ofdm basics file

8/9/2019 ofdm basics file

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30 www.rfdesign.com J anua ry

The principles of orthogonal frequency division

multiplexing (OFDM) modulation have been inexistence for several decades. However, in recentyears these techniques have quickly moved out oftextbooks and research laboratories and into prac-tice in modern communications systems. The tech-niques are employed in data delivery systems overt h e p h on e l i n e , d i g it a l r a d i o a n d t e l ev i s io n , a n dwireless networking systems. What is OFDM? Andwhy has it recently become so popular? This articlewill review the fundamentals behind OFDM tech-

niques, a nd also discuss common impairments a ndhow, in some cases, OFDM mitigates their effect .Where applicable, the impairment effects and tech-niques will be compared to those in a single carrier

system. A brief overview of some modern a pplica-tions will conclude the article.

The single-carriermodulation system

A typical single-carrier modulation spectrum isshown in Figure 1. A single carrier system modu-lates information onto one carrier using frequency,phase, or a mplitude adjustment of the carrier. For

digital signals, th e informa tion is in the form of or collections of bits called symbols, that are mlated onto the carrier. As higher bandwidths (rates) are used, the duration of one bit or symboinformation becomes smaller. The system beco

more susceptible to loss of information from impn oi se , s i g n al r e f l e c t i on s an d ot h e r i m p ai r m eThese impairments can impede the ability to recthe information sent. In a ddition, as the bandwused by a single carrier system increases, the ceptibility to interference from other continuousnal sources becomes greater. This type of interence is commonly labeled as carrier wave (CWfrequency interference.

Frequency division multiplexingmodulation system

Frequency division multiplexing (FDM) extethe concept of single carrier modulation by um u l t i p l e s u b c a r r i e r s w i t h i n t h e s a m e s i nc h a n n e l . Th e t o t a l d a t a r a t e t o b e s e n t i n

channel is divided between the various subcarrThe data do not have to be divided evenly not h e y h a ve t o or ig i n at e f r om t h e sam e i n for masource. Advantages include using separate modtion/demodulat ion customized to a par ticular t ypd at a , or sen d i ng ou t b an k s of d i ssim i lar d at a can be best sent using multiple, and possiblyferent, modulation schemes.

C u r r e n t n at i on al t e l evi sion sy st e m s c om m i(NTSC) television and FM stereo multiplex are ge xam p l es of F D M . F D M off er s a n ad van t ag e single-carrier modulation in terms of narrowbfrequency interference since th is int erference only a ffect one of t he frequency subban ds. The osubcarriers will not be affected by the interfereSince each subcarrier has a lower information

the data symbol periods in a digital system willonger, adding some a dditiona l immunity t o impnoise and reflections.

F D M sy st em s u su al l y r e q u ir e a g u ar d b an dtween modulated subcarriers to prevent the spt r u m of on e su b c ar r i e r f r om i n t e r f e r i n g w i t ho t h e r . Th e s e g u a r d b a n d s l o w e r t h e s y s t eeffective information ra te w hen compar ed to a sicarrier system with similar modulat ion.

Orthogonality and OFDMIf the FDM system above had been able to u

set of subcarriers th at were orthogonal to each ota higher level of spectral efficiency could have bac h i eve d. Th e g u ar d b a n d s t h at w e r e n e ce ssarallow individual demodulation of subcarriers in

FDM system w ould n o longer be necessary. Theof orthogonal subcarriers would allow the subcers’ spectra to overlap, thus increasing the specefficiency. As long as orthogonality is ma intaineis still possible to recover the individual subcarrsignals despite their overlapping spectrums.

If th e dot product of tw o deterministic signaequal to zero, these signals are said to be orthn a l t o e a c h o t h e r . O r t h o g o n a l i t y c a n a l s o

Figure 1. Single carrier spectrum example.

