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Discrete-Time Convolution Linear Systems and Signals Lecture 8 Spring 2008

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8 - 2 Linear Time-Invariant System Any linear time-invariant system (LTI) system, continuous-time or discrete-time, can be uniquely characterized by its –Impulse response: response of system to an impulse –Frequency response: response of system to a complex exponential e j 2 f for all possible frequencies f –Transfer function: Laplace transform of impulse response Given one of the three, we can find other two provided that they exist

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8 - 3 Example Frequency Response System response to complex exponential e j for all possible frequencies where = 2 f Passes low frequencies, a.k.a. lowpass filter |H( )| pp ss s p passband stopband H() H()

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8 - 4 Kronecker Impulse (Function) Let [n] be a discrete-time impulse function, a.k.a. the Kronecker delta function: Impulse response h[n]: response of a discrete- time LTI system to a discrete impulse function n [n][n] 1

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8 - 5 Discrete-time Convolution Output y[n] for input x[n] Any signal can be decomposed into sum of discrete impulses Apply linear properties Apply shift-invariance Apply change of variables y[n] = h[0] x[n] + h[1] x[n-1] = ( x[n] + x[n-1] ) / 2 n h[n]h[n] Averaging filter impulse response 0123

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8 - 6 Comparison to Continuous Time Continuous-time convolution of x(t) and h(t) –For each value of t, we compute a different (possibly) infinite integral. –Discrete-time definition is the continuous-time definition with integral replaced by summation LTI system –If we know impulse response and input, we can determine the output –Impulse response uniquely characterizes it

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8 - 7 Fundamental Theorem The Fundamental Theorem of Linear Systems –If one inputs a complex sinusoid into an LTI system, then the output will be a complex sinusoid of the same frequency that has been scaled by the frequency response of the LTI system at that frequency –Scaling may attenuate the signal and shift it in phase –Example in continuous time: see handout G –Example in discrete time. Let x[n] = e j n, H( ) is the discrete-time Fourier transform of h[n] and is also called the frequency response

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8 - 8 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

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8 - 9 Five-tap discrete-time (scaled) averaging FIR filter with input x[n] and output y[n] Lowpass filter (smooth/blur input signal) Impulse response is {1, 1, 1, 1, 1} First-order difference FIR filter Highpass filter (sharpens input signal) Impulse response is {1, -1} n h[n]h[n] First-order difference impulse response

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