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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 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 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
+;
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+#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 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 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 32 SYSTEM IDENTIFICATION IN A NOISE FREE ENVIRONMENT;
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