MP3 decoding tables

The fixed tables and formulas of MPEG audio decoding, as the explorer’s decoder uses them: the Layer III Huffman tables, the inverse MDCT and its windows, and the synthesis filterbank that all three layers end with. Every number on this page is checked by the site’s tests against the decoder, whose output matches mpg123’s. The other tables (bit rates, scalefactor bands, allocation) are on the pages for the frame header, main data and Layers I and II.

Huffman tables

Each big-value table codes a pair (x, y), each from 0 to 15; the value below is x × 16 + y. A table is written here as a decoding tree, the compact form used by mpg123 and by the explorer, which holds exactly the same codes as the standard’s lists of code words and lengths. To decode, start at the first entry, and repeat: read the entry and step past it; if it is negative, read one bit, and if the bit is 1 skip forward by minus that entry; if it is zero or more, it is the decoded value.

For example, table 1 is -5, -3, -1, 17, 1, 16, 0: the code 1 is (0, 0), 01 is (1, 0), 001 is (0, 1) and 000 is (1, 1).

Table	Tree	Largest value	Linbits
0	none: every value is 0 and no bits are read	0	0
1, 2, 3, 5 to 13, 15	its own, below	1, 2, 2, 3, 3, 5, 5, 5, 7, 7, 7, 15, 15	0
4, 14	not used
16 to 23	table 16’s	15, then linbits	1, 2, 3, 4, 6, 8, 10, 13
24 to 31	table 24’s	15, then linbits	4, 5, 6, 7, 8, 9, 11, 13

After the code: for x, if x is 15 and the table has linbits, that many bits giving the amount above 15; then a sign bit if x is not 0 (1 means negative); then the same for y.

Table 1

-5, -3, -1, 17, 1, 16, 0

Table 2

-15, -11, -9, -5, -3, -1, 34, 2, 18, -1, 33, 32, 17, -1, 1, 16, 0

Table 3

-13, -11, -9, -5, -3, -1, 34, 2, 18, -1, 33, 32, 16, 17, -1, 1, 0

Table 5

-29, -25, -23, -15, -7, -5, -3, -1, 51, 35, 50, 49, -3, -1, 19, 3, -1, 48, 34, -3, -1, 18, 33, -1, 2, 32, 17, -1, 1, 16, 0

Table 6

-25, -19, -13, -9, -5, -3, -1, 51, 3, 35, -1, 50, 48, -1, 19, 49, -3, -1, 34, 2, 18, -3, -1, 33, 32, 1, -1, 17, -1, 16, 0

Table 7

-69, -65, -57, -39, -29, -17, -11, -7, -3, -1, 85, 69, -1, 84, 83, -1, 53, 68, -3, -1, 37, 82, 21, -5, -1, 81, -1, 5, 52, -1,
80, -1, 67, 51, -5, -3, -1, 36, 66, 20, -1, 65, 64, -11, -7, -3, -1, 4, 35, -1, 50, 3, -1, 19, 49, -3, -1, 48, 34, 18,
-5, -1, 33, -1, 2, 32, 17, -1, 1, 16, 0

Table 8

-65, -63, -59, -45, -31, -19, -13, -7, -5, -3, -1, 85, 84, 69, 83, -3, -1, 53, 68, 37, -3, -1, 82, 5, 21, -5, -1, 81, -1, 52,
67, -3, -1, 80, 51, 36, -5, -3, -1, 66, 20, 65, -3, -1, 4, 64, -1, 35, 50, -9, -7, -3, -1, 19, 49, -1, 3, 48, 34, -1,
2, 32, -1, 18, 33, 17, -3, -1, 1, 16, 0

Table 9

-63, -53, -41, -29, -19, -11, -5, -3, -1, 85, 69, 53, -1, 83, -1, 84, 5, -3, -1, 68, 37, -1, 82, 21, -3, -1, 81, 52, -1, 67,
-1, 80, 4, -7, -3, -1, 36, 66, -1, 51, 64, -1, 20, 65, -5, -3, -1, 35, 50, 19, -1, 49, -1, 3, 48, -5, -3, -1, 34, 2,
18, -1, 33, 32, -3, -1, 17, 1, -1, 16, 0

