Digital Design of Signal Processing Systems: A Practical Approach

6

Multiplier-less Multiplication by Constants

6.1 Introduction

In many digital system processing (DSP) and communication algorithms a large proportion of multiplications are by constant numbers. For example, the finite impulse response (FIR) and infinite impulse response (IIR) filters are realized by difference equations with constant coefficients. In image compression, the discrete cosine transform (DCT) and inverse discrete cosine transform (IDCT) are computed using data that is multiplied by cosine values that have been pre-computed and implemented as multiplication by constants. The same is the case for fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) computation. For fully dedicated architecture (FDA), where multiplication by a constant is mapped on a dedicated multiplier, the complexity of a general-purpose multiplier is not required.

The binary representation of a constant clearly shows the non-zero bits that require the generation of respective partial products (PPs) whereas the bits that are zero in the representation can be ignored for the PP generation operation. Representing the constant in canonic sign digit (CSD) form can further reduce the number of partial products as the CSD representation of a number has minimum number of non-zero bits. All the constant multipliers in an algorithm are in double-precision floating-point format. These numbers are first converted to appropriate fixed-point format. In the case of hardware mapping of the algorithm ...