# Reducing weighted linear threshold gate to unweighted one

Reading "On the power of threshold circuits with small weights" by Siu and Bruck I have faced several problems understanding how unweighted linear threshold element can be built efficiently from the weighted one.

Let $$X=(x_1\ldots x_n), x_i\in \{-1, 1\}$$. We define linear weighted threshold element as $$f(X)=sgn(\sum_{i=1}^n w_i x_i + w_0)=sgn(F(X)), w_i\in Z$$. It has no restrictions on $$|w_i|$$, and the question is whether $$F(X)$$ can be represented as linear combination where $$|w_i|$$ is bounded by polynomial of $$n$$.

Authors say:

First observe that by considering the binary representation of the weights $$w_i$$, we can introduce more variables and assign some constant values to renamed variables in such way that any linear threshold function can be assumed to be of the following generic form:

$$f(X)=sgn(F(X))$$, where $$F(X)=\sum_{i=1}^{nlogn} 2^i(x_{1_i}+\ldots +x_{n_i})$$

• I don't get how binary representation of the weights relates to the $$logn$$ which is defined by $$X$$ only.
• Why summation is made for $$i=1\ldots nlogn$$?
• It's unclear to me what $$x_{k_i}$$ stands for.