The Decibel scale

decibel scale
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.

decibel scale circuit diagram for equation
Voltage-current relationships for a four-terminal network

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)

To know more >> Click here