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function [PY_x coeff] = bss_tvproj(x,Y,tvshape,tvstep)

> compute the orthogonal projection of x onto the space of shifted windowed versions of the row(s) of Y
>
> Usage: [PY_x coeff] = tvproj(x,Y,tvshape,tvstep)
>
> Input:
> - x: row vector of length T corresponding to the signal to be projected,
> - Y: vector or matrix of length T with n_rows rows which windowed rows span the projection space.
> - tvshape : row vector of length V corresponding to the window applied
> to Y to define the projection space
> - tvstep : number of samples between two adjacent windows
>
> Ouput:
> - PY_x: row vector of length T containing the orthogonal projection of
> x onto the range of the shifted windowed versions of the row(s) of Y.
> - coeff : matrix with n_rows rows and n_frames columns containing the coefficients
> of the projection
>
> Developers: - Cedric Fevotte (cf269@cam.ac.uk) - Emmanuel Vincent
> (vincent@ircam.fr) - Remi Gribonval (remi.gribonval@iri
... ...

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... ...
n_rows);
for f=1:n_frames
for f1 = 1:n_frames
> locate the range of the intersection between frames
first = max(frames_index(f),frames_index(f1));
last = min(frames_index(f),frames_index(f1))+V-1;
> if the intersection is non empty, fill it
if (first <= last)
trange = (first:last)-frames_index(f)+1;
trange1= (first:last)-frames_index(f1)+1;
Gram((f-1)*n_rows+(1:n_rows),(f1-1)*n_rows+(1:n_rows)) = reshape(Y_frames(f,trange,:),length(trange),n_rows)'*reshape(Y_frames(f1,trange1,:),length(trange1),n_rows); > shall we reshape Y_frames ????
end
end
end

> 4. Apply the inverse of the Gram matrix to ip to get coeff
coeff = Gram\ip;
coeff = reshape(coeff,n_frames,n_rows);
> 4. Reconstruct using coefficients alpha
PY_x = zeros(size(x));

for f=1:n_frames
PY_x(frames_index(f)+(0:(V-1))) = PY_x(frames_index(f)+(0:(V-1))) + coeff(f,:)*reshape(Y_frames(f,:,:),V,n_rows)';
end