EE235 Signals and Systems Leo Lam © 2010-2011

An ex and a Constant were… …walking down the street; when a Differentiator walked up to them. Constant started running away, and ex asked him, “what are you doing?!” Constant replied, “If I meet a Differentiator, I will disappear!” ex said, proudly, “I don’t care, I am ex!”, and walked up to the Differentiator. “Hi I am ex,” he said, thumbing his nose… “Hi,” said the Differentiator, “I’m d/dy.” Leo Lam © 2010-2011

Today’s menu Posted: Textbook chapters (on website/blog) To do: Sign up to Facebook Group Bookmark our website From Friday (Signals x and y relationships) End of hand-waving Describing Common Signals Periodicity Leo Lam © 2010-2011

Signals: Digging in Types of signals Some “standard” signals (alphabets!) Leo Lam © 2010-2011

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 © 2010-2011

Signal types Continuous time / Discrete time An x-axis relationship Discrete time = “indexed” time Leo Lam © 2010-2011

Signals: Notations A continuous time signal is specified at all values of time, when time is a real number. Leo Lam © 2010-2011

Signals: Notations A discrete time signal is specified at only discrete values of time (e.g. only on integers) Leo Lam © 2010-2011

What types are these? 90.3 FM radio transmitted signal Daily count of orcas in Puget Sound Muscle contraction of your heart over time A capacitor’s charge over time A picture taken by a digital camera Local news broadcast to your old TV Video on YouTube Your voice (continuous) (discrete) (c) ((c)) (d) (c) (d) (c) Leo Lam © 2010-2011

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

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

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

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

Common signals (memorize) Building blocks to bigger things constant signal t a unit step 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 unit ramp signal t 1 Leo Lam © 2010-2011

Sinusoids/Decaying sinusoids Leo Lam © 2010-2011

Decaying and growing Leo Lam © 2010-2011

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

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

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