Signals and Systems EE235 Leo Lam
Today s menu Lab detailed arrangements Homework vacation week From yesterday (Intro: Signals) Intro: Systems More: Describing Common Signals
Taking a signal apart Offset (atmospheric pressure) A sound signal Frequency Amplitude st ( ) = Asin(2 π ft) + a 0 0 A+a 0 a 0 T t (seconds)
Frequency g( t) = sin(2πf 0t) f = 0 196Hz = t (seconds) 196 f (Hz) time-domain frequency-domain
t to f 196 293.66 440 t (seconds) 659.26 F (Hz)
Combining signals
Summary: Signals Signals carry information Signals represented by functions over time or space Signals can be represented in both time and frequency domains Signals can be summed in both time and frequency domains
Systems A system describes a relationship between input and output Examples? v(t) g(t) y(t)
Definition: System A system modifies signals or extracts information. It can be considered a transformation that operates on a signal.
Motivation: Complex systems
Filters All kinds, and everywhere
Surprising high pass
Summary: System System transforms an input to an output System can extract information System can shape signals (filters)
Signals: Digging in Types of signals Some standard signals (alphabets!)
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
Signal types Continuous time / Discrete time An x-axis relationship Discrete time = indexed time
Signals: Notations A continuous time signal is specified at all values of time, when time is a real number. xt ( ), t R
Signals: Notations A discrete time signal is specified at only discrete values of time (e.g. only on integers) xn [ ], n Z
What types are these? 1) 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 ((c)) 5) A picture taken by a digital camera (d) 6) Local news broadcast to your old TV (c) 7) Video on YouTube (d) 8) Your voice (c) (continuous) (discrete) (c)
Analog / Digital values (y-axis) An analog signal has amplitude that can take any value in a continuous interval (all Real numbers) xn [ ] R, n Z xt () Rt, R 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). xt () G, t R xn [ ] G, n Z 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)
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 Building blocks to bigger things constant signal a xt () = a t unit step signal unit ramp signal 1 0 1 0 t t u(t)=0 for t<0 u(t)=1 for t 0 t r(t)=0 for t<0 r(t)=t = for t 0 r(t)=t*u(t) u(τ ) d for t 0 τ 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)=ce at, a=σ+jω Equivalently: x(t)=ce σt e jωt Remember Euler s Formula? e j ω t = cos( ωt) + j sin( ωt) x(t)=ce σt e jωt amplitude Sinusoidal with frequency ω (in radians) Exponential (3 types) What is the frequency in Hz? Leo Lam 2010-2011
Imaginary signals Remember how to convert between the two? imag z r φ b a real z=a+jb real/imaginary z=re jφ magnitude/phase Leo Lam 2010-2011
Describing signals Of interest? Peak value +/- time? Complex? Magnitude, phase, real, imaginary parts? Periodic? Total energy? Power? 0 s(t) t Time averaged * s(t)s (t)dt = 2 s(t) dt Leo Lam 2010-2011
Periodic signals Definition: x(t) is periodic if there exists a T (time period) such that: x ( t) x( t + nt ) = For all integers n The minimum T is the period Fundamental frequency f 0 =1/T Leo Lam 2010-2011
Periodic signals: examples Sinusoids x(t)=a cos(ω 0 t+φ) Complex exponential (non-decaying or increasing) y t Ae A ω t ja ω t jω0t ( ) = = cos( ) + sin( ) 0 0 Infinite sum of shifted signals v(t) (more later) T 0 z() t = v( t kt ) k = 0 Leo Lam 2010-2011