Discrete-time signal processing
Discrete-time signal processing
Signal Processing with Lapped Transforms
Signal Processing with Lapped Transforms
The integer transforms analogous to discrete trigonometric transforms
IEEE Transactions on Signal Processing
Integer fast Fourier transform
IEEE Transactions on Signal Processing
Integer DCTs and fast algorithms
IEEE Transactions on Signal Processing
Integer lapped transforms and their applications to image coding
IEEE Transactions on Image Processing
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This paper studies the round-off analysis, design and implementation, and applications of the multiplier-less Fast Fourier Transform-like (ML-FFT) transformation proposed by Chan et al. [1, 2]. The ML-FFT parameterizes the twiddle factors in the conventional FFT algorithm as certain rotation-like matrices and approximates the associated parameters inside these matrices by the sum-of-power-of-two (SOPOT) or canonical signed digits representations, hence avoiding expensive multiplications. The error due to the SOPOT approximation is called the coefficient round-off error and it has been studied in [1, 2]. This paper studies the signal round-off error arising from internal rounding and develops a recursive noise model for ML-FFT. Using this model, a random search algorithm is proposed to minimize the hardware resources for realizing the ML-FFT subject to a prescribed output bit accuracy. To address the irregular structure of the ML-FFT due to the varying number of SOPOT terms used, a framework for its software implementation is also developed. The resulting algorithm has a regular implementation structure and is shown to offer a good performance similar to their floating-point counterpart. Finally, a new ML-FFT for real-valued input, called the ML-RFFT, is proposed. Because of the symmetry in the algorithm, it only requires about half the number of additions as required by the ML-FFT. Using the mappings between the DFT and the DCTs and DWTs, new ML-FFT-based transformations called ML-DCTs and ML-DWTs are derived. Design examples are given to demonstrate the usefulness of the proposed methods.