# Digital circuit sizing example (GP)

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    % Boyd, Kim, Patil, and Horowitz, "Digital circuit optimization
    % via geometric programming"
    % Written for CVX by Almir Mutapcic 02/08/06
    %
    % Solves the problem of choosing gate scale factors x_i to give
    % minimum ckt delay, subject to limits on the total area and power.
    % Uses max gate arrival time T formulation that avoids evaluation
    % of the delay over all paths in the circuit.
    %
    %   minimize   T_bar
    %       s.t.   T_j <= T_bar      for j an output gate
    %              T_j + d_i <= T_i  for j in FI(i)
    %              P <= Pmax, A <= Amax
    %              x >= 1
    %
    % where variables are x and T.
    %
    % We use the circuit topology presented in figure 1 (page 902),
    % where we take gates 1, 3 and 6 to be inverters (INV),
    % gates 2 and 7 to be three input NANDs (NAND3),
    % and gates 4 and 5 to be two input NORs (NOR2).

    %********************************************************************
    % user specified data (specify problem constant and ckt topology)
    %********************************************************************
    m = 7;        % number of gates
    Vdd = 5;      % supply voltage
    Amax = 250;   % maximum area spec

    % gate specs
    INV   = struct('Cin',3, 'Cint',3, 'Rdrv',0.48, 'A',3,  'Ileak',0.006);
    NAND3 = struct('Cin',4, 'Cint',6, 'Rdrv',0.48, 'A',8,  'Ileak',0.007);
    NOR2  = struct('Cin',5, 'Cint',6, 'Rdrv',0.48, 'A',10, 'Ileak',0.009);

    clear gates;
    gates([1 3 6]) = INV;
    gates([2 7])   = NAND3;
    gates([4 5])   = NOR2;

    % primary inputs and primary outputs labels (start with m+1)
    primary_inputs = [8 9 10];
    primary_outputs = [11 12];
    M = m + length( primary_inputs ) + length( primary_outputs );

    % fan-in cell array
    FI = cell(M,1);
    FI{1} = [8];
    FI{2} = [8 9 10];
    FI{3} = [10];
    FI{4} = [1 2];
    FI{5} = [2 3];
    FI{6} = [4];
    FI{7} = [3 4 5];
    FI{8} = [];
    FI{9} = [];
    FI{10} = [];
    FI{11} = [6];
    FI{12} = [7];

    % primary output has Cin capacitance (but has no Cload)
    Cin_po = sparse(M,1);
    Cin_po(primary_outputs) = [10 10];

    % primary input has Cload capacitance (but has no Cin)
    Cload_pi = sparse(M,1);
    Cload_pi(primary_inputs) = [10 10 10];

    % activity frequency of gates and primary inputs
    f_gates = 0.001*ones(m,1);
    f_pi = sparse(M,1);
    f_pi(primary_inputs) = 0.001*[10 10 10];

    %********************************************************************
    % derived problem data (computed from user inputs)
    %********************************************************************
    % fan-out cell array (compute it from the fan-in cell array)
    FO = cell(M,1);
    for gate = [1:m primary_outputs]
      preds = FI{gate};
      for k = 1:length(preds)
        FO{preds(k)}(end+1) = gate;
      end
    end

    % input and internal capacitance of gates, and driving resistance
    Cin_norm  = [gates.Cin]';
    Cint_norm = [gates.Cint]';
    Rdrv_norm = [gates.Rdrv]';

    % area specification for each gate with unit scaling
    A_norm = [gates.A]';

    % leakage current of gate i with unit scaling
    Ileak_norm = [gates.Ileak]';

    %********************************************************************
    % optimization (with tradeoff curve generation)
    %********************************************************************
    % objective is the upper bound on the overall delay
    % and that is the max of arrival times for output gates
    output_gates = [FI{primary_outputs}];

    % varying parameters for the tradeoff curve
    N = 25;
    Pmax = linspace(10,20,N);
    min_delay = zeros(N,1);
    disp('Generating the optimal tradeoff curve...')
    for n = 1:N
      fprintf('Pmax = %6.2f: ',Pmax(n));
      cvx_begin gp quiet
        % optimization variables
        variable x(m)                 % scale factor
        variable T(m)                 % arrival times

        % input capacitance is an affine function of sizes
        Cin  = Cin_norm.*x;
        Cint = Cint_norm.*x;

        % driving resistance is inversily proportional to sizes
        R = Rdrv_norm./x;

        % gate delay is the product of its driving resistance and load cap.
        Cload = cvx( zeros(m,1) );
        for gate = 1:m
          if ~ismember( FO{gate}, primary_outputs )
            Cload(gate) = sum( Cin(FO{gate}) );
          else
            Cload(gate) = Cin_po( FO{gate} );
          end
        end

        % delay
        D = 0.69*ones(m,1).*R.*( Cint + Cload );

        % total area
        area = A_norm'*x;

        % total power calculation
        Pdyn = Vdd^2*sum( f_pi(primary_inputs).*Cload_pi(primary_inputs) ) + ...
               Vdd^2*(f_gates'*(Cint + Cload));
        Pstat = Vdd*Ileak_norm'*x;
        power = Pdyn + Pstat;

        minimize( max( T(output_gates) ) )
        subject to
          % constraints
          x >= 1;
          area <= Amax;
          power <= Pmax(n);

          % create timing constraints
          for gate = 1:m
            if ~ismember( FI{gate}, primary_inputs )
              for j = FI{gate}
                % enforce T_j + D_j <= T_i over all gates j that drive i
                D(gate) + T(j) <= T(gate);
              end
            else
              % enforce D_i <= T_i for gates i connected to primary inputs
              D(gate) <= T(gate);
            end
          end
      cvx_end
      fprintf( 'delay = %3.2f\n', cvx_optval );
      min_delay(n) = cvx_optval;
    end

    % plot the tradeoff curve
    figure, clf
    plot(Pmax,min_delay);
    xlabel('Pmax'); ylabel('Dmin');
    title(['Tradeoff curve for Amax = ' num2str(Amax)])
    disp('Optimal tradeoff curve plotted.')

    Generating the optimal tradeoff curve...
    Pmax =  10.00: delay = 14.20
    Pmax =  10.42: delay = 12.33
    Pmax =  10.83: delay = 11.51
    Pmax =  11.25: delay = 11.02
    Pmax =  11.67: delay = 10.72
    Pmax =  12.08: delay = 10.48
    Pmax =  12.50: delay = 10.27
    Pmax =  12.92: delay = 10.08
    Pmax =  13.33: delay = 9.92
    Pmax =  13.75: delay = 9.78
    Pmax =  14.17: delay = 9.66
    Pmax =  14.58: delay = 9.54
    Pmax =  15.00: delay = 9.44
    Pmax =  15.42: delay = 9.35
    Pmax =  15.83: delay = 9.26
    Pmax =  16.25: delay = 9.18
    Pmax =  16.67: delay = 9.15
    Pmax =  17.08: delay = 9.15
    Pmax =  17.50: delay = 9.15
    Pmax =  17.92: delay = 9.15
    Pmax =  18.33: delay = 9.15
    Pmax =  18.75: delay = 9.15
    Pmax =  19.17: delay = 9.15
    Pmax =  19.58: delay = 9.15
    Pmax =  20.00: delay = 9.15
    Optimal tradeoff curve plotted.