Talk:μ-law algorithm

Talk

This article should be called "μ-law algorithm" with a lower-case μ. Unfortunately, the software currently prevents this, capitalizing the mu. Capital mu looks just like "M", which just looks wrong. -- Karada 23:19, 29 October 2005 (UTC)

I changed it to "Mu-law algorithm", because a capital mu is evil. Not only does it look dumb, it isn't obvious to most people that it's a mu and not the Latin letter M. And nobody calls it M-law. - furrykef (Talk at me) 14:26, 8 April 2006 (UTC)

Graph/legend and text inconsistency

"Comparison with A-law: A special feature ... near zero sound pressure...": this is not expected from the analytical expression of F, and not visible on the graph (does it occur at a level below -80 dB ? In this case, a proper scaling of the axis should display the difference between A and Mu laws).

In addition, the green line on the graph is not "A-law", but "No companding"... --Dgcrete 12:27, 22 August 2006 (UTC)

I fixed the graph. Regarding the "special feature near zero" - this statement is false (assuming there is no mistake in the formulae). Near zero, the mu-law and A-law formulae differ only by a constant multiplier. This is because as x goes to zero, ln(1+x) becomes the same as x, hence the top halves of the Mu-law and A-law equations become the same. The same is true for the for the quantized versions of the equations.

I will remove the "special feature near zero" comment.

I've also added quantized points to the graph

Ozhiker 22:18, 22 August 2006 (UTC)

Table of 14 bit linear values incorrect?

I think the table of u-law to 14 bit linear is slightly incorrect. e.g. the first value should be -8031 rather than -8159.

-8159 is the maximum negative input to an encoder, which encodes to 0, but 0 should decode to -8031 (which is the centre of the range of values that encode to 0)

I think the 14 bit values should just be the given 16 bit values divided by 4. (The 16 bit values appear to be correct) TimMorley 16:34, 25 May 2007 (UTC)

Odd reference

One of the two references is from Wikipedia itself. That seems a bit sketchy...? 24.4.102.10 11:28, 23 September 2007 (UTC)

Who is right?

The table in this article differs from the code given in [1]: Both 0x7F and 0xFF decodes to 0. But the encoder in the article encodes 0 to 0xFF and -1 to 0x7F. --RokerHRO (talk) 08:11, 1 March 2010 (UTC)

Encoding table

I created an encoding table from "unsigned linear 16 bit" values into µ-law 8 bit values using some self-written programs for data generating and pretty printing and SoX for the µ-law encoding. As you can see the octet 0x7F does not apper in this µ-law encoding table.

