orthogonal frequency division multiplexin_paper18!3!12

7/30/2019 Orthogonal Frequency Division Multiplexin_paper18!3!12

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Orthogonal Frequency Division Multiplexing (OFDM)

Modulation: Modulation is the process of varying the high frequency carrier with respect to

the low frequency message (baseband signal).either the carriers phase, frequency or

amplitude, or combination is varied.

Multiplexing: Method of transmitting multiple signals over a channel.Or

It is a method of sharing the bandwidth by multiple data channels.

OFDM:- OFDM is a combination of both modulation and multiplexing, Here multiplexing

refers to the independent signals which are produced from different sources.

In OFDM the multiplexing is applied to each individual signal where there individual signals

are subset of the main signal.

In OFDM the signal itself is divided into multiple independent channels which are modulated

by data and re-multiplexed to create the OFDM carrier.

When we compare the OFDM and FDM we correlate the FDM channel is like water flowing

out of a tap. Whereas the OFDM signal is like a shower.

In the tap all water comes out like one big stream and cannot be subdivided, whereas theOFDM shower is made up of a lot of little stream. We can think here what the advantage of

one over the other.

If we put the thumb over the tap hole, we could stop the water flow whereas the same is not

possible with the shower.

Fig: 1(a)A Regular-FDM single carrier-A whole bunch of water coming all in one stream.

(b)Orthogonal-FDM-same amount of water coming from a lot of small streams.

So although both do the same thing but they respond in a different way to interference.

We can explain the above concept with another example.

Suppose we want to make a shipment using a truck. We have two options. {1}Hire one big

truck or {2}Hire few small ones.(four small trucks).By using both the method we are able to

carry the same amount of data,but in case of an accident only of data of the OFDM

trucking will suffer.

Fig 2:All cargo on one truck vs. splitting the shipment into more then one

When we see the four small trucks as signals, they are called sub carriers in OFDM system

and for to work this idea they must be orthogonal. The independent sub carriers can be

multiplexed by FDM which is called multicarrier transmission and if it is based on code

division multiplexing(CDM) then it is called multi code transmission.

In this paper we will explain only about multicarrier FDM or OFDM.


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Fig3.Multi-carrier FDM and Multi-code division multiplex

The importance of being orthogonal

The main fundamental concept of OFDM is orthogonality of the sub carriers. As the carriers

are all sine/cosine wave and the area under one sine cosine wave is zero which is shown in

below figure

Fig 4: The area under the sine and cosine wave over one period is always zero

If we take a sine wave of frequency m and multiply it by a sinusoid (sine or a cosine) of a

frequency n, where both m and n are integers. The integral or the area under this product is

given by

F (t) =sinmwt x sinnwt -------- (1)

F (t) = sinwt x sinnwt

Sine wave multiplied by another of a different harmonic.

Fig (5): The area under a sine wave multiplied by its own harmonic is always zero.

By simple trigonometric relationship, this is equal to a sum of two sinusoids of frequencies

(n-m) and (n+m)

=

=

=0-0

The above two components are each a sinusoid, so the integral is equal to zero over one

period.

We can come to the conclusion that when we multiply a sinusoid of frequency n by a

sinusoid of frequency m/n, the area under the product is zero. In general for all integers n and

m, sinmx, cosmx, cosnx, sinnx are all orthogonal to each another. These frequencies are

called harmonics.

By understanding OFDM we can come to the conclusion that orghogonality allows

simultaneous transmission of data over different sub-carriers in a tight frequency space

without interference from each other.

This is similar to CDMA were PN codes are used to make data sequences independent (alsoorthogonal) which allows many users to transmit in same space successfully.


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OFDM is a special case of FDM:

If we have a bandwidth which goes from a to b, first we will subdivide the given frequency

ie. a to b in to frequency space of four equal spaces in frequency spaces the modulated

carriers would look like this.

Fig(6):FDM carrier are placed to next to each other

The frequencies are not dependent a and b can have any integer or non integer value because

they are not dependent on one another and the same is with the carrier frequencies which are

based on frequencies and they do not have any special relationship with each other.

If the frequency C1 and Cn are such that for any n (an integer), we can write

Cn=n*c1 ---------- (2)Similarly,

C2=2*C1 =2 C1

C3=3*C1 =3C1C4=4*C1 =4C1

All the above frequencies are harmonic to C1And as all the above carriers are orthogonal to each other, when added together, they do not

interfere with each other.

