Number of subdivisions in a Riemann integral (DFT)

In summary, the conversation discusses calculating the discrete Fourier transform (DFT) of a given sequence and drawing the DFT on a complex plane. The DFT is calculated using a specific formula and the result is used to draw the DFT on the complex plane. In the second question, there is confusion about the value of N, which is used in the formula for calculating the DFT. The speaker suggests using the periodicity and symmetricity of the DFT to determine the values of G_n in the given sequence.
  • #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
H4_4Acompass.png

H4_4Aplot.png
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:
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  • #2
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?
 

FAQ: Number of subdivisions in a Riemann integral (DFT)

1. What is a Riemann integral in the context of a DFT?

A Riemann integral is a mathematical concept used in the Discrete Fourier Transform (DFT) to approximate the area under a curve. It is defined as the limit of the sum of rectangular areas under the curve, as the width of the rectangles approaches zero.

2. How do you calculate the number of subdivisions in a Riemann integral?

The number of subdivisions in a Riemann integral is determined by the chosen sampling rate and the desired frequency resolution. It is typically calculated by dividing the sampling rate by the frequency resolution and rounding up to the nearest integer.

3. How does the number of subdivisions affect the accuracy of the Riemann integral?

The number of subdivisions directly affects the accuracy of the Riemann integral. A higher number of subdivisions results in a more accurate approximation of the area under the curve. However, a very large number of subdivisions can also increase the computational complexity of the DFT.

4. Can the number of subdivisions be adjusted during a DFT?

Yes, the number of subdivisions can be adjusted during a DFT. This can be done by changing the sampling rate or the frequency resolution. However, it is important to note that changing the number of subdivisions during a DFT may also affect the accuracy of the results.

5. Is there an optimal number of subdivisions to use in a Riemann integral?

There is no one optimal number of subdivisions that applies to all cases. The ideal number of subdivisions will depend on the specific application and the desired level of accuracy. It may require some experimentation to find the optimal number for a particular use case.

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