# Mod-01 Lec-27 Introduction to OFDM and Multi-Carrier Modulation

Hello, welcome to another lecture, in the

course on 3G 4G wireless communication systems. In the last lecture, we discussed the non-linear

V blast receiver for MIMO communication systems; we also said that, this receiver is based

on the SIC principle; that is successive interference cancellation, where are symbols are estimated

and their impact is successively cancelled followed by detection of other symbols and

so on. This is an iterative procedure essentially. And, we had also seen that advant, we have

also seen that in this procedure, you consider the matrix q, which is the pseudo inverse

or left inverse of the matrix H. However, we do not employ 0 forcing, you only cancel,

one stream employing the row in the corresponding row in cube, and then you remove its impact

and followed by the detection of other schemes. And, we had also seen that the advantage of

this scheme; the significant advantage of this scheme is that the diversity order of

the successively decoded frames are higher; that is as you keep decoding frame symbols,

the diversity progressively increases for the later decoded symbols in this procedure,

thus enhancing reliability. And, we had also seen an example of a V blast

system of the simple V blast receiver for a 2 cross 2 MIMO system. Then, we had also talked it, talked about

beam forming; that is forming a beam in a MIMO communication system; that is a instead

of employing all the modes, we transmit only in a certain given direction in n dimensional

space, which is less beam forming. For this purpose, we said that the x bar vector

can be formed as at beam in the direction of v 1 bar, where v 1 is the dominant right

singular vector of the matrix H, employing was single transmitted symbol x 1 tilde. And corresponding to that, we said the received

SNR was sigma 1 square P over sigma n square, so it exact. So, the gain of this channel

is sigma 1 square, which extracts the dominant or which corresponds to the dominant singular

mode of this MIMO channel. And, further we said that, this scheme is

maximal ratio of transmission; it is termed as MRT, which essentially extracts the, which

essentially extracts corresponds to the yields. The channel gain corresponding to the largest

singular value or the larger the strongest mode in the MIMO channel; and at that point

with that essentially that concludes our discussion of the MIMO model models. So let us start

with summarizing, let me start summarizing MIMO model, and then I will continue with

the next module in this course on 3G 4G wireless communication systems. So, let me summarize MIMO module; in this

MIMO module, we had seen we had started with, so in the MIMO module, we had started with

first the MIMO architecture; we started with the MIMO architecture system model; we started

with the MIMO architecture with this multiple antennas at the transmitter, multiple antennas

at the receiver, we started with this schematic developed. The system model in terms of y

equals H x plus n, where x is now a vector, y is now vector, and H is a matrix channel.

We also looked at MIMO receivers. We first started with linear MIMO receivers. We said

for instance, MIMO linear receiver is the MIMO zero Forcing Receiver; simply abbreviated

as MIMO-ZF; that is the MIMO zero Forcing Receiver, it employs the left inverse of H

and the pseudo inverse of H, when the number of rows H of the, number of receive antennas

is essentially greater than or equal to the number of transmit antennas. And, we also

said that, one of these advantages of the MIMO zero Forcing; is that it results in noise

enhancement, hence we also proposed another MIMO receiver architecture; that is the MIMO

MMSE receiver. So, we also proposed a in fact, this is the

MIMO LLMSE receiver, I will simply call this the MIMO MMSE receiver, where MMSE stands

for Minimum Mean… So, essentially this MIMO MMSE receiver; it minimizes the mean square

of error of detection at the receiver, and we also saw that it has superior properties

in the sense; that it does not result in noise enhancement, because of its unique structure.

Then, we focused on something important, which is the singular value decomposition of the…

We focused on the singular value decomposition of the MIMO channel; this is also termed as

SVD, so we focused on the singular value decomposition of the MIMO channel; this is termed as the

SVD. We also said, this is based on a pre coding and receiving beam forming at the receiver.

This is based on, based on a pre coding structure essentially. This is also, this is also essentially

a transmit preprocessing that has to be done at the MIMO transmitter before transmission,

but the big advantage of this is as, we saw it results in parallelization of the MIMO

channel. This SVD results in parallelization of the

MIMO channel, or instead in other words decoupling of the different special modes of the MIMO

channel; so it represents a convenient frame work, so it results in… Or decoupling or decoupling of the special modes

of the MIMO channel; that is a big advantage, because now, there is no interference from

the co streams at the receive antennas. Once you push process using her mention, what

you have is each it appears as, if this MIMO channel consists of t independent channels,

the x 1 transmit across channel 1, x 2 transmit across channel 2 and so on. So, we had looked

as, how the MIMO channel can decoupled by transmit preprocessing and post processing?