µ-law encoding table

u16	µ-law	Range size
0x0000 ... 0x0487	00	1160 values
0x0488 ... 0x0887	01	1024 values
0x0888 ... 0x0C87	02	1024 values
0x0C88 ... 0x1087	03	1024 values
0x1088 ... 0x1487	04	1024 values
0x1488 ... 0x1887	05	1024 values
0x1888 ... 0x1C87	06	1024 values
0x1C88 ... 0x2087	07	1024 values
0x2088 ... 0x2487	08	1024 values
0x2488 ... 0x2887	09	1024 values
0x2888 ... 0x2C87	0A	1024 values
0x2C88 ... 0x3087	0B	1024 values
0x3088 ... 0x3487	0C	1024 values
0x3488 ... 0x3887	0D	1024 values
0x3888 ... 0x3C87	0E	1024 values
0x3C88 ... 0x4087	0F	1024 values
0x4088 ... 0x4287	10	512 values
0x4288 ... 0x4487	11	512 values
0x4488 ... 0x4687	12	512 values
0x4688 ... 0x4887	13	512 values
0x4888 ... 0x4A87	14	512 values
0x4A88 ... 0x4C87	15	512 values
0x4C88 ... 0x4E87	16	512 values
0x4E88 ... 0x5087	17	512 values
0x5088 ... 0x5287	18	512 values
0x5288 ... 0x5487	19	512 values
0x5488 ... 0x5687	1A	512 values
0x5688 ... 0x5887	1B	512 values
0x5888 ... 0x5A87	1C	512 values
0x5A88 ... 0x5C87	1D	512 values
0x5C88 ... 0x5E87	1E	512 values
0x5E88 ... 0x6087	1F	512 values
0x6088 ... 0x6187	20	256 values
0x6188 ... 0x6287	21	256 values
0x6288 ... 0x6387	22	256 values
0x6388 ... 0x6487	23	256 values
0x6488 ... 0x6587	24	256 values
0x6588 ... 0x6687	25	256 values
0x6688 ... 0x6787	26	256 values
0x6788 ... 0x6887	27	256 values
0x6888 ... 0x6987	28	256 values
0x6988 ... 0x6A87	29	256 values
0x6A88 ... 0x6B87	2A	256 values
0x6B88 ... 0x6C87	2B	256 values
0x6C88 ... 0x6D87	2C	256 values
0x6D88 ... 0x6E87	2D	256 values
0x6E88 ... 0x6F87	2E	256 values
0x6F88 ... 0x7087	2F	256 values
0x7088 ... 0x7107	30	128 values
0x7108 ... 0x7187	31	128 values
0x7188 ... 0x7207	32	128 values
0x7208 ... 0x7287	33	128 values
0x7288 ... 0x7307	34	128 values
0x7308 ... 0x7387	35	128 values
0x7388 ... 0x7407	36	128 values
0x7408 ... 0x7487	37	128 values
0x7488 ... 0x7507	38	128 values
0x7508 ... 0x7587	39	128 values
0x7588 ... 0x7607	3A	128 values
0x7608 ... 0x7687	3B	128 values
0x7688 ... 0x7707	3C	128 values
0x7708 ... 0x7787	3D	128 values
0x7788 ... 0x7807	3E	128 values
0x7808 ... 0x7887	3F	128 values
0x7888 ... 0x78C7	40	64 values
0x78C8 ... 0x7907	41	64 values
0x7908 ... 0x7947	42	64 values
0x7948 ... 0x7987	43	64 values
0x7988 ... 0x79C7	44	64 values
0x79C8 ... 0x7A07	45	64 values
0x7A08 ... 0x7A47	46	64 values
0x7A48 ... 0x7A87	47	64 values
0x7A88 ... 0x7AC7	48	64 values
0x7AC8 ... 0x7B07	49	64 values
0x7B08 ... 0x7B47	4A	64 values
0x7B48 ... 0x7B87	4B	64 values
0x7B88 ... 0x7BC7	4C	64 values
0x7BC8 ... 0x7C07	4D	64 values
0x7C08 ... 0x7C47	4E	64 values
0x7C48 ... 0x7C87	4F	64 values
0x7C88 ... 0x7CA7	50	32 values
0x7CA8 ... 0x7CC7	51	32 values
0x7CC8 ... 0x7CE7	52	32 values
0x7CE8 ... 0x7D07	53	32 values
0x7D08 ... 0x7D27	54	32 values
0x7D28 ... 0x7D47	55	32 values
0x7D48 ... 0x7D67	56	32 values
0x7D68 ... 0x7D87	57	32 values
0x7D88 ... 0x7DA7	58	32 values
0x7DA8 ... 0x7DC7	59	32 values
0x7DC8 ... 0x7DE7	5A	32 values
0x7DE8 ... 0x7E07	5B	32 values
0x7E08 ... 0x7E27	5C	32 values
0x7E28 ... 0x7E47	5D	32 values
0x7E48 ... 0x7E67	5E	32 values
0x7E68 ... 0x7E87	5F	32 values
0x7E88 ... 0x7E97	60	16 values
0x7E98 ... 0x7EA7	61	16 values
0x7EA8 ... 0x7EB7	62	16 values
0x7EB8 ... 0x7EC7	63	16 values
0x7EC8 ... 0x7ED7	64	16 values
0x7ED8 ... 0x7EE7	65	16 values
0x7EE8 ... 0x7EF7	66	16 values
0x7EF8 ... 0x7F07	67	16 values
0x7F08 ... 0x7F17	68	16 values
0x7F18 ... 0x7F27	69	16 values
0x7F28 ... 0x7F37	6A	16 values
0x7F38 ... 0x7F47	6B	16 values
0x7F48 ... 0x7F57	6C	16 values
0x7F58 ... 0x7F67	6D	16 values
0x7F68 ... 0x7F77	6E	16 values
0x7F78 ... 0x7F87	6F	16 values
0x7F88 ... 0x7F8F	70	8 values
0x7F90 ... 0x7F97	71	8 values
0x7F98 ... 0x7F9F	72	8 values
0x7FA0 ... 0x7FA7	73	8 values
0x7FA8 ... 0x7FAF	74	8 values
0x7FB0 ... 0x7FB7	75	8 values
0x7FB8 ... 0x7FBF	76	8 values
0x7FC0 ... 0x7FC7	77	8 values
0x7FC8 ... 0x7FCF	78	8 values
0x7FD0 ... 0x7FD7	79	8 values
0x7FD8 ... 0x7FDF	7A	8 values
0x7FE0 ... 0x7FE7	7B	8 values
0x7FE8 ... 0x7FEF	7C	8 values
0x7FF0 ... 0x7FF7	7D	8 values
0x7FF8 ... 0x7FFF	7E	8 values

