Orthogonal Frequency Division Multiplexing (OFDM)
OFDM is a special case of multi-carrier communication as opposed to a conventional single-carrier system. (OFDM is the technology behind many high speed systems such as WiFi (IEEE 802.11a, g, n, ac), WiMAX (IEEE 802.16), 4G and 5G mobile communications (LTE). )
1.A Wireless Channel(无线信道)
Large scale
fading, that arises from regular power decay with distance as well as shadowing caused by buildings and other obstacles affecting the wave propagation.
Small scale fading,
that arises from constructive and destructive interference of multi-path components and results in fast amplitude variations at the receiver. An example of these multi-path components is shown in Figure 1 below where many multi-path components arrive at the Rx of a hiker after being reflected from the nearby surfaces such as the aeroplane, houses, trees and the mountains.
2.A Low Data Rate Signal* （低数据率信号与数字调制之间的关系）
Suppose that the year is early 2000s and our hiker in Figure 1 only wants to check his email and/or messages (probably driven by what is available) and requires a data rate of only 100
kbps.That translates to a bit time (also known as bit duration) of
As 传一个字符需要10us.
described above, the first and second multi-path components arrive 1μs and 2μs after the direct path, respectively (ignoring the carrier). This is shown in Figure 4.
First and second multipath components arriving after 1 and2 μs after the direct path, respectively (carrier wave not shown)（只用基带信号的多路径情况说明）
Since nature adds the signals at the antenna, the Rx will have a summation of these three paths, effectively the same signal delayed by different amounts but with different attenuation and phase shifts of the carrier waves (not drawn in the figure).
This is a situation that can be handled without much effort in terms of computational resources. We claim that the wireless channel does not pose a significant problem to Rx processing time in this low data rate scenario.(低速率时，多路径衰减问题可以很容易被解决，所以不算是个问题。)
3.Moving towards High Data Rate
Fast forward to a decade, say early 2010s.Assume that a data rate of 10 Mbps is needed for this urpose, which translates to a bit duration of 0.1μs as shown in Figure .
A high rate10 Mbps signal with a bit duration equal to 0.1μs。（传递一个字符需要0.1us,与之前的10us,速度增加了100倍）。
The main point is that the environment is still the same and does not care about our data rates! Multi-path components previously arriving1 and2μs after the direct path will still arrive1 and2μs
after the direct path. This is illustrated in Figure 6 (again ignoring the carrier). The different path lengths will translate into different attenuations and phase shifts resulting in constructive and destructive interference throughout the signal span.（传输速度变快了，但是环境没变，所以传完以后就成了下图这样）
First and second multipath components still arrive after 1 and2 μs after the direct path, respectively (carrier wave not shown)
What does this imply for the high rate transmission? Notice that in this case, the initial bits of the transmission are interfering with many tens of bits in the future through the late arriving paths, a phenomenon known as Inter-Symbol Interference (ISI). In most cases of interest, this ISI could have been observed even extending to hundreds or even thousands of bits. We can conclude that the same harmless channel for low rate communication has become harsh for high rate communication!（这样会造成严重的码间干扰，为了解决这种问题，所以的找出解决办法。）
4.Solution – Equalizer
It turns out that a solution for this kind of problem was devised by Robert Lucky at Bell Labs in 1964: an adaptive equalizer. An equalizer is a filter that mitigates the effects of channel fading on the Rx signal and removes the Inter-Symbol Interference (ISI).
Equalizer input is a distorted waveform and its output is a clean bit stream
Can there be a technique to achieve fast communication with a simpler equalizer? The answer is yes and the technique is OFDM.
5…OFDM in Time Domain and Frequency
in time domain, OFDM breaks one serial fast bit stream into many parallel slow bit streams.
Then, these parallel slow bit streams are multiplied with orthogonal sinusoids, where orthogonality between two sinusoids is defined with a summation over a certain time interval as
An OFDM example in time domain.
The multi-path components for individually modulated subcarriers as well as for the composite signal are shown in Figure
Multi-path components for individually modulated subcarriers as well as the composite signal. First and second path respectively shown up and down for clarity.Hence, the equalizer design is easy having less spread paths and consequently less interference with future symbols,
provided that we find a way to separate the subcarriers at the Rx.
Signal bandwidth plays a central role in determining how it is treated by the channel
In frequency domain, OFDM slices the spectrum through using the subcarriers; now each spectral segment can be processed individually
ust like a whole bread needs to be sliced for eating convenience, OFDM slices the spectrum for communication convenience
It was difficult to process a whole bread before that invention. Similarly, it is difficult to process the collective spectrum for communication purposes. By slicing the spectrum, OFDM not only made it easier to equalize the wireless channel but also made it possible to send different modulation signals on different subcarriers (e.g., subcarriers experiencing ‘good’ channel can be used to transmit a higher-order modulation signal that translates into more bits within the same time). On a lighter note, now we have a formal proof that OFDM is the best thing since sliced bread.(正交调制不仅可以使得无线信号高数据率传递，而且在不同的子载波上可以发送不同的调制，这也是雷达通信一体化设计策略中用到的技术。)
_In time domain, OFDM converts one serial fast bit stream into many parallel slow bit streams.
_In frequency domain, OFDM segments one wide spectrum into many narrow spectra.
