In Orthogonal Frequency-Division Multiplexing (OFDM) the spectrum or channel is divided into a set of mutually orthogonal subcarriers, which are used to modulate the data to be transmitted. In Multiuser OFDM the subcarriers may be assigned to different communication paths where the spectrum are divided into blocks of subcarriers for each path.

__Figure: __*Multiuser Cooperative Radio Scenario. The communication path may be either directly from base station to mobile station or data can be relayed through other mobile*

*Stations to get to the destination.*

In OFDM the Discrete Fourier Transform (DFT) and it’s inverse counterpart plays a significant role, as it is used to modulate data symbols onto the subcarriers in the transmitter and demodulate the data at the receiver.

__Figure: __*Example of subcarrier allocation for each communication path, where each color is*

*assigned to a different path*

The goal of applying a power consumption measure to FPGA implementations of Fourier transform algorithms is to investigate how the theoretical measure of computational complexity compares to performance achievements of actual implementations. Evaluating the performance of these implementations, across a variable number of needed subcarriers and using a power measure, will show in which situation it is advantageous to use each of the investigated algorithms.

__DFT Algorithms:__

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For this project two Fast Fourier Transform (FFT) algorithms are chosen for comparison. The algorithms considered in this project are:-

__Split-Radix Fast Fourier Transform (SRFFT):__

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One approach is to calculate the full transform, regardless of how many subcarriers, that are of interest. Several algorithm exist which can perform this task. The radix-2 FFT is a recursive decomposition into two DFTs of N/2 length, the radix-4 makes use of decompositions into four N/4 DFT, and the Split-Radix FFT employs decomposition into one N/2 and two N/4 DFTs. The SRFFT has proven to feature one of the lowest computational complexities of the mentioned algorithms [Duhamel and Hollmann.

__“Sørensen” Fast Fourier Transform (SFFT):__

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As mentioned above, one user may only need to receive the data sent using a subset of the available subcarriers. Therefore methods calculating only a subset of the FFT have been developed, where the SFFT reduces the number of calculations by only calculating decomposition into a set of small FFTs and then recombining the results for only the subset of subcarrier that are of interest.

**Rama University UttarPradesh, Kanpur**

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