4G,3G networks,Orthogonal Frequency Division Multiplexing,Orthogonal Matrix and functions, Dirac delta,Principle behind orthogonal frequency division multiplexing,Implementation of Transceivers.

Even while 3G networks are still being deployed, the next generation (sometimes called 4G or
3.9G) has been developed. Most infrastructure manufacturers are concentrating on the Long-Term
Evolution (LTE) of the dominating 3G standard. An alternative standard, whose roots are based in
fixed wireless access systems, is also under deployment. In addition, access to TV programming
(either live TV or prerecorded episodes) from cellphones is becoming more and more popular.
For 4G networks as well as TV transmission, Multiple Input Multiple Output system-Orthogonal
Frequency Division Multiplexing (MIMO-OFDM) is the modulation method of choice, which has spurred research in this area.

What is Multiple input, multiple output-Orthogonal Frequency Division Multiplexing (MIMO-OFDM).
Multiple input, multiple output-orthogonal frequency division multiplexing (MIMO-OFDM) is the dominant air interface for 4G and 5G broadband wireless communications. It combines multiple input, multiple output (MIMO) technology, which multiplies capacity by transmitting different signals over multiple antennas, and orthogonal frequency division multiplexing (OFDM), which divides a radio channel into a large number of closely spaced subchannels to provide more reliable communications at high speeds. Research conducted during the mid-1990s showed that while MIMO can be used with other popular air interfaces such as time division multiple access (TDMA) and code division multiple access (CDMA), the combination of MIMO and OFDM is most practical at higher data rates.
MIMO-OFDM is the foundation for most advanced wireless local area network (Wireless LAN) and mobile broadband network standards because it achieves the greatest spectral efficiency and, therefore, delivers the highest capacity and data throughput. Greg Raleigh invented MIMO in 1996 when he showed that different data streams could be transmitted at the same time on the same frequency by taking advantage of the fact that signals transmitted through space bounce off objects (such as the ground) and take multiple paths to the receiver. That is, by using multiple antennas and precoding the data, different data streams could be sent over different paths. Raleigh suggested and later proved that the processing required by MIMO at higher speeds would be most manageable using OFDM modulation, because OFDM converts a high-speed data channel into a number of parallel, lower-speed channels.
Orthogonal frequency-division multiplexing (OFDM) is a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL Internet access, wireless networks, powerline networks, and 4G mobile communications.
OFDM is a frequency-division multiplexing (FDM) scheme used as a digital multi-carrier modulation method. A large number of closely spaced orthogonal sub-carrier signals are used to carry data on several parallel data streams or channels. Each sub-carrier is modulated with a conventional modulation scheme (such as quadrature amplitude modulation or phase-shift keying) at a low symbol rate, maintaining total data rates similar to conventional single-carrier modulation schemes in the same bandwidth.
The primary advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions (for example, attenuation of high frequencies in a long copper wire, narrowband interference and frequency-selective fading due to multipath) without complex equalization filters. Channel equalization is simplified because OFDM may be viewed as using many slowly modulated narrowband signals rather than one rapidly modulated wideband signal. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate intersymbol interference (ISI) and utilize echoes and time-spreading (on analogue TV these are visible as ghosting and blurring, respectively) to achieve a diversity gain, i.e. a signal-to-noise ratio improvement. This mechanism also facilitates the design of single frequency networks (SFNs), where several adjacent transmitters send the same signal simultaneously at the same frequency, as the signals from multiple distant transmitters may be combined constructively, rather than interfering as would typically occur in a traditional single-carrier system.

Let's understand Orthogonal Frequency DivisionMultiplexing (OFDM)  in somewhat simple and detail manner because it is at the heart of 4G networks.

Orthogonal Frequency Division Multiplexing (OFDM) is a modulation scheme that is especially
suited for high-data-rate transmission in delay-dispersive environments. It converts a high-rate data stream into a number of low-rate streams that are transmitted over parallel, narrowband channels that can be easily equalized.

Let us first analyze why traditional modulation methods become problematic at very high data
rates. As the required data rate increases, the symbol duration Ts has to become very small in
order to achieve the required data rate, and the system bandwidth becomes very large.1 Now, delay dispersion of a wireless channel is given by nature; its values depend on the environment, but not on the transmission system. Thus, if the symbol duration becomes very small, then the impulse response (and thus the required length of the equalizer) becomes very long in terms of symbol durations. The computational effort for such a long equalizer is very large and the probability of instabilities increases. For example, the Global System for Mobile communications (GSM) system which is designed for peak data rates up to 200 kbit/s, uses 200 kHz bandwidth, while the IEEE 802.11 system (see Chapter 29), with data rates of up to 55Mbit/s uses 20MHz bandwidth. In a channel with 1 μs maximum excess delay, the former needs a two-tap equalizer, while the latter needs 20 taps. OFDM, on the other hand, increases the symbol duration on each of its carriers compared to a single-carrier system, and can thus have a very simple equalizer for each subcarrier.

