# Digital Communication - M-ary Encoding

The word binary represents two bits. M represents a digit that corresponds to the number of conditions, levels, or combinations possible for a given number of binary variables.

This is the type of digital modulation technique used for data transmission in which instead of one bit, two or more bits are transmitted at a time. As a single signal is used for multiple bit transmission, the channel bandwidth is reduced.

## M-ary Equation

If a digital signal is given under four conditions, such as voltage levels, frequencies, phases, and amplitude, then M = 4.

The number of bits necessary to produce a given number of conditions is expressed mathematically as

$$N = \log_{2}{M}$$

Where

N is the number of bits necessary

M is the number of conditions, levels, or combinations possible with N bits.

The above equation can be re-arranged as

$$2^N = M$$

For example, with two bits, 22 = 4 conditions are possible.

## Types of M-ary Techniques

In general, Multi-level (M-ary) modulation techniques are used in digital communications as the digital inputs with more than two modulation levels are allowed on the transmitterâ€™s input. Hence, these techniques are bandwidth efficient.

There are many M-ary modulation techniques. Some of these techniques, modulate one parameter of the carrier signal, such as amplitude, phase, and frequency.

### M-ary ASK

This is called M-ary Amplitude Shift Keying (M-ASK) or M-ary Pulse Amplitude Modulation (PAM).

The amplitude of the carrier signal, takes on M different levels.

### Representation of M-ary ASK

$S_m(t) = A_mcos (2 \pi f_ct) \quad A_m\epsilon {(2m - 1 - M) \Delta, m = 1,2... \: .M} \quad and \quad 0 \leq t \leq T_s$

Some prominent features of M-ary ASK are −

• This method is also used in PAM.
• Its implementation is simple.
• M-ary ASK is susceptible to noise and distortion.

### M-ary FSK

This is called as M-ary Frequency Shift Keying (M-ary FSK).

The frequency of the carrier signal, takes on M different levels.

### Representation of M-ary FSK

$S_i(t) = \sqrt{\frac{2E_s}{T_s}} \cos \left ( \frac{\pi}{T_s}\left (n_c+i\right )t\right )$ $0 \leq t \leq T_s \quad and \quad i = 1,2,3... \: ..M$

Where $f_c = \frac{n_c}{2T_s}$ for some fixed integer n.

Some prominent features of M-ary FSK are −

• Not susceptible to noise as much as ASK.

• The transmitted M number of signals are equal in energy and duration.

• The signals are separated by $\frac{1}{2T_s}$ Hz making the signals orthogonal to each other.

• Since M signals are orthogonal, there is no crowding in the signal space.

• The bandwidth efficiency of M-ary FSK decreases and the power efficiency increases with the increase in M.

### M-ary PSK

This is called as M-ary Phase Shift Keying (M-ary PSK).

The phase of the carrier signal, takes on M different levels.

### Representation of M-ary PSK

$S_i(t) = \sqrt{\frac{2E}{T}} \cos \left (w_o t + \phi _it\right )$ $0 \leq t \leq T \quad and \quad i = 1,2 ... M$

$$\phi _i \left ( t \right ) = \frac{2 \pi i}{M} \quad where \quad i = 1,2,3 ... \: ...M$$

Some prominent features of M-ary PSK are −

• The envelope is constant with more phase possibilities.

• This method was used during the early days of space communication.

• Better performance than ASK and FSK.

• Minimal phase estimation error at the receiver.

• The bandwidth efficiency of M-ary PSK decreases and the power efficiency increases with the increase in M.

So far, we have discussed different modulation techniques. The output of all these techniques is a binary sequence, represented as 1s and 0s. This binary or digital information has many types and forms, which are discussed further.

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