Understanding Sinusoidal Signals and Complex Numbers

1. Signals and Systems Lecture 3: Sinusoids

2. Today's lecture • Sinusoidal signals • Review of the Sine and Cosine Functions • Examples • Basic Trigonometric Identities • Relation of Frequency to Period • Phase Shift to Time Shift • Example Sampling and Plotting Sinusoids • Complex Exponentials and Phasors • Complex Number Representation • Addition of Complex Numbers • Mathematical Addition • Graphical Addition

4. Fig. 2-6: x(t) = 20cos(2π(40)t - 0.4π)

5. Sinusoidal signal : x(t) = 10cos(2π(440)t - 0.4π)

6. MATLAB Demo of Tuning Fork • % TuningFork • t = 0:.0001:.01; • y = 10*cos(2*pi*440*t-0.4*pi); • plot(t,y) • grid • pause; • t = 0:.0001:1; • y = 10*cos(2*pi*440*t-0.4*pi); • sound (y)

7. Basic Properties of sine and cosine functions

8. Some basic trigonometric identities

9. Relation of Frequency to Period X(t)=A cos(0t+ ) x(t + T0) = x(t) A cos(0 (t + T0) +  )= A cos(0t+ ) cos(0 t + 0 T0+  )= cos(0t+ ) Since cosine function has a period of 2π 0 T0 =2π 2πf0 T0=2π T0 =1/f0

10. Fig 2-7: x(t) = 5cos(2πfot) for different values of fo

11. Phase Shift and Time Shift x0 (t - t1) = A cos(0 (t - t1) = A cos (0t + ) cos(0 t-0 t1)= cos(0t + ) t1 = -/ 0 = -/ 2πf0 Phase Shift is negative when time-shift is positive  = - 2πf0 t1 = - 2πt1 /T0

12. Phase Shift and Time Shift

13. Phase Shift is Ambiguous

14. X(t) =Acos(wt +Φ)

15. Sinusoid from a Plot

16. Represent following graph in form of X(t) =Acos(wt +Φ)

17. A=6 • T =6 • f=1/6 • tm=2; • Φ=-wtm • Φ=-2*pi*f*tm • -2pi/3; • X(t)=6cos(pi/3 -2pi/3)

18. Sampling and Plotting Sinusoids

19. Effect of Sampling Period

20. Sample Spacing

21. Complex Numbers

22. Plot Complex Numbers

23. Complex Addition = Vector Addition

24. Polar Form

25. Polar versus Rectangular

26. Practice

27. Solution

28. Complex Conjugation