DSP Trick: Magnitude Estimator

To compute the approximate absolute magnitude of a vector given the real and imaginary parts:

use the alpha max plus beta min algorithm. The approximation is expressed as:

For closest approximation, the use

and

giving a maximum error of 3.96%.


To improve accuracy at the expense of a compare operation, if Min ≤ Max/4, we use the coefficients, α = 1 and β = 0;
otherwise, if Min > Max/4, we use α = 7/8 and β = 1/2.

Division by powers of 2 can be easily done in hardware. Error comparisons for various min/max values:

		Largest error (%)	Mean error (%)
1/1	1/2	11.80	8.68
1/1	1/4	11.61	0.65
1/1	3/8	6.80	4.01
7/8	7/16	12.5	4.91
15/16	15/32	6.25	1.88
α0	β0	3.96	1.30

Sources:

• http://www.dspguru.com/book/export/html/62
• http://en.wikipedia.org/wiki/Alpha_max_plus_beta_min_algorithm
• http://www.eetimes.com/design/embedded/4007218/Digital-Signal-Processing-Tricks--High-speed-vector-magnitude-approximation/
• Mark Allie and Richard Lyons, "A Root of Less Evil," IEEE Signal Processing Magazine, March 2005, pp. 93-96.