MDCT

MDCT stands for Modified Discrete Cosine Transform. It is a lapped transform used in digital signal processing to convert time-domain signals into frequency-domain coefficients while using overlapping blocks. In its standard form, the transform operates on blocks of 2N samples and yields N coefficients per block. The 50% overlap between consecutive blocks, together with a suitable window function, allows adjacent blocks to share samples and enables perfect reconstruction when the inverse transform is applied with an appropriate overlap-add step.

The MDCT is closely related to the discrete cosine transform but uses a sine-type window and a specific cosine basis. A common formulation uses the input window w(n) and defines the coefficients X_k as X_k = sum_{n=0}^{2N-1} x[n] w(n) cos[ (π/N) (n+1/2)(k+1/2) ], for k = 0,...,N-1. The choice of the window is constrained to satisfy time-domain aliasing cancellation (TDAC) so that overlap-add in synthesis exactly reconstructs the original signal.

Inverse MDCT (IMDCT) maps N coefficients back to 2N time samples, which are then windowed and overlapped with adjacent blocks to form the continuous signal. Common windows include the sine window and the Kaiser-Bessel derived (KBD) window; these windows help ensure stability and reconstruction accuracy.

Applications: MDCT is widely used in audio compression standards and processing. It is a core component of many modern audio codecs, including AAC and Vorbis, and has been used in other formats as well. Beyond audio, MDCT-based lapped transform techniques appear in digital communications and multi-rate filter banks.