A 64 Point Fourier Transform Chip

Fourth generation wireless and mobile system are currently the focus of research and development. Broadband wireless system based on orthogonal frequency division multiplexing will allow packet based high data rate communication suitable for video transmission and mobile internet application. Considering this fact we proposed a data path architecture using dedicated hardwire for the baseband processor. The most computationally intensive part of such a high data rate system are the 64-point inverse FFT in the transmit direction and the viterbi decoder in the receiver direction. Accordingly an appropriate design methodology for constructing them has to be chosen a) how much silicon area is needed b) how easily the particular architecture can be made flat for implementation in VLSI c) in actual implementation how many wire crossings and how many long wires carrying signals to remote parts of the design are necessary d) how small the power consumption can be .This paper describes a novel 64-point FFT/IFFT processor which has been developed as part of a large research project to develop a single chip wireless modem.

ALGORITHM FORMULATION

The discrete fourier transformation A(r) of a complex data sequence B(k) of length N
where r, k ={0,1……, N-1} can be described as


Where WN = e-2?j/N . Let us consider that N=MT , ? = s+ Tt and k=l+Mm,where s,l ? {0,1…..7} and m, t ? {0,1,….T-1}. Applying these values in first equation and we get


This shows that it is possible to realize the FFT of length N by first decomposing it to one M and one T-point FFT where N = MT, and combinig them. But this results in in a two dimensional instead of one dimensional structure of FFT. We can formulate 64-point by considering M =T = 8

This shows that it is possible to express the 64-point FFT in terms of a two dimensional structure of 8-point FFTs plus 64 complex inter-dimensional constant multiplications. At first, appropriate data samples undergo an 8-point FFT computation. However, the number of non-trivial multiplications required for each set of 8-point FFT gets multiplied with 1. Eight such computations are needed to generate a full set of 64 intermediate data, which once again undergo a second 8-point FFT operation . Like first 8-point FFT for second 8-point again such computions are required. Proper reshuffling of the data coming out from the second 8-point FFT generates the final output of the 64-point FFT .

Fig. Signal flow graph of an 8-point DIT FFT.

For realization of 8-point FFT using the conventional DIT does not need to use any multiplication operation.

The constants to be multiplied for the first two columns of the 8-point FFT structure are either 1 or j . In the third column, the multiplications of the constants are actually addition/subtraction operation followed multiplication of 1/?2 which can be easily realized by using only a hardwired shift-and-add operation. Thus an 8-point FFT can be carried out without using any true digital multiplier and thus provide a way to realize a low- power 64-point FFT at reduced hardware cost. Since a basic 8-point FFT does not need a true multiplier. On the other hand, the number of non-trivial complex multiplications for the conventional 64-point radix-2 DIT FFT is 66. Thus the present approach results in a reduction of about 26% for complex multiplication compared to that required in the conventional radix-2 64-point FFT. This reduction of arithmetic complexity furthur enhances the scope for realizing a low-power 64-point FFT processor. However, the arithmetic complexity of the proposed scheme is almost the same to that of radix-4 FFT algorithm since the radix-4 64-point FFT algorithm needs 52 non-trivial complex multiplications.