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doc/PCM8-conversions: finish encoding description
author Mychaela Falconia <falcon@freecalypso.org>
date Mon, 08 May 2023 03:28:40 +0000
parents 0ee1a66c1846
children e4a4bf11f37c
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0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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1 What is the authoritatively correct, officially endorsed bidirectional mapping
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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2 between G.711 A-law and mu-law encodings on one side and 16-bit 2's complement
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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3 linear PCM on the other side? Surprisingly, there is no official answer to this
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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4 problem anywhere in the specs! Instead the specs provide the following partial
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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5 answers:
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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6
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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7 * The G.711 spec itself provides one mapping from A-law code octets to linear
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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8 numeric values in range [-4032,4032] and another mapping from mu-law code
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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9 octets to linear numeric values in range [-8031,8031]. The output from each
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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10 of these mapping is given in "pure mathematical" form, without specifying any
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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11 bit-level encoding, and furthermore, mu-law decoder output in its pure
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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12 "conceptual" form has both +0 and -0 values. (The same signed zero problem
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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13 does not occur in A-law because it's a mid-riser code rather than mid-tread,
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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14 and thus has no quantized values equal to 0.)
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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15
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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16 * If one takes the "pure mathematical" output from the spec-prescribed G.711
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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17 decoder and represents it in 2's complement form, squashing +0 and -0 outputs
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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18 from the canonical mu-law decoder into "plain 0" at this step, the result is
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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19 a 13 bits wide 2's complement value for A-law decoding and a 14 bits wide 2's
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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20 complement value for mu-law.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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21
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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22 * All GSM speech encoders take 13-bit 2's complement linear PCM samples as their
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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23 input. How should this 13-bit GSM codec input be derived from A-law or mu-law
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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24 code octets? GSM specs refer to ITU's G.726 spec for ADPCM - it just so
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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25 happens that inside the ADPCM algorithm of G.726 (a totally unrelated codec of
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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26 no relevance to GSM codec work outside of this reference) there is a pair of
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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27 functions for expanding A-law and mu-law to linear PCM and compressing linear
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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28 PCM back to A-law or mu-law.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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29
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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30 * Following this obscure G.726 reference, we eventually conclude that in the
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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31 case of A-law, GSM specs call for the obvious treatment: take the "natural"
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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32 output from the canonical A-law decoder, represent it in 2's complement form,
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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33 the result is 13 bits wide, and just feed that 13-bit 2's complement form to
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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34 the input of GSM speech encoders. However, in the case of mu-law the
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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35 "natural" G.711 decoder output is one sign bit plus 13 bits of magnitude,
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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36 requiring 14 bits in 2's complement representation - and none of the specs I
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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37 could find says anything about exactly how this 14-bit input should be reduced
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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38 to 13 bits for feeding to GSM speech encoders. Canonical C implementations
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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39 of all GSM speech encoders take their input in 16-bit words and clear the 3
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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40 least significant bits as their first step; if the 14-bit mu-law decoder
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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41 output is represented in 16-bit words by padding 2 zero bits on the right and
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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42 this output is then fed to GSM speech encoder functions, the end effect is
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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43 that the least-significant bit of the 14-bit decoder output is simply cut off.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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44 This form of mu-law-to-GSM transcoder implementation is consistent with
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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45 TESTx-U.INP and TESTx-U.COD sequences provided in the GSM 06.54 package for
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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46 EFR.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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47
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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48 Based on the above considerations, we have our answer for how we should convert
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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49 from G.711 to 16-bit 2's complement linear PCM:
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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50
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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51 * For A-law, we emit the "natural" output in 13-bit 2's complement form and
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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52 append 3 zero bits on the right; this transformation is fully lossless.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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53
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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54 * For mu-law, we emit the "natural" output in 14-bit 2's complement form and
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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55 append 2 zero bits on the right. This transformation is almost lossless,
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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56 with just one exception: the "pure" decoder's -0 output (resulting from PCMU
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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57 octet 0x7F) is squashed to "plain 0", and will be re-emitted as PCMU octet
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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58 0xFF rather than 0x7F on subsequent re-encoding to G.711 PCMU.
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Mychaela Falconia <falcon@freecalypso.org>
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59
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Mychaela Falconia <falcon@freecalypso.org>
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60 For anyone needing a G.711 to 16-bit linear PCM decoder, the present package
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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61 provides ready-made decoding tables (following the above rules) in
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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62 dev/a2s-regen.out and dev/u2s-regen.out, generated by dev/a2s-regen.c and
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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63 dev/u2s-regen.c programs.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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64
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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65 Now for the opposite problem: what is the most correct way to compress 16-bit
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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66 2's complement linear PCM to A-law or mu-law? In this direction the official
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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67 specs leave even more ambiguity than in the G.711 decoding direction:
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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68
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Mychaela Falconia <falcon@freecalypso.org>
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69 * The G.711 spec itself says: "The conversion to A-law or mu-law values from
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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70 uniform PCM values corresponding to the decision values, is left to the
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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71 individual equipment specification." The specific implementation used in the
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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72 guts of G.726 ADPCM codec is referred to only as a non-normative example.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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73
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Mychaela Falconia <falcon@freecalypso.org>
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74 * GSM specs likewise refer to this G.726 section 4.2.8 (for compression of
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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75 13-bit speech decoder output to G.711) with language that suggests a
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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76 non-normative example.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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77
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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78 After painstakingly comparing the C implementation of G.726 in the ITU-T G.191
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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79 STL against the language of G.726 spec itself and convincing myself that they
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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80 really do match, and then painstakingly comparing this approach against the one
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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81 implemented in the same G.191 STL for G.711 in alaw_compress() and
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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82 ulaw_compress() and against the table lookup method implemented in libgsm/toast
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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83 (my first reference, before I went down the rabbit hole of tracking down
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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84 official specs), I reached the following conclusions:
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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85
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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86 * For A-law encoding all 3 parties (G.191 STL alaw_compress() function, G.726
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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87 "compress" block and toast_alaw.c) agree on the same mapping. In this
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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88 mapping only the most significant 12 bits of the 2's complement input word
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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89 (equivalent to one sign bit and 11 bits of magnitude) are relevant, leading
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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90 to the following two interesting properties:
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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91
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Mychaela Falconia <falcon@freecalypso.org>
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92 - the least-significant bit of GSM speech decoder output is always discarded
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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93 when converting to A-law;
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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94
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Mychaela Falconia <falcon@freecalypso.org>
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95 - conversion can be easily implemented with a 4096-byte look-up table based
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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96 on the upper 12 bits of input, exactly as was done in toast_alaw.c in the
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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97 venerable libgsm source.
