- #1

TheSodesa

- 224

- 7

## Homework Statement

This is a combination of two questions, one being the continuation of the other

3) Calculate the DFT of the sequence of measurements

\begin{equation*}

\{ g \}_{k=0}^{5} = \{ 1,0,4,-1,0,0 \}

\end{equation*}

4a) Draw the DFT calculated in question 3 on the complex plane.

4b) What are the members of the sequence:

\begin{equation*}

G_7, G_{11}, G_{14}, G_{22},G_{-7}, G_{-11}, G_{-14}, G_{-22}

\end{equation*}

## Homework Equations

The assumed values of the unknown function are the sequence

\begin{equation}

g_k = f\left(\frac{kT}{N} \right), k = 0, 1, 2, \ldots, N-1,

\end{equation}

mentioned above, where ##N## is the number of subdivisions in the Riemann integral used to approximate the coefficients ##c_n## of the Fourier series of the unknown function ##f##.

Each member, ##G_n##, of the DFT is calculated as follows:

\begin{equation}

G_n = \sum_{k=0}^{N-1} g_k e^{-jnk\frac{2\pi}{N}}, n = 0, 1, 2, \ldots, N-1

\end{equation}

## The Attempt at a Solution

Questions ##3## and ##4a## were not problematic at all. The DFT was

\begin{align*}

G_0 = \sum_{k=0}^{5} g_k e^{0} &= 1 + 0 + 4 - 1 + 0 + 0 = 4,\\

%

G_1 = \sum_{k=0}^{5} g_k e^{-jk\frac{2\pi}{6}} &= 1 e^{0} + 0 + 4 e^{-j2\frac{2\pi}{6}} - 1 e^{-j3\frac{2\pi}{6}}\\

&= 1 + 4 e^{-j\frac{2\pi}{3}} + 1 \\

&= 2 + 4 e^{-j\frac{2\pi}{3}}\\

%

G_2 = \sum_{k=0}^{5} g_k e^{-j2k\frac{2\pi}{6}} &= 1 + 0 + 4 e^{-j2\frac{4\pi}{6}} - 1 e^{-j3\frac{4\pi}{6}}\\

&= 1 + 4 e^{j\frac{2\pi}{3}} - 1\\

&= 4 e^{j\frac{2\pi}{3}}\\

%

G_3 = \sum_{k=0}^{5} g_k e^{-j3k\frac{2\pi}{6}} &= 1 + 0 + 4 e^{-j2\pi} - 1 e^{-j3\pi}\\

&= 1 + 4 + 1 = 6\\

G_4 = \sum_{k=0}^{5} g_k e^{-j4k\frac{2\pi}{6}} &= 1 + 0 + 4 e^{-j2\frac{4\pi}{3}} - 1 e^{-j3\frac{4\pi}{3}}\\

&= 1 + 4 e^{-j\frac{8\pi}{3}} - 1\\

&= 4 e^{-j\frac{2\pi}{3}}\\

G_5 = \sum_{k=0}^{5} g_k e^{-j5k\frac{2\pi}{6}} &= 1 + 0 + 4 e^{-j2\frac{5\pi}{3}} - 1 e^{-j3\frac{5\pi}{3}}\\

&= 1 + 4 e^{-j\frac{10\pi}{3}} - 1 e^{-j5\pi}\\

&= 2 + 4 e^{j\frac{2\pi}{3}}

\end{align*}

Based on these, I drew the following images

##n## seems to go all the way up to ##22## and down to ##-22##, which to my mind seems to imply, that ##N=23## or ##N=46##.

I'm also assuming, that the sequence ##\{ g \}_{k=0}^{5}## doesn't change, since I'm not given any new information regarding this, but I'm not sure of this either.

So, what is ##N##, exactly?

EDIT: Added labels on the points in the pictures.

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