# Bit-reversal equivalence on IFFT (radix-2 Cooley-Tukey)

From everything that I read on the internet, I understood that when performing FFT or IFFT, one should perform bit-reversal to obtain a result in natural order (Assuming input is also in natural order). I also understood that this bit-reversal can be performed before or after the FFT or IFFT processing.

However, when interverting the order of operations, from :

bit_reversal();
IFFT();


to :

IFFT();
bit_reversal();


I don't get the same results. I would expect some difference due to rounding (I am working in fixed-precision here). But I get really completely different results.

Unfortunately, I am not allowed to share the code here, so my question is not about the implementation of the bit-reversal or IFFT, but more about fundamentals that I may have missed : can we operate bit-reversal equivalently before or after the radix-2 IFFT, without changing the radix-2 algorithm ?

• does the same happen on the forward FFT as well, or only on the inverse FFT? Commented Oct 17, 2017 at 9:20
• Hi Fat32, yes, the same happens. Commented Oct 17, 2017 at 10:18
• you need to take a close look at the documentation of both FFT and iFFT function calls. they could both be the same algorithm. and it might have the bit reversing built in. look for terms like "Decimation-in-Time" and "Decimation-in-Frequency". Commented Oct 17, 2017 at 10:57
• Hi Robert and all, Thank you for your answers. I have direct access to the code (but am not allowed to share it for confidientiality issues). We have investigated a bit further and found the probable root cause. I will post an ansser to my own question to share the findings with the community. In short : it was indeed a misunderstanding of the basics on my side. Commented Oct 17, 2017 at 11:26
• Fabien, now you have written your own answer and indeed I made a test with my own FFT code, so it made clear and I remembered that changing the place of bit-reveresal stage has an effect on the result. So we can say that the re-ordering algorithm depends on the place where it would take place. Which is also evindent from the butterfly nets you have provided. Commented Oct 17, 2017 at 12:50

To answer my own question : no you can't use the exact same algorithm.

In the case here of a decimation-in-time, radix-2 Cooley Tuckey, in-place FFT/IFFT algorithm, we have the choice between two implementations which are exactly equivalent :

The above uses bit-reversed inputs and produce natural order results.

The above uses natural order input and produces bit-reversed outputs.

You can see that the operations are the same : for each butterfly operators, same input data, same twiddle factors. It's just the address of that data (here, the number of lines from the top) that changes.

The same applies for IFFT. When applying the bit-reversal after the IFFT, I would have needed to change the IFFT algo so that it changes the addressing scheme to match the second algo.

To go a bit further on the topic : "ALGEBRAIC FORMULATION OF THE FAST FOURIER TRANSFORM" from Tran-Thong. (http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6323745)