Table 10

-125, -121, -111, -83, -55, -35, -21, -13, -7, -3, -1, 119, 103, -1, 118, 87, -3, -1, 117, 102, 71, -3, -1, 116, 86, -1, 101, 55, -9, -3,
-1, 115, 70, -3, -1, 85, 84, 99, -1, 39, 114, -11, -5, -3, -1, 100, 7, 112, -1, 98, -1, 69, 53, -5, -1, 6, -1, 83, 68, 23,
-17, -5, -1, 113, -1, 54, 38, -5, -3, -1, 37, 82, 21, -1, 81, -1, 52, 67, -3, -1, 22, 97, -1, 96, -1, 5, 80, -19, -11, -7,
-3, -1, 36, 66, -1, 51, 4, -1, 20, 65, -3, -1, 64, 35, -1, 50, 3, -3, -1, 19, 49, -1, 48, 34, -7, -3, -1, 18, 33, -1,
2, 32, 17, -1, 1, 16, 0

Table 11

-121, -113, -89, -59, -43, -27, -17, -7, -3, -1, 119, 103, -1, 118, 117, -3, -1, 102, 71, -1, 116, -1, 87, 85, -5, -3, -1, 86, 101, 55,
-1, 115, 70, -9, -7, -3, -1, 69, 84, -1, 53, 83, 39, -1, 114, -1, 100, 7, -5, -1, 113, -1, 23, 112, -3, -1, 54, 99, -1, 96,
-1, 68, 37, -13, -7, -5, -3, -1, 82, 5, 21, 98, -3, -1, 38, 6, 22, -5, -1, 97, -1, 81, 52, -5, -1, 80, -1, 67, 51, -1,
36, 66, -15, -11, -7, -3, -1, 20, 65, -1, 4, 64, -1, 35, 50, -1, 19, 49, -5, -3, -1, 3, 48, 34, 33, -5, -1, 18, -1, 2,
32, 17, -3, -1, 1, 16, 0

Table 12

-115, -99, -73, -45, -27, -17, -9, -5, -3, -1, 119, 103, 118, -1, 87, 117, -3, -1, 102, 71, -1, 116, 101, -3, -1, 86, 55, -3, -1, 115,
85, 39, -7, -3, -1, 114, 70, -1, 100, 23, -5, -1, 113, -1, 7, 112, -1, 54, 99, -13, -9, -3, -1, 69, 84, -1, 68, -1, 6, 5,
-1, 38, 98, -5, -1, 97, -1, 22, 96, -3, -1, 53, 83, -1, 37, 82, -17, -7, -3, -1, 21, 81, -1, 52, 67, -5, -3, -1, 80, 4,
36, -1, 66, 20, -3, -1, 51, 65, -1, 35, 50, -11, -7, -5, -3, -1, 64, 3, 48, 19, -1, 49, 34, -1, 18, 33, -7, -5, -3, -1,
2, 32, 0, 17, -1, 1, 16

