Talk:A-law algorithm

Talk

how does the a-law algorithm look like? What's the difference to the mu-law algorithm? --Abdull 17:08, 4 Apr 2005 (UTC)

equation

This needs finishing:

A-law expansion is then given by the inverse equation:

$F^{-1}(y)={\begin{cases}{\operatorname {sgn}(y)|y|[1+\ln(A)]/A},&0\leq |y|<{1 \over 1+\ln(A)}\\\operatorname {sgn}(y)e^{|y|[1+ln(A)]-1)}/[A+A\ln(A)],&{1 \over 1+\ln(A)}\leq |y|\leq 1\end{cases}}$

It's not quite right. Check the external links and such. - Omegatron 15:12, May 24, 2005 (UTC)

retransformation wrong  ?

I know these formulas are in many documents but I'm afraid the 2nd $F^{-1}$ is wrong.

I tried to build a VoIP (SIP) phone with A-Law audio codec. And I was very surprised because the decoded A-Law stream was very silent. I used exactly these formulas. And I think there is a mistake. Assume x = 1 then:

$y={\frac {1+\ln(A|x|)}{1+\ln(A)}}={\frac {1+\ln(A)}{1+\ln(A)}}=1$

But the retransformation gives

$x=F^{-1}(y)={\exp(|y|(1+\ln(A))-1) \over A(1+\ln(A))}={\exp((1+\ln(A))-1) \over A(1+\ln(A))}={A \over A(1+\ln(A))}=0.1827$

I think the A must be the denominator in both cases:

$x=F^{-1}(y)={\exp(|y|(1+\ln(A))-1) \over A}$

i.e.

$F^{-1}(1)={\exp(|1|(1+\ln(A))-1) \over A}={A \over A}=1$

Joerg Anders (GERMANY)

a or A

Is it a-law or A-law and what (if anything) does the a/A stand for? --Abdull 16:54, 11 August 2006 (UTC)

Legend of the graph uses wrong colours

The green straight line corresponds to no companding... 192.54.144.226 08:11, 22 August 2006 (UTC)Denis

Whoops - You are right - It's fixed now Ozhiker 08:35, 22 August 2006 (UTC)

The graph gives now the impression that not every linear signal value is mapped into a quantized value. This impression is incorrect because ofcourse for every input there will be a quantized output. Bob.v.R (talk) 05:14, 2 November 2019 (UTC)

Interoperability

The article states: "A-law is used for an international connection if at least one country uses it." Wouldn't both ends of the connection need to support the algorithm to send signals using it? Or am i missing something? Foobaz·o< 17:35, 17 May 2007 (UTC)

Maybe it could be worded more clearly - If either country uses A-law as their standard companding, then the companding rule when calls are made between the countries must be A-law. e.g. The UK uses A-Law for their standard companding, the USA uses u-Law. Hence because one uses A-Law, then all calls from the USA to UK or UK to USA must use A-Law. e.g.(2) Canada uses u-Law for their standard companding, the USA also uses u-Law. Hence they use u-Law companding on calls between the two countries becuase neither uses A-Law. --Ozhiker 22:19, 17 May 2007 (UTC)

Encoding table

I created an encoding table from "unsigned linear 16 bit" values into A-law 8 bit values. Some self-written programs generates the example data and does the pretty printing and SoX did the A-law encoding.
c is an intermediate value after the compression, that is given here to make the encoding better to understand. The final A-law-encoded value is the unintuitively result of: c XOR 55ₕₑₓ.

