Test Your EQ (Engineering Quotient)

EQ #31

a—The conventional way to calculate the magnitude (length) of a vector is to take the square root of the sum of the squares of its components. On small processors, this can be somewhat difficult (especially the square root operation), and various approximations are used instead.

One approximation that works surprisingly well for 2-D vectors and complex numbers is to take the absolute values of the two components, compare them, then add 1/3 of the smaller to the larger.

What is the maximum error using this method?

b—But doesn’t a true hybrid use transformers, or at least some tricky transformer simulation with op amps to ensure the transmitted signal does not appear on the receive port?

a—If we restrict the discussion to unit vectors at various angles A, the x component is cos(A) and the y component is sin(A), and the correct magnitude is 1.

Furthermore, let’s concentrate on angles between 0 and 45° — then we know that both cos(A) and sin(A) are positive and that cos(A) > sin(A). (The absolute value and compare operations provide the symmetry that covers the rest of the unit circle.) The approximation then gives the result

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Magnitude = cos(A) + sin(A)/3

Graphing this shows that this is most negative (0.943) at 45° and most positive (1.054) at approximately 18.4° (the actual angle is given by atan(1/3) —can you show why?). The peak error is therefore –5.7%, +5.4%.

Yes. One more multiplication operation gives a result that has less than 4% error:

Magnitude = 0.960433 × max(|x|, |y|) + 0.397826 × min(|x|, |y|) This function is most negative at 0° and 45°, and most positive at 22.5°. The error is ± 3.96%. This form is well-suited to DSPs that have multiply-accumulate units. The two constants can be expressed as 62943/65536 and 26072/65536, respectively.


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