µ-law encoding table

u16	µ-law	Range size
0x8000 ... 0x8003	FF	4 values
0x8004 ... 0x800B	FE	8 values
0x800C ... 0x8013	FD	8 values
0x8014 ... 0x801B	FC	8 values
0x801C ... 0x8023	FB	8 values
0x8024 ... 0x802B	FA	8 values
0x802C ... 0x8033	F9	8 values
0x8034 ... 0x803B	F8	8 values
0x803C ... 0x8043	F7	8 values
0x8044 ... 0x804B	F6	8 values
0x804C ... 0x8053	F5	8 values
0x8054 ... 0x805B	F4	8 values
0x805C ... 0x8063	F3	8 values
0x8064 ... 0x806B	F2	8 values
0x806C ... 0x8073	F1	8 values
0x8074 ... 0x807B	F0	8 values
0x807C ... 0x808B	EF	16 values
0x808C ... 0x809B	EE	16 values
0x809C ... 0x80AB	ED	16 values
0x80AC ... 0x80BB	EC	16 values
0x80BC ... 0x80CB	EB	16 values
0x80CC ... 0x80DB	EA	16 values
0x80DC ... 0x80EB	E9	16 values
0x80EC ... 0x80FB	E8	16 values
0x80FC ... 0x810B	E7	16 values
0x810C ... 0x811B	E6	16 values
0x811C ... 0x812B	E5	16 values
0x812C ... 0x813B	E4	16 values
0x813C ... 0x814B	E3	16 values
0x814C ... 0x815B	E2	16 values
0x815C ... 0x816B	E1	16 values
0x816C ... 0x817B	E0	16 values
0x817C ... 0x819B	DF	32 values
0x819C ... 0x81BB	DE	32 values
0x81BC ... 0x81DB	DD	32 values
0x81DC ... 0x81FB	DC	32 values
0x81FC ... 0x821B	DB	32 values
0x821C ... 0x823B	DA	32 values
0x823C ... 0x825B	D9	32 values
0x825C ... 0x827B	D8	32 values
0x827C ... 0x829B	D7	32 values
0x829C ... 0x82BB	D6	32 values
0x82BC ... 0x82DB	D5	32 values
0x82DC ... 0x82FB	D4	32 values
0x82FC ... 0x831B	D3	32 values
0x831C ... 0x833B	D2	32 values
0x833C ... 0x835B	D1	32 values
0x835C ... 0x837B	D0	32 values
0x837C ... 0x83BB	CF	64 values
0x83BC ... 0x83FB	CE	64 values
0x83FC ... 0x843B	CD	64 values
0x843C ... 0x847B	CC	64 values
0x847C ... 0x84BB	CB	64 values
0x84BC ... 0x84FB	CA	64 values
0x84FC ... 0x853B	C9	64 values
0x853C ... 0x857B	C8	64 values
0x857C ... 0x85BB	C7	64 values
0x85BC ... 0x85FB	C6	64 values
0x85FC ... 0x863B	C5	64 values
0x863C ... 0x867B	C4	64 values
0x867C ... 0x86BB	C3	64 values
0x86BC ... 0x86FB	C2	64 values
0x86FC ... 0x873B	C1	64 values
0x873C ... 0x877B	C0	64 values
0x877C ... 0x87FB	BF	128 values
0x87FC ... 0x887B	BE	128 values
0x887C ... 0x88FB	BD	128 values
0x88FC ... 0x897B	BC	128 values
0x897C ... 0x89FB	BB	128 values
0x89FC ... 0x8A7B	BA	128 values
0x8A7C ... 0x8AFB	B9	128 values
0x8AFC ... 0x8B7B	B8	128 values
0x8B7C ... 0x8BFB	B7	128 values
0x8BFC ... 0x8C7B	B6	128 values
0x8C7C ... 0x8CFB	B5	128 values
0x8CFC ... 0x8D7B	B4	128 values
0x8D7C ... 0x8DFB	B3	128 values
0x8DFC ... 0x8E7B	B2	128 values
0x8E7C ... 0x8EFB	B1	128 values
0x8EFC ... 0x8F7B	B0	128 values
0x8F7C ... 0x907B	AF	256 values
0x907C ... 0x917B	AE	256 values
0x917C ... 0x927B	AD	256 values
0x927C ... 0x937B	AC	256 values
0x937C ... 0x947B	AB	256 values
0x947C ... 0x957B	AA	256 values
0x957C ... 0x967B	A9	256 values
0x967C ... 0x977B	A8	256 values
0x977C ... 0x987B	A7	256 values
0x987C ... 0x997B	A6	256 values
0x997C ... 0x9A7B	A5	256 values
0x9A7C ... 0x9B7B	A4	256 values
0x9B7C ... 0x9C7B	A3	256 values
0x9C7C ... 0x9D7B	A2	256 values
0x9D7C ... 0x9E7B	A1	256 values
0x9E7C ... 0x9F7B	A0	256 values
0x9F7C ... 0xA17B	9F	512 values
0xA17C ... 0xA37B	9E	512 values
0xA37C ... 0xA57B	9D	512 values
0xA57C ... 0xA77B	9C	512 values
0xA77C ... 0xA97B	9B	512 values
0xA97C ... 0xAB7B	9A	512 values
0xAB7C ... 0xAD7B	99	512 values
0xAD7C ... 0xAF7B	98	512 values
0xAF7C ... 0xB17B	97	512 values
0xB17C ... 0xB37B	96	512 values
0xB37C ... 0xB57B	95	512 values
0xB57C ... 0xB77B	94	512 values
0xB77C ... 0xB97B	93	512 values
0xB97C ... 0xBB7B	92	512 values
0xBB7C ... 0xBD7B	91	512 values
0xBD7C ... 0xBF7B	90	512 values
0xBF7C ... 0xC37B	8F	1024 values
0xC37C ... 0xC77B	8E	1024 values
0xC77C ... 0xCB7B	8D	1024 values
0xCB7C ... 0xCF7B	8C	1024 values
0xCF7C ... 0xD37B	8B	1024 values
0xD37C ... 0xD77B	8A	1024 values
0xD77C ... 0xDB7B	89	1024 values
0xDB7C ... 0xDF7B	88	1024 values
0xDF7C ... 0xE37B	87	1024 values
0xE37C ... 0xE77B	86	1024 values
0xE77C ... 0xEB7B	85	1024 values
0xEB7C ... 0xEF7B	84	1024 values
0xEF7C ... 0xF37B	83	1024 values
0xF37C ... 0xF77B	82	1024 values
0xF77C ... 0xFB7B	81	1024 values
0xFB7C ... 0xFFFF	80	1156 values

