Recall that the discrete Fourier transform (DFT) of a vector x=[x[0],x[1],,x[N1]]T of length N is defined as X[k]=(Fx)[k]=n=0N1x[n]ej2kn/N,k{0,1,,N1}. The inverse DFT of a vector X=[X[0],X[1],,X[N1]]T as x[n]=(F1X)[n]=N1k=0N1X[k]ej2kn/N,n{0,1,,N1}. 1. We have seen in class that any linear operator in finite dimensions can be represented by a matrix. Fourier transform is a linear operator. Determine the matrices representing F and F1 such that X=Fx and x=F1X. 2. The inverse DFT is, not surprisingly, the inverse of the forward DFT operator. Show that F1(Fx)=x by directly multiplying the matrices. Hint: you might want to recall formula for geometric series. 3. There is another relation between DFT and its inverse. Show that F1=N1F where F is obtained from F by transposing and then applying complex conjugation to all entries. This implies that up to a rescaling, the DFT matrix is orthogonal. Had we defined a slightly different transform F~ where factors 1 and N1 in the forward and inverse transform are both replaced by 1/N then we would have that F~1=F~, which is to say that F~ is an orthogonal matrix. Check that this is indeed the case. 4. Show that k=0N1X[k]2=Nn=0N1x[n]2. This fact, that the 2-norm can be computed in either the spatial domain or the frequency domain, is called Parseval's theorem