sig nal pr oc essing

Theprinciples

ofOFDMM ul ti carr ier modulati on techni ques

are rapi dl y moving from the textbook

to the real world of m odern

com m un i cati on system s

By Louis Litwin andMichael Pugel


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viewed from the standpoint of stochas-tic processes. I f t wo ra ndom processesa r e u n c o r r e l a t e d , t h e n t h e y a r eor t h og on al . G i ve n t h e r an d om n at u r e

o f s i g n a l s i n a c o m m u n i c a t i o n s s y s -tem, t his probabil istic view of orthogo-n a l i t y p r o v id e s a n i n t u i t i v e u n d e r -standing of the implications of orthog-onali ty in OFDM. Later in this article,w e w i l l d i sc u ss h ow O F D M i s i m p l e -m e n t e d i n p r ac t i ce u si n g t h e d i scr e t efourier transform (DFT). Recall froms i g n a l s a n d s y s t e m s t h e o r y t h a t t h esinusoids of the DFT form an orthogo-n al b asi s se t , an d a s i g n al i n t h e ve c -tor space of the D FT can be represent-e d a s a l i n e a r c o m b i n a t i o n o f t h eorthogonal sinusoids. One view of theD F T i s t h at t h e t r an sf or m e sse n t i al l ycorrelates i ts input signal with each of

t h e s i n u s o id a l b a s i s f u n c t io n s . I f t h ei n pu t s i g n al h a s som e en e r g y at a c er -

tain frequency, there wil l be a peak inthe correlation of the input signal andthe basis sinusoid that is at that corre-sponding frequency. This tra nsform is

u se d at t h e O F D M t r an sm i t t e r t o m apan i n p u t s i g n al on t o a se t of or t h og o-n a l s u b ca r r i e r s , i . e ., t h e o r t h og on a lbasis functions of the DFT. Similarly,t h e t r a n s f o r m i s u s e d a g a i n a t t h eOFDM receiver to process the receivedsubcarriers. The signals from the sub-carriers are then combined to form ane st im at e of t h e sou r ce si g n al f r om t h et r a n s m i t t e r . Th e o r t h o g o n a l a n duncorrelated nature of the subcarriersi s e x p lo it e d i n O F D M w i t h p ow e r f u lr e s u l t s . S i n c e t h e b a s i s f u n c t i o n s o fthe DFT are uncorrelated, the correla-tion performed in the DFT for a givens u b c a r r i e r o n l y s e e s e n e r g y f o r t h a t

corresponding subcarrier. The energyf r om ot h e r su b c ar r i e r s d oe s n ot c on -tribute because it is uncorrelated. Thisse par a t i on of s i gn al e n er g y i s t h e r e a-

son t h a t t h e O F D M su b car r i e r s ’ st r u m s c a n o v e r l a p w i t h o u t c a u sinterference.

A simple OFDM exampleFigure 3 shows a simple represe

tion of an OFDM system. These tyo f s y s t e m s h a v e b e e n b u i l t b u t practicality of such construction quly diminishes a s th e number of subr i er s i n cr e a s e s . E a c h s u b c a r r i er r i e s o n e b i t o f i n f o r m a t i o n ( N tota l) by i ts presence or a bsence ino u t p u t s p e c t r u m . T h e f r e q u e n ceach subcarrier is selected to formo r t h o g o n a l s i g n a l s e t , a n d t h e s e q u e n c i e s a r e k n o w n a t t h e r e c e iN o t e t h a t t h e o u t p u t i s u p d a t e d periodic interval T that forms the sb ol p e r i od as w e l l as t h e t i m e b ouar y f or or t h og on al i t y . F i g u r e 4 sht h e r e su l t an t f r e q u e n c y sp e c t r u m

t h e f r e q u e n c y d o m a i n , t h e r e s u lsi n f u n c t i on si d e l ob es p r od u ce olapping spectra. The individual peo f s u bb a n d s a l l l in e u p w i t h t h e crossings of the other subbands. To v er l a p o f s p e ct r a l e n e r g y d o esi n t e r fe r e w i t h t h e s y s t e m ’s a b i l itrecover the original signal. The rece r m u l t i p l i e s (i . e . , c o r r e l a t e s )i n c o m i n g s i g n a l b y t h e k n o w n s esinusoids t o produce th e origina l seb i t s se n t . Th e d i g it a l i m p le m en t ao f a n O F D M s y s t e m w i l l e n h at h e s e s i m p l e p r i n c i p l es a n d p e rmore complex modulat ion.

Implementation ofan OFDM system

T h e i d e a b e h i n d t h e an al og i mmentation of OFDM can be extende

Figure 3. A Simple OFDM generator. N subcarri-ers transmitting 1 bit of information each, byturning on and off at time intervals T.