The table differs slightly from the more compact one in the article. Which is correct? --RokerHRO (talk) 22:05, 16 April 2010 (UTC)

What is the rationale behind the choice μ = 255?

Why is the value of μ chosen to be 255? The article seems to suggest that this is because 255 = 2ⁿ - 1, where n is the number of bits used to store the encoding, but I see no reason for this.

I guess that evidently, choosing μ = 255 works well, but to me, this seems like an ad hoc choice rather than a choice that has anything to do with the number of bits used. I think it seems more reasonable that the encoding (i.e., F(x)) should be independent of the number of bits used to store the encoding and rather be based on the maximum sound pressure you want to support and the sensitivity of the human auditory system as a function of the sound pressure (such that the encoding has more densely packed quantization levels for sound pressures where the auditory system is more sensitive to small changes in the sound pressure).

I can think of two reasons that led to the choice μ = 255:

1. The expressions for the encoding and decoding can be evaluated efficiently when μ = 255 — especially, ln(1 + μ) = ln(1 + 255) = 8 log₂(1 + μ) = log₂(1 + 255) = 8, so the division in the encoding can be preformed very efficiently by just preforming an arithmetic shift (presuming that the base of the logarithms is changed from e to 2) — however, this is true for any μ = 2^2m - 1, where m is a non-negatve integer
2. Choosing μ = 255 happens to give a reasonably good sound quality for all sound pressures that it should be possible represent

However, I see no connection between μ and the number of bits used to store the encoding, which the article suggests that there is.

So, is there such a connection? Or is the value 255 an ad hoc value chosen mostly for the reasons I listed? Or in other words, what is the rationale behind choosing μ = 255? —Kri (talk) 22:50, 17 June 2018 (UTC); edited 06:32, 21 June 2018 (UTC)

Isn't that just to get the resulting 8-bit value of the input of the function? In other words μ specifies the maximum value of the output (minimum being 0), with the input's range being -1 to 1. —DIYeditor (talk) 01:55, 18 June 2018 (UTC)

No, it's not. The output of F is still in the range -1 to +1, no matter what value you choose for μ. —Kri (talk) 19:27, 18 June 2018 (UTC)

I tried to look through some technical papers and had trouble finding a straight answer for this. I think maybe part of it is as you say that 255 gives approximately the desired curve. As for the main reason: does the value 255 make the segments as defined in the g.711 recommendation (tables 2a and 2b) turn out as approximately powers of two intervals (less ~31)? This was a good call on your part, we need to clarify the article and address the part implying this is directly related to the 8-bit value. We need a RS that directly addresses this choice. —DIYeditor (talk) 01:05, 19 June 2018 (UTC)

I also sifted through the G.711 recommendation but didn’t find any explanation for why μ = 255 is used.

