EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Discrete-Time Convolution

7 - 2 Discrete-time Convolution Output y[n] for input x[n] Any signal can be decomposed into sum of discrete impulses Apply linearity properties of homogeneity then additivity Apply shift-invariance Apply change of variables

7 - 3 Discrete-time Convolution Filtering viewpoint Hold impulse response h[n] in place and change variables Flip and slide input signal x[n] about impulse response Example of finite impulse response (FIR) filter Impulse response has finite extent (non-zero duration) x[n]x[n] y[n]y[n] 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

7 - 4 Convolution in Both Domains 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 Linear time-invariant (LTI) system Output signal in time domain is convolution of impulse response and input signal Impulse response uniquely characterizes the LTI system

7 - 5 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 sum of extents of two signals being convolved minus one

7 - 6 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

7 - 7 Frequency Response For continuous-time LTI system For discrete-time LTI system Note: Identity for cosine input assumes a real- valued impulse response

7 - 8 Example Frequency Response System response to complex exponential e j  n for all possible frequencies  in rad/sample Passes low frequencies, a.k.a. lowpass filter  |H(  )| pp ss  s  p passband stopband   H() H()

7 - 9 Differentiator/Difference Operation ContinuousDiscrete f(t)f(t)y(t)y(t)f[n]f[n]y[n]y[n] We can remove scaling by 1/T s without changing lowpass response

First-Order FIR Filters signal = [ ]; figure(1); stem(signal); averagingFilter = [ ]; average = conv(averagingFilter, signal); figure(2); stem(average); differenceFilter = [ ]; difference = conv(differenceFilter, signal); figure(3); stem(difference); Signal with a spike Output of averaging filter Output of difference filter y[n] = ½ x[n] + ½ x[n-1] y[n] = ½ x[n]  ½ x[n-1] x[n]x[n] n nn

Mandrill Demo (DSP First) First-order difference FIR filter Highpass filter (sharpens input signal) Impulse response is {1, -1} 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} n h[n]h[n] First-order difference impulse response

Mandrill Demo (DSP First) DSP First, Ch. 6, Freq. Response of FIR Filters, From lowpass filter to highpass filter original image  blurred image  sharpened/blurred image From highpass to lowpass filter original image  sharpened image  blurred/sharpened image Frequencies that are zeroed out can never be recovered (e.g. DC is zeroed out by highpass filter) Order of two LTI systems in cascade can be switched under the assumption that computations are performed in exact precision

Mandrill Demo (DSP First) Input image is 256 x 256 matrix Each pixel represented by eight-bit number in [0, 255] 0 is black and 255 is white for monitor display Each filter applied along row then column Averaging filter adds five numbers to create output pixel Difference filter subtracts two numbers to create output pixel Full output precision is 16 bits per pixel Demonstration uses double-precision floating-point data and arithmetic (53 bits of mantissa + sign; 11 bits for exponent) No output precision was harmed in making of this demo

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 two- sided complex exponential input signal for all possible frequencies Transfer function: Laplace transform (or z-transform) of impulse response Given one of the three, we can find other two provided that they exist