Shift‐Add Circuits for Constant Multiplications

Using Transposition to Efficiently Solve Constant Matrix-Vector Multiplication and Sum of Product Problems

“…The second, denoted SES2, gives priority to minimum adder depth in the selection, therefore guaranteeing a minimum adder depth for the complete solution, while still sharing sub-expressions. For an explanation of subexpression sharing algorithms we refer the reader to [2].…”

“…Hence, we only provide limited results based on random coefficients here and instead provide some synthesis results later in this section and two real-world examples in the next section. Figure 11a shows the average required adder cost for solving constant 2 × N matrices C that N ∈ [2,8] with random 12-bit coefficients. As expected, the proposed algorithm provides the lowest adder cost as it uses an adder graph algorithm which solve the problem without depending on the number representation.…”

“…In many applications, primarily within the field of digital signal processing (DSP), computations are performed with constant coefficient multiplication. Hence, one may replace the general multiplier with a network of adders, subtracters, and shifts [1,2]. The computations may have a different number of inputs and outputs and allow to reduce the number of adders and subtracters by utilizing common partial results.…”

“…The CMM problem is defined as finding a solution using adders, subtracters, and shifts that realizes the computation using a few adders and subtracters as possible [1][2][3][4][5][6][7]. As adders and subtracters have about the same complexity, we will from here refer to both as adders, and the number of adders as the adder cost.…”

“…a) Average required adder cost, (b) Average required adder depth, of solutions provided with the proposed algorithm, SES and SES2 to solve 2 × N matrices C that N ∈[2,8].…”