# Bels and Decibels

1. Mar 22, 2004

### sk381

I know that

1 Bel = Log (Power1/Power2)

and 1 decibel = 0.1 bel

then why is 1 decibel = 10 Log (Power1/Power2)

and not 0.1 Log (Power1/Power2)

Thanks

SK

2. Mar 22, 2004

### Staff: Mentor

Not true. See below.
It isn't.

1 dB = 0.1 Bel (this is true)

This is what Bel and dB mean:

Power difference (measured in Bels) = Log (Power1/Power2)
Power difference (measured in dB) = 10 Log (Power1/Power2)

Make sense?

3. Mar 22, 2004

### sk381

Hmm.. not clear yet...

I didn't get it..

Can you give some example to further elucidate?

4. Mar 22, 2004

### Chen

Re: Re: Bels and Decibels

If:
Power difference (measured in Bels) = Log (Power1/Power2)
Power difference (measured in dB) = 10 Log (Power1/Power2)
Then:
Power difference (measured in dB) = 10 * Power difference (measured in Bels)

But you say that:
1 dB = 0.1 Bel

So I don't get it either.

5. Mar 22, 2004

### Staff: Mentor

Re: Re: Re: Bels and Decibels

Absolutely. If you measure power in dB your answer will be 10 times bigger than if you measured in Bels.
Yep.

I'm not sure what you guys don't get.

The key point is that Log(P1/P2) does not equal 1 Bel, it is a measurement in units of Bels.

Example: Say P1 = 1000; P2 = 10;
Log (1000/10) = 2 Bels
10 Log (1000/10) = 20 dB

6. Mar 22, 2004

### Chen

lol, never mind. I'm not supposed to be awake anyway.

7. Mar 22, 2004

### sk381

Ok...
but then how do we arrive at the conclusion that 1 dB = 0.1 Bell

8. Mar 22, 2004

### Chen

Just combine the two statements:
Power difference (measured in dB) * 1 dB = 10 * Power difference (measured in Bels) * 1 Bel.
The power difference is the same so:
1 dB = 10 Bel.

9. Mar 22, 2004

### Staff: Mentor

First realize that 1 dB = 0.1 Bel by definition. (The prefix 'deci' means 1/10.)

But it all makes sense. A given power level $P$, measured with respect to the reference power level $P_{ref}$, would equal $\log_{10}(\frac{P}{P_{ref}}) \b{Bel} = 10 \log_{10}(\frac{P}{P_{ref}}) \b{dB}$. So, 1 Bel = 10 dB.

10. Mar 22, 2004

### Staff: Mentor

Snap out of it, Chen. You're still not quite awake.

The power difference is the same, just measured using different units.
[Log(Power1/Power2)]Bels = [10 Log(Power1/Power2)]dB