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Leo Lam © 2010-2012 Signals and Systems EE235

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Why did the statistician drown in the river? Because it had an average depth of 6 inches. Leo Lam © 2010-2012

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Today’s menu To Do: –Join Facebook Group –Read Lab 1 Intro: Signals Intro: Systems More: Describing Common Signals

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Definition: Signal A signal is a set of information or data that can be modeled as a function of one or more independent variables. Leo Lam © 2010-2012

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Taking a signal apart Leo Lam © 2010-2012 a0a0 T t (seconds) A+a 0 A sound signal Offset (atmospheric pressure) Frequency Amplitude

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Frequency Leo Lam © 2010-2012 196 t (seconds)f (Hz) = time-domainfrequency-domain

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t to f Leo Lam © 2010-2012 293.66 t (seconds) 196 440 659.26 F (Hz)

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Combining signals Leo Lam © 2010-2012

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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 Leo Lam © 2010-2012

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Systems A system describes a relationship between input and output Examples? Leo Lam © 2010-2012 v(t)y(t)g(t)

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Definition: System A system modifies signals or extracts information. It can be considered a transformation that operates on a signal. Leo Lam © 2010-2012

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Motivation: Complex systems Leo Lam © 2010-2012

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Filters All kinds, and everywhere Leo Lam © 2010-2012

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Surprising high pass Leo Lam © 2010-2012

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Summary: System System transforms an input to an output System can extract information System can “shape” signals (filters) Leo Lam © 2010-2012

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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 © 2010-2012

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Signal types Continuous time / Discrete time –An x-axis relationship Discrete time = “indexed” time Leo Lam © 2010-2012

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Signals: Notations A continuous time signal is specified at all values of time, when time is a real number. Leo Lam © 2010-2012

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Signals: Notations A discrete time signal is specified at only discrete values of time (e.g. only on integers) Leo Lam © 2010-2012

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What types are these? Leo Lam © 2010-2012 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 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)

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Analog / Digital values (y-axis) An analog signal has amplitude that can take any value in a continuous interval (all Real numbers) Leo Lam © 2010-2012 Where Z is a finite set of values

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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 © 2010-2012 Where Z and G are finite sets of values

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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 © 2010-2012

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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 © 2010-2012

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Common signals (memorize) Building blocks to bigger things Leo Lam © 2010-2012 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

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Sinusoids/Decaying sinusoids Leo Lam © 2010-2012

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Decaying and growing Leo Lam © 2010-2012

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Generalizing the sinusoids Leo Lam © 2010-2012 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?

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Imaginary signals Leo Lam © 2010-2012 z r a b z=a+jb real/imaginary z=re jΦ magnitude/phase real imag Remember how to convert between the two?

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