Whar Part Of The Ofdm Orthogonal Frequency Division Multiplexing Works In Frequency Domain Thank

What is orthogonality in the context of OFDM and how Orthogonality  makes these sub-carriers devoid of inter carrier interference?

Orthogonality of sub-carriers simply means their correlation is zero.[math]\ = \sum\limits_{n=0}^{N-1}r_{1}(n)r_{2}(n) = 0[/math]where [math]r_{i}[/math] refers to the [math]i^{th}[/math] subcarrier at the ReceiverUsually, the subcarriers are finite-length sinusoids of containing N samples of the waveform. DFT of a finite-length sinusoid is a sinc-like function, which you see in the examples. For two sinusoids to be orthogonal, they must be harmonics of some base frequency. Which is just a fancy way of saying that they are of the type:[math]r_{i}(n) = \cos(2\pi\dfrac{i}{N}n), i = 0,1,..,N/2[/math] and [math]n = 0,1,...,N-1[/math]The base frequency in this case is [math]2\pi\dfrac{1}{N}[/math]. We can use ANY set of values of [math]i[/math], and the generated signals will be orthogonal to each other. Post in comments if you need rigorous proof for that.Clearly, OFDM is not affected by inter carrier interference (ICI) if the subcarrier are strictly orthogonal as effect of any other subcarrier will be nullified by correlation. For this, OFDM requires precise frequency synchronization.

Long Term Evolution (LTE): What is OFDM? Also Its comparison with FDM and TDM?

To begin with, bandwidth(band of frequencies) is an extremely expensive resource. Also the amount of bandwidth available for an application is always limited. So, to make proper utilisation of this resource (especially in a cellular network), multiplexing techniques such as FDM, TDM were introduced.FDM (Frequency Division Multiplexing) involves splitting up a given bandwidth into smaller subcarriers. For example: say we have a bandwidth of 20MHz. In FDM, we will split this 20MHz into (say) 9 smaller subcarries of 2MHz each + 2 MHz of total guard band (to avoid interference amongst adjacent subcarriers). So, now we effectively have 9 small channels instead of 1 big channels. These 9 channels can be allocated to 9 different users, thus enhancing the capacity (no. of users supported) of the system. If we split the bandwidth into N subcarriers, the capacity of the system increases N times.TDM (Time Division Multiplexing) involves allocating the same frequency to different users at different time intervals. For example: say we have a frequency channel of 2MHz. Beginning at time t=0, we will allocate this frequency channel to user 1 for the first 2 seconds*, then the same frequency channel is allocated to user 2 for the next 2 seconds. Then, again the repeat the cycle. So, user 1 transmits for t=1,2,5,6,9,10,…… and user 2 trabsmits for t=3,4,7,8,11,12…….So, now we have effectively doubled the capacity of the system. If we split the same frequency between N users using TDM, the capacity of the system increases N times.OFDM (Orthogonal FDM) is a special case of FDM where the smaller subcarrier channels are chosen in such a way that they are orthogonal to each other. This implies that there won’t be any interference amongst the subcarriers and hence there is no need of a guard band (which was 2Mhz in FDM example) in OFDM. For same example as FDM: say we have a bandwidth of 20MHz. In OFDM, we will split this 20MHz into 10 smaller subcarries of 2MHz. So now we have any additional subcarrier as compared to FDM.Infact in OFDM the subcarriers can even partially overlap, but we’ll skip that part here.LTE uses a combination of OFDM and TDM.*the actual time interval is in milliseconds.

Why are mathematicians so obsessed with proving conjectures when it is of little practical value? Wouldn’t applying the conjectures be of greater practical value?

There are two ways to address this question that I see. One is to answer it at face value. That is, I can tell you there are plenty of people who like to prove things just because the topic or method of proof involved is of interest to them. You might as well ask why certain people like playing basketball rather than getting a second job where they can earn more money. The vast majority of people don't distill all their life choices down to a ranking of things that are practical for them or others. A large portion of what people do is simply due to fact that they enjoy doing them.The second is to point out that there is often more "practical value" to various things (including proving conjectures) than may be immediately obvious on cursory examination. I will go ahead and skip the more-or-less obvious point that applying an idea that is not provably true can be a risky proposition. Instead, I'll point out that the techniques involved in the proof itself can sometimes increase understanding of the topic and this understanding can lead to improvements in the "practical" techniques in the topic. For instance, I have been reading lately about methods in inverse scattering of acoustical / electromagnetic waves which are more-or-less derived from proofs of certain topics in that field. In these cases, the conjectures were already well-known or obvious, but not necessarily practical. However the proofs themselves led to new techniques for solving real-world problems. One last point which is mostly related to the last one is that often times a proof of some concept, even if it doesn't lead directly to something "practical", can improve the understanding of that topic. This improved understanding can inspire practitioners to develop new methods that they may not have developed on their own.In summary:1) Human beings are human beings complete with their own sets of personal preferences and interests which have nothing to do with how practical those preference/interests are.2) Things that don't seem practical aren't always what they seem. 3) No practitioner ever suffered for understanding their field-of-interest more completely.

Why does OFDM involve the FFT? I find this confusing, as the FFT seems like a coarse tool, which would severely limit the data rate because of the large number of samples required to do an FFT.

OFDM (Orthgonal Frequency Domain Multiplexing) refers to the concept of using multiple, non-overlapping, carriers at once in order to exchange communication between two endpoints. Non-overlapping is meant in the spectrum sense, here.Among the benefits is higher throughput or more robust communication or both, thanks to the use of different modulation techniques on each of these carriers depending on their current channel characteristics, instead of using the entire band (covered by all the carriers combined) as a single channel with a single modulation technique.For instance, and within a given spectrum band, a "clean" carrier (with low error rates) may allow a high-rate modulation such as Octal Phase Shift Keying or high-order QAM (Quadrature and Amplitude Modulation) such as 64QAM or 128QAM resulting in high effective throughput, while another carrier with poorer transmission characteristics can be modulated on more basic Binary Phase Shift Keying (with lower throughput but more robustness in the face of noise and errors).This ability to spread the parts of a message over multiple narrow sub-carriers within a given spectrum band, and to tailor the data communication methods to the properties of each of these carriers, and doing so simultaneously in time, can deliver superior aggregate performance when, for instance, noisy conditions are specific to narrow ranges within the spectral band.OFDMA refers to Orthogonal Frequency Domain Multiple Access and is an "access method" for multiple parties to simultaneously share the spectrum and "talk over one another" so to speak. They do so by talking over non-overlapping spectral bands. This is in contrast to other methods such as TDMA for instance (Time Division Multiple Access) where, when one party is talking others must back off, resulting in degradation of throughput. Of course this means that the communicating devices must be able to "tune into" multiple carriers at once, but these trade-offs of complexity in return for added throughput or robustness are increasingly favorable with advances in semiconductor technology versus cost.