supposed that the measurement RSS is "-70dBm" and the predicted RSS is "-68dBm, the transmission power of antenna is "-12dBm"，
then if the following equation is right? if not, how to calculate it?
Error = |10 * log10 (70/12) - 10 * log10 (68/12)| = 10 * log10 (70/68)
now my measurement is the RSS in dBm, how to convert it into dB?
This often confuses folks in my experience, and as such warrants a thorough explanation.
The "m" in "dBm" means relative to 1 milliwatt. It is typically used for absolute measurements, whereas "regular" dBs are typically used for power gains/losses/diffs.
Example:
in/tx out/rx
1w .5w
(1 milliwatt = 0.001)
10log(1/0.001) = 30dBm
10log(0.5/0.0001) = 27dBm
loss = 3dB
10log(1) = 0dB
10log(0.5) = 3dB
loss still is 3dB
(note there is an implied /1w here since the argument to log must be unit-less, e.g. 0.5w/1w = 0.5 "flat" (aka no units))
So in the context of power differences, the m does not matter.
Things to note:
1/2 of power lost == -3dB gain (or +3dB loss)
power gains/losses in series are added/subtracted when in dBs -vs- multipled/divided when in watts
0 watts == -infinity dB
0 dBm == 1 milliwatt
log here is base 10 (not 2 nor e)
reletiveGainOrLoss = 10^(valueOfGainOrLossInDb/10)
valueOfPowerInMilliwatts = 10^(valueOfPowerInDbm/10)
In your example, I'll presume by error you mean the error of the predicted loss relative measured loss:
predicted loss =
known transmission power - predicted RSS =
-12dBm - -68dBm =
56dB
measured loss =
known transmission power - measured RSS =
-12dBm - -70dBm =
58dB
error of predicted loss relative to measured loss =
|predicted loss - measured loss| =
|56dB - 58dB| =
2dB (==2dBms, but for diffs we should drop the m) =
Or more directly: 70 - 68 (so easy with dBs!)
This equates to a 63% error (or 58, depending on how it is done):
10^(-2dB/10) =
0.63
(10^(2dB/10) =
1.584893192461113
In millwatts:
10^(valueInDbm/10) =
10^(-70/10) = 0.0000001 milliwatts
10^(-68/10) = 0.000000158489319 milliwatts
10^(-12/10) = 0.063095734448019 milliwatts
As a sanity check:
(0.0000001/0.063095734448019 - 0.000000158489319/0.063095734448019) / (0.0000001/0.063095734448019) =
(0.000001584893192 - 0.000002511886428) / 0.000001584893192 =
-0.000000926993236 / 0.000001584893192 =
-0.584893190707832
(note that doing it in watts is much more laborious! (not to even mention float errors))
To answer your other question regarding:
Error = |10 * log10 (70/12) - 10 * log10 (68/12)| = 10 * log10 (70/68)
The first equation is nonsensical; as discussed above: for dBs we add/subtract -vs- multiply/divide. The second equation is however true, based on one of the rules of logs:
log a + log b = log ab