Can anyone give me a simple explanation for why doubling the power leads to a 3dBSPL increase, whereas doubling the number of bits in digital audio leads to a 6dBSNR increase? This is confusing because I thought dBs were a dimensionless scale that should work the same no matter what property was being measured.
After reading this page: How to Calculate the Signal to Noise Ratio | eHow.com I think I have an idea of how to answer my question.
The reason comes down to the different quantities being measured. SPL measures power, but it's the signal level (expressed in Voltage) that doubling bit depth doubles. The relationship between the two is P=V^2/R, or in other words Power is proportional to Voltage squared. That power of 2 is the reason for the 6db digital versus 3db SPL.
Another perhaps simpler way of looking at this is to take an example.
Lets compare two systems:
System A is a 16 bit system with a max voltage of 1 Vrms full scale.
System B is a 17 bit system with a max voltage of 2 Vrms full scale.
Both systems have the same quantization error noise level because they both represent values up to 1 volt with the first 16 bits, exactly the same.
System B doubles the full scale Voltage, and Quadruples the full scale Power (because Power is proportional to 1V squared in System A, and proportional to 2V squared in system B). The effect where a single bit quadruples the power is the reason for the 6db (not 3db) increase per bit in dbSNR for digital.
Yet another thing that helped jog my memory on this is the two formulas:
dB in power terms = 10 log (P1/P2) -> doubling increases by 3db for Power
dB in Voltage terms = 20 log (V1/V2) -> doubling increases by 6db for Voltage
The reason comes down to the different quantities being measured. SPL measures power, but it's the signal level (expressed in Voltage) that doubling bit depth doubles. The relationship between the two is P=V^2/R, or in other words Power is proportional to Voltage squared. That power of 2 is the reason for the 6db digital versus 3db SPL.
Another perhaps simpler way of looking at this is to take an example.
Lets compare two systems:
System A is a 16 bit system with a max voltage of 1 Vrms full scale.
System B is a 17 bit system with a max voltage of 2 Vrms full scale.
Both systems have the same quantization error noise level because they both represent values up to 1 volt with the first 16 bits, exactly the same.
System B doubles the full scale Voltage, and Quadruples the full scale Power (because Power is proportional to 1V squared in System A, and proportional to 2V squared in system B). The effect where a single bit quadruples the power is the reason for the 6db (not 3db) increase per bit in dbSNR for digital.
Yet another thing that helped jog my memory on this is the two formulas:
dB in power terms = 10 log (P1/P2) -> doubling increases by 3db for Power
dB in Voltage terms = 20 log (V1/V2) -> doubling increases by 6db for Voltage
I have no idea why they adopted the term "power" to specify amplifiers and speakers.
Voltages and currents would have done better and avoided all the confusion that has resulted from power specifications.
I would love to see a speaker specified as being matched up or requiring an amplifier with a 20Vac to 30Vac maximum continuous output.
I would love to see amplifiers specified by their output volts into a range of standard load resistances, eg. 2r0, 4r0, 8r0. And it would be nice to see how long each load can be driven. Better than nice. It would differentiate the good from the bad.
Voltages and currents would have done better and avoided all the confusion that has resulted from power specifications.
I would love to see a speaker specified as being matched up or requiring an amplifier with a 20Vac to 30Vac maximum continuous output.
I would love to see amplifiers specified by their output volts into a range of standard load resistances, eg. 2r0, 4r0, 8r0. And it would be nice to see how long each load can be driven. Better than nice. It would differentiate the good from the bad.
And that's the problem. Even people buying cheap gear want to believe they are not getting crap. Maybe not top flight, but not crap. Truly useful specs would force too many manufacturers to decide between making good equipment or marketing crap.
The way it stands, the philosophy remains "caveat emptor".
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