Table 13

-509, -503, -475, -405, -333, -265, -205, -153, -115, -83, -53, -35, -21, -13, -9, -7, -5, -3, -1, 254, 252, 253, 237, 255, -1, 239, 223, -3, -1, 238,
207, -1, 222, 191, -9, -3, -1, 251, 206, -1, 220, -1, 175, 233, -1, 236, 221, -9, -5, -3, -1, 250, 205, 190, -1, 235, 159, -3, -1, 249,
234, -1, 189, 219, -17, -9, -3, -1, 143, 248, -1, 204, -1, 174, 158, -5, -1, 142, -1, 127, 126, 247, -5, -1, 218, -1, 173, 188, -3, -1,
203, 246, 111, -15, -7, -3, -1, 232, 95, -1, 157, 217, -3, -1, 245, 231, -1, 172, 187, -9, -3, -1, 79, 244, -3, -1, 202, 230, 243, -1,
63, -1, 141, 216, -21, -9, -3, -1, 47, 242, -3, -1, 110, 156, 15, -5, -3, -1, 201, 94, 171, -3, -1, 125, 215, 78, -11, -5, -3, -1,
200, 214, 62, -1, 185, -1, 155, 170, -1, 31, 241, -23, -13, -5, -1, 240, -1, 186, 229, -3, -1, 228, 140, -1, 109, 227, -5, -1, 226, -1,
46, 14, -1, 30, 225, -15, -7, -3, -1, 224, 93, -1, 213, 124, -3, -1, 199, 77, -1, 139, 184, -7, -3, -1, 212, 154, -1, 169, 108, -1,
198, 61, -37, -21, -9, -5, -3, -1, 211, 123, 45, -1, 210, 29, -5, -1, 183, -1, 92, 197, -3, -1, 153, 122, 195, -7, -5, -3, -1, 167,
151, 75, 209, -3, -1, 13, 208, -1, 138, 168, -11, -7, -3, -1, 76, 196, -1, 107, 182, -1, 60, 44, -3, -1, 194, 91, -3, -1, 181, 137,
28, -43, -23, -11, -5, -1, 193, -1, 152, 12, -1, 192, -1, 180, 106, -5, -3, -1, 166, 121, 59, -1, 179, -1, 136, 90, -11, -5, -1, 43,
-1, 165, 105, -1, 164, -1, 120, 135, -5, -1, 148, -1, 119, 118, 178, -11, -3, -1, 27, 177, -3, -1, 11, 176, -1, 150, 74, -7, -3, -1,
58, 163, -1, 89, 149, -1, 42, 162, -47, -23, -9, -3, -1, 26, 161, -3, -1, 10, 104, 160, -5, -3, -1, 134, 73, 147, -3, -1, 57, 88,
-1, 133, 103, -9, -3, -1, 41, 146, -3, -1, 87, 117, 56, -5, -1, 131, -1, 102, 71, -3, -1, 116, 86, -1, 101, 115, -11, -3, -1, 25,
145, -3, -1, 9, 144, -1, 72, 132, -7, -5, -1, 114, -1, 70, 100, 40, -1, 130, 24, -41, -27, -11, -5, -3, -1, 55, 39, 23, -1, 113,
-1, 85, 7, -7, -3, -1, 112, 54, -1, 99, 69, -3, -1, 84, 38, -1, 98, 53, -5, -1, 129, -1, 8, 128, -3, -1, 22, 97, -1, 6,
96, -13, -9, -5, -3, -1, 83, 68, 37, -1, 82, 5, -1, 21, 81, -7, -3, -1, 52, 67, -1, 80, 36, -3, -1, 66, 51, 20, -19, -11,
-5, -1, 65, -1, 4, 64, -3, -1, 35, 50, 19, -3, -1, 49, 3, -1, 48, 34, -3, -1, 18, 33, -1, 2, 32, -3, -1, 17, 1, 16, 0