What is an RS? —Kri (talk) 07:10, 21 June 2018 (UTC)

WP:RS. —DIYeditor (talk) 19:28, 21 June 2018 (UTC)

So is it okay if I remove the parenthesis "(8 bits)" since the choice μ = 255 and the number of bits used don't really seem to be related? —Kri (talk) 07:32, 2 July 2018 (UTC)

Yes I agree with that. Does not appear to be directly related, it was original research by someone. —DIYeditor (talk) 08:30, 2 July 2018 (UTC)

Done. —Kri (talk) 14:13, 5 July 2018 (UTC)

Origin of μ and A in names?

I think it would be interesting for readers to learn about the origin / history of the µ and A in the terms µ-law and A-law. Do we have any WP:RS for this? --Matthiaspaul (talk) 18:01, 8 June 2021 (UTC)

Comparison with A-law

The "Comparison with A-law" section states that "The μ-law algorithm provides a slightly larger dynamic range than the A-law at the cost of worse proportional distortions for small signals." I think this may be backwards -- that is, A-law provides larger dynamic range and worse small-signal distortion than μ-law. The same (possibly erroneous) statement is in the A-law article. --99.121.214.126 (talk) 02:41, 18 December 2021 (UTC)

x-Axis of comparison plot confusingly labeled "Linear" but actually *logarithmic* (dBm0)

I'm looking at the comparison chart that is in both this μ-law page and the A-law page:

While it is an otherwise excellent chart, I have a frustration with the creator's (@Ozhiker) word choice of "Linear" in the x-Axis label, because really the x-axis is a *log* scale (dBm0 is a logarithmic unit). Hence the "No companding" graph is a straight line. Maybe the creator was trying to express something else with "Linear" (or am I misunderstanding something), but I think "Linear" just confuses readers. Better readability would be simply "Input Signal (dBm0)". It seems the chart was made with gnuplot and has instructions, so I could easily edit it or the svg if the creator is not around. But I want to make sure I'm not misunderstanding. Em3rgent0rdr (talk) 23:18, 26 June 2023 (UTC)

I agree, "Input Signal (dBm0)" would be a better label. Linear in the current label likely refers to the fact that compression is a non-linear process so the input in that sense is linear and the output non-linear. ~Kvng (talk) 12:27, 29 June 2023 (UTC)

fixed. Em3rgent0rdr (talk) 19:46, 29 June 2023 (UTC)

Additional problem: The red (and blue?) line(s) seem(s) to extend beyond 0dB, up to maybe +2 or +3dB? That might be fine for an analogue amplification system or recording to tape, perhaps (like using dolby companding to extend the effective dynamic range of audio cassette), but it's not possible in the digital domain. 0dB is the maximum possible representable signal strength, both for input and output. Even if the encoding means you end up with internally calculated values exceeding this, wouldn't they either be clipped to 0dB, or the entire output scale normalised to shift the maximum output to be such (and all quieter values moving down a little in kind)? — Preceding unsigned comment added by 92.12.87.15 (talk) 17:53, 31 October 2023 (UTC)

Audio sample quality mismatch

The two samples given for "original" and "encoded" are so far apart in overall quality that it's impossible to make a judgement of the effect on the sound caused by the A-law encoding. The "encoded" version seems to be recorded at a much lower sample rate / frequency for a start, which compromises the apparent quality much more than the companding, and isn't an inherent effect of the codec either. Possibly somebody fed a CD quality voice synth output into something that encodes into Phone quality mu-law, without considering that it also reduces the sample rate to about 1/6th as part of the process?

Either the "encoded" version needs to be remade with all other aspects being the same as the original one, which isn't hard to achieve via the export / save menu options of any half decent audio processing program (and would sound somewhere between the original and a plain 8-bit PCM conversion), or the "original" needs to be preprocessed and presented in a state otherwise matching the current encoded one, except for being 16 (or 14?) bit PCM not 8-bit mu-law encoded (so would also sound rather muffled etc).

...and if whoever's supplying the audio samples doesn't understand what any of that means or why it's a problem, they should refrain from any further uploads until they've studied up.

(I have no account so can't do that I'm afraid, otherwise I'd have already replaced it myself, it's been more than 25 years since the first time I experimentally messed around with format conversions of that type so it'd be a piece of cake) 92.12.87.15 (talk) 18:00, 31 October 2023 (UTC)