A-law encoding table

u16	c	A-law	Range size
0x0000 ... 0x03FF	7F	2A	1024 values
0x0400 ... 0x07FF	7E	2B	1024 values
0x0800 ... 0x0BFF	7D	28	1024 values
0x0C00 ... 0x0FFF	7C	29	1024 values
0x1000 ... 0x13FF	7B	2E	1024 values
0x1400 ... 0x17FF	7A	2F	1024 values
0x1800 ... 0x1BFF	79	2C	1024 values
0x1C00 ... 0x1FFF	78	2D	1024 values
0x2000 ... 0x23FF	77	22	1024 values
0x2400 ... 0x27FF	76	23	1024 values
0x2800 ... 0x2BFF	75	20	1024 values
0x2C00 ... 0x2FFF	74	21	1024 values
0x3000 ... 0x33FF	73	26	1024 values
0x3400 ... 0x37FF	72	27	1024 values
0x3800 ... 0x3BFF	71	24	1024 values
0x3C00 ... 0x3FFF	70	25	1024 values
0x4000 ... 0x41FF	6F	3A	512 values
0x4200 ... 0x43FF	6E	3B	512 values
0x4400 ... 0x45FF	6D	38	512 values
0x4600 ... 0x47FF	6C	39	512 values
0x4800 ... 0x49FF	6B	3E	512 values
0x4A00 ... 0x4BFF	6A	3F	512 values
0x4C00 ... 0x4DFF	69	3C	512 values
0x4E00 ... 0x4FFF	68	3D	512 values
0x5000 ... 0x51FF	67	32	512 values
0x5200 ... 0x53FF	66	33	512 values
0x5400 ... 0x55FF	65	30	512 values
0x5600 ... 0x57FF	64	31	512 values
0x5800 ... 0x59FF	63	36	512 values
0x5A00 ... 0x5BFF	62	37	512 values
0x5C00 ... 0x5DFF	61	34	512 values
0x5E00 ... 0x5FFF	60	35	512 values
0x6000 ... 0x60FF	5F	0A	256 values
0x6100 ... 0x61FF	5E	0B	256 values
0x6200 ... 0x62FF	5D	08	256 values
0x6300 ... 0x63FF	5C	09	256 values
0x6400 ... 0x64FF	5B	0E	256 values
0x6500 ... 0x65FF	5A	0F	256 values
0x6600 ... 0x66FF	59	0C	256 values
0x6700 ... 0x67FF	58	0D	256 values
0x6800 ... 0x68FF	57	02	256 values
0x6900 ... 0x69FF	56	03	256 values
0x6A00 ... 0x6AFF	55	00	256 values
0x6B00 ... 0x6BFF	54	01	256 values
0x6C00 ... 0x6CFF	53	06	256 values
0x6D00 ... 0x6DFF	52	07	256 values
0x6E00 ... 0x6EFF	51	04	256 values
0x6F00 ... 0x6FFF	50	05	256 values
0x7000 ... 0x707F	4F	1A	128 values
0x7080 ... 0x70FF	4E	1B	128 values
0x7100 ... 0x717F	4D	18	128 values
0x7180 ... 0x71FF	4C	19	128 values
0x7200 ... 0x727F	4B	1E	128 values
0x7280 ... 0x72FF	4A	1F	128 values
0x7300 ... 0x737F	49	1C	128 values
0x7380 ... 0x73FF	48	1D	128 values
0x7400 ... 0x747F	47	12	128 values
0x7480 ... 0x74FF	46	13	128 values
0x7500 ... 0x757F	45	10	128 values
0x7580 ... 0x75FF	44	11	128 values
0x7600 ... 0x767F	43	16	128 values
0x7680 ... 0x76FF	42	17	128 values