Table 15

-495, -445, -355, -263, -183, -115, -77, -43, -27, -13, -7, -3, -1, 255, 239, -1, 254, 223, -1, 238, -1, 253, 207, -7, -3, -1, 252, 222, -1, 237,
191, -1, 251, -1, 206, 236, -7, -3, -1, 221, 175, -1, 250, 190, -3, -1, 235, 205, -1, 220, 159, -15, -7, -3, -1, 249, 234, -1, 189, 219,
-3, -1, 143, 248, -1, 204, 158, -7, -3, -1, 233, 127, -1, 247, 173, -3, -1, 218, 188, -1, 111, -1, 174, 15, -19, -11, -3, -1, 203, 246,
-3, -1, 142, 232, -1, 95, 157, -3, -1, 245, 126, -1, 231, 172, -9, -3, -1, 202, 187, -3, -1, 217, 141, 79, -3, -1, 244, 63, -1, 243,
216, -33, -17, -9, -3, -1, 230, 47, -1, 242, -1, 110, 240, -3, -1, 31, 241, -1, 156, 201, -7, -3, -1, 94, 171, -1, 186, 229, -3, -1,
125, 215, -1, 78, 228, -15, -7, -3, -1, 140, 200, -1, 62, 109, -3, -1, 214, 227, -1, 155, 185, -7, -3, -1, 46, 170, -1, 226, 30, -5,
-1, 225, -1, 14, 224, -1, 93, 213, -45, -25, -13, -7, -3, -1, 124, 199, -1, 77, 139, -1, 212, -1, 184, 154, -7, -3, -1, 169, 108, -1,
198, 61, -1, 211, 210, -9, -5, -3, -1, 45, 13, 29, -1, 123, 183, -5, -1, 209, -1, 92, 208, -1, 197, 138, -17, -7, -3, -1, 168, 76,
-1, 196, 107, -5, -1, 182, -1, 153, 12, -1, 60, 195, -9, -3, -1, 122, 167, -1, 166, -1, 192, 11, -1, 194, -1, 44, 91, -55, -29, -15,
-7, -3, -1, 181, 28, -1, 137, 152, -3, -1, 193, 75, -1, 180, 106, -5, -3, -1, 59, 121, 179, -3, -1, 151, 136, -1, 43, 90, -11, -5,
-1, 178, -1, 165, 27, -1, 177, -1, 176, 105, -7, -3, -1, 150, 74, -1, 164, 120, -3, -1, 135, 58, 163, -17, -7, -3, -1, 89, 149, -1,
42, 162, -3, -1, 26, 161, -3, -1, 10, 160, 104, -7, -3, -1, 134, 73, -1, 148, 57, -5, -1, 147, -1, 119, 9, -1, 88, 133, -53, -29,
-13, -7, -3, -1, 41, 103, -1, 118, 146, -1, 145, -1, 25, 144, -7, -3, -1, 72, 132, -1, 87, 117, -3, -1, 56, 131, -1, 102, 71, -7,
-3, -1, 40, 130, -1, 24, 129, -7, -3, -1, 116, 8, -1, 128, 86, -3, -1, 101, 55, -1, 115, 70, -17, -7, -3, -1, 39, 114, -1, 100,
23, -3, -1, 85, 113, -3, -1, 7, 112, 54, -7, -3, -1, 99, 69, -1, 84, 38, -3, -1, 98, 22, -3, -1, 6, 96, 53, -33, -19, -9,
-5, -1, 97, -1, 83, 68, -1, 37, 82, -3, -1, 21, 81, -3, -1, 5, 80, 52, -7, -3, -1, 67, 36, -1, 66, 51, -1, 65, -1, 20,
4, -9, -3, -1, 35, 50, -3, -1, 64, 3, 19, -3, -1, 49, 48, 34, -9, -7, -3, -1, 18, 33, -1, 2, 32, 17, -3, -1, 1, 16, 0

Table 16 (and 17 to 23)

