Okay. We need to talk about a slightly esoteric
invention that's used in Electrical Engineering all the time, and that's the
use of the word decibels. So, I'm sure you all know what the prefix
deci means, it means a tenth of something, but the question is, a tenth
of what? And apparently, it is a tenth of a bel.
And we'll talk about the specific definition in a second, but who was the
bel named for? Think about that for a second.
Any idea who it might be? Of course, it was named for Alexander Graham Bell.
Well, the definition of a decibel scale, what it, going into the details, well
what it means is that you express a value logarithmically relative to a reference.
So, we're going to see how that works. Here's the explicit definition power with
signal s in decibels. So, we've got our signal s and it has
some power. There's a reference signal s0 and we
compute its power. It could be a standard like it's a 1 watt
signal or 1 milliwatt signal. Any kind of standard reference value that
is convenient for your application. You take the ratio between the two, you
evaluate the log base 10. That ratio would multiply by 10, the 10
is the deci part. And that is the definition of the power
of a signal expressed in decibels. Now this second definition down here for
specifing the amplitude of a signal in decibels looks, is different.
Notice that it's 20 sitting out here, but it turns out that 20 is there because
it's consistent with our relationship between power and amplitude.
Because power is proportional to amplitude squared.
If you insert this formula up into the first definition, you get amplitude
squared divided by the amplitude squared of the reference.
Those are squares because the log come out and multiply and give you a 20 log 10
amplitude of the ratio. So, these are just different ways of
thinking about the same thing. You can express the power of a signal in
decibels or you can express its amplitude in decibels.
And the main reason why we use decibels is because adding decibel values
corresponds to multiplying either power or amplitude values,
right? Because the log a times b, that's equal to the log(a)+log(b) and this has
to do with transfer functions. We know that when you put systems in
cascade, their transfer functions multiply.
And so, you can talk about what the overall transfer function is in decibels
by just adding the transfer function at each, expressed in decibels.
Now, let's go over some interesting values for decibels that come up all the
time. This is something everybody remembers.
So, clearly, if the power ratio is 1, that is equivelent to 0 dB.
If your power ratio was signal to noise ratio, signal power divided by reference
noise power, 0 dB means there's just as much power in the signal as there is in
the noise, and that's not good. That's a very low signal to noise ratio
If the power ratio is 10, well that turns out to correspond to 10 dB.
If the power ration is a tenth, the signal power is 1/10 the power of the
reference, that's -10 dB. And, of course, the log of 1/x is the
negative log of x, so that explains the minus sign.
And here it is, we multiply two things together.
Well, their decibel value is added. So, that's 10+10, gives you 20.
Now, what's interesting is that when the power ratio 2 corresponds almost exactly
to 3 dB. That turns out to be very, very close
approximation so that, it turns out to be a very good one to memorize, that 3 dB
corresponds to a factor of 2. Now, how do you get this, that a power
ration of 5 has to be 7 dB? Think about that for a second.
How would you get that? The answer, of course, is that 2*5, is
10, well, since they multiply here, they add here and the sum has to be 10 because
we've already seen that 10 is 10 dB. So right away, you can derive other
decibel values from other ones by using the properties of the log.
And then finally, I want to mention a very interesting and important one.
2^10, 2^10 power is just about a 1000. The log base 10 of a 1000, is 3, which
makes this decibel value very close to 30 dB, turns out to be, or exactly 30.1 dB.
But you can roughly say that if you have a power ratio of 2^10, that's about 30
dB, that's going to be very very important to remember as we go along.
So, that's all there is to decibels. It's very important thing we use in
Electrical Engineering to summarize how systems work and how they're behaving.
Once you get used to it, you'll use it all the time, too.