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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98
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Mychaela Falconia <falcon@freecalypso.org>
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99 * Mu-law encoding is the real hair-raiser: if the input to the to-be-implemented
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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100 encoder has 14 or more bits (including the most practical problem of 16-bit
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
parents:
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101 2's complement input), there are no less than 3 different ways to implement
0ee1a66c1846 doc/PCM8-conversions: beginning of document
Mychaela Falconia <falcon@freecalypso.org>
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102 this encoder!
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Mychaela Falconia <falcon@freecalypso.org>
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103
236
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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104 Let us now look at the 3 different ways of encoding a 14-bit or 16-bit 2's
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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105 complement linear PCM input to G.711 mu-law. In this analysis we shall use
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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106 14-bit notation, with 2's complement inputs contained in the domain
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parents: 235
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107 [-8192,8191]. The difference between the 3 identified ways of mapping from this
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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108 domain to mu-law have to do with boundaries between quantization intervals.
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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109 Tables 2a and 2b in the G.711 spec list all defined quantization intervals and
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parents: 235
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110 all decision values that mark boundaries between them; here is a digested form
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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111 of the beginning of the canonical quantization table for either side of zero:
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parents: 235
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112
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113 Quantization interval Quantized value
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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114 (range of magnitudes) (absolute)
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Mychaela Falconia <falcon@freecalypso.org>
parents: 235
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115 ---------------------------------------
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Mychaela Falconia <falcon@freecalypso.org>
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116 0-1 0
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Mychaela Falconia <falcon@freecalypso.org>
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117 1-3 2
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118 3-5 4
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119 ...
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120 29-31 30
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121 31-35 33
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122 35-39 37
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123 ...
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124 91-95 93
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125 95-103 99
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126 ...
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127
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128 This canonical quantization table is defined in terms of absolute values, and
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129 is therefore fully symmetric around zero. A careful look at the above table
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130 raises a question: which quantization interval (and thus which PCMU octet
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131 output) should be selected if the input value to the encoder has a magnitude
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132 exactly equal to one of the threshold points, or decision values as they are
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133 officially called? In other words, what should the encoder do if the magnitude
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134 of the 14-bit input value equals 1, 3, 5, ..., 31, 35, 39 etc? The answer to
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135 this question is where the 3 candidate mappings under our consideration differ:
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136
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137 * The "compress" function of G.726 operating in mu-law mode selects the higher
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138 (in absolute value) quantization interval at every decision value threshold,
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139 on both sides of zero: see Table 15/G.726. PCMU octet 0x7F (meaning -0) will
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140 never be emitted by this version, and an input sequence of -3, -2, -1, 0, 1,
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141 2, 3 will map to quantized values -4, -2, -2, 0, 2, 2, 4.
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142
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143 * The ulaw_compress() function in G.191 STL behaves like the G.726 version for
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144 positive values, but selects the smaller-absolute-value quantization interval
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145 for negative inputs. Given the same input sequence as above, the output will
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146 correspond to quantized values -2, -2, -0, 0, 2, 2, 4. (Quantized value -0
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147 is PCMU octet 0x7F.)
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148
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149 * The s2u[] table in toast_ulaw.c in libgsm source is flat-out wrong and should
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150 not be used or considered further (and because those authors did not include
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151 the source for whatever program they used to generate their broken s2u[] and
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152 u2s[] tables, we have no way to really analyze them), but one CAN construct a
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153 new table for the same function, using the upper 13 bits of 16-bit 2's
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154 complement input to generate PCMU output - see our dev/s2u-regen.c program
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155 and its output table in dev/s2u-regen.out. The resulting mapping is
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156 "mirrored" around zero compared to G.191 STL ulaw_compress(): for the same
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157 input sequence as in the previous two examples, the output will correspond to
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158 quantized values -4, -2, -2, 0, 0, 2, 2. Just like the G.726 version, this
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159 look-up table version will never emit PCMU octet 0x7F for -0.
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160
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161 It is important to note that all GSM speech decoders produce 2's complement
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162 outputs that are only 13 bits wide, not 14 - therefore, when the input to the
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163 G.711 encoder comes from the output of a GSM speech decoder, the difference
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164 between all 3 alternatives listed above is masked, with all 3 producing
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165 identical output.
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166
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167 For production software, our (Themyscira) recommendation is to use look-up
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168 tables (dev/s2a-regen.out and dev/s2u-regen.out) for both A-law and mu-law
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169 encoding, using the upper 12 bits from 16-bit 2's complement input for A-law
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170 encoding and the upper 13 bits for mu-law encoding.