-509, -503, -461, -323, -103, -37, -27, -15, -7, -3, -1, 239, 254, -1, 223, 253, -3, -1, 207, 252, -1, 191, 251, -5, -1, 175, -1, 250, 159, -3,
-1, 249, 248, 143, -7, -3, -1, 127, 247, -1, 111, 246, 255, -9, -5, -3, -1, 95, 245, 79, -1, 244, 243, -53, -1, 240, -1, 63, -29, -19,
-13, -7, -5, -1, 206, -1, 236, 221, 222, -1, 233, -1, 234, 217, -1, 238, -1, 237, 235, -3, -1, 190, 205, -3, -1, 220, 219, 174, -11, -5,
-1, 204, -1, 173, 218, -3, -1, 126, 172, 202, -5, -3, -1, 201, 125, 94, 189, 242, -93, -5, -3, -1, 47, 15, 31, -1, 241, -49, -25, -13,
-5, -1, 158, -1, 188, 203, -3, -1, 142, 232, -1, 157, 231, -7, -3, -1, 187, 141, -1, 216, 110, -1, 230, 156, -13, -7, -3, -1, 171, 186,
-1, 229, 215, -1, 78, -1, 228, 140, -3, -1, 200, 62, -1, 109, -1, 214, 155, -19, -11, -5, -3, -1, 185, 170, 225, -1, 212, -1, 184, 169,
-5, -1, 123, -1, 183, 208, 227, -7, -3, -1, 14, 224, -1, 93, 213, -3, -1, 124, 199, -1, 77, 139, -75, -45, -27, -13, -7, -3, -1, 154,
108, -1, 198, 61, -3, -1, 92, 197, 13, -7, -3, -1, 138, 168, -1, 153, 76, -3, -1, 182, 122, 60, -11, -5, -3, -1, 91, 137, 28, -1,
192, -1, 152, 121, -1, 226, -1, 46, 30, -15, -7, -3, -1, 211, 45, -1, 210, 209, -5, -1, 59, -1, 151, 136, 29, -7, -3, -1, 196, 107,
-1, 195, 167, -1, 44, -1, 194, 181, -23, -13, -7, -3, -1, 193, 12, -1, 75, 180, -3, -1, 106, 166, 179, -5, -3, -1, 90, 165, 43, -1,
178, 27, -13, -5, -1, 177, -1, 11, 176, -3, -1, 105, 150, -1, 74, 164, -5, -3, -1, 120, 135, 163, -3, -1, 58, 89, 42, -97, -57, -33,
-19, -11, -5, -3, -1, 149, 104, 161, -3, -1, 134, 119, 148, -5, -3, -1, 73, 87, 103, 162, -5, -1, 26, -1, 10, 160, -3, -1, 57, 147,
-1, 88, 133, -9, -3, -1, 41, 146, -3, -1, 118, 9, 25, -5, -1, 145, -1, 144, 72, -3, -1, 132, 117, -1, 56, 131, -21, -11, -5, -3,
-1, 102, 40, 130, -3, -1, 71, 116, 24, -3, -1, 129, 128, -3, -1, 8, 86, 55, -9, -5, -1, 115, -1, 101, 70, -1, 39, 114, -5, -3,
-1, 100, 85, 7, 23, -23, -13, -5, -1, 113, -1, 112, 54, -3, -1, 99, 69, -1, 84, 38, -3, -1, 98, 22, -1, 97, -1, 6, 96, -9,
-5, -1, 83, -1, 53, 68, -1, 37, 82, -1, 81, -1, 21, 5, -33, -23, -13, -7, -3, -1, 52, 67, -1, 80, 36, -3, -1, 66, 51, 20,
-5, -1, 65, -1, 4, 64, -1, 35, 50, -3, -1, 19, 49, -3, -1, 3, 48, 34, -3, -1, 18, 33, -1, 2, 32, -3, -1, 17, 1, 16, 0

Table 24 (and 25 to 31)

-451, -117, -43, -25, -15, -7, -3, -1, 239, 254, -1, 223, 253, -3, -1, 207, 252, -1, 191, 251, -5, -1, 250, -1, 175, 159, -1, 249, 248, -9,
-5, -3, -1, 143, 127, 247, -1, 111, 246, -3, -1, 95, 245, -1, 79, 244, -71, -7, -3, -1, 63, 243, -1, 47, 242, -5, -1, 241, -1, 31,
240, -25, -9, -1, 15, -3, -1, 238, 222, -1, 237, 206, -7, -3, -1, 236, 221, -1, 190, 235, -3, -1, 205, 220, -1, 174, 234, -15, -7, -3,
-1, 189, 219, -1, 204, 158, -3, -1, 233, 173, -1, 218, 188, -7, -3, -1, 203, 142, -1, 232, 157, -3, -1, 217, 126, -1, 231, 172, 255, -235,
-143, -77, -45, -25, -15, -7, -3, -1, 202, 187, -1, 141, 216, -5, -3, -1, 14, 224, 13, 230, -5, -3, -1, 110, 156, 201, -1, 94, 186, -9,
-5, -1, 229, -1, 171, 125, -1, 215, 228, -3, -1, 140, 200, -3, -1, 78, 46, 62, -15, -7, -3, -1, 109, 214, -1, 227, 155, -3, -1, 185,
170, -1, 226, 30, -7, -3, -1, 225, 93, -1, 213, 124, -3, -1, 199, 77, -1, 139, 184, -31, -15, -7, -3, -1, 212, 154, -1, 169, 108, -3,
-1, 198, 61, -1, 211, 45, -7, -3, -1, 210, 29, -1, 123, 183, -3, -1, 209, 92, -1, 197, 138, -17, -7, -3, -1, 168, 153, -1, 76, 196,
-3, -1, 107, 182, -3, -1, 208, 12, 60, -7, -3, -1, 195, 122, -1, 167, 44, -3, -1, 194, 91, -1, 181, 28, -57, -35, -19, -7, -3, -1,
137, 152, -1, 193, 75, -5, -3, -1, 192, 11, 59, -3, -1, 176, 10, 26, -5, -1, 180, -1, 106, 166, -3, -1, 121, 151, -3, -1, 160, 9,
144, -9, -3, -1, 179, 136, -3, -1, 43, 90, 178, -7, -3, -1, 165, 27, -1, 177, 105, -1, 150, 164, -17, -9, -5, -3, -1, 74, 120, 135,
-1, 58, 163, -3, -1, 89, 149, -1, 42, 162, -7, -3, -1, 161, 104, -1, 134, 119, -3, -1, 73, 148, -1, 57, 147, -63, -31, -15, -7, -3,
-1, 88, 133, -1, 41, 103, -3, -1, 118, 146, -1, 25, 145, -7, -3, -1, 72, 132, -1, 87, 117, -3, -1, 56, 131, -1, 102, 40, -17, -7,
-3, -1, 130, 24, -1, 71, 116, -5, -1, 129, -1, 8, 128, -1, 86, 101, -7, -5, -1, 23, -1, 7, 112, 115, -3, -1, 55, 39, 114, -15,
-7, -3, -1, 70, 100, -1, 85, 113, -3, -1, 54, 99, -1, 69, 84, -7, -3, -1, 38, 98, -1, 22, 97, -5, -3, -1, 6, 96, 53, -1,
83, 68, -51, -37, -23, -15, -9, -3, -1, 37, 82, -1, 21, -1, 5, 80, -1, 81, -1, 52, 67, -3, -1, 36, 66, -1, 51, 20, -9, -5,
-1, 65, -1, 4, 64, -1, 35, 50, -1, 19, 49, -7, -5, -3, -1, 3, 48, 34, 18, -1, 33, -1, 2, 32, -3, -1, 17, 1, -1, 16, 0