0x7700 ... 0x777F	41	14	128 values
0x7780 ... 0x77FF	40	15	128 values
0x7800 ... 0x783F	3F	6A	64 values
0x7840 ... 0x787F	3E	6B	64 values
0x7880 ... 0x78BF	3D	68	64 values
0x78C0 ... 0x78FF	3C	69	64 values
0x7900 ... 0x793F	3B	6E	64 values
0x7940 ... 0x797F	3A	6F	64 values
0x7980 ... 0x79BF	39	6C	64 values
0x79C0 ... 0x79FF	38	6D	64 values
0x7A00 ... 0x7A3F	37	62	64 values
0x7A40 ... 0x7A7F	36	63	64 values
0x7A80 ... 0x7ABF	35	60	64 values
0x7AC0 ... 0x7AFF	34	61	64 values
0x7B00 ... 0x7B3F	33	66	64 values
0x7B40 ... 0x7B7F	32	67	64 values
0x7B80 ... 0x7BBF	31	64	64 values
0x7BC0 ... 0x7BFF	30	65	64 values
0x7C00 ... 0x7C1F	2F	7A	32 values
0x7C20 ... 0x7C3F	2E	7B	32 values
0x7C40 ... 0x7C5F	2D	78	32 values
0x7C60 ... 0x7C7F	2C	79	32 values
0x7C80 ... 0x7C9F	2B	7E	32 values
0x7CA0 ... 0x7CBF	2A	7F	32 values
0x7CC0 ... 0x7CDF	29	7C	32 values
0x7CE0 ... 0x7CFF	28	7D	32 values
0x7D00 ... 0x7D1F	27	72	32 values
0x7D20 ... 0x7D3F	26	73	32 values
0x7D40 ... 0x7D5F	25	70	32 values
0x7D60 ... 0x7D7F	24	71	32 values
0x7D80 ... 0x7D9F	23	76	32 values
0x7DA0 ... 0x7DBF	22	77	32 values
0x7DC0 ... 0x7DDF	21	74	32 values
0x7DE0 ... 0x7DFF	20	75	32 values
0x7E00 ... 0x7E0F	1F	4A	16 values
0x7E10 ... 0x7E1F	1E	4B	16 values
0x7E20 ... 0x7E2F	1D	48	16 values
0x7E30 ... 0x7E3F	1C	49	16 values
0x7E40 ... 0x7E4F	1B	4E	16 values
0x7E50 ... 0x7E5F	1A	4F	16 values
0x7E60 ... 0x7E6F	19	4C	16 values
0x7E70 ... 0x7E7F	18	4D	16 values
0x7E80 ... 0x7E8F	17	42	16 values
0x7E90 ... 0x7E9F	16	43	16 values
0x7EA0 ... 0x7EAF	15	40	16 values
0x7EB0 ... 0x7EBF	14	41	16 values
0x7EC0 ... 0x7ECF	13	46	16 values
0x7ED0 ... 0x7EDF	12	47	16 values
0x7EE0 ... 0x7EEF	11	44	16 values
0x7EF0 ... 0x7EFF	10	45	16 values
0x7F00 ... 0x7F0F	0F	5A	16 values
0x7F10 ... 0x7F1F	0E	5B	16 values
0x7F20 ... 0x7F2F	0D	58	16 values
0x7F30 ... 0x7F3F	0C	59	16 values
0x7F40 ... 0x7F4F	0B	5E	16 values
0x7F50 ... 0x7F5F	0A	5F	16 values
0x7F60 ... 0x7F6F	09	5C	16 values
0x7F70 ... 0x7F7F	08	5D	16 values
0x7F80 ... 0x7F8F	07	52	16 values
0x7F90 ... 0x7F9F	06	53	16 values
0x7FA0 ... 0x7FAF	05	50	16 values
0x7FB0 ... 0x7FBF	04	51	16 values
0x7FC0 ... 0x7FCF	03	56	16 values
0x7FD0 ... 0x7FDF	02	57	16 values
0x7FE0 ... 0x7FEF	01	54	16 values
0x7FF0 ... 0x7FFF	00	55	16 values

