Leo Lam © Signals and Systems EE235

An e x and a Constant were… …walking down the street; when a Differentiator walked up to them. Constant started running away, and e x asked him, “what are you doing?!” Constant replied, “If I meet a Differentiator, I will disappear!” e x said, proudly, “I don’t care, I am e x !”, and walked up to the Differentiator. “Hi I am e x,” he said, thumbing his nose… “Hi,” said the Differentiator, “I’m d/dy.” Leo Lam ©

Today’s menu Textbook Chapter 1, Schaum’s Chapter 1 To do: –Sign up to Facebook Group –Bookmark our website From yesterday: definitions End of hand-waving Describing Common Signals –Type of signals –Some standard signals Periodicity

Signals: A signal is a mathematical function –x(t) –x is the value (real, complex)  y-axis –t is the independent variable (1D, 2D etc.)  x-axis –Both can be Continuous or Discrete –Examples of x… Leo Lam ©

Signal types Continuous time / Discrete time –An x-axis relationship Discrete time = “indexed” time Leo Lam ©

Signals: Notations A continuous time signal is specified at all values of time, when time is a real number. Leo Lam ©

Signals: Notations A discrete time signal is specified at only discrete values of time (e.g. only on integers) Leo Lam ©

What types are these? Leo Lam © )90.3 FM radio transmitted signal 2)Daily count of orcas in Puget Sound 3)Muscle contraction of your heart over time 4)A capacitor’s charge over time 5)A picture taken by a digital camera 6)Local news broadcast to your old TV 7)Video on YouTube 8)Your voice (c) ((c)) (c) (continuous) (c) (d) (discrete)

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

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 © Where Z and G are finite sets of values

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 ©

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 ©

Common signals (memorize) Building blocks to bigger things Leo Lam © 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

Sinusoids/Decaying sinusoids Leo Lam ©

Decaying and growing Leo Lam ©

Generalizing the sinusoids Leo Lam © 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?

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

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 © For all integers n

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

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 ©

Rephrasing in math Leo Lam © Goal: find T such that

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

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 © T 1 T 2

Your turn! Find the period of: Leo Lam © No LCM exists! Why? Because LCM exists only if T 1 /T 2 is a rational number

A few more Leo Lam © Not rational, so not periodic Decaying term means pattern does not repeat exactly, so not periodic

Summary Description of common signals Periodicity Leo Lam ©