EE 313 Linear Systems and Signals Fall 2010

EE 313 Linear Systems and Signals Fall 2010 Continuous-Time Convolution Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Initial conversion of content to Power. Point by Dr. Wade C. Schwartzkopf

Convolution Integral • Commonly used in engineering, science, math • Convolution properties – – Commutative: f 1(t) * f 2(t) = f 2(t) * f 1(t) Distributive: f 1(t) * [f 2(t) + f 3(t)] = f 1(t) * f 2(t) + f 1(t) * f 3(t) Associative: f 1(t) * [f 2(t) * f 3(t)] = [f 1(t) * f 2(t)] * f 3(t) Shift: If f 1(t) * f 2(t) = c(t), then f 1(t) * f 2(t - T) = f 1(t - T) * f 2(t) = c(t - T). – Convolution with impulse, f(t) * d(t) = f(t) – Convolution with shifted impulse, f(t) * d(t-T) = f(t-T) 2 important later in modulation

Graphical Convolution Methods • From the convolution integral, convolution is equivalent to – Rotating one of the functions about the y axis – Shifting it by t – Multiplying this flipped, shifted function with the other function – Calculating the area under this product – Assigning this value to f 1(t) * f 2(t) at t 3

Graphical Convolution Example • Convolve the following two functions: f(t) g(t) 3 2 * t t 2 -2 2 • Replace t with t in f(t) and g(t) • Choose to flip and slide g(t) since it is simpler and symmetric 3 g(t-t) • Functions overlap like this: 2 f(t) -2 + t 2+t 2 t 4

Graphical Convolution Example • Convolution can be divided into 5 parts I. t < -2 • • Two functions do not overlap Area under the product of the functions is zero 3 2 -2 + t • • f(t) 3 Part of g(t) overlaps part of f(t) Area under the product of the functions is t 2 2+t -2 t < 0 II. g(t-t) 2 -2 + t f(t) 2+t t 2 5

Graphical Convolution Example III. 0 t < 2 • • Here, g(t) completely overlaps f(t) Area under the product is just 3 2 f(t) 2 2+t -2 + t IV. 2 t < 4 • • Part of g(t) and f(t) overlap Calculated similarly to -2 t < 0 t 4 V. • • g(t) and f(t) do not overlap Area under their product is zero g(t-t) t 3 g(t-t) 2 f(t) -2 + t 2+t 6

Graphical Convolution Example • Result of convolution (5 intervals of interest): y(t) 6 t -2 0 2 4 7

Convolution Demos • Johns Hopkins University Demonstrations http: //www. jhu. edu/~signals Convolution applet to animate convolution of simple signals and hand-sketched signals Convolve two rectangular pulses of same width gives a triangle (see handout E) • Some conclusions from the animations Convolution of two causal signals gives a causal result Non-zero duration (called extent) of convolution is the sum of extents of the two signals being convolved 8

Transmit One Bit • Transmission over communication channel (e. g. telephone line) is analog receive ‘ 0’ bit input Tp t x(t) -A A t Tp ‘ 1’ bit t Communication Channel Model channel as LTI system with impulse response h(t) Assume that Th < Tp output Th y(t) t -A Th receive ‘ 1’ bit A Th 1 Th Th+Tp t Th Th+Tp 9 t

Transmit Two Bits (Interference) • Transmitting two bits (pulses) back-to-back will cause overlap (interference) at the receiver * A = 1 Th+Tp 2 Tp Tp ‘ 1’ bit t Th t -A Th ‘ 0’ bit Assume that Th < Tp • How do we prevent intersymbol interference at the receiver? t Tp ‘ 1’ bit ‘ 0’ bit intersymbol interference 10

Transmit Two Bits (No Interference) • Prevent intersymbol interference by waiting Th seconds between pulses (called a guard period) * A = 1 Th+Tp Tp ‘ 1’ bit Th+Tp Th t t -A Th ‘ 0’ bit Assume that Th < Tp t Th Tp ‘ 1’ bit ‘ 0’ bit • Disadvantages? 11

Discrete-time Convolution Preview • Discrete-time convolution • Continuous-time convolution • For every value of n, we compute a new summation • For every value of t, we compute a new integral x[n] h[n] LTI system represented by its impulse response y[n] x(t) h(t) LTI system represented by its impulse response y(t) 12

Discrete-time Convolution Preview • Assuming that h[n] has finite duration from n = 0, …, N-1 • Block diagram of an implementation (finite impulse response digital filter): see slide 2 -4 x[n] h[0] z-1 … z-1 h[1] h[2] … h[N-1] S y[n] 13

Philosophy • Pillars of electrical engineering (related) • Finite-state machines for digital input/output Fourier analysis Probability and random processes • Pillars of computer engineering (related) System state Complexity EE 313 is pre-requisite for EE 351 K Finite number of states Models all possible inputoutput combinations Can two outputs be true at the same time? • Given output observation, work backwards to inputs to see if output is possible • This is called observability 14

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