So as to decouple the channels and essentially make them interference free from the other

symbols. And in this context, we have also seen the

SVD based optimal power allocation. We have also seen SVD we have also seen the SVD based

optimal power allocation for capacity maximization, optimal power allocation for…In fact, we

can, we also said that, this is derived as the water filling based power allocation solution.

This we derived a water filling, we also derived a water filling based power allocation solution

and then, we looked at an interesting framework to characteristic as the asymptotic MIMO capacity,

we characterized the asymptotic, we characterized asymptotic MIMO capacity alright.

And, further we also said that the MIMO capacity linearly increases as the minimum of r for

the same transmit power. And that is in fact, what results in the high through put up, these

MIMO wireless communication systems. Then, we also looked at the unique framework of

orthogonal space time. We also looked at orthogonal space time block codes or OSTBC. We looked

at orthogonal space time block codes or OSTBC. In this context, we had looked at the in detail,

at the Alamouti space time block code, the Alamouti OSTBC, which is essentially intended

to 1 cross 2 MIMO system. And, we also saw an example of a rate half. We said Alamouti,

in fact full rate or r equals 1 OSTBC and, we also saw an example of a rate half another

OSTBC for 1 cross 3 MIMO systems, which essentially means 1 transmit antenna, 1 receive antenna,

and 3 transmit, it essentially has 1 receive antenna, and 3 transmit antennas. And we said

this is a rate half code; this not a full rate code, but this is a rate half code or

in fact, this is a rate half orthogonal space time block code. And then finally, we had earlier seen; non-linear

receiver v BLAST, which is vertical BLAST; which it

stands for vertical bell labs layered space time architecture alright. This is a linear

receiver, in fact as have also summarized today’s based on, the successive interference

cancellation. And finally, we had looked at, beam forming in MIMO systems, and it is also

looked at… And, this is based on principal of Maximum Ratio Transmission. We have introduced

the concept of MRT, which show a Maximum… So we have introduce the principal of MRT,

which stands for Maximum Ratio Transmission. Already, we introduce the last topic talked

about, beam forming in MIMO system, so with that we conclude the conclusion of MIMO module;

this is summary of different topics; that we covered in this module, so please again

go over the the understand each of this topics. And now, I move on to the next module in the

course, which is which is attractive, which is probably, which has come to attract a lot

of attention; this is OFDM. We will explain, what OFDM stands for? So, the new module that,

we are going to start talking about is OFDM, which stands for orthogonal frequency division,

so OFDM basically stands for orthogonal frequency division multiplexing alright. And, this is

a revolutionary communication technology and why is this a revolutionary communication

technology? OFDM essentially forms the bases for all 4G wireless communication systems

for instance, if you remember, what we had talked about earlier in one of the very first

lectures of the course. We had looked at 3G technologies, we said CDMA from the bases

put 3D technologies over all 4 technologies are based on OFDM. So, OFDM is very important, because OFDM forms

the basis for… it forms the basis for futuristic 4G or fourth generation or 4G, which is essentially

fourth generation wireless communication systems alright. So, OFDM or orthogonal frequency

division multiplexing is a very important technology, revolutionary radical technology

in wireless communication systems, because it is a key technology for 4G wireless communication

systems. For instance, we would also let me briefly

summarize what kind of 4G wireless communications systems looked at we looked at the next generation

cellular networks based on the LTE this is the 4G cellular expected to be 4G cellular

standard, which stands for long term stands for this is LTE, which also stands

for long term evolution alright. So this is basis of OFDM and this is based

on OFDM LTE this is based on o f d m and in fact o f d m is also based on the basis for

the competing cellular standard of WiMAX this is also another 4 g cellular standards in

fact 4 g cellular and also 4 g 4 g 4 g cellular standard and this stands for worldwide

interoperability for microwave access in fact wimax also has

capabilities for fixed wireless access wireless access and other modes of wire not just simply

cellular but, also fixed broadband wireless access essentially for mile wireless access

using from the tower to the home and so the hub to the home.

And so, alright so OFDM is the basis for these two key; next generation, wireless communication

technology. And, the reason is because, OFDM is a broadband wireless technology, and there

are several advantages; In fact, it supports data rates, which are in excess of 100 mbps.

