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

2. 2 Lecture 3 Phasor Addition Theorem Digital Signal Processing

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4. 4 READING ASSIGNMENTS This Lecture: –Chapter 2, Section 2-6 Other Reading: –Appendix A: Complex Numbers –Appendix B: MATLAB –Next Lecture: start Chapter 3

5. 5 LECTURE OBJECTIVES Phasors = Complex Amplitude –Complex Numbers represent Sinusoids Develop the ABSTRACTION: –Adding Sinusoids = Complex Addition –PHASOR ADDITION THEOREM

6. 6 Z DRILL (Complex Arith) MATLAB Command Name: spfirst zdrill Complex drill ZDrill is a program that tests the users ability to calculate the result of simple operations on complex numbers. The program emphasizes the vectorial view of a complex number. The following six operations are supported: Add Subtract Multiply Divide Inverse Conjugate

7. 7 AVOID Trigonometry Algebra, even complex, is EASIER !!! Can you recall cos(  1 +  2 ) ? Use: real part of e j  1  2  = cos(  1 +  2 )

8. 8 Euler’s FORMULA Complex Exponential –Real part is cosine –Imaginary part is sine –Magnitude is one

9. 9 Real & Imaginary Part Plots PHASE DIFFERENCE =  /2

10. 10 COMPLEX EXPONENTIAL Interpret this as a Rotating Vector  t Angle changes vs. time ex:  rad/s Rotates  in 0.01 secs

11. 11 Rotating Phasor See Demo on CD-ROM Chapter 2

12. 12 Cos = REAL PART Real Part of Euler’s General Sinusoid So,

13. 13 COMPLEX AMPLITUDE General Sinusoid Complex AMPLITUDE = X Sinusoid = REAL PART of (Ae j  )e j  t

14. 14 POP QUIZ: Complex Amp Find the COMPLEX AMPLITUDE for: Use EULER’s FORMULA:

15. 15 WANT to ADD SINUSOIDS ALL SINUSOIDS have SAME FREQUENCY HOW to GET {Amp,Phase} of RESULT ?

16. 16 ADD SINUSOIDS SAMESum Sinusoid has SAME Frequency

17. 17 PHASOR ADDITION RULE Get the new complex amplitude by complex addition

18. 18 Phasor Addition Proof

19. 19 POP QUIZ: Add Sinusoids ADD THESE 2 SINUSOIDS: COMPLEX ADDITION:

20. 20 POP QUIZ (answer) COMPLEX ADDITION: CONVERT back to cosine form:

21. 21 ADD SINUSOIDS EXAMPLE t m1 t m2 t m3

22. 22 Convert Time-Shift to Phase Measure peak times: t m1 =-0.0194, t m2 =-0.0556, t m3 =-0.0394 Convert to phase (T=0.1)  1 =-  t m1 = -2  (t m1 /T) = 70  /180,  2 = 200  /180 Amplitudes A 1 =1.7, A 2 =1.9, A 3 =1.532

23. 23 Phasor Add: Numerical Convert Polar to Cartesian X 1 = 0.5814 + j1.597 X 2 = -1.785 - j0.6498 sum = X 3 = -1.204 + j0.9476 Convert back to Polar X 3 = 1.532 at angle 141.79  /180 This is the sum

24. 24 ADD SINUSOIDS VECTOR (PHASOR) ADD X1X1 X2X2 X3X3