Count1 tables

Above the big values, each code stands for four lines v, w, x and y, each 0 or 1 in size; a sign bit follows each that is 1, in that order. count1table_select chooses the table.

v w x y	Table A	Table B
0 0 0 0	`1`	`1111`
0 0 0 1	`0101`	`1110`
0 0 1 0	`0100`	`1101`
0 0 1 1	`00101`	`1100`
0 1 0 0	`0110`	`1011`
0 1 0 1	`000101`	`1010`
0 1 1 0	`00100`	`1001`
0 1 1 1	`000100`	`1000`
1 0 0 0	`0111`	`0111`
1 0 0 1	`00011`	`0110`
1 0 1 0	`00110`	`0101`
1 0 1 1	`000000`	`0100`
1 1 0 0	`00111`	`0011`
1 1 0 1	`000010`	`0010`
1 1 1 0	`000011`	`0001`
1 1 1 1	`000001`	`0000`

The inverse MDCT and its windows

For each of the 32 subbands, Layer III turns 18 frequency lines into 36 time samples. A long block uses one transform with n = 36; a short block uses three with n = 12, one for each window. With X_k the inputs and n the transform size:

x_i = Σ_{k = 0}^{n/2 − 1} X_k cos( π / (2n) × (2i + 1 + n/2) × (2k + 1) ), i = 0 … n − 1

Each output is then multiplied by the window for the block type, z_i = x_i × w_i:

Block type	i	w_i
0, normal	0 to 35	sin( π / 36 × (i + ½) )
1, start	0 to 17	sin( π / 36 × (i + ½) )
	18 to 23	1
	24 to 29	sin( π / 12 × (i − 18 + ½) )
	30 to 35	0
2, short	0 to 11, for each window	sin( π / 12 × (i + ½) )
3, stop	0 to 5	0
	6 to 11	sin( π / 12 × (i − 6 + ½) )
	12 to 17	1
	18 to 35	sin( π / 36 × (i + ½) )

For a short block, window w of the three (w = 0, 1, 2) takes the inputs X_k = line 3k + w after reordering, and its 12 windowed outputs are added into the 36 at positions 6 + 6w to 17 + 6w; positions 0 to 5 and 30 to 35 stay 0. In a mixed block, subbands 0 and 1 use the normal window.

