EECE 301 Signals & Systems Prof. Mark FowlerEECE 301 Signals & SystemsProf. Mark FowlerDiscussion #3b• DT Convolution ExamplesConvolution Example “Table view”h(-m)h(1-m)Discrete-Time Convolution Example: “Sliding Tape View”D-T Convolution Examples()]4[][][][21][ −−== nununhnunxn-3 -2 -1 1 2 3 4 5 6 7 8 9 ][ihi…-3 -2 -1 1 2 3 4 5 6 7 8 9 ][ixi…-3 -2 -1 1 2 3 4 5 6 7 8 9 ]0[ ih −iChoose to flip and slide h[n] This shows h[n-i] for n = 0…For n < 0 h[n-i]x(i) = 0 ∀i00][<= nforny⇒][][ ixih −iFor n = 0⇒ y[0] = 1-3 -2 -1 1 2 3 4 5 6 7 8 9 ][ixiNotice that for n = 0, n = 1, …, n = 3The general result is: ()3,2,1,021][0==∑=nfornynii())(2112111SumGeometricn−−=+()3,2,1,02112][1=⎥⎦⎤⎢⎣⎡−=+nfornyn-2 -1 1 2 3 4 5 6 7 8…]1[][ ihinh−=−i⇒ y(1) = 1 + ½ = 3/2 For n = 1-3 -2 -1 1 2 3 4 5 6 7 8 9 ][ixin = 4 case-3 -2 -1 1 2 3 4 5 6 7 8 9 ]4[][ ihinh…−=−iNow for n = 4, n = 5, …13 =−= ni 4==nin = 5 case-3 -2 -1 1 2 3 4 5 6 7 8 9 …]5[][ ihinh−=−i23 =−= ni 5==niNotice that: for n = 4, 5, 6, …()() ()!2112121,...6,5,421][133simplifythennfornynninni−−===+−−=∑Then we can write out the solution as:()[]() ()[]⎪⎪⎩⎪⎪⎨⎧=−=−<=+−+...,6,5,4,2/12/123,2,1,0,2/1120,0][131nnnnynnn-3 -2 -1 1 2 3 4 5 6 7…][nhn122. Same Impulse Response: h[n] nietc()][2cos][ nunnxπ=][ix][ ih −iAgain y[n] = 0 for n < 0:00101]5[01010]4[00101]3[0101]2[101]1[1]0[=+++−==++−==+−+==−+==+==yyyyyyNotice: y[n] = 0 ∀ n = 2, 3, 4, 5, …!So suppose we had a desired part of our signal as:But say we “receive” our desired pulse signal with an “interfering” sinusoid:()[]nunnpnx2cos][][π+=][ˆ][ npny=][nh][nxFrom above we know that system “zeros out” (or suppresses) the sinusoid…We also know that the system will “pass” the pulses, although their edges will be smoothed. 9-point pulse keeps repeating][np9 points0.25nMatlab Explorations%%% Matlab exploration for Pulses with Interfering Sinusoidp=[ones(1,9) zeros(1,6)]; %%% Create one pulse and zerosp=[p p p p p]; %%% stack 5 of them togetherp=0.25*p; %%% adjust its amplitude to be 0.25subplot(3,1,1)stem(0:74,p) %%% look at the sequence of pulsesxlabel('Sample Index, n')ylabel('Pulsed Signal p[n]')x=p+cos((pi/2)*(0:74)); % add in an interfering sinusoidsubplot(3,1,2)stem(0:74,x)xlabel('Sample Index, n')ylabel('x[n] Input = pulse + sinusoid')y=conv(x,ones(1,4)); %% filter out sinusoid with DT Conv.subplot(3,1,3)stem(0:77,y)xlabel('Sample Index, n')ylabel('y[n] = Output')%%% Note that pulses are free of sinusoidal interference but have been
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