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Corso di Comunicazioni Mobili

Data Transmission By OFDM Modulation

Prof. Carlo Regazzoni

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References[1] M.L. Doelz, E.T. Heald, D. Martin, “Binary Data Transmission Techniquesfor Linear Systems”, Proceedings of IRE, Maggio 1957, pp. 656-661.

[2] B. Hirosaki, “An Orthogonally Multiplexed QAM System Using the Discrete Fourier Transform”, IEEE Trans on Comm, Vol. 29, No. 7, Luglio 1981, pp. 982-989.

[3] J.A.C. Bingham, “Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come”, IEEE Comm. Magazine, Maggio 1990, pp. 5- 14.

[4] T. De Cousanon, R. Monnier, J.B. Rault, “OFDM for digital TV broadcasting”; Signal Processing, Vol. 39 (1994), pp. 1-32.

[5] B. Le Floch, M. Alard, C. Berrou, “Coded Orthogonal Frequency Division Multiplex”, Proceedings of IEEE, Vol. 83, No.6, Giugno 1995, pp. 982-996.

[6] E. Ayanoglu, et al, “VOFDM Broadband Wireless Transmission and ItsAdvantages over Single Carrier Modulation”, Proc of ICC 2001 Conference, Helsinki (SF) 11-14 Giugno 2001, Vol. 6, pp. 1660- 1664.

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IntroductionThe OFDM modulation (Orthogonal Frequency Division Multiplexing) is the basic technique among the multi-carrier modulations a clear example of multi-carrier modulation is the DMT that is employed in the ADSL standard for transmissions on twisted pair with high bit-rate.

Another example is comprised by the MC-CDMA techniques (Multi-carrier CDMA) which are the spread-spectrum version of the OFDM modulation.

The basic concept of multi-carrier modulation is the transmission in diversity, i.e. the transmission of information on sub-channels with different bandwidth, where the distortion effects of the channel are different.

As contrary to the single-carrier techniques, transmitted message will suffer, in a different measure, of the frequency selective effects of the channel. But a substantial improvement of performances in terms of BER is possible.

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Historical MentionsMulti-carrier modulation techniques are considered the fourth generation (4G)communication systems used for fixed and mobile digital transmission.The idea of multi-carrier modulation dates at the end of fifties (Doelz, Heald, Martin, Procedings of IRE, May 1957, pp. 656-661).This work showed a practical implementation of a digital transmission system (called KINEPLEX) with bit multiplexing on orthogonal carriers, which is the basis principles of the OFDM.

KINEPLEX demodulator

KINEPLEX S/P converter

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Historical MentionsThe main problem of KINEPLEX lies in the totally analogical implementation of multiplexer (RLC resonance oscillators) which involves the huge dimension of the equipments.

The theoretical evolution, which allowed the practical implementation of an OFDM system, was studied by B. Hirosaki in 1981 (IEEE Trans on Comm, July 1981, pp. 982-989), i.e. the multiplexing mapping in frequency on a structure like Fast Fourier Transform (FFT).

An FFT structure can be implemented in a totally digital and software way.

Ten years after the evolution of digital processing technology (such as DSP) allowed the practical implementation of an OFDM system (see J.C. Bingham, IEEE Comm. Magazine, May 1990, pp. 5-14).

The first commercial prototypes of OFDM systems were announced in 2001(Cisco’s VOFDM presented in the conference ICC2001) and their commercialization is expected for 2005

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OFDM Modulation: IntroductionThe OFDM is a multi-carrier modulation in which carriers are frequency spaced by a multiple of 1/T, where T is the modulation period, and it is characterized by an overlap of the spectrum of the signals transmitted on different carriers.

A possible OFDM modulator could be the following:

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OFDM Modulation: Introduction

The previous figure shows a symbol stream, codified in-phase and in-

quadrature components (an , bn) is cyclically multiplexed on N branches

containing a QAM digital modulator.

The output of the k-th modulation branch is an M-QAM signal, modulated on

carrier frequency fk which is orthogonal to each other.

In this way it is possible, at the receiver, to recover the symbol streams

transmitted in every branch and to rebuild, after a de-multiplexing operation,

the original symbol stream.

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OFDM Modulation: Transmitted SignalEvery QAM modulator has an assigned constellation that can be equal in every branch.