In FDM as the orthogonality is not maintained with the adjacent carriers we get interference

from neighbour carriers. To avoid interference in FDM the signals are moved apart i.e. the

guard bands are inserted between two carriers.The symbol rate that can be carried by a PSK carrier of bandwidth b is given by

Rs=2BL =BpRs-------Symbol rate

BL ----------low pass bandwidth

Bp --------------------passband bandwidth

The above relationship assumes a perfect Nyquist filtering with roll off =0.0. As this is

unachievable, we use root raised cosine filtering for which the roll off of gives the

relationship as below.

Rs= Bp1+

If we need three carriers, each of data rate = 20Mbps then we might place the BPSK carriers

as below.

With Rs=20 and B= 20 x 1.25 = 25 MHZ

Each carrier may be placed (25+ 2.5) 27.5 MHZ apart allowing for a 10% guard band.

The frequencies need not be orthogonal but in FDM we dont care about this where as the

Guard band helps keep interference under control.

An example of OFDM using 4 sub-carriers

In OFDM technology we can have N carriers where can be from 16 to 1024 in the present

technology and depends on the environment in which the system will be used.Let us see an example to study the OFDM signal using 4 sub carriers.


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The symbol rate of the signal is 1 and sampling frequency also 1 sample per symbol, hence

each transmission is a bit.

Fig 7:Abit stream that will be modulated using a 4 carrier OFDM.

First few bits are 1,1,-1,-1,1,1,1,-1,1,-1,-1,-1,-1,-,-1,-1,-1,1. . . . . .

We can represent there bits in rows of fours. Since in this demonstration we are using only

four subcarriers here we have effective achieved serial to parallel conversion

Table 1,

Serial to parallel conversion of data bits

C1 C2 C3 C4

1 1 -1 -11 1 1 -1

1 -1 -1 -1

-1 1 -1 -1

-1 1 1 -1

-1 -1 1 1

Here each coloumn represents the bits that will be carried by one sub-carrier.

In the Nyquist sampling theorem the information rate is twice for the smallest frequency

which can convey information.

In the above example the information rate is or 1 symbol/sec totally for all 4 carriers.

If I had picked Hz as the starting frequency then the harmonics would be 1,3/2 and 2 Hz.Ifwe pick BPSK as a modulation scheme for this example, note that we can pick any other

modulation method such as QPSK, 8 PSK, 32-QAM or we can use TCM (Trellis

Code modulation) which provides coding in addition to modulation.

Carrier 1: if i need to transmit 1,1,1,-1,-1,-1, what I saw from the below figure is

superimposed on the BPSk carrier of frequench 1 Hz.

First three bits are 1 and the last three -1. If I show the Q channel of this carrier (i.e. which

could be a cosine) then this could be QPSK modulation.

Fig 8:Sub-carrier 1 and the bits it is modulating(the first column of table 1)

Carrier 2: The next carrier frequency of 2 Hz, It will be the next orthogonal/harmonic to the

first carrier of 1 Hz. Here take the bits from second column which is marked c 2 1,1,-1,1,1,-1

and modulate this carrier with these bits.


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Fig 9:Sub-carrier 2 and the bits that it is modulating (the 2nd column of Table 1)

Carrier 3: The frequency for carrier 3 is 3 Hz and fourth is 4 Hz the third modulated with

-1,1,1,-1,-1,1 and the fourth with -1,-1,-1,-1,-1,-1,1 from the table 1.

Fig 10:Sub-carrier 3and 4 and the bits that they modulating(3rd and 4th column of Table1).

If we modulate all the bits using four independent carriers of orthogonal frequencies 1 to 4

Hz.Here we have taken the bit stream distirbuted the bits, one bit at a time to the four subh-

carrier as shown in figure below.

Fig 11:OFDM signal in time and frequency domain.

Add all the modulated carriers to create the OFM signal,. Often produced by a block called

the IFFT.

Fig:12 Fig:13

Fig 12 :Functional diagram of an OFDM signal creation. The outlined part is often called an

IFFT block.


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Fig 13:The generated OFDM signal .Note how much it varies compared to the underlying

constant amplitude sub-carrier

The above process can be represented as

---------(4)

he above eqn 4 is for IFFT

Fig14:The two views of a signal.

