Center for New Music and Audio Technologies

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Proof

The sequence of directions $ d$ is completely defined in Equation 1.12

$\displaystyle z_{i+1}=z_i-d_i\cdot\arctan\left(2^{-i}\right)$

and Equation 1.14:

\begin{displaymath}
d_{i}=
\begin{cases}
-1, &\text{if $z_{i} < 0$}\\
+1, &\text{if $z_{i} \geq 0$}
\end{cases} \text{.}
\end{displaymath}

$\displaystyle \left(x_n, y_n\right)=\left(K\left(x_0\cdot P(d)+y_0\cdot Q(d)\right), K\left(y_0\cdot P(d)-x_0\cdot Q(d)\right)\right)$ (1.17)

be the general term of the iterative system determined by Equation 1.10 and 1.11, where $ P(d)$ and $ Q(d)$ are polynomials in the variables d.
Setting the initial vector $ V_0$ to $ \left(x_0,0\right)$ we get:
$\displaystyle \left(x_n, y_n\right)$ $\displaystyle =$ $\displaystyle \left(K \cdot x_0\cdot P(d), -K\cdot x_0\cdot Q(d)\right)$ (1.18)
  $\displaystyle =$ $\displaystyle K x_0 \left(P(d), -Q(d)\right)$ (1.19)

Now define a new initial Vector $ V_0'=\left(\lambda
x_0,0\right)$:
$\displaystyle \left(x_n', y_n'\right)$ $\displaystyle =$ $\displaystyle K \lambda x_0 \left(P(d), -Q(d)\right)$ (1.20)
  $\displaystyle =$ $\displaystyle \lambda (x_n, y_n)$ (1.21)

From Equation 1.21 it follows that the gain inherent in the CORDIC algorithm is a constant for a constant number of iterations and independent of the initial values if coordinate $ y_0 =
0$. Consequently, multiplication can be performed by initializing the rotation with the control parameter amplitude. This eliminates the need for an additional multiplier and reduces the design complexity enormously.


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Next: Implementation Up: Amplitude Previous: Amplitude
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Norbert Lindlbauer
2000-01-19