If you look at and WiMAX for instance claims, that it can support data rates in excess of

100 mbps. So, OFDM is a key broadband

wireless technology, which supports data rates in excess of… So OFDM is a key

broadband wireless technology, and it helps, it supports essentially one of the key technologies,

which supports data rates in excess of 100 mbps, which is crucial and next generation

broadband wireless communication systems and communication networks alright. and also further

not only cellular standard, OFDM is also the basis for land standards. So, if you looked at 11 a b not b I am sorry11

a g and n standards; these are also based on OFDM; in fact these in fact, dot 11 n can

progressively support data rates up to around 200 mbps. So, 8 dot 11 n can support data

rates, around 200 mbps and and more. So, OFDM is also a basis for these wireless LAN standards;

these are the as we all know 8 o 2 dot 11 suite of standards suite of standards; it

is essentially the dominating set of wireless, the dominating wireless LAN standards. And

as we said so these are based on some very interesting and convenient properties of OFDM,

which we are going to start looking at in this module. So, in OFDM for instance, even

before going to OFDM, let us try to understand, what is what is the scenario for communications?

they are existed previously to OFDM. So, we can consider a band width. Let us,

start by considering a band width of B; available for communication. So, consider

a bandwidth B available for, when I say a

bandwidth B. I am considering the two sided bandwidth, here I am considering the…I am

considering the two sided bandwidth B; that is available in a traditional communication

system. Typically, when bandwidth B is available for

communication for instance, when I look at this two sided bandwidth B; that is available

for communication. Typically, the employ symbol time T; this is the symbol time, which is

equal to 1 over B, so I employ a symbol time T, which is equal to 1 over 1 over B, and

I sent symbols every T. So I sent 1 symbol every T. Technically, if you have the bandwidth

of B, you can sent two symbols on different cosign and sign carrier. But for the sake

of simplicity, I am just simplifying it; say, we are using one of the orthogonal carriers,

and I am sending symbols every T seconds, so send So let me summarize that idea bandwidth b transmit one symbol every T seconds where T is in fact one over b so every T equals one over b you

transmit every T equals one over B you transmit a symbol and in fact the rate the symbol rate

was 1 symbol every one over B second which is nothing but, B and that is what is so you

have the bandwidth of for instance let us say 100 megahertz you can roughly transmit

symbols at megabits per second alright that is the essential idea in a communication system

that is t equals 1 over B,which is 1 over hundred megahertz which is her n to the power

minus two micro seconds we transmit one symbols that is t equals one over hundred megahertz.

So, every n to the power minus 2 micro seconds, you transmit 1 symbol, which essentially gives

you a single rate of 100 megabits per second alright. And, this is traditionally; what

you do is? You have base band, you modulate symbols on the base band and employ a single

carrier to modulate this stream and transmit it in the allocated pass band bandwidth alright.

So, this is the philosophy or this is the conventional communication system, which is

which is known as single carrier communication, this is a single carrier communication system,

this is a single carrier communication system. As, we have seen; this is a single carrier

communication system; what it means is? There is one carrier, this is a

which means; there is a unique carrier; that

is one carrier and that consumes the entire bandwidth, that occupies

the entire communication bandwidth. There

is a single communication carrier; and that occupies, the entire communication bandwidth

of B and that is the philosophy of a conventional single carrier system alright. And, if you look at the transmission itself

the transmission can roughly be expressed as follows as s t equals x k for k T less

than equal to t less than equal to k plus one T alright for instance between if k equals

0 between 0 and T you transmit x 0 so x 0 between 0 less than equal to T less than equal

to T you transmit x one between t less than equal to t less than equal to k plus two k

b which is two t you transmit x two in two t less than or equal to t less than equal

to k plus one which is three t and so on so you have symbols essentially you have a slots

of duration capital t which is one over b that is as we said before one symbol

every you are transmitting one symbol every

every T seconds. So, you are transmitting one symbol every

capital T seconds, while T itself; this is equal to 1 over B. So, that is what, we said

is essentially the principle of a single carrier system and the rate obviously 1 over 1 over

B, which is B symbols per seconds. Now, as against this let us consider something different,

I will not explain, why need to consider some different? I will not the motivation at this

point, but let us start by considering it, and I will explain the motivation once, we

go a little bit further into this thing alright. Let me start by explaining the motivation. Let me, consider a system, in which I have

a bandwidth of B. Again remember, I am talking about the two sided bandwidth, and what I

am going to do is instead of having one single carrier, I am going to divide it into multiple

carriers divided into multiple carriers at a spacing of B over N.

