1. Leo Lam © 2010-2011 Signals and Systems EE235 Leo Lam

2. Leo Lam © 2010-2011 Today’s menu From yesterday (Signals x and y relationships) More: Describing Common Signals Periodicity

3. Analog / Digital values (y-axis) An analog signal has amplitude that can take any value in a continuous interval (all Real numbers) Leo Lam © 2010-2011 Where Z is a finite set of values

4. Analog / Digital values (y-axis) An digital signal has amplitude that can only take on only a discrete set of values (any arbitrary set). Leo Lam © 2010-2011 Where Z and G are finite sets of values

5. Nature vs. Artificial Natural signals mostly analog Computers/gadgets usually digital (today) Signal can be continuous in time but discrete in value (a continuous time, digital signal) Leo Lam © 2010-2011

6. Brake! X-axis: continuous and discrete Y-axis: continuous (analog) and discrete (digital) Our class: (mostly) Continuous time, analog values (real and complex) Clear so far? Leo Lam © 2010-2011

7. Common signals Building blocks to bigger things Leo Lam © 2010-2011 constant signal t a 0 unit step signal t 1 0 unit ramp signal t 1 u(t)=0 for t<0 u(t)=1 for t≥0 r(t)=0 for t<0 r(t)=t for t≥0 r(t)=t*u(t) for t≥0

8. Sinusoids/Decaying sinusoids Leo Lam © 2010-2011

9. Decaying and growing Leo Lam © 2010-2011

10. Generalizing the sinusoids Leo Lam © 2010-2011 General form: x(t)=Ce at, a=σ+jω Equivalently: x(t)=Ce σt e jωt Remember Euler’s Formula? x(t)=Ce σt e jωt amplitude Exponential (3 types) Sinusoidal with frequency ω (in radians) What is the frequency in Hz?

11. Imaginary signals Leo Lam © 2010-2011 z r a b z=a+jb real/imaginary z=re jΦ magnitude/phase  real imag Remember how to convert between the two?

12. Describing signals Of interest? –Peak value –+/- time? –Complex? Magnitude, phase, real, imaginary parts? –Periodic? –Total energy? –Power? Leo Lam © 2010-2011 0 s(t) t Time averaged

13. Periodic signals Definition: x(t) is periodic if there exists a T (time period) such that: The minimum T is the period Fundamental frequency f 0 =1/T Leo Lam © 2010-2011 For all integers n

14. Periodic signals: examples Sinusoids Complex exponential (non-decaying or increasing) Infinite sum of shifted signals v(t) (more later) Leo Lam © 2010-2011 x(t)=A cos(   t+  ) T0T0

15. Periodicity of the sum of periodic signals Question: If x 1 (t) is periodic with period T 1 and x 2 (t) is periodic with period T 2 –What is the period of x 1 (t)+x 2 (t)? Can we rephrase this using our “language” in math? Leo Lam © 2010-2011

16. Rephrasing in math Leo Lam © 2010-2011 Goal: find T such that

17. Rephrasing in math Leo Lam © 2010-2011 Goal: find T such that Need:  T=LCM(T 1,T 2 ) Solve it for r=1, true for all r

18. Periodic sum example If x 1 (t) has T 1 =2 and x 2 (t) has T 2 =3, what is the period of their sum, z(t)? LCM (2,3) is 6 And you can see it, too. Leo Lam © 2010-2011 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 1 1 T 1 T 2

19. Your turn! Find the period of: Leo Lam © 2010-2011 No LCM exists! Why?