# 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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