ミニマックス最適化

この例では次を使用します:

この例では、Optimization Toolbox™ のミニマックス最適化アルゴリズムfminimaxを使用して非線形フィルターの設計問題を解く方法を説明します。この例を実行するには、Signal Processing Toolbox™ がインストールされていなければなりません。

有限精度パラメーターの設定

有限精度フィルターの設計の例を考えます。そのためには、フィルター設計パラメーター (カットオフ周波数や係数の数など) を指定するだけでは不十分です。設計が有限精度であるため、使用可能なビット数も指定しなければなりません。

nbits = 8;% How many bits have we to realize filtermaxbin = 2^nbits-1;% Maximum number expressable in nbits bitsn = 4;% Number of coefficients (order of filter plus 1)Wn = 0.2;% Cutoff frequency for filterRp = 1.5;% Decibels of ripple in the passbandw = 128;% Number of frequency points to take

まず連続設計

これは連続フィルター設計です。cheby1を使用しますが、ellip、yulewalk、またはremezを使用することもできます。

[b1,a1] = cheby1(n-1,Rp,Wn); [h,w] = freqz(b1,a1,w);% Frequency responseh = abs(h);% Magnitude responseplot(w, h) title('Frequency response using non-integer variables')

Figure contains an axes object. The axes object with title Frequency response using non-integer variables contains an object of type line.

x = [b1,a1];% The design variables

フィルター係数の範囲の設定

最大値と最小値に範囲を設定します。

if(any(x < 0))% If there are negative coefficients - must save room to use a sign bit% and therefore reduce maxbinmaxbin = floor(maxbin/2); vlb = -maxbin * ones(1, 2*n)-1; vub = maxbin * ones(1, 2*n);else% otherwise, all positivevlb = zeros(1,2*n); vub = maxbin * ones(1, 2*n);end

係数のスケーリング

maxbin に等しい最大値を設定し、他のフィルター係数を適切にスケーリングします。

[m, mix] = max(abs(x)); factor = maxbin/m; x = factor * x;% Rescale other filter coefficientsxorig = x; xmask = 1:2*n;% Remove the biggest value and the element that controls D.C. Gain% from the list of values that can be changed.xmask(mix) = []; nx = 2*n;

最適化基準の設定

optimoptionsを使用して、実行時間を短縮するために終了基準を適度に高い値に調整します。また、反復ごとに結果の表示をオンにします。

options = optimoptions('fminimax',...'StepTolerance', 0.1,...'OptimalityTolerance', 1e-4,...'ConstraintTolerance', 1e-6,...'Display','iter');

最大絶対値の最小化

最大絶対値を最小にしなければならないため、options.MinAbsMax を周波数点の数に設定します。

if长度(w) = = 1 = optimoptions(选项,选项'AbsoluteMaxObjectiveCount',w);elseoptions = optimoptions(options,'AbsoluteMaxObjectiveCount',length(w));end

最適化のための最初の値の削除

FMINIMAX を呼び出すことで、最初の値の離散化と削除を行い、最適化を実行します。

[x, xmask] = elimone(x, xmask, h, w, n, maxbin)

x =1×80.5441 1.6323 1.6323 0.5441 57.1653 -127.0000 108.0000 -33.8267

xmask =1×61 2 3 4 5 8

硝石=长度(xmask);disp (sprintf ('Performing %g stages of optimization.\n\n', niters));

Performing 6 stages of optimization.

form = 1:niters fun = @(xfree)filtobj(xfree,x,xmask,n,h,maxbin);% objectiveconfun = @(xfree)filtcon(xfree,x,xmask,n,h,maxbin);% nonlinear constraintdisp (sprintf ('Stage: %g \n', m)); x(xmask) = fminimax(fun,x(xmask),[],[],[],[],vlb(xmask),vub(xmask),...confun,options); [x, xmask] = elimone(x, xmask, h, w, n, maxbin);end

Stage: 1

Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 8 0 0.00329174 1 17 0.0001845 3.34e-07 1 0.0143 Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.

Stage: 2

Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 7 0 0.0414182 1 15 0.01649 0.0002558 1 0.261 2 23 0.01544 6.126e-07 1 -0.0282 Hessian modified Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.

Stage: 3

Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 6 0 0.0716961 1 13 0.05943 -1.156e-11 1 0.776 Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.

Stage: 4

Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 5 0 0.129938 1 11 0.04278 -3.871e-10 1 0.183 Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.

Stage: 5

Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 4 0 0.0901749 1 9 0.03867 -4.838e-11 1 0.256 Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.

Stage: 6

Objective Max Line search Directional Iter F-count value constraint steplength derivative Procedure 0 3 0 0.11283 1 7 0.05033 9.714e-17 1 0.197 Local minimum possible. Constraints satisfied. fminimax stopped because the size of the current search direction is less than twice the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.

最も近い整数値のチェック

最も近い値によってフィルターの機能が向上するかどうか確認します。

xold = x; xmask = 1:2*n; xmask([n+1, mix]) = []; x = x + 0.5;fori = xmask [x, xmask] = elimone(x, xmask, h, w, n, maxbin);endxmask = 1:2*n; xmask([n+1, mix]) = []; x = x - 0.5;fori = xmask [x, xmask] = elimone(x, xmask, h, w, n, maxbin);endifany(abs(x) > maxbin) x = xold;end

周波数応答の比較

まず、フィルターの周波数応答をプロットし、そのフィルターを、係数が切り上げまたは切り捨てられるフィルターと比較します。

subplot(211) bo = x(1:n); ao = x(n+1:2*n); h2 = abs(freqz(bo,ao,128)); plot(w,h,w,h2,'o') title('Optimized filter versus original') xround = round(xorig)

xround =1×81 2 2 1 57 -127 108 -34

b = xround(1:n); a = xround(n+1:2*n); h3 = abs(freqz(b,a,128)); subplot(212) plot(w,h,w,h3,'+') title('Rounded filter versus original')

Figure contains 2 axes objects. Axes object 1 with title Optimized filter versus original contains 2 objects of type line. Axes object 2 with title Rounded filter versus original contains 2 objects of type line.

fig = gcf; fig.NextPlot ='replace';

参考

fminimax