# MDCT

It means the “Modified Discrete Cosine Transform” and does the same as the “Discrete Fourier Transform” (DFT).

It is a newer mathematical algorithm for lossy audio bit-reduction in most audio bit-reductions.

The basis of audio data compression or decreasing the number of bits in the audio stream is to convert the discrete audio “time samples” to “discrete frequency samples”, meticulously encode the latter and then save or send them as what we know, for example, MP3 and AAC file or streams.

We could also use The Discrete Fourier Transform (DFT). However, the FFT was a bit of a show-stopper. Minimizing audio artefacts needed windowing and 50/50 overlapping of the sample window. However, the 50/50 overlapping of the samples means that the DFT took time to execute two times on the same samples with a double output of frequency samples.

When we look under the hood of the Discrete Fourier Transform (DFT), it does a Discrete Cosine Transform (real part) or DCT, but on top of that, it also does a Discrete Sine Transformation (DST). Both together then produce a complex result with the cosine as the real part and the sine as the imaginary part. Therefore, due to the lapped sampling, the Discrete Fourier Transform (DFT) results in double the calculation work with double the number of bits as output. It is not precisely what we are looking for when we want to reduce the number of audio bits.

In short, we can say that the DFT does two transformations, and the audio conversion needs a double-lapped sampling. There are too many bits and too many calculations.

Why do approximately two times the same thing, resulting in a double of bits? It was probably the question that some mathematicians wondered about. In the seventies, they came up with a solution called MDCT.

What if, instead of the Fourier transform, you do only the real part transformation (the Discrete Cosine Transform)? You try to retrieve the missing information through the overlap without doing the Discrete Sine Transform in the Fourier transformation.

They came up with solely the Discrete Cosine Transform combined with specific overlapping, resulting in half of the bits. They called it the Modified Discrete Cosine Transform (MDCT).

It was an opportunity to achieve a substantial bit reduction in digital audio coding when suddenly the double sampling had the advantage that it was sufficient to reduce the DFT to a simpler Discrete Cosine Transform (DCT).

However, the advantage of the necessary overlap can ensure that only the Discrete Cosine Transform is sufficient and that the overlap resulting in a double conversion can still preserve the phase between the frequencies. When encountering MDCT in DAB literature, you can assume this is an improved DFT version suitable for audio data compression or bit reduction. Many audio bit reductions include MP3 (2 steps FFT + MDCT), AAC (MDCT only), Ogg Vorbis, LDAC, Dolby AC-4, and Windows Media Audio using MDCT.

Back