The forward FFT takes a random signal and is multiplied successively by complex

exponentials over the range of frequencies.

Sums each product and plot the results as a coefficient of that frequency the coefficients are

called the spectrum and represent how much of that frequency is present in the input signal.

The results of the FFT in common understanding is a frequency domain signal.

The FFT in sinusoids can be represented as

-------------- (5)

Where(n) are the co-efficient of the sines and cosines of frequency 2k/N.

Where k is the index of the frequency over the n frequencies and n is the time index. X(k) is

the value of the spectrum for the kth frequency and x(n) is the value of the signal at time n.

In the above figure the x(k=1)=1.0 is one such value.

The IFFT takes this spectrum and converts the whole thing back to time domain signal by

again successively multiplying it by a range of sinusoids.

The equation of IFFT is

--------------- (6)

The difference between eqn 3 and 6 is that the type of coefficients the sinusoids are taking

and the minus sign and thats all.

The co-efficient by convention are defined as time domain samples x(k) for FFT and x(n)

frequency bin values for the IFFT.

The two processes FFT and IFFT in sequence will give the original result back.


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Fig 15:FFT and IFFT are a matched linear pair

Column 1 of table 1 the single bits an be considered the amplitudes of a certain range of

sinusoids.

We can use the IFFT to produce the time domain signal.If you say that they are already in time domain, how we can process a time domain signal to

produce as another time domain signal.

If you say that they are already in time domain how we can process a time domain signal to

produce as another time domain signal.

The answer for the above question is that we presented the input bits are not itme domain

representations but are frequency amplitudes which you are thinking clearly and you will see

that is what they are.

In this way we can take these bits and by using IFFT we can create an output signal which is

actually a time domain OFDM signal.The IFFT is a mathematical concept and does not really care what is the Input and what is the

output.

Both FFT and IFFT will produce the identical results on the same input. But we do not use

FFT/IFFT this way we insist that only spectrums go inside the IFFT. This way we insist that

only spectrums go inside the IFFT.

Each row of table 1 can be considered a spectrum on plotted below in fig 16 these row

spectrum has 4 frequencies which are 1,2,3 and 4 Hz.

These spectrums can be converted to time domain signal which is exactly what an IFFT does.

Fig.16:The incoming block of bits can be seen as as a four bin spectrum,The IFFT converts

this spectrum to a time domain OFDM signal for one symbol ,which actually has four bits

in itFFT and IFFT are linear processes and completely reversible


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It should be called FFT instead of a IFFT. The results are the same whether you do FFT or

IFFT.

The function block diagram of how the signal is modulated/demodulated ins h\shown in

below fig

Fig 17:The OFDM link function.

Fading:

If we transmit a signal from the transmitter to a receiver if in between the tx to rx reflections

or any obstructions occur we get the fading effects.

Here is our example shown below in fig 18 the signal reaches the receiver from many

different routes, each is a copy of the original and each of the rays has slightly different delay

and slightly different gain.

The time delay results in phase shifts which is added to the main signal component (hereassuming there is one) which causes the signal to be degraded.

Fig 18:Fading is a big froblem for signals.The signal is lost and demodulation must have a

way of dealing with it.Fading is a particular problem when the link path is changing ,such as

for a moving car or inside a building or in a populated urban area with tall building.

If we draw the interference inpulses which look as below in fig 19

Fig 19:Reflected signals arrive at a delayed time period and interfere with the main line of

sight signal, if their is one .In pure Raleigh fading ,we have no main signal, all components

are reflected.

Fading is process where in the reflected signals are delayed and added to the main ksignal

which can cause either gain in the signal strength or deep fades.

By deep fade we mean that the signal is nearly wiped out,. The signal strength become so

weak that the receiver cannot decide what was there on the channel.

The maximum time delay that occurs is called the delay spread of the signal is thatenvironemtn. The delay spread can be short so that is is less then symbol time or larger.


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The above both the causes cause different types of degradations to the signal.

All cell phone users know that the delay spread of a signal changes as the environment

changes.

The above fig 19 shown the spectrum of the signal.

The dark line shown the response we wish the channel to have, Its like a door through which

the signal has to pass.The door is large enough that it allows the signals to go through without bending or

distortion.

The fading response of the channel is shown in fig 20 b.

At some frequencies in the band the channel does not allow any information to go through so

called deep fades frequencies.