Given and , polar coordinates of the transmitted symbol in the QAM constellation relative to the k-th carrier in the interval [(j - 1)T, jT ], the transmitted signal can be written as:

jkr jkϕ

∑+∞

−∞=−Π+−=

jjkkjkk jTtjTtfrts )())(2cos()( ϕπ ∑

−

=

=1

0

)()(N

kk tsts

Signal transmitted by the k-th carrier Signal transmitted on the channel

Tkffk += 0

⎩⎨⎧ <≤

=Πaltrimenti ,00 ,1

)(Tt

twhere

Fundamental frequencyRectangular Pulse

N : number of sinusoidal carriers

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OFDM Modulation: Transmitted Signal

The signal transmitted on the channel is a summation of a huge number of sinusoidal carriers, modulated with arbitrary phase and amplitude.

The result, in the time domain, is a noise-like signal:

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OFDM Modulation: Transmitted SignalThe duration T of an OFDM modulation impulse is fixed and it’s equal to:

sNTDNaT ==

where

•D is the source bit-rate;

•a is the number of bit for each transmitted symbol;

•Ts is the duration time of a symbol ( N multiplexed symbols are transmitted in the duration time of an OFDM modulation pulse);

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OFDM Modulation: Transmitted SignalThe spectrum of the signal transmitted by every carrier assumes the shape of the function and zero crossing every 1/T Hz.

The spectrum of the overall transmitted signal is a succession of functions spaced by 1/T Hz.

The spectral components given by the single carriers are overlapped, as shown in figure:

)( fSffsin )(

)( fSk

ffsin )(

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OFDM Modulation: Bandwidth OccupationThe spectrum of the OFDM signal has theoretically unlimited bandwidth. But it needs a truncation to compute the significant bandwidth occupied by the signal.

The truncation is performed to remove all the components of the power spectrum which are at least 20 dB under the amplitude of main lobe.

In this case only two side lobes are conserved, as shown in the following figure:

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OFDM Modulation: Bandwidth Occupation

The bandwidth occupied by the N carriers of the OFMD signal is equal to:

TN

TTNW 5321 +

=+−

=

It can be interesting to compute the spectral efficiency of the QAM modulation, given by the ratio (source bit-rate)/(occupied bandwidth).

Supposing to have an M-QAM constellation (or M-PSK) in two dimensions with 2a points (where a is the number of bits for every transmitted symbol); since in T are transmitted N symbols, source bit-rate can be expressed as:

TNaD =

NaDNW )5( +

= Hz.Bit/sec.

aN

NWD

5+==ηNow it’s easy to compute the spectral efficiency

of the QAM modulation, that is given by:

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OFDM Modulation: Bandwidth OccupationIn the following figure the power spectral density of an OFDM modulated signal with T = 125 nsec, f0= 8 MHz, number of carriers N = 32, 128, 512 is shown.

Signal spectrum is inclined to become ideal (without using spectral shaping filters, like Nyquist filter with low roll-off) when N is high.

In this case the spectral efficiency can be written as:

The figure points out that (ideal value, difficult to reach with a QAM modulation, even using Nyquist filters) for high values (but finite) of N.

( ) aNN

=∞→ηlim

aη

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OFDM Demodulation

In the following figure a possible

modulator/demodulatorschematic is shown. The

demodulator is based on the

orthogonality of the carriers. It

is composed by a bench of demodulators with matched filter used both for in-phase

and in-quadrature components.

DemodulatorModulator

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OFDM DemodulationThese orthogonality conditions on the carriers, both for in-phase and in-quadrature components, allow to demodulate the signal, as pointed out in the following formulas:

h k t)dtf()tf(rT

hkkk ≠=+∫ 02cos2cos0

πϕπ

h kaT) (rTkkk ===

2cos

2ϕ

h k t)dtf)sin(tf(rT

hkkk ≠=+∫ 022cos0

πϕπ

h kbT) sin(rTkkk ===

22ϕ

To have a correct demodulation process, other two conditions, which are considered guaranteed for simplicity, are necessary:

• strict synchronization on the carrier (coherent demodulation);

• strict synchronization of the clock on the receiver side (clock recovery)

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OFDM ModulationA modulation/demodulation schematic, such as the previous one, cannot be implemented via hardware with analog oscillators: it would be too much expensive and the imperfections of the oscillators (frequency drift, phase noise) would cause critical malfunctions.

