DFT spectrum estimation from critically sampled lapped transforms

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

“…Then let us substitute time shifts of F 0 , F, and F 1 in place of n 0 , and times shifts of P 0 , P, and P 1 in place of n 1 for matrices U and V defined in (6). From (8), we have the phase terms:…”

“…As already mentioned in the introduction, this is a notable extension to the previous work[6] where a rectangular window function was considered for the DFT. Note also that a specific matrix structure was also exploited in[6] but this was within the MDCT-to-CLT conversion, although we consider here direct MDCT-to-DFT conversion matrices.…”

“…However, the MDCT is poorly appropriate for spectral analysis and signal manipulation in the TF domain, for several reasons [4], [5], [6]: Its basis vectors are not shift-invariant, it does not conserve the energy, and MDCT coefficients, which are real-valued, cannot be easily interpreted in terms of magnitude and phase. All this contrasts with the widely used Discrete Fourier Transform (DFT) or Short-Term Fourier Transform (STFT) 1 .…”

“…For all those reasons, the DFT/STFT are used in most audio/speech (TF) processing systems. Therefore, if one wants to apply some TF-domain signal processing on signals that are coming from perceptual audio decoders, one has the following two possibilities: 1) develop (often tricky and over-specific) MDCT-domain processing (e.g., [8] for instantaneous frequency estimation; see [6] for a review of several other examples of MDCT-domain processing), or 2) go to the DFT domain. The latter possibility is the more general, and currently, most audio processing systems that are cascaded with audio coders consider this solution.…”

“…Indeed, in practice different window functions are very often used for the MDCT (e.g., the Kaiser-Bessel-Derived (KBD) window [15] or the sine window that both ensure the perfect reconstruction property, for coding [1]) and the DFT (e.g., Hamming or Hanning windows, for spectral analysis and processing). In [6], an intermediate transform called Circulant Lapped Transform (CLT) has been proposed to convert the MDCT to the DFT, i.e., MDCT-to-CLT followed by CLT-to-DFT. The overall process is shown to be efficiently approximated by a complex-valued Finite Impulse Response (FIR) filtering applied on MDCT coefficients.…”