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# AR# 21924

## Description

Why can I not meet the data sheet timing numbers for Virtex-4 implementations of the xFFT?

## Solution

It is important to remember that the FFT will scale the results.

Typically by 1/N (N = number of points).

In the xFFT, the scaling is set by the user.

The following is a supplement for the Scaling section of the datasheet, called Finite Word Length Considerations.

2. Next you will need to determine the number of butterfly stages.

This is based on the number of points in the FFT.

- radix-4 - # of stages is log-base4(N) + 1 radix-2 stage at the output, for point sizes not divisible by 4 (N/4 != 0)

For example 512/4 != 0, 4 radix-4 stages and 1 radix-2 stage.

- radix-2 = # of stages is log-base2(N)

3. After you have the number of stages, you can calculate the scaling factor.

S = 2^((sum from i=0 to i=logN-1)*bi)

The notation might be confusing, but it is just a sum of the number of stages using the scaling value for each stage to get the overall scaling value S.

(Note: The equations are from the Scaling section of the datasheet, called Finite Word Length Considerations.)

This process states that we are going to do a binary shift by a certain number of bits in order to account for the scaling.

Since a shift by x bits equals a multiplication of 2^x, the equations are simply defining the number of bits that we shift for any given scaling schedule.

Here is an example of this process for a 256 pt FFT:

1. We choose a radix-4 implementation.

2. log-base4(256) = 4 stages

e.g. log(256) / log(4) = 4

3. If each stage is scaled by 2 , then we could calculate S by:

S = 2^(2 + 2 + 2 + 2) = 256.

Therefore the FFT output would be scaled by 1/256.

For more information, Please see the reference material mentioned at the end of the xFFT datasheet under the References section.
AR# 21924
Date 03/06/2015
Status Active
Type General Article
Devices
• FPGA Device Families
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