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#X text 116 249 (for visualization);
#X obj 26 113 fexpr~ $f2*$x1 + $f3*$x1[-1] + $f4*$x1[-2];
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#X restore 245 268 pd unknown_system~;
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#X text 330 436 <- Audio IO;
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#X text 22 15 input signal;
#X obj 143 35 inlet~;
#X text 122 15 desired signal;
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#X msg 249 63 adaptation 1;
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#X text 224 234 (for visualization);
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#X msg 248 162 clear;
#X obj 37 156 lms2~ 3 0.01;
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#X msg 249 116 mu \$1;
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#X restore 117 337 pd adaptive_filter~;
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#X text 700 769 -- 512 samples ---;
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#X text 125 241 x[n];
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#X text 698 597 -- 512 samples ---;
#X text 76 438 <- Visualization IO;
#X text 481 137 (1);
#X text 481 338 (2);
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#X text 582 510 (3);
#X text 581 686 (4);
#X text 86 760 (3) d[n] in time domain;
#X text 86 777 (4) y[n] in time domain;
#X text 86 742 (2) amplitude of the output signal y[n];
#X text 86 711 (1) amplitude of the desired signal d[n] (= output of
the unknown system);
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#X text 113 601 <- clear coefficients \, so adaptation will start again
;
#X floatatom 46 471 5 0 100 0 - init_tlp tlp;
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#X obj 685 238 r tlp;
#X text 90 469 <- temporal lowpass for spectrum view (0...100);
#X text 54 689 VISUALIZATIONS:;
#X text 42 522 unknown system:;
#X text 72 546 d[n] =;
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#X text 40 578 adaptive filter:;
#X floatatom 84 627 8 0 0 0 - mur mu;
#X text 146 627 <- step size parameter mu (learning rate);
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#X text 33 89 In this example the adaptive filter tries to identify
an unknown system.;
#X text 34 124 The unknown system is a FIR filter of order 3 and the
adaptive system is an adaptive transversal filter using the LMS algorithm
(see lms~ help-patch) with 3 coefficients.;
#X text 75 224 d[n] = h0*x[n] + h1*x[n-1] + h2*x[n-2];
#X text 35 188 unknown system:;
#X text 74 209 FIR Filter \, order = 3;
#X text 35 259 adaptive system:;
#X text 77 278 LMS \, 3 coefficients (c0 \, c1 \, c2);
#X text 35 342 The System Identification problem is sucessfull \, if
c0=h0 \, c1=h1 and c2=h2. Then the unknown and the adaptive system
have the same behavior.;
#X text 60 40 SYSTEM IDENTIFICATION IN A NOISE FREE ENVIRONMENT;
#X text 77 292 step-size parameter mu;
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#X text 26 95 e.g. mu = 0.5:;
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#X text 26 179 e.g. mu = 1e-05:;
#X text 25 59 1) influenc of mu:;
#X text 84 119 -> very fast adaptation \, but c0 \, c1 \, c2;
#X text 104 134 are not so precise;
#X text 83 201 -> slower adaptation \, but very precise;
#X text 27 430 2) influenc of white vs. non-white input signal:;
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#X text 28 469 white signal:;
#X text 30 550 non-white signal:;
#X text 87 569 -> c0 \, c1 \, c2 not precise;
#X text 90 489 -> c0 \, c1 \, c2 precise;
#X text 29 274 how to choose mu ?;
#X text 72 300 0 < mu < 2/(abs(x[n])^2);
#X text 72 321 -> abs(x[n])^2 is the tap-input energy;
#X text 94 336 at time n (lenght of x[n] is PDs;
#X text 46 373 Note: this only ensures "stability on average";
#X text 94 351 blocksize - so use block~ to change it!);
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#X text 33 73 ReadMe:;
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#X text 292 63 non white signal:;
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#X text 145 183 non white signal (filtered noise);
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#X text 33 32 SYSTEM IDENTIFICATION IN A NOISE FREE ENVIRONMENT;
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