# The Decibel scale

The characteristics of audio signals and noise are often specified in decibel (dB).

The bel is defined as the base ten logarithm of a power ratio.

The logarithm compresses the numerical range of its argument;

This is often a convenient feature when one must deal with numbers differing over several orders of magnitude.

We have,

#### Logarithm properties

1.log P 1 P 2 = log P 1 + log P 2

2. log P 1  / P 2 = log P 1 – log P 2

3. log P n = n log P

4. log 1 = 0

In communications systems, gain is measured in bels.

The bel is used to measure the ratio of two levels of power or power gain G; i.e.,

G = Number of bels = log 10 (P 2  / P 1 )    …(1)

The decibel (dB) provides us with a unit of less magnitude.

It is 1/10th of a bel and is given by

G dB = 10 log 10 (P 1  / P 2 )       …(2)

When P 1 = P 2, there is no change in power and the gain is 0 decibel (dB). If

P 2 = 2P 1, the gain is

G dB = 10 log 10 2 = 3 dB …(3)

and when P 2 = 0.5P 1 , the gain is

G dB = 10 log 10 0.5 = −3 dB   …(4)

Equations (3) and (4) show another reason why logarithms are

greatly used:

#### “The logarithm of the reciprocal of a quantity is simply negative the logarithm of that quantity.”

Further, the gain G can be expressed in terms of voltage or

current ratio.

To do so, consider the network shown in Fig.

If P 1 is the input power, P 2 is the output (load) power, R 1 is the input resistance,

and R 2 is the load resistance, then P 1 = 0.5V 1 2 /R 1 and P2 = 0.5 V2 2 /R 2 ;

and Eq. (2) becomes

GdB = 10 log 10 P2/P1 = 10 log 10 (V2 2/R2) / (V1 2/R1) …(5)

= 10 log 10 (V2/V 1)2 + 10 log 10 R1/R2

= 20 log 10 (V2/V 1) – 10 log 10 R2/R1 …(6)

#### Other cases

When R 2= R 1, after comparing voltage levels, Eq. (6) becomes

GdB = 20 log 10 V2/V 1       …(7)

Instead, if P1 = I2 1/R1  and P2 = I2 2/R2, for R1 =  R2, we obtain

GdB = 20 log 10 I 2/I 1         …(8)