Overlap and add: the first 18 values of z, plus the last 18 values kept from the same subband in the previous granule, are this granule’s 18 output samples; the last 18 of z are kept for the next. Then every odd sample of every odd subband is negated (frequency inversion), and the 18 time slots of 32 subband samples go to the synthesis filterbank.

The synthesis filterbank

All three layers rebuild sound from 32 subband samples at a time, S₀ to S₃₁, which give 32 output samples. The decoder keeps a vector V of 1024 values, and for each new set of 32:

Shift V along by 64: V_i = V_{i − 64} for i from 1023 down to 64.
Matrixing: V_i = Σ_{k = 0}³¹ N_i,k × S_k for i = 0 … 63, with N_i,k = cos( (16 + i)(2k + 1) π / 64 ).
Build U of 512 values: for i = 0 … 7 and j = 0 … 31, U_{64i + j} = V_{128i + j} and U_{64i + 32 + j} = V_{128i + 96 + j}.
Window: W_i = U_i × D_i for i = 0 … 511.
Output sample j = Σ_{i = 0}¹⁵ W_{j + 32i}, for j = 0 … 31.

The matrixing is a 32-point DCT in disguise, which fast decoders, the explorer’s included, compute as one. The outputs are in the range −1 to 1 for full scale.

The window D

The 512 coefficients are fixed. The first 257, D₀ to D₂₅₆, are the numbers below divided by 65,536. The rest follow from them: for i = 1 … 255, D_{512 − i} = −D_i, except that when i is a multiple of 64 (64, 128, 192), D_{512 − i} = D_i. So D₁ = −0.000015259 and D₂₅₆ = 1.144989014, the largest.

0, -1, -1, -1, -1, -1, -1, -2, -2, -2, -2, -3, -3, -4, -4, -5,
-5, -6, -7, -7, -8, -9, -10, -11, -13, -14, -16, -17, -19, -21, -24, -26,
-29, -31, -35, -38, -41, -45, -49, -53, -58, -63, -68, -73, -79, -85, -91, -97,
-104, -111, -117, -125, -132, -139, -147, -154, -161, -169, -176, -183, -190, -196, -202, -208,
213, 218, 222, 225, 227, 228, 228, 227, 224, 221, 215, 208, 200, 189, 177, 163,
146, 127, 106, 83, 57, 29, -2, -36, -72, -111, -153, -197, -244, -294, -347, -401,
-459, -519, -581, -645, -711, -779, -848, -919, -991, -1064, -1137, -1210, -1283, -1356, -1428, -1498,
-1567, -1634, -1698, -1759, -1817, -1870, -1919, -1962, -2001, -2032, -2057, -2075, -2085, -2087, -2080, -2063,
2037, 2000, 1952, 1893, 1822, 1739, 1644, 1535, 1414, 1280, 1131, 970, 794, 605, 402, 185,
-45, -288, -545, -814, -1095, -1388, -1692, -2006, -2330, -2663, -3004, -3351, -3705, -4063, -4425, -4788,
-5153, -5517, -5879, -6237, -6589, -6935, -7271, -7597, -7910, -8209, -8491, -8755, -8998, -9219, -9416, -9585,
-9727, -9838, -9916, -9959, -9966, -9935, -9863, -9750, -9592, -9389, -9139, -8840, -8492, -8092, -7640, -7134,
6574, 5959, 5288, 4561, 3776, 2935, 2037, 1082, 70, -998, -2122, -3300, -4533, -5818, -7154, -8540,
-9975, -11455, -12980, -14548, -16155, -17799, -19478, -21189, -22929, -24694, -26482, -28289, -30112, -31947, -33791, -35640,
-37489, -39336, -41176, -43006, -44821, -46617, -48390, -50137, -51853, -53534, -55178, -56778, -58333, -59838, -61289, -62684,
-64019, -65290, -66494, -67629, -68692, -69679, -70590, -71420, -72169, -72835, -73415, -73908, -74313, -74630, -74856, -74992,
75038

Other Layer III constants

Requantisation raises each value to the power 4/3 and scales by 2^{(global_gain − 210)/4}; see main data. The alias-reduction coefficients and the intensity-stereo ratios are given there too.

.MP3 File Format – Decoding tables