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 P_{2} = 0.5 V_{2 }^{2} /R _{2} ;

and Eq. (2) becomes

G_{dB} = 10 log _{10} P_{2}/P_{1} = 10 log _{10} (V_{2 }^{2}/R_{2}) / (V_{1 }^{2}/R_{1}) …(5)

= 10 log _{10} (V_{2}/V _{1})^{2} + 10 log _{10} R_{1}/R_{2}

= 20 log _{10} (V_{2}/V _{1}) – 10 log _{10} R_{2}/R_{1} …(6)

#### Other cases

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

G_{dB} = 20 log _{10} V_{2}/V _{1} …(7)

Instead, if P_{1 = }I^{2}_{ 1}/R_{1 }and P_{2 = }I^{2}_{ 2}/R_{2, }for R_{1 = }R_{2}, we obtain

G_{dB} = 20 log _{10} I _{2}/I _{1} …(8)

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