So, I have a bandwidth of two sided bandwidth B. I am going to divide this into multiple

carriers at spacing B over N, that is what I am going to divide I am going to divide

it into multiple sub carriers. Precisely, how many carriers are there? There are N carriers.

In fact, these carriers occupies small band, so these are known as sub carriers. So, I

am dividing I am dividing the available bandwidth B into

n sub carriers, in fact that we finish drawing this diagonal this is minus B by N; this minus

N by 2 minus 1 B by… And in fact, so if you look at this; this

is a the sub carrier spacing is obviously B over n. I am using N sub carrier, so these

are all this together are N sub carriers. why sub carriers because dividing the bandwidth

into smaller bands and e in sub bands and in each of them. I am using a carrier; each

carrier represents this smaller sub band. Hence, it is a sub carrier, in fact this is

a sub band this is a sub band of the… So, I am de compositing into multiple sub

bands. I am using sub carriers in each of these bands, and the net bandwidth is allocated

to B, so I am dividing it into N sub carriers and the spacing is B by N; this is in fact,

the sub carrier this is in fact a multi-carrier system; so

this is an example of a this is an example of a multi-carrier system. In fact, in this

system each carrier each of the N sub carriers has a much smaller bandwidth, compared to

the bandwidth of a single carrier system alright. This is a, what we have, here is a multi-carrier

or a multiple carrier system. Now for instance, let me give you an example:

Let me consider a bandwidth B, consider B, which is equal to 256 kilo hertz. Typically,

N the number of sub carriers. Typically, n either equals values such as 256,512,1024.

Typically, the number of sub carriers is a power of 2, typically N, which is the number

of sub carriers, is a power of 2 alright. So, typically the number of sub carriers is

a power of 2.We will see the reason for this; now let us see since, it is typically a power

of 2. Why it is the power of 2? you are going to see it, shortly in the this lecture subsequently

lecture. For instance, let us have a bandwidth 256 kilo hertz.

Let me, consider n equals, let us say 64 sub carriers, then the bandwidth per sub carrier

is B over N, which is equal to 256 hertz divided by 256 kilo hertz divided by 64 equals 4 kilo

hertz; so in this example, where I am employing a total bandwidth of 256 kilo hertz, I am

dividing it into 256 smaller sub bands; and each of those sub bands is a bandwidth of

256 divided by 64, which is equal to 4 kilo hertz alright.

Each of those has sub bands has a bandwidth of 4 kilo hertz, and that is what we are saying

here essentially. Now fine, I have divided it into several sub bands of much smaller

bandwidth; how am I going transmit? How am I going to transmit information symbols on

these sub carriers? So, we have to consider the multi-carrier transmission scheme. So, let us consider the multi… So, let us

consider the multi-carrier transmission scheme; so this multi-carrier in this multi-carrier

transmission, I have seen the sub carrier is given as the that is given as i times B

over N; that has a center frequency i times B over N; this is the center frequency of

of the

sub carrier, where this index i. If such that minus N by 2 or minus of N by 2 minus 1 is

less than or equal to i is less than or equal to N by 2 alright. So, we are transmitting

the sub carrier. In fact, I am going to denote this by f of i, so f of I, which is the sub

carrier has a bandwidth of B over N and it is centered at i times B over N, where the

index i has this range as given below. Hence, let me consider X i, which is the data;

that has transmitted on this sub carrier, so let denoting by X i, the data X i is the data; that has transmitted ith

sub carrier. In fact, previously remember, we had only one carrier, so we are transmitting

only one data stream. Now, I have N subcarrier, so I can transmit N data stream. So, I am

denoting by i the essentially the ith data stream; this is the data; that is transmitted

on the ith sub carrier. In fact, the modulated signal on this sub carrier is given as S i

t equals X i e to the power of j 2 pi f i t; this is the modulated data stream; so this

as we see is the data; that has transmitted on the ith sub carrier.

So, this is the this is the data; that is transmitted on the

ith sub carrier and this f i; this is the center frequency of the ith sub carrier, which

is i times B over N alright, f i is essentially, i times B over N; this is the center frequency

of the ith sub carrier. And in fact, I have N such sub carrier, so I have N such data

streams alright. In fact, I am going to write this as x i times e power j 2 pi i B by N

over t alright, because remember the center the center frequency of this sub carrier is

i times B over N t. So, and as we said again, let me rewrite again.

