Why is 1 Decibel = 10 Log (Power1/Power2)?

[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

 

[Doc Al]

Originally posted by sk381

1 Bel = Log (Power1/Power2)

Not true. See below.

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

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?

 

[sk381]

Hmm.. not clear yet...

I didn't get it..

Can you give some example to further elucidate?

 

[Chen]

Originally posted by Doc Al
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)

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.

 

[Doc Al]

Originally posted by Chen
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)

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

But you say that:
1 dB = 0.1 Bel

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

 

[Chen]

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

 

[sk381]

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

 

[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.

 

[Doc Al]

Originally posted by sk381
Ok...
but then how do we arrive at the conclusion that 1 dB = 0.1 Bell

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.

 

[Doc Al]

Originally posted by 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.

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