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Continuous-Time Convolution

Spring 2008. Continuous-Time Convolution. Linear Systems and Signals Lecture 5. Convolution Demos. Johns Hopkins University Demonstrations http://www.jhu.edu/~signals Convolution applet to animate convolution of simple signals and hand-sketched signals

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Continuous-Time Convolution

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  1. Spring 2008 Continuous-Time Convolution Linear Systems and SignalsLecture 5

  2. 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

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

  4. 1 Th t Transmit Two Bits (Interference) • Transmitting two bits (pulses) back-to-back will cause overlap (interference) at the receiver • How do we prevent intersymbolinterference at the receiver? * = A Th+Tp 2Tp t Tp Tp t -A Th Assume that Th < Tp ‘0’ bit ‘1’ bit ‘0’ bit ‘1’ bit intersymbol interference

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

  6. 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 x[n] h[n] y[n] x(t) h(t) y(t) LTI systemrepresentedby its impulseresponse LTI system representedby its impulseresponse Discrete-time Convolution Preview

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

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