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Fourier analysis of periodic signals

Suppose $X[n]$ is a complex-valued signal that repeats every $N$ samples. (We are continuing to use complex-valued signals rather than real-valued ones to simplify the mathematics.) Because of the period $N$, the values of $X[n]$ for $n=0,\ldots,N-1$ determine $X[n]$ for all integer values of $n$.

Suppose further that $X[n]$ can be written as a sum of complex sinusoids of frequency $0$, $2\pi/N$, $4\pi/N$, $\ldots$, $2(N-1)\pi/N$. These are the partials, starting with the zeroth, for a signal of period $N$. We stop after the $N$th partial because the next one would have frequency $2\pi $, equivalent to frequency $0$, which is already on the list.

Given the values of $X$, we wish to find the complex amplitudes of the partials. Suppose we want the $k$th partial, where $0 \leq k < N$. The frequency of this partial is $2\pi k / N$. We can find its complex amplitude by modulating $X$ downward $2\pi k / N$ radians per sample in frequency, so that the $k$th partial is modulated to frequency zero. Then we pass the signal through a low-pass filter with such a low cutoff frequency that nothing but the zero-frequency partial remains. We can do this in effect by averaging over a huge number of samples; but since the signal repeats every $N$ samples, this huge average is the same as the average of the first $N$ samples. In short, to measure a sinusoidal component of a periodic signal, modulate it down to DC and then average over one period.

Let $\omega=2\pi/N$ be the fundamental frequency for the period $N$, and let $U$ be the unit-magnitude complex number with argument $\omega $:

\begin{displaymath}
U = \cos(\omega) + i \sin(\omega)
\end{displaymath}

The $k$th partial of the signal $X[n]$ is of the form:

\begin{displaymath}
{P_k}[n] = {A_k}{{\left [ {U^k} \right ]} ^ {n}}
\end{displaymath}

where ${A_K}$ is the complex amplitude of the partial, and the frequency of the partial is:

\begin{displaymath}
\angle({U^k}) = k \angle(U) = k\omega
\end{displaymath}

We're assuming for the moment that the signal $X[n]$ can actually be written as a sum of the $n$ partials, or in other words:

\begin{displaymath}
X[n] =
{A_0}{{\left [ {U^0} \right ]} ^ {n}}
+ {A_1}{{\l...
...n}}
+ \cdots
+ {A_{N-1}}{{\left [ {U^{N-1}} \right ]} ^ {n}}
\end{displaymath}

By the heterodyne-filtering argument above, we expect to be able to measure each $A_k$ by multiplying by the sinusoid of frequency $-k\omega$ and averaging over a period:

\begin{displaymath}
{A_k} = {1\over N} \left (
{{\left [ {U^{-k}} \right ]} ^ ...
...dots +
{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
\right )
\end{displaymath}

This is such a useful formula that it gets its own notation. The Fourier transform of a signal $X[n]$, over $N$ samples, is defined as:

\begin{displaymath}
{\cal FT}\left \{ X[n] \right \} (k) =
{V ^ {0}} X[0] +
{V ^ {1}} X[1] +
\cdots +
{V ^ {N-1}} X[N-1]
\end{displaymath}

where $V = {U^{-k}}$. The Fourier transform is a function of the variable $k$, equal to $N$ times the amplitude of the input's $k$th partial. So far $k$ has taken integer values but the formula makes sense for any value of $k$ if we define $V$ more generally as:

\begin{displaymath}
V = \cos(-k\omega) + i\sin(-k\omega)
\end{displaymath}

where, as before, $\omega=2\pi/N$ is the (angular) fundamental frequency associated with the period $N$.



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Next: Periodicity of the Fourier Up: Fourier analysis and resynthesis Previous: Fourier analysis and resynthesis   Contents   Index
Miller Puckette 2006-09-24