Principle of Orthogonal Frequency Division Multiplexing:

If you have read Linear Algebra very well the you must be familiar with the concept of orthogonality. If A and B are two symmetric squate matrices having same dimension ,then we A and B are orthogonal to each other if  Transpose(A)*B=Identity Matrix

Orthogonal matrix
In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.

where I is the identity matrix.
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

An orthogonal matrix Q is necessarily invertible (with inverse Q⁻¹ = Q^T), unitary (Q⁻¹ = Q*) and therefore normal (Q*Q = QQ*) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.
The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.
The complex analogue of an orthogonal matrix is a unitary matrix.
This is the concept of orthogonality in linear Algebra we can also define orthogonality in coordinate geometry as two lines are orthogonal if they fotm 90 degree angle with respect to each other.
In mathematics, orthogonality is the relation of two lines at right angles to one another (perpendicularity), and the generalization of this relation into n dimensions; and to a variety of mathematical relations thought of as describing non-overlapping, uncorrelated, or independent objects of some kind.
"Orthogonality and rotation" by Maschen - Own work. Licensed under CC0 via Wikimedia Commons.

Principle of Orthogonal Frequency Division Multiplexing:
OFDM splits a high-rate data stream into N parallel streams, which are then transmitted by modulating N distinct carriers (henceforth called subcarriers or tones). Symbol duration on each subcarrier thus becomes larger by a factor of N. In order for the receiver to be able to separate signals carried by different subcarriers, they have to be orthogonal. Conventional Frequency Division Multiple Access (FDMA), as described in Section 17.1 and depicted again in Figure 19.1, can achieve this by having large (frequency) spacing between carriers. This, however, wastes precious spectrum. A much narrower spacing of subcarriers can be achieved. Specifically, let subcarriers be at the frequencies fn = nW/N, where n is an integer, and W the total available bandwidth; in the most
simple case, W = N/Ts. We furthermore assume for the moment that modulation on each of the
subcarriers is Pulse Amplitude Modulation (PAM) with rectangular basis pulses. We can then easily
see that subcarriers are mutually orthogonal, since the relationship

holds true.shows this principle in the frequency domain. Due to the rectangular shape of pulses
in the time domain, the spectrum of each modulated carrier has a sin(x)/x shape. The spectra of
different modulated carriers overlap, but each carrier is in the spectral nulls of all other carriers.
Therefore, as long as the receiver does the appropriate demodulation (multiplying by exp(−j2πfnt )
and integrating over symbol duration), the data streams of any two subcarriers will not interfere.

Try correlating it to orthogonal matrix multiplication above you can see that I identity matrix reduces here to Delta,Whose integral is unity.
The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

and which is also constrained to satisfy the identity

This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties. The Dirac delta function can be rigorously defined either as a distribution or as a measure.

Implementation of Transceivers
OFDM can be interpreted in two ways: one is an “analog” interpretation following from the picture
of Figure below:

Transceiver structures for orthogonal frequency division multiplexing in purely analog technology (a), and using inverse fast Fourier transformation (b)as shown in the following figure.

we first split our original data stream into N parallel
data streams, each of which has a lower data rate. We furthermore have a number of local oscillators
(LOs) available, each of which oscillates at a frequency fn = nW/N, where n = 0, 1, . . .,N − 1.
Each of the parallel data streams then modulates one of the carriers. This picture allows an easy
understanding of the principle, but is ill suited for actual implementation – the hardware effort of
multiple local oscillators is too high.An alternative implementation is digital . It first divides the transmit data into blocks of N symbols. Each block of data is subjected to an Inverse Fast Fourier Transformation (IFFT), andthen transmitted as shown in the above figure b.

This approach is much easier to implement with integrated circuits. In the following, we will show that the two approaches are equivalent. Let us first consider the analog interpretation. Let the complex transmit symbol at time instant i on the nth carrier be cn,i . The transmit signal is then:

where the basis pulse gn(t ) is a normalized, frequency-shifted rectangular pulse:

Let us now – without restriction of generality – consider the signal only for i = 0, and sample it
at instances tk = kTs/N:

Now, this is nothing but the inverse Discrete Fourier Transform (DFT) of the transmit symbols.
Therefore, the transmitter can be realized by performing an Inverse Discrete Fourier Transform
(IDFT) on the block of transmit symbols (the blocksize must equal the number of subcarriers).
In almost all practical cases, the number of samples N is chosen to be a power of 2, and the
IDFT is realized as an IFFT. In the following, we will only speak of IFFTs and Fast Fourier
Transforms (FFTs).

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