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Sawtooth wave

First we apply this to the sawtooth wave $s[n]$. For $0 \le n < N$ we have:

\begin{displaymath}
s[n] - s[n-1] = -{1 \over N} +
\left \{
\begin{array}{ll}
{1} & {n = 0} \\
0 & \mbox{otherwise}
\end{array} \right .
\end{displaymath}

Ignoring the DC offset of $-{1 \over N}$, this gives an impulse, zero everywhere except one sample per cycle. The summation in the Fourier transform only has one term, and we get:

\begin{displaymath}
{\cal FT}\{ s[n] - s[n-1] \} (k) = 1 , \; k \neq 0, \; -N < k < N\
\end{displaymath}

We then apply the difference formula backward to get:

\begin{displaymath}
{\cal FT}\{ s[n] \} (k) \approx {1 \over {i \omega k}}
= {{-iN} \over {2 \pi k}}
\end{displaymath}

valid for integer values of $k$ with $k \neq 0$ and $k << N$. (To get the second form of the expression we plugged in $\omega=2\pi/N$ and $1/i = -i$.)

This analysis doesn't give us the DC component ${\cal FT}\{ s[n] \}(0)$, because we would have had to divide by $k=0$. Instead, we can evaluate the DC term directly as the sum of all the points of the waveform: it's approximately zero by symmetry.

To get a Fourier series in terms of familiar real-valued sine and cosine functions, we combine corresponding terms for negative and positive values of $k$. The first harmonic ($k = \pm 1$) is:

\begin{displaymath}
{1 \over N}\left [
{\cal FT}\{ s[n] \} (1) \cdot {U^n} +
{\cal FT}\{ s[n] \} (-1) \cdot {U^{-n}}
\right ]
\end{displaymath}


\begin{displaymath}
\approx {{-i} \over {2 \pi}} \left [ U^n - U^{-n} \right ]
\end{displaymath}


\begin{displaymath}
= {{\sin ( \omega n)} \over {\pi}}
\end{displaymath}

and similarly for other values of $k$, the $k$th harmonic is

\begin{displaymath}
{{\sin ( 2 \omega n)} \over {k \pi}}
\end{displaymath}

and the entire Fourier series is:

\begin{displaymath}
s[n] \approx {1 \over \pi} \left [
{\sin ( \omega n )}
+ ...
...\over 2}
+ {{\sin ( 3 \omega n)} \over 3}
+ \cdots
\right ]
\end{displaymath}


next up previous contents index
Next: Parabolic wave Up: Fourier series of the Previous: Fourier series of the   Contents   Index
Miller Puckette 2006-03-03