I have Bob Katz' book. Absolutely great stuff and very readable.
Yes, after normalisation the steps are bigger. The signal itself is, too, so the signal to noise ratio won't change. However, the need for normalisation lies in levels that were too low, so the signal to noise ratio already was bad.
Suppose you're recording at 24 bit and you keep 18 dB headroom.
Every 6 dB is one bit. I explained in another article why that is.
So, 18 dB headroom is 3 bits that are not being used. That still leaves 21 bits. That means you're using 2^21 levels = 2 million levels. The range from minimum signal value to maximum (that is actually being used) is divided into 2 million steps.
So, each step is 1/2.000.000 of the range.
Now we're going to normalize. That means in this case we'll add 18 dB. The whole story shifts 3 bits into the direction of the most significant bit. That means that the three least significant bits are zero - the steps are 2³ times as big as they need to be.
In other words: we add 18 dB to the signal, so we add 18 dB to the quantization error.
Let's visualize. I like pictures and tables.
24 bit information looks like this:
100011101110110110101110
A recorded stream of 24 bit info may look like this (a 7,3 KHz sine wave recorded in 24 bits, close to no headroom):
Code:
23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 dec value samplenr
0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 8000000 0
1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 14731767 1
1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 15274379 2
1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 9128960 3
0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1945580 4
0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 328605 5
0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 5764676 6
1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 13255892 7
1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 0 1 15914865 8
1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 11296947 9
0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 3647831 10
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 78 11
0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 3707416 12
1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 11361336 13
1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 15924858 14
1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 13202302 15
0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 1 0 1 5696773 16
0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 1 0 0 308820 17
0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 1992102 18
1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 1 9199017 19
1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 15303562 20
1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 1 14693245 21
0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 7929189 22
0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1230236 23
0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 755373 24
0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 6941185 25
1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 1 14100467 26
1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1 1 1 15651007 27
1 0 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 10167246 28
0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 2690928 29
0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 95747 30
0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 1 0 4767698 31
1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 1 12411413 32
1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 0 15999294 33
1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1 0 1 12232661 34
0 1 0 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 4574538 35
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 65769 36
0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 2851694 37
1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 10370948 38
1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 15710363 39
1 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 13960905 40
0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 6731018 41
0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 667827 42
0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 1345802 43
0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 8141615 44
1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 14807228 45
1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 1 0 15214306 46
1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 8988584 47
0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1853962 48
You see that the largest value (see decimal column) is about 16.000.000, so we're using the whole range. Sample nr. 33 comes closest, and this gives us a headroom of less than half a dB. So, we're actually recording 24 bit of information.
You also see that the least significant bit (on the right) has both values (0 and 1) and therefore carry information. The least significant bit that is actually used determines the step size, which is actually one bit.
Now, lets use the same signal but with 18 dB of headroom (which is 3 bit - the 3 most significant bits are not being used):
Code:
23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 dec value samplenr
0 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1000000 0
0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1841470 1
0 0 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1909297 2
0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 1141120 3
0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 1 243197 4
0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 41075 5
0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 720584 6
0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1656986 7
0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1989358 8
0 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1412118 9
0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 455978 10
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 9 11
0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 463427 12
0 0 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 1420167 13
0 0 0 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1990607 14
0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1650287 15
0 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 712096 16
0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 38602 17
0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 249012 18
0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1149877 19
0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1912945 20
0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1836655 21
0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 991148 22
0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 153779 23
0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 94421 24
0 0 0 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 867648 25
0 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 1762558 26
0 0 0 1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1956375 27
0 0 0 1 0 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1270905 28
0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0 336366 29
0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 0 11968 30
0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 595962 31
0 0 0 1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 1551426 32
0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1999911 33
0 0 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1529082 34
0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 571817 35
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 8221 36
0 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 356461 37
0 0 0 1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1296368 38
0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 1963795 39
0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 1745113 40
0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 841377 41
0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 83478 42
0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 168225 43
0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1017701 44
0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1850903 45
0 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 1901788 46
0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 1123573 47
0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 231745 48
You see that the three most significant bits (21, 22 and 23) are not being used. You can also see, looking at sample 33, that the highest value is close to 2 million, which is 1/8 of the first stream (close to 16 million). That means the amplitude of the wave is only 1/8 of what it could be.
OK let's correct this by normalizing: we add 18 dB of gain. We multiply every number by 8.
Code:
23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 dec value samplenr
0 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 8000000 0
1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 1 1 0 0 0 0 14731760 1
1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 15274376 2
1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 9128960 3
0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1945576 4
0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 328600 5
0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 5764672 6
1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 13255888 7
1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 0 0 15914864 8
1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 11296944 9
0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 3647824 10
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 72 11
0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 3707416 12
1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 0 11361336 13
1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 15924856 14
1 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 13202296 15
0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 5696768 16
0 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0 0 0 308816 17
0 0 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1992096 18
1 0 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 9199016 19
1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 15303560 20
1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 0 0 0 14693240 21
0 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 7929184 22
0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1230232 23
0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 755368 24
0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6941184 25
1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 14100464 26
1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 15651000 27
1 0 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 10167240 28
0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 2690928 29
0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 95744 30
0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 4767696 31
1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 12411408 32
1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 15999288 33
1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 12232656 34
0 1 0 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1 0 0 0 4574536 35
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 65768 36
0 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 0 2851688 37
1 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 10370944 38
1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 15710360 39
1 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 0 13960904 40
0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 0 0 6731016 41
0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 667824 42
0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1345800 43
0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 0 0 8141608 44
1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 14807224 45
1 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 0 15214304 46
1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 8988584 47
0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1853960 48
Now compare the second table with the third. You see that all decimal values are exactly 8x bigger now. (18 dB = 3 x 6 dB = 3 bits = a factor 2³ = a factor 8 ).
Now comes the important part! Look at the three least significant bits (on the right). They are al zero! So the least significant bit that is actually used is bit nr. 3. That means the steps are also 2³ = 8 as big as they should be. There are no levels in between (well, there are but they're not being used as that info was not recorded - the signal was too weak so all variations were in between levels and therefore rounded off to the same level), so you go immediately from level 0 to level 8, from 8 to 16, from 16 to 24 etcetera, without using 1, 2, 3, 4, 5, 6, 7, 9, 10, ...
That means the average quantization error is 4 levels (2 bits) as opposed to an average quantization error of ½ bit.
If you have a 24 bit recording, which uses only the 16 least significant bits, you had a headroom of 8 bits x 6 dB = 48 dB!! You still have a CD quality recording though. Mathematically you could chop the 8 most significant bits off, keeping you with 16 bit of info. In practice you have to add 48 dB of gain, shifting all bits 8 places to the left. The 8 least significant bits will all be 0 (or close to it - 1 bit is not exactly 6 dB but very very close). You can shop those bits off by using any conversion tool, or render it to 16 bit keeping you with a CD quality, 16 actually used bits, recording.
Promise me that if you have any more questions, you
will ask.