DSP butterfly Unit

Hello friends. You must be aware of the applications of Verilog coding & its significance in different domain. From the last few months i was working on one such use of Verilog in DSP.  Signals & systems   is my all time favorite subject & its the one with lot of applications & used in many products like mobiles, tv, image processing etc. The list is long though  :) We will talk about one such DSP module today " The FFT Butterfly unit " . Before you read this post i suggest you to go through the FFT algorithm (DIT/DIF) so that it will be easy for you to understand the code.  Fast Fourier transform is used to convert a signal from time domain to frequency & this is needed so that you can view the frequency components present in a signals . If you know the frequency components present in a signals you can play with the signals :) Lets say, u want to design a low pass filter and want to decide on the cut off frequency of the filter . If you have the frequency domain details for a signals u can clearly identify the frequency components which u want to retain & the ones which u want to take out. Well let us not discuss this topic in detail here , but if you are still interested than you can refer to my short notes on FFT here:
 Fourier analysis

The figure below shows the FFT implementation using radix 2 algorithm.

this is a 8 point FFT implementation using the butterfly unit, The butterfly unit is the heart of FFT algorithm . From the figure u can see that if we are done with the butterfly unit we are 70% done with the FFT coding.
Okie now lets start the coding for butterfly unit . The back box model of the butterfly will have 2 complex inputs & 2 complex outputs

The second input gets multiplied with the twiddle factor i.e (wr + j wi ) first. So before you start coding you must have the code written for complex number multiplication. Once you are done with the multiplication the next step is to do addition . You can do this by simple CLA (carry look ahead adders) . Output Z1 is obtained by adding the result of multiplication i.e(b1+jb2) * (Wr+jWi) to the first number i.e a1+ja2. And output Z2 is obtained by subtracting the multiplied product with the first number . To put it in mathematical form

Z1r+jZ1i= (b1+jb1)*(Wr+jWi) + (a1+ja2)

Z2r+jZ2i= (b1+jb1)*(Wr+jWi) - (a1+ja2)

If the result is analysed proerly u will discover that we need a complex multiplier ( which has 4 normal multipliers ) & 4 CLA units . To put the process in orderly manner we have to proceed in steps like this

1> read 2 complex numbers & the twiddle factor 

2>multiply 2nd number with the twiddle factor

3> add the product to 1st number to get the first o/p

4>Subtract the product to 1st number to get the first o/p

We are done. Here is the code for the Butterfly unit & the results of simulation & the RTL schematic

CODE:

RTL SCHEMATIC FOR THE CODE:

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Black Box model:

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