A-law encoding table

u16	c	A-law	Range size
0x8000 ... 0x800F	80	D5	16 values
0x8010 ... 0x801F	81	D4	16 values
0x8020 ... 0x802F	82	D7	16 values
0x8030 ... 0x803F	83	D6	16 values
0x8040 ... 0x804F	84	D1	16 values
0x8050 ... 0x805F	85	D0	16 values
0x8060 ... 0x806F	86	D3	16 values
0x8070 ... 0x807F	87	D2	16 values
0x8080 ... 0x808F	88	DD	16 values
0x8090 ... 0x809F	89	DC	16 values
0x80A0 ... 0x80AF	8A	DF	16 values
0x80B0 ... 0x80BF	8B	DE	16 values
0x80C0 ... 0x80CF	8C	D9	16 values
0x80D0 ... 0x80DF	8D	D8	16 values
0x80E0 ... 0x80EF	8E	DB	16 values
0x80F0 ... 0x80FF	8F	DA	16 values
0x8100 ... 0x810F	90	C5	16 values
0x8110 ... 0x811F	91	C4	16 values
0x8120 ... 0x812F	92	C7	16 values
0x8130 ... 0x813F	93	C6	16 values
0x8140 ... 0x814F	94	C1	16 values
0x8150 ... 0x815F	95	C0	16 values
0x8160 ... 0x816F	96	C3	16 values
0x8170 ... 0x817F	97	C2	16 values
0x8180 ... 0x818F	98	CD	16 values
0x8190 ... 0x819F	99	CC	16 values
0x81A0 ... 0x81AF	9A	CF	16 values
0x81B0 ... 0x81BF	9B	CE	16 values
0x81C0 ... 0x81CF	9C	C9	16 values
0x81D0 ... 0x81DF	9D	C8	16 values
0x81E0 ... 0x81EF	9E	CB	16 values
0x81F0 ... 0x81FF	9F	CA	16 values
0x8200 ... 0x821F	A0	F5	32 values
0x8220 ... 0x823F	A1	F4	32 values
0x8240 ... 0x825F	A2	F7	32 values
0x8260 ... 0x827F	A3	F6	32 values
0x8280 ... 0x829F	A4	F1	32 values
0x82A0 ... 0x82BF	A5	F0	32 values
0x82C0 ... 0x82DF	A6	F3	32 values
0x82E0 ... 0x82FF	A7	F2	32 values
0x8300 ... 0x831F	A8	FD	32 values
0x8320 ... 0x833F	A9	FC	32 values
0x8340 ... 0x835F	AA	FF	32 values
0x8360 ... 0x837F	AB	FE	32 values
0x8380 ... 0x839F	AC	F9	32 values
0x83A0 ... 0x83BF	AD	F8	32 values
0x83C0 ... 0x83DF	AE	FB	32 values
0x83E0 ... 0x83FF	AF	FA	32 values
0x8400 ... 0x843F	B0	E5	64 values
0x8440 ... 0x847F	B1	E4	64 values
0x8480 ... 0x84BF	B2	E7	64 values
0x84C0 ... 0x84FF	B3	E6	64 values
0x8500 ... 0x853F	B4	E1	64 values
0x8540 ... 0x857F	B5	E0	64 values
0x8580 ... 0x85BF	B6	E3	64 values
0x85C0 ... 0x85FF	B7	E2	64 values
0x8600 ... 0x863F	B8	ED	64 values
0x8640 ... 0x867F	B9	EC	64 values
0x8680 ... 0x86BF	BA	EF	64 values
0x86C0 ... 0x86FF	BB	EE	64 values
0x8700 ... 0x873F	BC	E9	64 values
0x8740 ... 0x877F	BD	E8	64 values
0x8780 ... 0x87BF	BE	EB	64 values
0x87C0 ... 0x87FF	BF	EA	64 values
0x8800 ... 0x887F	C0	95	128 values