But it can easily implemented via software, in a totally digital way, using the FFT (Fast Fourier Transform).

In fact, if the k-th symbol (k = 1,…,N , where N is the number of transmitted symbols in a modulation period), mapped in the chosen M-QAM constellation, is written as:

kkj

kk jbaerS k +== ϕ

{ }10 ,....., −NSSSet of symbols

transmitted in T sec.

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OFDM ModulationThe OFDM signal transmitted on the channel can be obtained by the following

steps:

1. computing the Inverse FFT (IFFT) on a set of symbols transmitted in a

modulation period T

2. performing the digital-to-analog conversion (D/A) of the signal

obtained in the previous step.

In fact a sequence s(n) is generated by using an IFFT operation performed on the set of symbols transmitted in the modulation period T, with a number of samples NFFT (generally it is a power of 2).

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OFDM ModulationThe operation, previous described, is the following:

( )∗− = knnNk WW )(

( ) ( )( )

( ) ( )

∑

∑ ∑

∑ ∑

−

=

−

=

−

=

−

−

=

−

=

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=+=

=+=

1

0

1

0

1

0

)(2*

2

1

0

1

0

**

2cos1

11

11)(

N

kk

FFTk

FFT

N

k

N

k

nkNN

j

FFT

knN

j

FFT

N

k

N

k

knN

FFT

knN

FFT

knN

rN

ekSN

ekSN

WkSN

WkSN

ns

FFTFFTFFT

NFFTNFFT

ϕπ

ππ

1,..,0 −= FFTNn

This result is obtained remembering one of the fundamental properties of the W coefficients of the FFT, i.e. :

Since the set of N symbols has to be transmitted every T seconds, the sampling frequency has to be:

FFTs NT

f 11=

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OFDM ModulationThen, the s(n) sequence is sent to a D/A converter which works with a smapligfrequency equal to .

The s(t) signal, which is the output of the converter, can be written as: sf

∑−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

1

0

2cos1)(N

kks

FFTk

FFT

tkfN

rN

ts ϕπ

s

FFT

fNt ,....,0 =

( )∑−

=

+=1

0

2cos1)(N

kkkk

FFT

tfrN

ts ϕπ Tt ,....,0 =Tkfk =

s

FFT

fNT=

but it can also rewritten as:

This is the base band OFDM signal!

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OFDM ModulationThe following figure shows the complete schematic of and OFDM modulator/demodulator system which uses the FFT

modulator

demodulator

k0=0 for simplicity

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OFDM Modulation

This schematic can be implemented in a totally software way on a DSP

architecture because an FFT (or IFFT) structure can be mapped on this signal

processing architecture with well known algorithms.

With the actual technology a full-digital implementation is impossible; only the

base band stage and intermediate frequency stage are developed. The radio

frequency stage is still implemented with analogical components.

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OFDM: performances on AWGN channelThe performances of the OFDM modulation/demodulation systems on a noisy channel depend on the chosen M-QAM constellation.

Supposing to transmit data on an AWGN channel, the signal received by the matched filter associated to the k-th carrier, is the following:

An error occurs when the noise components are greater than the half of the distance d between two points of the constellation. Then the error probability on the symbol in an OFDM system has the following expression:

d depends on the chosen constellation and can be expressed as a function of the average energy of the constellation.

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=⎟

⎠⎞

⎜⎝⎛ >= 2

2

821

2 k

ik

derfcdnperrP

σ

ikkk naa +=' (in-phase component)qkkk nbb +=' (in-quadrature component)

cE

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OFDM: performances on AWGN channel

Practically the error probability of an OFDM system is the same of an M-QAM system on a single carrier.

108 2

2 cEd =16 points QAM Constellation

208 2

2 cEd =32 points QAM Constellation

428 2

2 cEd =64 points QAM Constellation

828 2

2 cEd =128 points QAM Constellation

∑−

=

+=

12

0

22

221

ˆc

kc

kkc

baE Number of points in the constellation

c2where

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OFDM: performances with pulse noiseThe OFDM modulation offers remarkable robustness properties against impulsive noises, where impulsive noise is defined as a rectangular pulsewith limited amplitude and duration (generally inferior than the symbol time). Hence it is a wide band disturb.

