Orthogonal Frequency Division Multiplexin_paper

7/30/2019 Orthogonal Frequency Division Multiplexin_paper

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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 in one by stream and cannot be subdivided, whereas the OFDMshower is made up of a lot of little stream.

We cn think here what the advantage of one over the other.

If I put my thumb over the tap hole, I can stop the water flow where as the same I cannot do

with the shower. Fig 1(a) (b)

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 ODM trucks will suffer.

Fig 2

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 an be multiplex by FDM which is called multicarrier

transmission and if it is based on code division multiplexing(CDM) then it is called multi

code modulation,

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

Fig3.


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The importance of being orthogonal

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

carriers are all klsine/cosine wavbe and the area under one sine cosine wave is zero which

is shown in below figure fig 4

If we take a sine wave of frequency in 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)

By simple trigonometric relationship, this is equal to a sum of two sinusoids offrequencies (n-m) and (n+m)

= cos(m-n)- cos(m+n)

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

period.

=0-0

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,sinnx are all orthogonal to one 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(also orthogonal) which allows many users to transmit in same space

successfully.

OFDM is a special case of FDM:If I have a bandwidth which goes from a to b. First i will subdivide the given frequency

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

carriers would look like them.

Fig(6)

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 wigth the carrier

frequencies which are based on frequencies and they do not have any special relationship

with each other.

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

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

Similarly,C2=2xC1 =2 C1


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C3=3xC1 =3C1C4=4xC1 =4C1All 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= Bp

1+

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.

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

hence each transmission is a bit.

Fig 7

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 -1

1 1 1 -1

1 - 1 -1 -1


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-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.If we 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

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 c2 1,1,-

1,1,1,-1 and modulate this carrier with these bits.

Fig 9

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

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

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

called the IFFT.

Fig 12 Fig 13


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The above process can be represented as

The above eqn 4 is for IFFT

Fig 14

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

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

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.

Fig 15

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.


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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 rowspectrum 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.

FFT and IFFT are linear processes and completely reversible

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

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 (here

assuming there is one) which causes the signal to be degraded.

Fig 18

Hc(t)

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

Fig 19

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 that

environemtn. The delay spread can be short so that is is less then symbol time or larger.

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.


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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 doesnot 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

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 hechannel 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 whic are fading, lthe ofdm offs for better BER than a witly the same

modulation.

The advant

Orthogonal Frequency Division Multiplexin_paper

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