0x8880 ... 0x88FF	C1	94	128 values
0x8900 ... 0x897F	C2	97	128 values
0x8980 ... 0x89FF	C3	96	128 values
0x8A00 ... 0x8A7F	C4	91	128 values
0x8A80 ... 0x8AFF	C5	90	128 values
0x8B00 ... 0x8B7F	C6	93	128 values
0x8B80 ... 0x8BFF	C7	92	128 values
0x8C00 ... 0x8C7F	C8	9D	128 values
0x8C80 ... 0x8CFF	C9	9C	128 values
0x8D00 ... 0x8D7F	CA	9F	128 values
0x8D80 ... 0x8DFF	CB	9E	128 values
0x8E00 ... 0x8E7F	CC	99	128 values
0x8E80 ... 0x8EFF	CD	98	128 values
0x8F00 ... 0x8F7F	CE	9B	128 values
0x8F80 ... 0x8FFF	CF	9A	128 values
0x9000 ... 0x90FF	D0	85	256 values
0x9100 ... 0x91FF	D1	84	256 values
0x9200 ... 0x92FF	D2	87	256 values
0x9300 ... 0x93FF	D3	86	256 values
0x9400 ... 0x94FF	D4	81	256 values
0x9500 ... 0x95FF	D5	80	256 values
0x9600 ... 0x96FF	D6	83	256 values
0x9700 ... 0x97FF	D7	82	256 values
0x9800 ... 0x98FF	D8	8D	256 values
0x9900 ... 0x99FF	D9	8C	256 values
0x9A00 ... 0x9AFF	DA	8F	256 values
0x9B00 ... 0x9BFF	DB	8E	256 values
0x9C00 ... 0x9CFF	DC	89	256 values
0x9D00 ... 0x9DFF	DD	88	256 values
0x9E00 ... 0x9EFF	DE	8B	256 values
0x9F00 ... 0x9FFF	DF	8A	256 values
0xA000 ... 0xA1FF	E0	B5	512 values
0xA200 ... 0xA3FF	E1	B4	512 values
0xA400 ... 0xA5FF	E2	B7	512 values
0xA600 ... 0xA7FF	E3	B6	512 values
0xA800 ... 0xA9FF	E4	B1	512 values
0xAA00 ... 0xABFF	E5	B0	512 values
0xAC00 ... 0xADFF	E6	B3	512 values
0xAE00 ... 0xAFFF	E7	B2	512 values
0xB000 ... 0xB1FF	E8	BD	512 values
0xB200 ... 0xB3FF	E9	BC	512 values
0xB400 ... 0xB5FF	EA	BF	512 values
0xB600 ... 0xB7FF	EB	BE	512 values
0xB800 ... 0xB9FF	EC	B9	512 values
0xBA00 ... 0xBBFF	ED	B8	512 values
0xBC00 ... 0xBDFF	EE	BB	512 values
0xBE00 ... 0xBFFF	EF	BA	512 values
0xC000 ... 0xC3FF	F0	A5	1024 values
0xC400 ... 0xC7FF	F1	A4	1024 values
0xC800 ... 0xCBFF	F2	A7	1024 values
0xCC00 ... 0xCFFF	F3	A6	1024 values
0xD000 ... 0xD3FF	F4	A1	1024 values
0xD400 ... 0xD7FF	F5	A0	1024 values
0xD800 ... 0xDBFF	F6	A3	1024 values
0xDC00 ... 0xDFFF	F7	A2	1024 values
0xE000 ... 0xE3FF	F8	AD	1024 values
0xE400 ... 0xE7FF	F9	AC	1024 values
0xE800 ... 0xEBFF	FA	AF	1024 values
0xEC00 ... 0xEFFF	FB	AE	1024 values
0xF000 ... 0xF3FF	FC	A9	1024 values
0xF400 ... 0xF7FF	FD	A8	1024 values
0xF800 ... 0xFBFF	FE	AB	1024 values
0xFC00 ... 0xFFFF	FF	AA	1024 values

