Signal Processing First

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Signal Processing First Lecture 19 Continuous-Time Signals and Systems 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer READING ASSIGNMENTS This Lecture: Chapter 9, Sects 9-1 to 9-5 Other Reading: Recitation: Ch. 9, all Next Lecture: Chapter 9, Sects 9-6 to 9-8 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer LECTURE OBJECTIVES Bye bye to D-T Systems for a while The UNIT IMPULSE signal Definition Properties Continuous-time signals and systems Example systems Review: Linearity and Time-Invariance Convolution integral: impulse response 9/3/2018 © 2003, JH McClellan & RW Schafer

D-T Filtering of C-T Signals C-to-D D-to-C LTI System H(z) LTI ANALOG System 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer ANALOG SIGNALS x(t) INFINITE LENGTH SINUSOIDS: (t = time in secs) PERIODIC SIGNALS ONE-SIDED, e.g., for t>0 UNIT STEP: u(t) FINITE LENGTH SQUARE PULSE IMPULSE SIGNAL: d(t) DISCRETE-TIME: x[n] is list of numbers 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer CT Signals: PERIODIC Sinusoidal signal Square Wave INFINITE DURATION 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer CT Signals: ONE-SIDED Unit step signal One-Sided Sinusoid “Suddenly applied” Exponential 9/3/2018 © 2003, JH McClellan & RW Schafer

CT Signals: FINITE LENGTH Square Pulse signal Sinusoid multiplied by a square pulse 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer What is an Impulse? A signal that is “concentrated” at one point. 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Defining the Impulse Assume the properties apply to the limit: One “INTUITIVE” definition is: Concentrated at t=0 Unit area 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Sampling Property 9/3/2018 © 2003, JH McClellan & RW Schafer

General Sampling Property 9/3/2018 © 2003, JH McClellan & RW Schafer

Properties of the Impulse Concentrated at one time Unit area Sampling Property Extract one value of f(t) Derivative of unit step 9/3/2018 © 2003, JH McClellan & RW Schafer

Continuous-Time Systems Examples: Delay Modulator Integrator Output Input 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer CT BUILDING BLOCKS INTEGRATOR (CIRCUITS) DIFFERENTIATOR DELAY by to MODULATOR (e.g., AM Radio) MULTIPLIER & ADDER 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Ideal Delay: Mathematical Definition: To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so 9/3/2018 © 2003, JH McClellan & RW Schafer

Output of Ideal Delay of 1 sec 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Integrator: Mathematical Definition: To find the IMPULSE RESPONSE, h(t), let x(t) be an impulse, so Running Integral 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Integrator: Integrate the impulse IF t<0, we get zero IF t>0, we get one Thus we have h(t) = u(t) for the integrator 9/3/2018 © 2003, JH McClellan & RW Schafer

Graphical Representation 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Output of Integrator 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Differentiator: Mathematical Definition: To find h(t), let x(t) be an impulse, so Doublet 9/3/2018 © 2003, JH McClellan & RW Schafer

Differentiator Output: 9/3/2018 © 2003, JH McClellan & RW Schafer

Linear and Time-Invariant (LTI) Systems If a continuous-time system is both linear and time-invariant, then the output y(t) is related to the input x(t) by a convolution integral where h(t) is the impulse response of the system. 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Testing for Linearity 9/3/2018 © 2003, JH McClellan & RW Schafer

Testing Time-Invariance 9/3/2018 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Integrator: Linear And Time-Invariant 9/3/2018 © 2003, JH McClellan & RW Schafer