A 1D Hermitian array of Gaussian Fourier coefficients

randgpn1.m

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+function varargout=randgpn1(k,dc,dcn,xver)
+% [Zk,Zx]=randgpn1(k,dc,dcn,xver)
+% 
+% Returns a set of (complex proper) normal variables suitable for
+% IFFT, most notably this works for even and/or odd length vectors
+%
+% INPUT:
+%
+% k       A wavenumber vector (DC component in center)
+% dc      The index to the DC component in k
+% dcn     The index to the Myquist components in k
+%         --> Note that these three are straight out of KNUM1, and that
+%         the actual wavenumbers are not used, only size(k) is needed. 
+% xver    1 Checks the Hermiticity of the result by inverse transformation
+%         0 No such check is being conducted 
+%
+% OUTPUT:
+%
+% Zk      A size(k) vector with (complex proper) normal Fourier variables
+% Zx      A size(k) vector with a real-valued random field
+%
+% EXAMPLE 1
+%
+% [k,kx,dci,dcn]=knum1(round(rand*100),100);
+% [Zk,Zx]=randgpn1(k,dci,dcn); plot(Zx)
+%
+% EXAMPLE 2 
+% 
+% [k,kx,dci,dcn]=knum1(200,100);
+% F=abs(randn*10); [Zk,Zx]=randgpn1(k,dci,dcn); Zf=ifft(ifftshift(Zk.*exp(-F*k)));
+% plot(Zf); title(sprintf('exp(%3.3gk)',-F));
+%
+% SEE ALSO: KNUM2, KNUM1
+%
+% Last modified by fjsimons-at-alum.mit.edu, 03/18/2019
+
+defval('xver',0)
+
+% Make a receptacle with the dimension of the wavenumbers
+n=length(k);
+Zk=zeros(1,n); clear k
+
+% Determine the parity
+nodd=mod(n,2);
+
+% Define the ranges to pick out the halves
+lhn=1:dc;
+rhn=dc+1:n;
+
+% Now make some complex proper normal random variables of half variance
+ReZk=randn(1,dc)/sqrt(2);
+ImZk=randn(1,dc)/sqrt(2);
+% And make a handful real normal random variables of unit variance
+realZk=randn(2,1);
+
+% Fill the entire left half plane with these random numbers
+lh=ReZk+sqrt(-1)*ImZk;
+Zk(1,lhn)=lh;
+
+% Fill the center with a real, where the DC component goes 
+Zk(1,dc)=realZk(1);
+
+% Fill the Nyquist with a real, if we capture them exactly 
+for ind=1:size(dcn,1)
+  Zk(dcn)=realZk(1+ind);
+end
+
+% And now symmetrize the right half plane so that output is Hermitian
+Zk(1,rhn)=conj(fliplr(lh(2-nodd:dc-1)));
+
+if nargout>1 || xver==1
+  Zx=ifft(ifftshift(Zk));
+  % You could now check that the IFFT2 is real (don't forget the "2"!!)
+  if ~isreal(Zx) ; error(sprintf('Not Hermitian by %5g',mean(imag(Zx(:))))); end
+  % You can perform this check more directly as well
+  hermcheck(Zk)
+else
+  Zx=NaN;
+end
+
+% Output
+varns={Zk,Zx};
+varargout=varns(1:nargout);