"CS" multipliers

Partial product of "CS" operands

Multiplier X and multiplicand A are both in "CS" notation, i.e. with digits values Î {'0', '1', '2'}.
An array of "xCS" cells output a set of bits whose weighted sum is the product A × X in constant time (no propagation). To make sure that the "xCS" cells never overflow, it is necessary that either in A or in X every digit value '2' is preceded by a '0' at its right. Either A or X requires a rewriting. In the applet the double arrow at right shows or hides the wires distributing A and X through the "xCS" cells.

Partial products reduction

The partial product of two "CS" digits is a simple bit, reduced by Wallace trees in the very same way as conventional binary fast multiplication.

"xCS" cell

The "xCS" cell computes the product of two digits "a" and "x" in "CS" notation and gives a bit "i". Its arithmetic equation is "2 × b + 2 × y + i = a × x + z + c". Furthermore the outputs "b" and "y" does not depend on "c" or "z" (no propagation). Actually "c + z" takes only the values '0' or '1' (it is impossible that "c + z" be 2) and "a × x" takes the values 0, 1, 2 or 4. Only the value 1 might cause a propagation but this value is only possible when "a" and "x" are both '1' implying that "c" and "z" are both '0'.

Coding circuit "CS2CS"

The transcoder "CS2CS" rewrites from "CS" to "CS" while making sure that in the output with value '2' is always preceded (at the right) by a '0'.

This permits to generate the partial product of a multiplicand A by a multiplier X both in "CS", with no overflow.

Coding circuit "CS2BC"

The transcoder "CS2BC" passes from "CS" to "BC", that is from the "Carry-Save" code to the "Booth Code" ( symmetric minimally redundant radix-4 code ).

This transcoder replaces the "Booth code" generator whenever the multiplier X is in "CS" instead of the usual conventional binary. The multiplicand A must be in 2's complement.

Integer constant multiplication

Multiplication of a number by a series of constants

The discrete Fourier transform, the discrete cosine transform or inverse, digital filters, and so on all contains the multiplication of a variable X by several constants C1, C2, .. Cn. The factorization of those constants permits a dramatic reduction of the number of additions/subtractions demanded by those multiplications. The following applet computes
Y1 = X × C1, Y2 = X × C2, .. Yn = X × Cn.

Use the arrow at the left to browse the optimization steps.