# 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 transmitter’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.

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.

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

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.