# M-ary Encoding

The word binary represents two-bits. M simply 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, (M-ary) multi-level modulation techniques are used in digital communications as the digital inputs with more than two modulation levels allowed on the transmitter’s input. Hence, these techniques are bandwidth efficient.

There are many different 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.

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

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

## M-ary FSK

This is called as M-ary Frequency Shift Keying.

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

### Representation of M-ary FSK

$$S_{i} (t) = \sqrt{\frac{2E_{s}}{T_{S}}} \cos\lgroup\frac{\Pi} {T_{s}}(n_{c} + i)t\rgroup \:\:\:\:0\leq t\leq T_{s}\:\:\:and\:\:\:i = 1,2.....M$$

where $f_{c} = \frac{n_{c}}{2T_{s}}$ for some fixed integer n.

This is 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 an 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.

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

### Representation of M-ary PSK

$$S_{i}(t) = \sqrt{\frac{2E}{T}} \cos(w_{0}t + \emptyset_{i}t)\:\:\:\:0\leq t\leq T_{s}\:\:\:and\:\:\:i = 1,2.....M$$

$$\emptyset_{i}t = \frac{2\Pi i} {M}\:\:\:where\:\:i = 1,2,3...\:...M$$

Here, the envelope is constant with more phase possibilities. This method was used during the early days of space communication. It has 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.