ECE 352 Systems II Manish K. Gupta, PhD Office: Caldwell Lab Home Page: TA: Zengshi Chen Chen Office Hours for TA : in CL 391: Tu & Th 1:00-2:30 pm Home Page:

Acknowledgements Various graphics used here has been taken from public resources instead of redrawing it. Thanks to those who have created it. Thanks to Brian L. Evans and Mr. Dogu Arifler Thanks to Randy Moses and Bradley Clymer

ECE 352 Fourier Transform

ECE 352 REVIEW SOME CONCEPTS Slides edited from: Prof. Brian L. Evans and Mr. Dogu Arifler Dept. of Electrical and Computer Engineering The University of Texas at Austin course: EE 313 Linear Systems and Signals Fall 2003

Signals A function, e.g. sin(t) or sin(2  n / 10), useful in analysis A sequence of numbers, e.g. {1,2,3,2,1} which is a sampled triangle function, useful in simulation A collection of properties, e.g. even, causal, and stable, useful in reasoning about behavior A piecewise representation, e.g. A functional, e.g.  (t)

T{} y(t)y(t)x(t)x(t) y[k]y[k]x[k]x[k] Systems Systems operate on signals to produce new signals or new signal representations For a single-input one-dimensional continuous- time system, we can represent it –As an operator –As a block diagram

System Properties Let x[k], x 1 [k], and x 2 [k] be inputs to a linear system and let y[k], y 1 [k], and y 2 [k] be their corresponding outputs A linear system satisfies –Additivity: x 1 [k] + x 2 [k]  y 1 [k] + y 2 [k] –Homogeneity:  x[k]   y[k] for any constant  Let x[k] be the input to time-invariant system and y[k] be its corresponding output. Then, x[k - m]  y[k - m], for any integer m

s(t)s(t) t TsTs Sampling Many signals originate as continuous-time signals, e.g. conventional music or voice By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers k  {…, -2, -1, 0, 1, 2,…} T s is the sampling period. Sampled analog waveform

Optical Disk Writer Optical Disk Reader D/AA/D x(t)x(t)x(t)x(t) CDv[k]v[k]v[k]v[k] Recording StudioStereo System / PC F s = 44.1 kHz T s = ms F s = 44.1 kHz T s = ms Sampling Consider audio compact discs (CDs) Analog-to-digital (A/D) conversion consists of filtering, sampling, and quantization Digital-to-analog (D/A) conversion consists of interpolation and filtering

Laplace Transform

Forward Laplace Transform Decompose a signal f(t) into complex sinusoids of the form e s t where s is complex: s =  + j2  f Forward (bilateral) Laplace transform f(t): complex-valued function of a real variable t F(s): complex-valued function of a complex variable s Bilateral means that the extent of f(t) can be infinite in both the positive t and negative t direction (a.k.a. two-sided)

Inverse (Bilateral) Transform  is a contour integral which represents integration over a complex region– recall that s is complex c is a real constant chosen to ensure convergence of the integral Notation F(s) = L{f(t)}variable t implied for L f(t) = L -1 {F(s)}variable s implied for L -1

L F(s)f(t)f(t) Laplace Transform Properties Linear or nonlinear? Linear operator

Laplace Transform Properties Time-varying or time-invariant? This is an odd question to ask because the output is in a different domain than the input.

Example

Convergence The condition Re{s} > -Re{a} is the region of convergence, which is the region of s for which the Laplace transform integral converges Re{s} = -Re{a} is not allowed (see next slide)

Regions of Convergence What happens to F(s) = 1/(s+a) at s = -a? (1/0) -e a t u(-t) and e -a t u(t) have same transform function but different regions of convergence Im{s} Re{s} = -Re{a} Re{s} t f(t)f(t) f(t) = -e a t u(-t) anti-causal 1 t f(t)f(t) f(t) = e -a t u(t) causal

Review of 0 - and 0 +  (t) not defined at t = 0 but has unit area at t = refers to an infinitesimally small time before refers to an infinitesimally small time after 0

Unilateral Laplace Transform Forward transform: lower limit of integration is 0 - (i.e. just before 0) to avoid ambiguity that may arise if f(t) contains an impulse at origin Unilateral Laplace transform has no ambiguity in inverse transforms because causal inverse is always taken: –No need to specify a region of convergence –Disadvantage is that it cannot be used to analyze noncausal systems or noncausal inputs

Existence of Laplace Transform As long as e -  t decays at a faster rate than rate f(t) explodes, Laplace transform converges for some M and  0, there exists  0 >  to make the Laplace transform integral finite We cannot always do this, e.g. does not have a Laplace transform

Key Transform Pairs

Inverse Laplace Transform Definition has integration in complex plane –We can use lookup tables instead – See Tables 8.1 page 375 and Table 8.2 page 376, Text Book Many Laplace transform expressions are ratios of two polynomials, a.k.a. rational functions Convert complicated rational functions into simpler forms –Apply partial fractions decomposition –Use lookup tables

Partial Fractions Example #1

Partial Fractions Example #2

Partial Fractions Example #3

Use Matlab to find Laplace Transform, Inverse Laplace Transform ?

Laplace Transform Properties Linearity Time shifting Frequency shifting Differentiation in time

Differentiation in Time Property

Laplace Transform Properties Differentiation in frequency Integration in time –Example: f(t) =  (t) Integration in frequency

Laplace Transform Properties Scaling in time/frequency –Under integration, Convolution in time Convolution in frequency t f(t) 2-2 t f(2 t) 1 Area reduced by factor 2

Example Compute y(t) = e a t u(t) * e b t u(t), where a  b If a = b, then we would have resonance What form would the resonant solution take?

Linear Differential Equations Using differentiation in time property we can solve differential equations (including initial conditions) using Laplace transforms Example: (D 2 + 5D + 6) y(t) = (D + 1) f(t) With y(0 - ) = 2, y’(0 - ) =1, and f(t) = e - 4 t u(t) So f ’(t) = -4 e -4 t u(t) + e -4 t  (t), f ’(0 - ) = 0 and f ’(0 + ) = 1

3rd Day Class Home Work No Submission Read Pages (Chapter 8) Play with Matlab !

Any Questions ?