What do you think about such a table? --RokerHRO (talk) 21:53, 16 April 2010 (UTC)

I think such a table makes the article longish and hard to read. If a table is to be included perhaps in 8×8 form of somekind for compactness. Also might useful to contrast with u-law values for same codepoint. In general I think a more graphical / illustrative approach showing the quantized code points vs continuous 16 bit would be more educational than a full table. Jaxbax7 (talk) 21:15, 7 January 2014 (UTC)

and what would be the decoding table?!

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) 18:02, 8 June 2021 (UTC)

Origin point for A/mu Law lines on chart questionable?

As I also just commented on the Mu-Law talk page - shouldn't the blue and red lines also start at 0:0, same as the green one, as these are all digital systems? An analogue setup doing the same may produce over-unity signals, but in this context either anything over 0dB will get clipped, or the whole line needs to be shifted to the right (on this chart) to normalise 0dB "encoded" with 0dB input/output and avoid that? 92.12.87.15 (talk) 18:04, 31 October 2023 (UTC)

Overall quality of sound samples don't match

And again, did just point this out on Mu-Law as well - the general quality of the "original" sample is basically CD grade, but the "encoded" one is phone grade. That decimation is not part of the standard itself and makes the encoding seem much lower quality than it actually is... I expect whoever prepared the samples didn't appreciate that point and used a simple generic converter which does all the parts in one (companding to 8 bits, reducing sample rate, merging channels etc).

Either a new "encoded" version needs to be prepared from the original with the only change being from "14" bit PCM (actually 16, but not hard to flatten the two LSBs by normalising to 25% then back to 100%) to 8-bit A-law, or the "original" needs to be decimated in the same way as the encoded one except for it remaining full precision PCM, so the actual effect of the encoding on noise and dynamics can be demonstrated. (I can't do it due to a lack of WP account, else I'd have just done it - it's not complicated to export something suitable from e.g. Audacity)

...and as a further point I *didn't* put on mu-law - perhaps a third sample could be provided as a comparison, that being straight 8-bit PCM, to show how the perceptual quality of the companded version sits somewhere between that and the original? (Maybe a fourth ADPCM or GSM one too to show an alternative method of crushing the data even further, if that's not excessive - it's not like they take up much page space or even a lot of data on the server... though I don't know of a GSM encoder that goes above 16kHz so that'd demand everything being reduced to that level as a maximum...) 92.12.87.15 (talk) 18:12, 31 October 2023 (UTC)

@Morn: These audio examples here and at µ-law are useless and misleading in their current form. The only noticeable difference between the original and the one labelled A-law (resp. µ-law) is that the latter is low-pass filtered.

This is, of course, an antialiasing filter used when downsampling to 8kHz, but if we want to show the effect of the companding itself (as the captions seem to imply), then the sample encoding should be the only difference between them and both should have the same sampling rate. However, this wouldn't be very illustrative either, because the difference between 8-bit A-/µ-law and the original 16-bit linear PCM would not be audible to most readers (if it's perceptible at all with speech). I think the IP above has a good idea re including 8-bit linear PCM for comparison, as this would necessarily have raised quantization noise to easily audible levels while A-/µ-law would keep it much lower at the same bitrate. – MwGamera (talk) 16:03, 25 November 2023 (UTC)

I modified the comparison to include A-/µ-law-companded samples (expanded without dithering back to 16-bit with 13/14 significant bits, respectively, because Commons doesn't allow A- or µ-law encoded WAVs) and with 8- and 16-bit linear PCM for comparison, all sampled at 8 kHz. I used the original audio that Morn provided, but perhaps it might be a good idea to have some jingle or something else besides speech to make the differences stand out more. – MwGamera (talk) 00:59, 27 November 2023 (UTC)

A-law as defined in G.711

Article neglects to mention that A-law as defined in G.711 is not actually defined in terms of the continuous equation but a piecewise linear approximation of it. This is mentioned in the readme of the linked C code as well as in various application notes, e.g. https://www.nxp.com/docs/en/application-note/ANE408.pdf: "Mathematically speaking, MU & A-laws are piece-wise linear approximations of the above equations." 2601:644:8100:168D:CDF5:858A:C61:502A (talk) 00:53, 26 November 2024 (UTC)

Value of A

The constant A appears to come from nowhere. In fact, it is chosen precisely so the encoding increases by 16 if you double the input, making it easy to encode and decode. To be precise, A is the solution to the equation A(1+ln(A))=16, giving 87,5565596972731855380546823552 if you want a few more digits :-) Jurjen B (talk) 14:13, 6 June 2026 (UTC)