AMAT 415 — Week 4 problems

P15) Finite, sampled signals x and their discrete Fourier transforms \hat{x} both live in \mathbf{C}^n — a vector space. Show that the DFT respects basic vector space operations — addition and scalar multiplication.  More precisely, show that (a) if x and y are signals in \mathbf{C}^n and z=x+y, then \hat{z}=\hat{x}+\hat{y}, and (b) if x \in\mathbf{C}^n, a\in\mathbf{C} is a scalar, and y=ax, then \hat{y}=a\hat{x}.

P16) Let x=(-1,2,5) and y=(2,3,5,7).  Calculate \hat{x} and \hat{y}.

P17) Let \omega= e^{2\pi i/N} and let A be the N\times N matrix whose (i,j)-th entry is \omega^{-(i-1)(j-1)}, i,j=1,\ldots,N. Show that (a) A is a symmetric matrix; (b) A^{-1}=N^{-1}\bar{A}, where \bar{A} is the matrix obtained by conjugating each entry of A; (c) if we view x\in \mathbf{C}^N as a column vector, then \hat{x}=Ax.

P18) Write down the matrix A of the previous problem in the case N=4. (Each entry should be expressed in standard for x+iy.) Use it to compute \hat{y} and \hat{z}, where y=(2,3,5,7) and z=(2,4,6,8).

P19) Make some stem plots of the modulus of \hat{x}, where x=(1,\ldots,1,0,\ldots,0) where the string of ones has length m and the string of zeroes has length N-m. (We computed \hat{x} in class, but you should check it with MATLAB.)

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