes

11. 10 Speech Signal: BAT PeriodicNearly Periodic in Vowel Region –Period is (Approximately) T = 0.0065 sec

12. 11 Euler’s Formula Reversed Solve for cosine (or sine)

13. 12 INVERSE Euler’s Formula Solve for cosine (or sine)

14. 13 SPECTRUM Interpretation Cosine = sum of 2 complex exponentials: One has a positive frequency The other has negative freq. Amplitude of each is half as big

15. 14 NEGATIVE FREQUENCY Is negative frequency real? Doppler Radar provides an example –Police radar measures speed by using the Doppler shift principle –Let’s assume 400Hz  60 mph –+400Hz means towards the radar –-400Hz means away (opposite direction) –Think of a train whistle

16. 15 SPECTRUM of SINE Sine = sum of 2 complex exponentials: –Positive freq. has phase = -0.5  –Negative freq. has phase = +0.5 

17. 16 GRAPHICAL SPECTRUM EXAMPLE of SINE AMPLITUDE, PHASE & FREQUENCY are shown  7-70

18. 17 SPECTRUM ---> SINUSOID Add the spectrum components: What is the formula for the signal x(t)? 0100250–100–250 f (in Hz)

19. 18 Gather (A,  ) information Frequencies: –-250 Hz –-100 Hz –0 Hz –100 Hz –250 Hz Amplitude & Phase –4-  /2 –7+  /3 –100 –7-  /3 –4+  /2 DC is another name for zero-freq component DC component always has  or  (for real x(t) ) Note the conjugate phase

20. 19 Add Spectrum Components-1 Amplitude & Phase –4 -  /2 –7 +  /3 –100 –7 -  /3 –4 +  /2 Frequencies: –-250 Hz –-100 Hz –0 Hz –100 Hz –250 Hz

21. 20 Add Spectrum Components-2 0100250–100–250 f (in Hz)

22. 21 Use Euler’s Formula to get REAL sinusoids: Simplify Components

23. 22 FINAL ANSWER So, we get the general form:

24. 23 Summary: GENERAL FORM

25. 24 Example: Synthetic Vowel Sum of 5 Frequency Components

26. 25 SPECTRUM of VOWEL –Note: Spectrum has 0.5X k (except X DC ) –Conjugates in negative frequency

27. 26 SPECTRUM of VOWEL (Polar Format) kk 0.5A k

28. 27 Vowel Wavefor (sum of all 5 components)