Now, there are N sub carriers and there are N data streams; so there are now N sub carriers.

Hence, we have N data streams, hence there are N sub carriers, hence there are data streams;

these are the index by x of i; basically x i denotes; this denotes the ith or essentially

the ith set of symbols, which can be modulated on to these sub carriers alright. So, that

is what we have seen. So, now how does multi-carrier modulation work, I mean what a how do we transmit

the signal? we simply sum up these signal corresponding to this N data streams, in fact

the modulated N capital N data streams and transmit them together. So, multi-carrier transmission, so let us

look at, so we have S i of t, which is ith transmitted stream. So, what I am going to

do is, I am going to form a composite transmit signal S t equals summation over i of S i

of t, so what I am going to…This is essentially the composite signal. This is the composite

transmit signal ka transmit signal; comprising of the N different transmit stream, which

has been modulated on to the N different sub carriers. In fact, you can see this is modulated

on to the ith stream, ith data stream modulated on to the ith; this is the ith data stream;

that is modulated on to the ith sub carrier. In fact, I can represent this as s t equal

to summation of pi summation over i S i of t, which is nothing but, if I expand this

out, this is X i e to the power j 2 pi f i of t, where this x i is the data; that is

the ith data stream, and f i is the sub carrier corresponding to the ith sub band; and this

is equal to summation over i X i e to the power of j 2 pi i B over N alright; so this

is the composite transmitted multi-carrier signal. Remember, we are still not talking

about OFDM, we are still talking about multi-carrier transmission only alright. So, one should

not confuse this with OFDM will be introduced subsequently alright. So, this is the multi-carrier

composite; this is the multi-carrier composite transmitted signal alright.

Now, this is the transmission scheme, so we illustrated transmission scheme now; this

is of course, only complete, if we illustrate a corresponding detection scheme, because

fine we can sum all this data signals and trans all these information streams and transmit

them over sub-carriers but, this is only going to be complete, if we illi [inter/introduce]

introduce a compatible or some symbols, some scheme for detection of the composite signal

at the receiver alright, only then I can claim this use this transmission scheme meaningfully. So, let us talk about the data detection;

the corresponding data detection at receivers. So, the multi-carrier, so let us talk about

multi carrier data detection; let me now for the moment ignore noise; I am not going to

consider noise, just to illustrate the detection scheme. So, I going to consider a receive

signal y t, which is in fact equal to the transmitted signal s t equal to summation

X i e to the power j 2 pi f i of t alright. So, this is the received signal s of t. Of

course, this is valid in the absence of noise; this is valid in the absence of noise, and

remember I am only assuming this at this point, to simplify the illustration alright, even

with the noise present this scheme is valid alright. So, let me just for the movement

to illustrate this module for illustration assuming the absence of noise.

Now, what I am going to do is, I am going to coherently demodulate each stream with

the corresponding sub carrier. So, I want the receiver, the detection scheme is coherent

demodulate each stream with. So, what I am going to do? I am going to coherently demodulate

each stream; that is I want to take this composite signal; that is received and before each stream

to recover each stream. I am going to coherently demodulate it with the corresponding with

the corresponding sub carrier. What I mean by this is for instance, let me

take this composite signal y t, which is the composite received signal and I am going to correlate it with e to the power j 2 pi f l of t, conjugate that is this is this is

nothing but, the l sub carrier in, and I am coherently multiplying with the conjugate,

which is I am in fact, I am doing the match filtering kind of operation; this is nothing

but, a correlation, so this is also nothing but, a correlation. I am going to correlate

from 0 to the duration is N over B and I have a scaling factor, which is B over N, and I

am going to motivate. Why this integral of duration 0 to N over

t slightly later? but essentially. What I am doing here is? I am doing a matched filtering

sort of operation, where I am correlating the composite signal with the coherent sub

carrier, here in this case I am correlating it with the coherent sub carrier, so this

I am going to represent this as y t. Remember now, I am going to substitute the expression for y t as X I, the summation over i X i e to the power of j 2 pi i B by N over t, I am taking this whole signal, which is

the composite signal; I am multiplying this by e to the power of minus j 2 pi, in fact

f of l and replace it as l times I am going to replace f of l as l times B over N t alright.

