Сигналды өңдеудегі модифицирленген дискретті косинус түрлендіруі

Mathematical transform using in signal processing

The modified discrete cosine transform (MDCT) is a transform based on the type IV discrete cosine transform (DCT IV), with the additional property of being lapped: it is designed to be performed on consecutive blocks of a larger dataset, where subsequent blocks are overlapped so that the last half of one block coincides with the first half of the next block. This overlapping, in addition to the energy compaction qualities of the DCT, makes the MDCT especially attractive for signal compression applications, since it helps to avoid artifacts stemming from the block boundaries. As a result of these advantages, the MDCT is the most widely used lossy compression technique in audio data compression. It is employed in most modern audio coding standards, including MP3, Dolby Digital (AC 3), Vorbis (Ogg), Windows Media Audio (WMA), ATRAC, Cook, Advanced Audio Coding (AAC), High Definition Coding (HDC), LDAC, Dolby AC 4, and MPEG H 3D Audio, as well as speech coding standards such as AAC LD (LD MDCT), G.722.1, G.729.1, CELT, and Opus. The discrete cosine transform (DCT) was first proposed by Nasir Ahmed in 1972, and demonstrated by Ahmed with T. Natarajan and K. R. Rao in 1974. The MDCT was later proposed by John P. Princen, A. W. Johnson and Alan B. Bradley at the University of Surrey in 1987, following earlier work by Princen and Bradley (1986) to develop the MDCT's underlying principle of time domain aliasing cancellation (TDAC), described below. (There also exists an analogous transform, the MDST, based on the discrete sine transform, as well as other, rarely used, forms of the MDCT based on different types of DCT or DCT/DST combinations.) In MP3, the MDCT is not applied to the audio signal directly, but rather to the output of a 32 band polyphase quadrature filter (PQF) bank. The output of this MDCT is postprocessed by an alias reduction formula to reduce the typical aliasing of the PQF filter bank. Such a combination of a filter bank with an MDCT is called a hybrid filter bank or a subband MDCT. AAC, on the other hand, normally uses a pure MDCT; only the (rarely used) MPEG 4 AAC SSR variant (by Sony) uses a four band PQF bank followed by an MDCT. Similar to MP3, ATRAC uses stacked quadrature mirror filters (QMF) followed by an MDCT.

As a lapped transform, the MDCT is a bit unusual compared to other Fourier related transforms in that it has half as many outputs as inputs (instead of the same number). In particular, it is a linear function (where R denotes the set of real numbers). The 2N real numbers x0, , x2N 1 are transformed into the N real numbers X0, , XN 1 according to the formula:

(The normalization coefficient in front of this transform, here unity, is an arbitrary convention and differs between treatments. Only the product of the normalizations of the MDCT and the IMDCT, below, is constrained.)

The inverse MDCT is known as the IMDCT. Because there are different numbers of inputs and outputs, at first glance it might seem that the MDCT should not be invertible. However, perfect invertibility is achieved by adding the overlapped IMDCTs of subsequent overlapping blocks, causing the errors to cancel and the original data to be retrieved; this technique is known as time domain aliasing cancellation (TDAC). The IMDCT transforms N real numbers X0, , XN 1 into 2N real numbers y0, , y2N 1 according to the formula:

(Like for the DCT IV, an orthogonal transform, the inverse has the same form as the forward transform.) In the case of a windowed MDCT with the usual window normalization (see below), the normalization coefficient in front of the IMDCT should be multiplied by 2 (i. e., becoming 2/N).

Although the direct application of the MDCT formula would require O(N2) operations, it is possible to compute the same thing with only O(N log N) complexity by recursively factorizing the computation, as in the fast Fourier transform (FFT). One can also compute MDCTs via other transforms, typically a DFT (FFT) or a DCT, combined with O(N) pre and post processing steps. Also, as described below, any algorithm for the DCT IV immediately provides a method to compute the MDCT and IMDCT of even size.