Figure 4. Overall spectrum of the simple OFDM signal shown with four subcarriers within. that the zero crossings all correspond to peaks of adjacent subcarriers.

Figure 2. FDM signal spectrum example.


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the digital domain by using the discreteFourier Transform (DFT) and its coun-t e r p a r t , t h e i n v er s e d i s c r et e F o u r i erTransform (IDFT). These mathematicalop e r at i on s ar e w i d e l y u se d f or t r an s-forming data between the time-domaina n d f r e q u e n cy -d o m a i n . T h es e t r a n s -f or m s a r e i n t e r est i n g f r om t h e O F D M

perspective because they can be viewedas m ap p in g d a t a on t o or t h ogon al su b -carriers. For example, the I DFT is usedt o t ak e i n f r e q u e n c y - d om ai n d at a an dconvert it to time-domain data. In orderto perform that operation, the IDFTcor-r e l a t e s t h e f r e q u e n cy -d o m a i n i n p u tdata with its orthogonal basis functions,which are sinusoids at certain frequen-cies. This correlation is equivalent tomapping the input data onto the sinu-soidal basis functions.

I n p r a c t i c e, O F D M s y s t e m s a r ei m p l e m e n t e d u si n g a c om b i n at i on off a s t F o u r i e r Tr a n s f o r m ( F F T ) a n dinverse fast Fourier Tra nsform (IFFT)

b l oc ks t h at ar e m a t h e m at i c al l y e q u iv-al e n t ve r si on s of t h e D F T an d I D F T ,r e s p ec t i v el y , b u t m o r e e f f i ci e n t t oi m p l e m e n t . A n O F D M s y s t e m t r e a t sthe source symbols (e.g. , the QPSK orQ A M sy m b ol s t h at w ou l d b e p r e se n ti n a s i n g l e c a r r i e r s y s t e m ) a t t h et r a n s m i t t er a s t h o u g h t h e y a r e i n t h efrequency-domain. These symbols areu s e d a s t h e i n p u t s t o a n I F F T b l o c kt h a t b r i n g s t h e s i g n a l i n t o t h e t i m e -domain. The IFFT takes in N symbolsa t a t i m e w h e re N i s t h e n u m b e r o fs u b c a r r i e r s i n t h e s y s t e m . E a c h o ft h e s e N i n p u t s y m b o l s h a s a s y m b o lp e r i o d o f T s e c o n d s . R e c a l l t h a t t h e

b a s i s f u n c t i o n s f o r a n I F F T a r e Northogonal sinusoids. These sinusoidse a c h h a v e a d i f f e r e n t f r e q u e n c y a n dt h e l o w e s t f r e q u e n c y i s D C . E a c hi n p u t s y m b o l a c t s l i k e a c o m p l e xw e i g h t f o r t h e c o r r e s p o n d i n g s i n u -soid al b asi s f u n ct i on . S i n c e t h e i n p u tsymbols are complex, the value of thes y m b o l d e t e r m i n e s b o t h t h e a m p l i -

t u d e a n d p h a s e o f t h e s i n u s o i d f o rt h a t s u b c a r r i e r . T h e I F F T o u t p u t i st h e s u m m a t i o n o f a l l N s i n u s o i d s .Thus, the IFFT block provides a sim-p l e w a y t o m o d u l a t e d a t a o n t o Northogonal subcarriers. The block of No ut p u t s a m p l es f r om t h e I F F T m a k eup a single OFDM sym bol. The length

of the OFD M symbol is NT where T ist h e I F F T i n p u t s y m b o l p e r i o d m e n -tioned a bove.

A f t e r s o m e a d d i t i o n a l p r o c e s s i n g ,t h e t i m e-d om a i n s i gn a l t h a t r e s u lt sf r o m t h e I F F T i s t r a n s m i t t e d a c r o s st h e c h an n e l . A t t h e r e c e i ve r , an F F Tb l oc k i s u se d t o p r oce ss t h e r e ce i vedsi g n al a n d b r i n g i t i n t o t h e f r e q u e n c y -d om ai n . I d e al l y , t h e F F T ou t p u t w i l lb e t h e o r i g i n a l s y m b o l s t h a t w e r es en t t o t h e I F F T a t t h e t r a n s m i t t e r.Wh e n p l ot t e d i n t h e c o m p le x p l a n e ,t h e F F T o u t p u t s a m p l e s w i l l f o r m ac o n s t e l l a t i o n , s u c h a s 1 6 -Q A M .H ow e ve r , t h e r e i s n o n ot i on of a c on -

