1 Prof. Nizamettin AYDIN Digital Signal Processing

2 Lecture 2 Phase & Time-Shift Complex Exponentials

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4 READING ASSIGNMENTS This Lecture: –Chapter 2, Sects. 2-3 to 2-5 Appendix A: Complex Numbers Appendix B: MATLAB Next Lecture: finish Chap. 2, –Section 2-6 to end

5 LECTURE OBJECTIVES Define Sinusoid Formula from a plot Relate TIME-SHIFT to PHASE Introduce an ABSTRACTION: Complex Numbers represent Sinusoids Complex Exponential Signal

6 SINUSOIDAL SIGNAL FREQUENCY –Radians/sec –or, Hertz (cycles/sec) PERIOD (in sec) AMPLITUDE –Magnitude PHASE

7 PLOTTING COSINE SIGNAL from the FORMULA Determine period: Determine a peak location by solving Peak at t=-4

8 ANSWER for the PLOT Use T=20/3 and the peak location at t=-4

9 TIME-SHIFT In a mathematical formula we can replace t with t-t m Then the t=0 point moves to t=t m Peak value of cos(  (t-t m )) is now at t=t m

10 TIME-SHIFTED SINUSOID

11 PHASE TIME-SHIFT Equate the formulas: and we obtain: or,

12 TIME-SHIFT Whenever a signal can be expressed in the form x 1 (t)=s(t-t 1 ), we say that x 1 (t) is time shifted version of s(t) –If t 1 is a + number, then the shift is to the right, and we say that the signal s(t) has been delayed in time. –If t 1 is a - number, then the shift is to the left, and we say that the signal s(t) was advanced in time.

13 SINUSOID from a PLOT Measure the period, T –Between peaks or zero crossings –Compute frequency:  = 2  /T Measure time of a peak: t m –Compute phase:  = -  t m Measure height of positive peak: A 3 steps

14 (A, ,  ) from a PLOT

15 SINE DRILL (MATLAB GUI) Sine drill 1 st, type spfirst in matlab, then run sindrill.

16 PHASE is AMBIGUOUS The cosine signal is periodic – Period is 2  –Thus adding any multiple of 2  leaves x(t) unchanged

17 COMPLEX NUMBERS To solve: z 2 = -1 – z = j – Math and Physics use z = i Complex number: z = x + j y x yz Cartesian coordinate system 12Ekim2k11

18 PLOT COMPLEX NUMBERS

19 COMPLEX ADDITION = VECTOR Addition

20 Complex drill 1 st, type spfirst in matlab, then run zdrill.

21 *** POLAR FORM *** Vector Form –Length =1 –Angle =  Common Values 1 has angle of  j has angle of 0.5   1 has angle of   j has angle of 1.5  also, angle of  j could be  0.5  1.5  2  because the PHASE is AMBIGUOUS

22 POLAR RECTANGULAR Relate (x,y) to (r,  ) Need a notation for POLAR FORM Most calculators do Polar-Rectangular

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

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

25 Rotating phasors demo

26 cos = REAL PART Real Part of Euler’s General Sinusoid So,

27 REAL PART EXAMPLE Answer: Evaluate:

28 COMPLEX AMPLITUDE Then, any Sinusoid = REAL PART of Xe j  t General Sinusoid Complex AMPLITUDE = X