How were the MP 3 filter coefficients produced ? ∗ Tom

The audio coding scheme MPEG 1 Layer-3, which was standardized in 1993 and which is popularly known as MP3, has had an enormous impact for facilitating electronic distribution of music over the internet. Although various other more efficient coding schemes have been developed and implemented since then, the mp3 file format is still the only format for compressed audio files which is available on all platforms. A lot of mathematics is being used in the MP3 coding. Various popularizing papers have appeared on this. However, it is not easy to get access to the precise formulation of the coding scheme and the precise mathematics behind it. Reasons for this are, on the one hand that the algorithm was developed by a community of signal analysts and computer scientists who use a terminology not familiar to mathematicians, on the other hand that this was developed in a commercial world, where access is far from free. The MPEG 1 audio coding scheme [1] has started with Layer-1 and was soon extended to Layer-2, and next Layer-3. The higher Layers brought further compression while keeping a reasonable sound quality, among others by implementing the so-called psycho-acoustical model together with a better frequency resolution. Already in Layer-1, and preserved in the higher Layers, there is a splitting of the input signal into 32 subbands of equal frequency width. Reconstruction from the signals in these 32 subbands to the full signal is not completely exact, but by a very clever subband encoding/decoding scheme it is almost exact. The most detailed mathematical description of this scheme can be found in lecture notes by Schniter [4], available online but not formally published. Another reference is Pan [2]. See also the Remark at the end of this note. The subband encoding/decoding scheme uses a prototype low-pass filter given by a list of 513 coefficients C[i] (i = 0, 1, . . . , 512) for the analysis filter (see [1, Annex C, Table C.1]), which are between −0.04 and 0.04 and which are given with 9 decimals. Similarly, for the synthesis filter there is a list (see [1, Annex B, Table B.3]) of 513 coefficients D[i] (i = 0, 1, . . . , 512), for which it turns out that D[i] = 32C[i]. There is more redundancy in the two tables because it turns out that C[i] = −C[512 − i] if 64 is not a divisor of i, and C[i] = C[512 − i] otherwise. Furthermore, C[0] = C[512] = 0.