1. ECE 734 Project Implementation of Multiple Constant Multiplication Algorithms for FIR Filters Hamid Shojaei

2. Motivation
Digital filters are widely used in digital communications and audio/video processing. They play a crucial role in digital communication chips such as Ethernet transceivers, cable modems, DSL modems, satellite receivers, mobile phones, etc.
In particular, finite impulse response (FIR) filters are used for their ease of implementation and stability.

3. Motivation An FIR filter implements a convolution in the time-domain
Critical path of N-tap filter:
N-1 adds + 1 multiply
Arithmetic complexity of N-tap filter modeled as:
N multiplications+ N-1 adds
Multipliers are expensive both in area and delay
Full flexibility of general purpose multipliers are not required
Implement multipliers by add/sub-shift operation
Minimize adders  multiple constant multiplication

4. Multiple Constant Multiplication(MCM) Algorithms
Graph based algorithms
BH algorithm
BHM algorithm
RAG-n algorithm
Common Subexpression Elimination (CSE) algorithms
Hybrid Algorithms

5. Graph based algorithms
Bull Horrocks (BH) Algorithm 9
17x[n] h1 17 1 2 4 8
1, 9, 16, 26 and 51 w1 w0 w2 w3 w4 9 h1 17 26 51 1 2 4 8 43
23 operations  8 operations w1
Using shift w0 w2 w3 w4 9 17 26 h1 51 1 3 43
23 operations  5 operations
Figures adapted from Horrocks et. al 1995

6. MCM Algorithms BHM Algorithm
In BH, partial sums are generated with values only up to the coefficient. (7 = rather than 7 = 1+2+4)
The n-dimensional reduced adder graph (RAG-n) algorithm
Similar to dynamic programming
CSE
Matrix splitting

7. Initial Results
# of operations
# of coefficients

8. Initial Results
# of operations
# of coefficients

9. Future Works Applying algorithms on some real FIR filters
A framework for generating coefficients and applying MCM algorithms
Based on the filter we can decide which algorithm or combination of them can applied.