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# AR# 4427

## Description

Keywords: parallel, distributed, arithmetic, filter, output width, full precision, pda

The way the PDA FIR Filter module calculates full precision output width may not be what the user expects.

## Solution

Maximum output width for the PDA FIR filter is the number of bits required to rerpresent the largest magnitude result that may appear at the filter output.

The process used to determine the filter's output port bit-widthis as follows:

Largest Coefficient Magnitude (given the coefficient bit-width provided by the user) =

2 ^ (Coefficient Bit Width - 1)

Largest Sample Magnitude (given the data bit-width provided by the user) =

2 ^ (Data Bit Width - 1) - 1 for signed data,
2 ^ (Data Bit Width) -1 for unsigned data

Largest Filter Output Magnitude (given the number of taps provided by the user) =

(Largest Sample Magnitude) x (Largest Coefficient Magnitude) x (Number Of Taps)

Number Of Bits Required At Output Port =

Log 2 ( Largest Filter Output Magnitude) + 2, rounded down to the nearest whole number

Given that not all coefficients will be as large as the Largest Coefficient Value calculated above, the filter's RSLT output port may, at times, be wider than necessary. Given a specific set of coefficients, the ideal output port bit-width may be found manually by substituting the sum of the magnitudes of the coefficients for the product of the Largest Coefficient Magnitude and the Number Of Taps.

That is,

Largest Sample Magnitude (Given the data bit-width provided by the user) =

2 ^ (Data Bit Width - 1) - 1 for signed data,
2 ^ (Data Bit Width) - 1 for unsigned data

Largest Filter Output Magnitude (Given the coefficients provided by the user) =

(Largest Sample Magnitude) x ( |C 0 | + |C 1 | + |C 2 | + ... + |C n | )

Number Of Bits Required At Output Port =

Log 2 ( Largest Filter Output Magnitude) + 2 , rounded-down to the previous whole number

The difference between the conservative bit-width calculated by the Core Generator and the ideal bit-width calculated manually represents the number of unnecessary MSBs that are included in the filter?s output port. These bits may safely be dropped from the filter?s output, and a small amount of logic trimming may occur as a result.

As an example, take a symmetric 6-tap filter, signed input data, and coefficients 1,-3,12,12,-3 and 1. The center coefficients require 5 bits to express, but the other coefficients require only 2 and 4 bits. Assuming 6-bit sample data, the Core Generator conservatively calculates that this filter requires a 12-bit RSLT port. Given the specific coefficients, however, only 11 bits are actually necessary. As a result, for this particular application, the MSB of the filter?s output may be discarded.

NORMALIZING COEFFICIENTS

Some users may normalize their coefficients to reduce the amount of logic required to implement their filter. Doing this guarantees that the filter will never overflow. In
this case, if full precision in the output is maintained, this means that the overflow bits from each summation will all end up as sign extension bits.

Users who normalize their coefficients may assume that the PDA FIR module does an analysis and recognizes that the overflow bits are unnecessary, but in actuality, the module does NOT do such an analysis.

If the output grows by 3 bits for a given PDA FIR filter whose coefficients have been normalized, this means that the 3 MSB's of their result will always be sign extension bits. If the result is positive, this means that the 3 MSB's of the result will be 0's; if the result is negative, all 3 MSB's will be 1's.

Now if the user selects an output width "W" which is LESS than the calculated full precision output width for the filter, only the "W" MSB's will be selected, and these will always include the sign bits.
AR# 4427
Date 04/01/2009
Status Archive
Type General Article
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