Digital Fourier Transform - create frequency axis

  1. 1. The problem statement, all variables and given/known data
    Using Matlab, I need to record vowels (a, e, o, i, etc) of my voice and apply a digital fourier transform (DFT) to the recorded sound signal.

    The resulting signal should be a periodic train of 'spikes', with modulated amplitude. The modulation of the amplitude is what forms the sound; different vowels produce different amplitude modulations. The highest point of a peak in the amplitude modulation is called a formant.

    I need to produce several of these DFT signals, and plot their powerspectrum ([itex]P = F F^* = |F|^2[/itex], where F is the (complex) signal) against the frequency.

    To do this, I record a vowel using a samplefrequency [itex]f_s[/itex] of 11025 Hz. I record a total number of 2^14 = 16384 samples, which results in about 1.5 seconds of recording time.

    Then, I calculate the DFT using the built-in "fft" Matlab function (which uses the equations below), and calculate it's powerspectrum.

    I can then plot the powerspectrum against the number of samples (from n = 0 To 16384), but what I really need is a plot of the powerspectrum against the frequency.

    Now, I am unsure how to convert from the number of samples to the frequency, so this is where I need your help.

    2. Relevant equations
    Matlab uses the following DFT:
    [tex]X(k) = \displaystyle \sum_{n=1}^N x(n) \exp \left( \frac{-2\pi j (n-1)(k-1)}{N}\right)[/tex]
    [tex]x(n) = \frac{1}{N} \sum_{k=1}^N X(k) \exp \left( \frac{2 \pi j (n-1)(k-1)}{N}\right)[/tex]

    3. The attempt at a solution
    Once I have the powerspectrum of a record signal, I need to plot it against the frequency [itex]f[/itex] (not angular frequency [itex]\omega[/itex]).

    My reasoning was that the last sample (n = 2^14 = 16384) should correspond to the sample frequency [itex]f_s = 11025[/itex]. Therefore, I created the axis (let's call it an array [itex]x[/itex]) by first creating an array of integers 1 to 2^14. I then divide each element by 2^14, and multiply each element by the sample frequency 11025 Hz:
    Code (Text):
    n = 2^14;
    Fs = 11025;

    x = 1:n;
    x = x ./ n;
    x = x .* Fs;
    This results in a frequency axis ranging from 0 Hz to 11025 Hz.

    Furthermore, because the powerspectrum appears to be symmetric about the center of the graph (n = 2^14 / 2 = 2^13, or at frequency 11025/2, assuming I am correct), I can discard half of the graph, so I only use the first half:
    Code (Text):
    x = x(1:n/2);

    I asked my teacher, and he told me it was OK, but to be honest I don't really trust him. My class mate (using the same technique) has already had his report graded, and he got a 4 out of 10, mainly because his frequency axis was apparently wrong.

    So this means the way I 'define' my frequency axis is wrong?

    How should I do it instead?

    I was maybe thinking about the Nyquist theorem; should the last sample correspond to only half the sample frequency perhaps?

    Thanks for any insights!

    Why is my Latex not working? Even simple things such as y = x^2 don't work:
    [tex]y = x^2[/tex]

  2. jcsd
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