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Number of subdivisions in a Riemann integral (DFT)

  1. Feb 4, 2017 #1
    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
    \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}

    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
    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: Feb 4, 2017
  2. jcsd
  3. Feb 4, 2017 #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?
     
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