Low Complexity Fourier Transforms using Multiple Square Waves

Abstract

Fourier Transform (FT) is widely applied in digital mobile cellular radio systems. The implementation requires low power consumption and smaller chip size. The primary factor of the FT applications is its chip complexity. The complexity is typically expressed in terms of number of adders, the number of multiplier, data storage and control complexity rather than the speed of operation. The current divide and conquer technique in fast Fourier transform (FFT) reduces the number of operations in conventional discrete Fourier transform (DFT) by utilizing the advantage of complex twiddle factors instead of matrix multiplications (Oppenheim, 1990). The computation of DFT is decomposed into nested smaller DFTs which are computed separately and combined to give the final results. FFT reduces the number of multiplier which account of much of the chip area and power consumption in digital hardware design. However, a pipeline FFT processor is characterized by real time continuous processing of an input data sequence. It is difficult to initiate the FFT operation until all of the N sampled data are taken. Another complexity issue is the arithmetic unit, especially multipliers, that requires larger area than a digital register. To meet real-time processing in FFT with size of N, the multiplicative complexity of N logr N is required (r is generally the radix). It contributes the complexity of the processor and power consumption. Another consideration of FFT is the data storage or memory for buffering the data and intermediate results of the real time computations. The butterfly at the first stage has to take the input data elements separated by N/r from the sequence. The required memory becomes another major chip area issue especially for large Fourier transform. The facts expressed above need to be improved so that the amounts of power consumption, chip area and complexity are suitable especially for handheld transceiver. Since the power consumption is directly related to the number of complex multiplications, an algorithm to reduce or replace these multiplications is important. In (Shattil and Nassar, 2002), a simple computation of Fourier transform using a squarewave is introduced. A mathematical derivation shows that it is possible to replace the