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I understand that when we compute a DFT, there are $N$ data points and the indices $n$ and $m$ go from zero to $N-1$ where $y_m$ are the data points in the time domain, and $Y_n$ are the amplitudes in the frequency domain.

My first question is more straightforward. Why do the indices count from zero rather than from one? If this is true, it seems to me that $Y_0$ is the amplitude of a zero Hz signal, which I cannot make sense of.

My second question probably reflects an even greater lack of understanding, but: If we are able to obtain the amplitudes for $N-1$ frequency components, and the Nyquist Frequency is half the sampling rate, do we not in practice take $0\le m\le N/2$ for $N$ even, and $0\le m\le N\pm 1$ for $N$ odd? Furthermore is this the reason that $Y_{N/2-n}=\overline{Y_{N/2+n}}$?

Also if it looks like I am missing any key concepts please let me know.

Thanks

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The key thing you are missing is the following: Conceptually the domain of DFT is not the set of integers $0$, $1$, $\ldots$, $N-1$ but the set ${\mathbb Z}/(N{\mathbb Z})$ of equivalence classes mod $N$. Geometrically this set can be viewed as the set of $N$th roots of $1\in{\mathbb C}$, so there is in fact a "circular symmetry". –  Christian Blatter Mar 25 '11 at 13:07
    
How can I learn more about equivalence classes? That's a bit beyond the math I've learned so far. The expression ${\mathbb Z}/(N{\mathbb Z})$ looks impressive but I'm not sure what it means. –  qweilun Mar 30 '11 at 22:36
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1 Answer

up vote 1 down vote accepted

To your first question: Yes, $Y_0$ is the amplitude of the component with zero frequency, which is a constant: $Y_0 \mathrm{e}^{2\pi\mathrm{i}m0/N}=Y_0 \mathrm{e}^0=Y_0$. You can see that you need a component like this because all the non-constant components have an average of $0$, so $Y_0$ is the only component that can give the sum a non-zero average, and in fact it simply measures the average of your signal.

To your second question: I'm not sure I understand it correctly. First, we don't obtain amplitudes for $N-1$ frequency components, but for $N$ frequency components -- this might be related to your first question. Second, why do you write $N/2$ for $N$ even but $N\pm1$ for $N$ odd? Did you mean $(N\pm1)/2$? The reason that $Y_{N/2-n}=\overline{Y_{N/2+n}}$ is that $\mathrm{e}^{2\pi\mathrm{i}m(N/2-n)/N}=\mathrm{e}^{2\pi\mathrm{i}m(N/2-n)/N}\mathrm{e}^{-2\pi\mathrm{i}m}=\mathrm{e}^{2\pi\mathrm{i}m(-N/2-n)/N}=\overline{\mathrm{e}^{2\pi\mathrm{i}m(N/2+n)/N}}$.

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oh, thanks. I understand about $Y_0$ being the average now. As for my second question, I think I mixed up the indices' evenness and oddness. Thanks for the tips. –  qweilun Mar 25 '11 at 16:22
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