I am going to…I am taking this composite signal, i am correlating with the lth coherent

sub carrier, integrating it over 0 to N by band, this B by N is a simple scaling factor

ok. Now, let see what happens in this scenario, this can simply be represented as I will take

the summation out of the integral now. And this can simply be represented as B over

N summation i integral 0 to N over B summa X i e e to the power of j 2 pi i minus l B

over N t alright. Now, look at this, this is B over N summation i zero to N or B x i

e for j 2 pi i minus l b over n t; let me call this as a let me call this as sum k let

me call this as sum k, in fact this is equal to B over N summation i 0 to N over B or now

that, because I am calling that by k, it becomes x k e power or if you just let this remain

as i j 2 pi k f k naught of t d 2, here f naught is now the fundamental frequency, look

at this f naught is being defined as f naught is being defined as N, which is the fundamental

frequency. If you are familiar with series, you know that and you can see here that all

the other frequencies are in fact the multiples, they are all sum i minus l f naught, which

are essentially the multiples of these fundamental frequency.

So, if I integrate over 1 over the period, if I integrate 1 over the f naught, the period

t naught equals 1 over f naught equals n over B alright; this is the fundamental period;

this is nothing but, the fundamental period of this system; of this which is the and,

you can see here in this scenario, when I integrate this from 0 to N over B; all these

terms go to 0, except when k equals 0, which corresponds to i equals l. So, this integral so let me again, simplify

this integral e for j 2 pi i minus l i minus l b over N t from 0 to B over N, which is

N or you can write it, sum 0 to N over B; this integral e equals this integrals equals

0, if i knot equals l and this integral e equals, in fact if you look at this, if i

equals l; this integral becomes 0 to N over B e for j 2 pi i minus l i equals l, since

this is 0 e power j 2 e power 0 is 1. Hence, this integral becomes 0 1 d t equals

N over B equals, so this integral equals N over B, if i equals l, hence what we are saying

is? when a coherently demodulated with the lth sub carrier, all the other sub carriers

are orthogonal to this sub carrier; that is why coherent demodulation receive results

in an output of 0, while only the lth sub carrier survives; so this is an important

idea, in fact what we are seeing here is essentially. Let summarize this, here all sub carries except lth, all sub carriers except lth sub carrier all sub carriers except lth sub carrier or essentially; this results in

an output of 0; they are orthogonal. Hence, I can essentially coherently demodulate by

correlating with e power j 2 pi f l of t; that is essentially. I am multiplying by e

power minus j 2 pi f l of t and integrating over the fundamental period, which is 0 to

N over B. And now, what we have is obvious, we can write

from here, let us go back to see, what we have had earlier? What we had here is, I can

write this B by N summation of I, I am going to write this as what we have? After demodulation is after demodulation in

the coherent, lth after demodulation with lth coherent sub carrier, what I have is,

I have this is equal to B over N into x l into the integral N over B plus rest all sub

carrier gives 0, hence this results in x l. So, I can employ coherent demodulation, at

the receiver and coherently demodulate with corresponding sub carrier for each sub carrier

to receive each symbol. So, I do this with N different sub carries, and demodulate with

N different sub carriers, and I can recover the N different symbols; this is the principle

of multi carrier transmission alright, multi carrier modulation multi carrier transmission

or multi carrier detection or reception. So, at this point, we so shortage of time, I will

end this lecture at this point, and we will be continuing with this idea in the next lecture.

Thank you.

It is very fluent and teaching is really step by step which allow you to understand very easy.

awesome teacher salute u

Amazing ! Indian professors are the best in teaching. Easy to understand and there are no missing links.

awesome knwodelge hats off sir

Excellent!! presentation so clear and precise love the step by step details.

why is the number of subcarriers always a power of 2

Hi, excellent professor. on around 53 min of this video, the result of the integration is equal to zero when l is not equal to i. I tried to solve the integral but i did not get zero! please advise

Hi

On about 53 minutes the integral is equal to zero when l does not equal to i. I tried to the integration but not able to get zero. please advise

The most meticulous, simple and efficient explanation I have studied about OFDM. Thank you Sir for your wonderful job.

Best series of lectures in communication. Great job and carry on.

im working on ofdm communication technology.thank you sir for ur lecture. easy to understand ur lesson good job sir…

detailed introduction， thx

OFDM starts @ 14:30 . Theory start @ 21:30

what about ISI and ICI distortion in multicarrier modultion

Are we transmitting n data streams simultaneously or just transmitting composite signal ..If we are transmitting composite signal then what is advantage of breaking the bandwidth into smaller bands..May be this is basic question but it will help me lot …

24:58 bandwidth should be the passband bandwidth