Since the information is coded in the frequency domain, the energy of the noise pulse is distributed on the entire bandwidth of the spectrum. This fact reduce its effect on the signal sub-carriers.

The following figure shows the error probability versus the impulsive signal-to-noise ratio.

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OFDM: performances with jamming noiseOFDM is much vulnerable respect to the narrow band interferences like jamming ones, composed by sinusoidal tones which interfere with the signal (man-made noise, ingress-noise); it offers worse performances than a normal QAM (see the following figure with N=512 carriers).

The power of the OFDM signal is

concentrated in a reduced portion of the spectrum; a noisy pulse which hits

in the bandwidth of the signal is able to

alter the bits transmitted by various sub-

carriers.

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OFDM: performances with jamming noise

A possible solution consist in switching off the sub-carriers corrupted by the

jamming pulse. This solution succeed if the position of the interfering tone in

the frequency domain is fixed and known. It can be implemented without any

additional hardware, by using the IFFT properties.

If the position of the interfering tone is not fixed known, a FEC coding has to

be introduced before the modulator (called Coded-OFDM or COFDM)

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OFDM: behave on multipath channelAfter a multipath channel, a series of delayed and out of phase replica of the desired signal are received.

Hence, sampling the signal on a certain instant, a linear combination of the previous symbol of the current symbol, and of the following symbol is obtained (ISI); then the channel behaves like a linear filter.

Theoretically, it could be possible to use the inverse equalizers, which are digital filters whose the coefficients are dynamically updated by proper algorithms.

The coefficients updating is a computationally heavy operation and substantially inefficient if the delay spread of the channel is greater than the modulation period (this condition is equivalent to the frequency selectivity for the multipath fading)

The OFDM techniques allow to increase the modulation period and to generate a modulated signal for which the channel is not frequency selective; but the bit-rate remain the same.

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OFDM: behave on multipath channelSupposing to introduce a multipath channel with discreet paths, it can be represented by the following impulse response:

( )∑=

−=P

ppp tAth

0

)( τδ

( ) ( )τδδ −+= ttth )(

Without losing generality, an extremely simplified model of the multipath channel can be assumed:

The signal received by every single carrier can be written as a summation of the LoS signal and its delayed echo.

( ) ( )τ−+= tststs )('

It is impossible to insulate (see figure) an interval of T seconds which contains only one symbol: ISI appears.

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OFDM: behave on multipath channelA possible solution to remove this kind of interference consists in increasing the duration of the OFDM symbol to (T+τ) where τ is called cyclic prefix.Sampling properly the received signal between the instants T1 and T2 , it’s possible to extract a portion of the signal, with duration T seconds, which contains information only about the j-th symbol, which is the desired; ISI is removed.To realize this mechanism, keeping the orthogonality, an expansion of the duration of the modulated carriers on the left (as shown in the figure) of τ seconds is needed.

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OFDM: behave on multipath channelTo use the OFDM on a channel where the delay spread is equal to τp , this process has to be applied to every single sub-carrier with a cyclic prefix of τp.

The ISI-free demodulation is possible if the received signal is processed in an interval comprised between τp and (T+ τp), where T is the modulation period.

The cyclic prefix can be inserted by modulating quickly the N carriers during a modulation period: before inserting, afterwards, the cyclic prefix.

To keep orthogonality, N carriers spaced by

should be used, before inserting, afterwards, the cyclic prefix.

The transmitted signal will be the following:

pTT τ−='

( ) 1,..,0 ˆ'' −=

−== Nk

Tk

Tkf

pk τ

∑∑+∞

−∞=

−

=

−Π−+=j

N

kpkjkkjk jTtftfrts

1

0

'' )()22cos()( τπϕπ

The demodulation will be performed during the period T’.

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OFDM: behave on multipath channel

( )( ) a

TNTN

WD

p

p

ττ

η65' −+

−==

The use of the cyclic prefix can limit the spectral efficiency of the modulation. It can be shown that the spectral efficiency of an OFDM system with cyclic prefix τp is equal to:

Where T is fixed and it’s equal to:

DNaT =

The following figure shows the required bandwidth as a function of the number of the orthogonal carriers, with D=34Mb/s, a=3,4,5,6 e τp = 8msec.