# Number of subdivisions in a Riemann integral (DFT)

Tags:
1. Feb 4, 2017

### TheSodesa

1. The problem statement, all variables and given/known data
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*}

2. Relevant equations
The assumed values of the unknown function are the sequence

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

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:

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

3. 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

In $4b$, however, I ran into a bit of a problem. I don't know what $N$ is supposed to be.

$n$ 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.

Last edited: Feb 4, 2017
2. Feb 4, 2017

### TheSodesa

It just occurred to me, that maybe I should be using the periodicity and symmetricity of the DFT, to find out the values of $G_n$ in 4b. Any comment on this?