I recently started reading Complex Digital Circuits by Jean-Pierre Deschamps and ran into a mathematical curiosity that has stalled me on making progress.
For context, the author is describing the underlying theory needed to design a circuit that can compute the base-2 logarithm of a real number with an accuracy of $p$ fractional bits.
The input, $x$, is an n-bit normalized fractional number:
$x = 1.x_{-1}x_{-2}...x_{-n}$
As such, $x$ will be bounded within the interval $[1,2)$.
With $x$ properly bounded, the base-2 logarithm can then be computed:
$y = log_2(x)$
As $x$ is restricted to the interval $[1,2)$, $y$, with an accuracy of $p$ fractional bits, will then be restricted to the interval $[0,1)$. In other words:
$y = 0.y_{-1}y_{-2}...y_{-p}$
Solving the base-2 logarithm equation for $x$:
$y = log_2(x)$
$x = 2^{y}$
$x = 2^{0.y_{-1}y_{-2}...y_{-p}...}$
However, the author next squares both sides of the above equation:
$x^2 = (2^{0.y_{-1}y_{-2}...y_{-p}...})^2$
And then states the following:
$x^2 = 2^{y_{-1}y_{-2}...y_{-p}...}$
I don't understand how he arrived at this conclusion and why the decimal point goes away. Also, I don't understand why the ellipsis, ..., appears after the base-2 logarithm equation is solved for $x$. Could someone please explain this to me?
For a bit more context, after this step, the author reasons the following:
- if $x^2 \geq 2: y_{-1} = 1, x' = \frac{x^2}{2} = 2^{0.y_{-2}...y_{-p}...}$
- if $x^2 < 2: y_{-1} = 0, x' = x^2 = 2^{0.y_{-2}...y_{-p}...}$
Unless there is a typo in the text, it should say $$x^2 = 2^{y_{-1}\color{red}{.}y_{-2}...y_{-p}...}$$ This is because $$(2^{0.y_{-1}y_{-2}...y_{-p}...})^2= 2^{2\cdot0.y_{-1}y_{-2}...y_{-p}...}= 2^{y_{-1}.y_{-2}...y_{-p}...}\tag{1}$$
The first equal sign in $(1)$ is just $(a^b)^2=a^{2b}$ and the second because we're using binary. As for the ellipsis, he jsut means that we can't solve for the logarithm exactly. It will have infinitely many bits, in general.