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.