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Numeric Formats
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DSP systems use a standardized numeric format. Fixed-point numbers are formatted A.B, where A is the number of bits to the left of the decimal point (the integer part) and B is the number of bits to the right of the decimal point (the fractional part).
The AD1940/AD1941 use the same numeric format for both the coefficient values (stored in the parameter RAM) and the signal-data values, as follows:
Inputs to the SigmaDSP cores are 24 bits. In the core, the DSP adds 4 additional zeros for additional headroom. The result is a 28 bit number representation.
So, in the case of most SigmaDSPs, audio representation is represented with A = 5 and B = 23. In other words, the number format is 5.23. Control signals and index table values generally require integer representation, and are therefore represented with A = 28 and B = 0. In other words, the format is 28.0.
28.0 (Integer) Format
Signals that are in integer format follow standard binary rules for representation. 0 = 0, 1 = 1, 10 = 2, 11 = 3, 100 = 4, et cetera. Including zero padding, positive integers in the DSP are represented as follows:
0000 0000 0000 0000 0000 0000 0000 = 0
0000 0000 0000 0000 0000 0000 0001 = 1
0000 0000 0000 0000 0000 0000 0010 = 2
0000 0000 0000 0000 0000 0000 0011 = 3
0000 0000 0000 0000 0000 0000 0100 = 4
…
0000 1000 0000 0000 0000 0000 0000 = 8388608 (0 dB full scale represented in 28.0 format)
…
0111 1111 1111 1111 1111 1111 1111 = 134217727 (2^27 - 1)
Negative numbers are represented in two's complement. For more information on two's complement, please read the very comprehensive article on Wikipedia here: http://en.wikipedia.org/wiki/Two%27s_complement
The corresponding 28-bit two's complement integers are represented as follows:
1000 0000 0000 0000 0000 0000 0000 = -134217728 (2^27)
…
1111 1111 1111 1111 1111 1111 1100 = -4
1111 1111 1111 1111 1111 1111 1101 = -3
1111 1111 1111 1111 1111 1111 1110 = -2
1111 1111 1111 1111 1111 1111 1111 = -1
In general, negative integers are not used in SigmaStudio or SigmaDSP algorithms.
When outputting to the serial ports or DACs, signals will saturate at 0 dBFS. This means any signal exceeding 8388608 in 28.0 format will be limited to full-scale on the outputs.
5.23 (Decimal) Format
Audio, unlike control signals, is not represented as an integer, but rather as a decimal value. A full-scale audio signal has an amplitude of 1. If -3 dB of gain is applied, it should have an amplitude of approximately 0.707. If -6 dB of gain is applied, the signal has an amplitude of 0.5. So, it is easy to see that audio signals need to be represented as decimal numbers.
A full scale 24-bit input signal would have a positive peak of 1000 0000 0000 0000 0000 0000 in unsigned, 24-bit representation. If we add 4 bits of headroom, the 28-bit representation becomes 0000 1000 0000 0000 0000 0000 0000. If two's-complement representation is used, then the negative peak of a full-scale signal becomes 1111 1000 0000 0000 0000 0000 0000.
Moving the leading '1' to the left or right will double or halve the value, respectively.
0000 0000 0000 0000 0000 0000 0000 = 0.0
0000 0010 0000 0000 0000 0000 0000 = 0.25
0000 0100 0000 0000 0000 0000 0000 = 0.5
0000 1000 0000 0000 0000 0000 0000 = 1.0 (0 dB full scale)
0001 0000 0000 0000 0000 0000 0000 = 2.0
0010 0000 0000 0000 0000 0000 0000 = 4.0
0111 1111 1111 1111 1111 1111 1111 = (16.0 - 1 LSB)
For negative numbers, signed two's-complement is used.
1000 0000 0000 0000 0000 0000 0000 = -16.0
1110 0000 0000 0000 0000 0000 0000 = -4.0
1111 0000 0000 0000 0000 0000 0000 = -1.0
1111 1000 0000 0000 0000 0000 0000 = -1.0
1111 1100 0000 0000 0000 0000 0000 = -0.5
1111 1110 0000 0000 0000 0000 0000 = -0.25
1111 1111 1111 1111 1111 1111 1111 = (1 LSB below 0.0)
When outputting to the serial ports or DACs, signals will saturate at 0 dBFS. This means any signal with a peak exceeding 1.0 in 5.23 format will be limited to full-scale on the outputs.
5.19 (Hardware readback) Format
Some cells in SigmaStudio may use slightly different number formats. For example, since the hardware-based DSP readback registers in the ADAU1701 only have 24 bits, the lower 4 bits from the 5.23 signal are truncated and the number is represented in 5.19 format.
So, a full-scale signal that was represented in 5.23 format as
0000 1000 0000 0000 0000 0000 0000
would have its lower 4 bits truncated for 5.19 representation:
0000 1000 0000 0000 0000 0000
The result is that very small amplitude signals will be truncated and therefore cannot be read back from the DSP on the older generation of SigmaDSP cores.
Newer cores, such as the ADAU1761 and ADAU144x, have full 5.23 readback capabilities implemented in software.
Concept for dB conversion:
- Count the number of leading 0s (minus 4) and multiply by 6; this is the integer part of the dB scale.
- Use the bits after the leading 1 to linearly interpolate between the 6 dB points.
In order to get a linear value into a hex/binary number that can be written to the SigmaDSP, and vice versa, it is important to understand number conversions.
The actual level parameters that you need to write are only the last 28 bits, and this value is simply a linear gain value. Using -40 dB as an example, convert this value to a linear value using the standard dB equation:
20log10 (x/1) = -40dB x = .01
Take this linear value and multiply by 223 in order to get the decimal representation of the value in hex, because it is a 5.23 value.
.01 * 223 = 83886.08.
Now take the integer part of this result and convert 83886 to hex, and you will get 0x00147AE.
The output of some blocks is a 5.19 number. The formula to determine the dB-output value from the readback value is the following:
dB_value = 96.32959861 * (readback_value / 219 - 1)
resources/tools-software/sigmastudio/usingsigmastudio/numericformats.1340054396.txt.gz · Last modified: 18 Jun 2012 23:19 by William Jahn