Here is an examle, using d=[1 2 3 4 5], which you suggested. Since d has length 5, then the convolution d*w = p (where * indicates convolution) will be 4 elements longer than w. So if w has length 5, p will have length 9, etc. For this example, I will assume w has length 6, in order to demonstrate that the length of w does not have to equal the length of d. I will do the forward convoltion (compute p=d*w) first. Then I will do the inverse convolution.
Compute Toeplitz matrix:
w=[-1 0 1 2 1 3]';
d=[1 2 3 4 5]';
%nr=length(d)+length(w)-1; % rows in Toeplitz matrix
%nc=length(w); % columns in Toeplitz matrix
c=[d;zeros(length(w)-1,1)]; % column 1 of Toeplitz matrix
r=[1,zeros(1,length(w)-1)]; % row 1 of Toeplitz matrix
D=toeplitz(c,r);
disp(D)
Use D to compute p=convolution of d with w:
p=D*w;
disp(p')
Compute estimate of w, from p, using the Toeplitz matrix:
wEst=inv(D'*D)*D'*p;
disp(wEst')
The result shows that the estimate of w equals the original w.