This form of channel frequency response is called frequency selective fading because it does

not occur uniformly across the band it occurs. If selected frequencies. This frequency

selection is based on environment ie if the environment is changing such as a moving car then

the response is also changing land the receiver must have some way to deal it.

Rayleigh fading is a term used when their is no direct signal reaching the receiver but the

reflected ovens are reaching the receiver, This type of environment is called Rayleigh fading.When the delay spread is less than one symbol, we get what is called flat fading when the

delay jspread is much larger than one symbol then it is called frequency selective fading.

Fig 20:(a)The signal we want to send and the channel frequency response are wellmatched.

(b)A fading channel has frequencies that do not allow anything to pass. Data is lost

sporadically. (c) With OFDM,where we have many little sub-carriers, only a small sub-set of

the data is lost due to fading.

An OFDM Signal offers an advantage in a channel that has a frequency selective fading

response.

When we lay on OFDM signal spectrum against the frequency selective response of he

channel only tow subcarriers are affected while all others are perfectly ok instead of loosign

whole symbol we hust lose a small subset of the (1/w0 bit with proper coding again this can

be recovered.The BER performance of an OFDM signal in a fading channels is far better compared to the

QPSK/FDM which is a simple carrier wideband signal.

The BER of an OFDM is smae as underlying modulation, ie QPSK signal is a guassian

channel.

but in case of channel which are fading, the OFDM offers far better BER than a wide band

signal of exactly the same modulation.

The advantage of multicarrier is suc that the fading applies only to a small subset.

FDM carrier often uses the root raised cosine shape to reduce its bandwidth where as in case

of OFDM the spacing of the carrier is optimal hence their is a natural bandwidth advantage.


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Delay Spread and the Use of Cyclic prefix to mitigate it:

If we are driving in rain and the car in front of you splashes a bunch of water on you what

will you do. The first of all you move further back. You have to maintain a little distance

between you and the front car far enough so that the splash would not reach you.

If you equate the reach of a splash to the delay spread of a splashed signal then we have a

better picture of the phenomena and how to avoid it.

Fig 21:Delay spread is like the undesired splash you might get from the car ahead of you . In

fading, the front symbol similarly throws asplash backwards which we wish to avoid.

We have to increase the distance from our vehicle to the car in the front to avoid splash.

The reach of a splash is same as the delay spread of a signal.

Fig.22(a) Fig.22(b)

Fig 22(a) The PSK symbol and its delayed version.22(b)Move the symbol back so the

arriving delayed signal peters out in the gray region.No interference to the next signal.

Fig 22(a)shows the symbol and its splash. In composite these splashes become more and

effect the beginning of the next symbol as shown in fig 22 b.

To overcome this more at the front of the symbol we move our symbol away from the regionof delay spread as shown in figure below.

A little bit blank space has been added between symbols to catch the delay spread.

But we cant leave the empty space in signals which wont work for the hardware so we just

let the symbol run longer as a first choice, ie we entext the symbol into the empty space so

the actual symbol is more than one cycle.

Fig 23.: If we just extend the symbol ,then the front of the symbol which is important to ussince it allows figuring out what the phase of this symbol is more then one cycle.

Here the start of the symbol is in danger zone which is the most important thing about our

symbol the slicer needs it in order to make a decision about the bit.

Wo we dont want to the start of the symbol to fall in this region.

So lets slide the symbol back wards so that the start of the original symbol lands out

side this zone.


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Fig 24 we move the symbol back and just put in convenient filter in this area ,then not onlywe have a continuous signal but one that can get corrupted and we dont care s:ince we will

just cut it out any way before demodulating.

Slide the symbol to start of the edge of the delay spread time and then fill the finaled space

with a copy of what turns out to be tail end of the symbol.

The start of the symbol should be out of the delay zone so that it is not corrupted.

Start the signal; at the new boundary such that the actual symbol edge falls out side this

zone.

We are extending the symbol to 1-25 times long to achieve this we are coping the back of the

symbol and glue it in the front.

In reality the symbol source is continuous what we are doing in adjusting the starting phaseand making the symbol period longer.

Fig.25:Cyclic prefix is this superfluous bit of signal we add to the front of our precious

cargo,the symbol.

The procedure is called adding a cyclic prefix.

