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#### AWR Version 13

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### Design Notes

__Fixed-Point Magnitude Calculation__

This example demonstrates a magnitude estimator whose hardware complexity is much simpler and faster than that of the full-precision exact calculation using a square-root operation. Magnitude estimators are used in many fixed-point ASIC designs for received signal strength indicators (RSSI) and demodulation algorithms.

The magnitude of a complex number, I + jQ, is defined as sqrt(I^2 + Q^2) and is approximated as:

Mag ~= A * max(|I|,|Q|) + B * min(|I|,|Q|)

where A and B are two constants whose values can be set such that either the RMS error, peak error and/or implementation complexity is reduced within acceptable levels. Some of the most common values for A and B are given below, together with the associated errors:

A B Average Error RMS Error A B Average Error RMS Error Peak Error

(linear) (dB) (dB)

1.0 0 0.5 5 -0.086775 086775 -20.7 7 -18.6

1.0 0 0.25 25 0.006456 006456 -27.6 6 -18.7

1.0 0 0.4 4 -0.049482 049482 -25.1 1 -22.3

0.94754 94754 0.39248 39248 0.000547 000547 -32.6 6 -25.6 (min RMS error)

0.96043 96043 0.39782 39782 -0.013049 013049 -31.4 4 -28.1 (min peak error)

Note that the first two choices are very simple to implement in hardware (using shifts and adds), while the others are more complex.

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3. The fixed-point approximation uses fixed-point arithmetic for the algorithm above. In this case, bit and decimal widths should be selected such that loss of information due to the real to fixed-point conversion is within acceptable limits. For example, the bit and decimal widths for the SCALE parameter in the SCALE_FP block should be set such that the selected values for A and B are represented by these widths.

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