Filter design by convex iteration

From Wikimization

 where  denotes impulse response.

For a low pass filter, frequency domain specifications are:

To minimize peak magnitude of  , the problem becomes

But this problem statement is nonconvex.

So instead, a new vector

is defined by concatenation of time-shifted versions of  .

Then

is a positive semidefinite rank 1 matrix.

Summing along each of subdiagonals gives entries of the autocorrelation function of  .

In particular, the main diagonal of holds squared entries of  .

Minimizing is therefore equivalent to minimizing  .

Define ...

By spectral factorization,  , an equivalent problem is:

Excepting the rank constraint, this problem is convex.

N = 32;
delta1 = 1.01;
delta2 = 0.01;
w = linspace(0,pi,N);

cvx_begin  
  variable h(N,1);
  variable r(2*N-1,1);
  variable g(N^2,1);
  variable G(N^2,N^2);
  variable R(2*N-1,1);
  
  for I=1:N
    g((I-1)*N+1:I*N,1) = circshift(h,[0,I-1]);
  end
  G = g*g';
%   G >= 0;
  minimize(max(diag(G)));
  for I=1:N
      r(I) = 1/I*trace(diag(ones(1,N),I-N)*G);
  end
  for I=N+1:2*N-1
      r(I) = 1/(I-N)*trace(diag(ones(1,N),I-N)*G);
  end
  for I=1:N
      temp = 0;
      for J=1:N-1
          temp = temp + 2*r(J)*cos(w(J)*J);
      end
      R(I) = r(1) + temp;
  end
  for I=1:N/2
      1/delta1^2 < R(I);
      R(I) < delta1^2;
  end
  for I=N/2+1:N
      R(I) < delta2^2;
  end

  R > 0;
  
  
cvx_end