Continuous-Time Convolution Linear Systems and Signals Lecture 5 Spring 2008

5 - 2 Convolution Demos Johns Hopkins University Demonstrations 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

5 - 3 Transmit One Bit Transmission over communication channel (e.g. telephone line) is analog hh t 1 pp t A ‘1’ bit t pp -A-A ‘0’ bit Model channel as LTI system with impulse response h(t) Communication Channel inputoutput x(t)x(t)y(t)y(t) t t receive ‘1’ bit -A T h receive ‘0’ bit h+ph+p t h+ph+p hh hh Assume that T h < T p A T h

5 - 4 Transmit Two Bits (Interference) Transmitting two bits (pulses) back-to-back will cause overlap (interference) at the receiver How do we prevent intersymbol interference at the receiver? hh t 1 Assume that T h < T p t pp A ‘1’ bit ‘0’ bit pp *= -A T h t pp ‘1’ bit ‘0’ bit h+ph+p intersymbol interference

5 - 5 Transmit Two Bits (No Interference) Prevent intersymbol interference by waiting T h seconds between pulses (called a guard period) Disadvantages? hh t 1 Assume that T h < T p *= t pp A ‘1’ bit ‘0’ bit h+ph+p t -A T h pp ‘1’ bit ‘0’ bit h+ph+p hh

5 - 6 h[n]h[n] y[n]y[n]x[n]x[n] LTI system represented by its impulse response h(t)h(t) y(t)y(t)x(t)x(t) Discrete-time Convolution Preview Discrete-time convolution For every value of n, we compute a new summation Continuous-time convolution For every value of t, we compute a new integral

5 - 7 z -1 … … x[n]x[n]  y[n]y[n] h[0]h[1]h[2]h[N-1] 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

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