As the OFDM has a lot of carriers we would do this to each and every carrier. OFDM signal

is a linear combination. we can add cyclic prefix once to the composite OFDM signal.

The prefix is anywhere from 10% to 25% of the symbol time.

Here in our example of OFDM signal with period equal to 32 samples, if we want to add a

25% cyclic shift to this signal.

1) First we cut pieces that are 32 samples long.

2) Then we take the last 0.25 (32) = 8 samples, copy and append them to the front asshown in fig below.

Fig 26 :The whole process can be done only once to the OFDM signal,rather then doing it to

each and every sub-carrier.


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Add the prefix after doing the IFFT just once to the composite signal after the signal has

arrived at the receiver, first remove this prefix to get back the perfectly periodic signal so it

can be FFTD to get back the symbol on each carrier.

We can overcome the effects of link fading and intersymbol interference but increase the

bandwidth by adding the cyclic prefix.

Fig 27:Addition of cyclic prefix to the OFDM signal further improves its ability to deal with

fading and interference.

Properties of OFDM

Spectrum and performance

Fig 28:The spectrum of an OFDM signal(without addition of cyclic prefix) is much more

bandwidth efficient than QPSK.

As more and more carriers are added the bandwidth approaches N+1 bits per Hz.

Fig:29 Fig:30

Fig 29 :The spectrum of OFDM signal with 1024 and Fig .30:The spectrum of QPSK signal.

Comparision of the spectrum of an OFDM signal with the spectrum of a

QPSK signal is shown in above figure.29 and 30

Bit error rate(BER) performance:

The BER of an OFDM is only exemplary in a fading environment.

OFDM cannot be used as a straight line of sight link such as a satellite link.

Due to variation in amplitude OFDM signal does not behave well in a non-linear channel

such as created by high power amplifiers on board satellites

Using OFDM for a satellite would require a fairly large back off, on the order of 3 dB, there

is some other compelling reason for its use such as when the signal is to be used for a moving

user.Peak to average power ratio(PAPR):


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For an OFDM signal that has 128 carriers each with normalised power of 1w, then the max

PAPR can be as large as log(128) or 21 dB. This is an instant when all 128 carriers combine

at their maximum point. The PAPR is defined as follows. Here R is the PAPR

R=

The RMS PAPR will be around half this number or 10 21dB. This same PAPR is seen inCDMA signals as well.

To overcome large PAPR several methods are used to overcome it .

[1].Clipping.

Here we can just clip the signal at a desired power level .This reduces the PAPR but

introduces other distortion and ICI.

[2]. Selective mapping.

Multiply the data signal by a set of codes ,perform the IFFT on each and then pick the one

with the least PAPR.

[3]. Partial IFFT.

Divide the signal in clusters perform IFFT on each and then combine these. So that if we

subdivide 128 carrier into a group of 32 carriers each, then the maximum PAPR of each willbe 12 dB instead of 21dB for the full. Combine these four sequences to create the transmit

signal.

[4].Synchronization.

One more problem is that tight synchronization is needed, often pilot tones are served in the

sub carrier space. These are used to lock on phase and to equalize the channel.

[5]. Coding.

The sub carrier are typically coded with convolution coding prior to going through IFFT.

The coded version of OFDM is called COFDM or Coded OFDM.

Parameters of real OFDM.

The OFDM use has increased greatly in the last 10 years. It is now proposed for Radiobroadcasting such as in Eureka 147 standard and Digital Radio Mondiale (DRM).OFDM is

used for Modem / ADSL application where it coexist with phone line. For ADSL use ,the

channel ,the phone line, is filtered to provide high SNR. The OFDM here is called Discrete

Multi Tone (DMT).OFDM is also used for wireless internet modem and this usage is called

802.11a.

The parameters of the OFDM application which are as bellow

Data rates:

6Mbps to 48 Mbps.

Modulation:

BPSK, QPSK,16 QAM and 64 QAM.

Coding:

Convolution concatenated with Reed Solomon

FFT size:

64 with 52 sub carriers uses 48 for data and 4 for pilots

Sub carrier frequency spacing:

20 Mhz divided by 64 carriers or 0.3125 Mhz.

FFT Period:

Also called symbol period , 3.2 Micro sec = 1/f.

Guard duration :

One Quater of symbol time ,0.8 micro sec .

Symbol time:4 Micro sec.

orthogonal frequency division multiplexin_paper18!3!12

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