Modified discrete cosine transform
The modified discrete cosine transform (English modified discrete cosine transform, in short: MDCT) is a real-valued, discrete, linear, orthogonal transformation, which is one of the group of discrete Fourier transforms ( DFT) and a modification of the eponymous discrete cosine transform ( DCT) is.
The MDCT was developed in the years 1986, 1987 by John P. Princen, AW Johnson and Alan B. Bradley. The scope of the MDCT transformation is as central in the audio compression data such as MP3, Vorbis, Advanced Audio Coding (AAC) or Dolby Digital (AC -3). In addition, the similar structure modified discrete sine transformation ( MDST ) exists which is based on the discrete sine transform, but in the digital signal processing does not have any substantial importance.
- 2.1 Inverse transformation
- 2.2 Calculation effort
Motivation
MDCT based on the type IV discrete cosine transform, referred to as the DCT -IV, and is used at the beginning to be transformed input signal sequence, for example, this is a finite number of samples of an audio signal, a straight continuation of and at the end of the signal sequence is an odd continued. The input signal is divided into consecutive blocks, wherein each block is separately subjected to transformation. In the MDCT, the signal sequences to form the individual blocks are partially overlapped with each other to compensate for the even and odd of the block forming continuations. In most English-language literature, this is referred to as time-domain aliasing cancellation ( TDAC ). Similar methods can be found in the framework of DFT at the overlap-add method and the overlap-save method application to transfer the there periodic continuation of the DFT in the aperiodic convolution operation.
The reason for the use of MDCT in the field of audio compression is that a periodic extension of the signal sequence is linked as in the DFT generally with cracks at the edge points of a block due to unfavorable sequels. This can be in the field of data compression upper spectral components corresponding roughly or not quantize this would be connected with the DFT with a greater loss of information or loss of quality in the audio signal. And with the DCT by the nature of the continuation, the signal energy is primarily concentrated in the lower frequency ranges. Resulting in higher spectral components with the same quality can be more quantized, to achieve a better compression efficiency, resulting in the DCT. And modifying the method of overlap extension as the DCT-IV is necessary to be able to transform any long signal sequences by the formation of the DCT block may be, but the overlap does not change the compression efficiency of transformation per se.
Application Examples
In applications such as MP3, the audio signal is not directly subjected to the MDCT. Rather, first a division of the input signal into different spectral frequency ranges by means of sets of band-pass filters that optimizes in the form of so-called polyphase quadrature filter ( PQF ) or Quadrature Mirror Filter ( QMF ), can be realized. Their output signals are fed to the MDCT. The spectral components produced by the MDCT are then quantized to different degrees by means of a psychoacoustic model.
In processes such as Advanced Audio Coding (AAC ), however, there will be no bandwidth allocation before the MDCT, and the input signal sequence is subjected directly to the MDCT.
Definition
The overlap, the amount of input samples from the time domain is twice as large as the spectral output data therefrom formed in the MDCT and, in contrast to the symmetrical frequency transformations. Are formally in the transformation of 2N real numbers x0, ..., X2N -1 to N real numbers X0, ..., XN- 1 shown by the following relationship:
In the literature are sometimes introduced additional constant factors in non- uniform shape and for normalization, in this respect, but which do not alter the transformation principle.
Inverse transformation
The inverse MDCT, the IMDCT abbreviated, is the inverse of the above transformation dar. Since the input or output sequence comprise a different number, is for reversing an addition in the time domain the consecutive blocks, and the time- overlapping areas in the time-domain aliasing cancellation ( TDAC ) is necessary.
Formal real numbers X0, ..., XN -1 in 2N real numbers y0, ..., Y2N -1 will be placed at the IMDCT N:
As with the DCT-IV, as a form of orthogonal transformation, the inverse transformation is identical to the forward transform to a factor.
Computational effort
The direct calculation of the MDCT according to the above formula requires O ( N2) operations. Similar to the Fast Fourier Transform (FFT), as a form of efficient computation of the DFT, also exist in the MDCT algorithm that are similar in structure to the radix-2 algorithm, the number of arithmetic operations to O ( N log N) to be reduced.