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.

17 thoughts on “Mod-01 Lec-27 Introduction to OFDM and Multi-Carrier Modulation

  1. 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

  2. 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

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

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

  5. 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 …

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