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Fast and Accurate Direct MDCT to DFT Conversion With Arbitrary Window Functions

In this paper, we propose a method for direct conversion of MDCT coefficients to DFT coefficients, without passing through time signal reconstruction. In contrast to previous works, this method is valid for any pair of MDCT and DFT window functions. It is based on the decomposition of the MDCT-to-DFT conversion matrices into a Toeplitz part plus a Hankel part. The latter is split, then mirrored and combined with the former to construct a global Toeplitz matrix. This leads to a fast FIR filtering implementation of the conversion process. The filter taps are DFT coefficients of window functions products, and concentrate most of their energy in a few low-frequency taps. The conversion can thus be efficiently approximated by keeping only a few most significant taps, as confirmed by numerical experiments: For example, for frame size of 2048, Hanning-windowed DFT is obtained from KBD-windowed MDCT with SNR over 60 dB when keeping only 20 taps.