st e l lat i on f or t h e t i m e -d om ai n si g n a l .W h e n p l ot t e d on t h e c om p l e x p l an e ,t h e t i m e - d om ai n si g n al f or m s a sc at -t e r p l ot w i t h n o r e g u l ar sh ap e . T h u s,an y r e c e i ve r p r oc e ssi n g t h at u se s t h ec o n c ep t o f a c o n s t e l l a t i o n ( s u c h a ssymbol sl icing) must occur in the fre-q u e n c y -d o m a i n . Th e b l oc k d ia g r a mi n F i g u r e 5 i l l u s t r a t e s t h e s w i t c hb e t w e e n f r e q u e n c y - d om ai n an d t i m e -d om ai n i n an O F D M sy st e m .

Multipath channels andthe use of cyclic prefix

A m aj or p r ob l e m i n m ost w i r esystems is the presence of a multic h an n e l . I n a m u l t i p at h e n vi r on mthe t ran smitted signal r eflects off of

e r a l o b j e ct s . A s a r e s u l t , m u l tdelayed versions of the transmittednal arrive at the receiver. The multversions of the signal cause th e recesignal to be distorted. Many wired tems also have a similar problem wreflections occur due to impedance ma tches in the tr an smission line.

A m u l t ip at h c h an n e l w i ll c au sep r o b l e m s f o r a n O F D M s y s t e m . f i r st p r obl e m i s i n t e r sy m b ol i n t e re n c e . T h i s p r ob l e m oc c u r s w h e nreceived OFDM symbol is distortethe previously transmitted OFDM sbol. The effect is similar to the insymbol interference that occurs in a

g l e - c ar r i e r sy st e m . H ow e ve r , i n ssy st e m s, t h e i n t e r f e r e n c e i s t y p i cdue to several other symbols insteaj u st t h e p r e vi ou s sy m b ol ; t h e sy mperiod in single carrier systems is tcally much shorter than the t ime so f t h e c h a n n e l , w h e r e a s t h e t y pO F D M sy m b ol p e r i od i s m u c h l ontha n t he t ime span of the channel. second problem is unique to mu ltice r s y s t e m s a n d i s c a l le d I n t r a s y mInterference. It is the result of interence amongst a given OFDM symbown subcarriers. The next sections it r at e h ow O F D M d e al s w i t h t h e setypes of interference.

Intersymbol interferenceAs s u m e t h a t t h e t i m e s p a n o f

channel is L C samples long. Insteadsingle carrier with a data rate of R sb ols/se con d , a n O F D M sy st e m h as u b c a r r i e r s , e a c h w i t h a d a t a r a tR/N symbols/second. B eca use the r a t e i s r e d uc ed b y a f a c t or o f N ,OFDM symbol period is increased factor of N. By choosing an appropr

Figure 5. Block diagram of a simple OFDM system.

Figure 6. Example of intersymbol interference. The green symbol was transmitted first, followed bblue symbol.


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v a l u e f or N , t h e l en g t h o f t h e O F D Msy m b ol b e com es l on g e r t h a n t h e t i m es p a n o f t h e c h a n n e l . B e c a u s e o f t h i sconfiguration, the effect of intersymbolinterference is the distortion of the firstLC samples of the received OFDM sym-bol. An example of this effect is shownin Figure 6. By noting tha t only the firstfew sa mples of the symbol are distorted,o n e ca n c on s i d er t h e u s e o f a g u a r di n t e r val t o r e m ove t h e e f fe ct of i n t e r -symbol interference. The gua rd intervalcould be a section of al l zero samples

t r a n s m i t t e d i n f r o n t o f e a c h O F D Msy m b ol . S i n c e i t d oe s n ot c on t ai n an yu sef u l i n f or m at i on , t h e g u ar d i n t e rvalw ou l d b e d i scar d e d at t h e r e c ei ver . I fthe length of the guard interval is prop-erly chosen such that i t is longer thant h e t i m e s p a n o f t h e c h a n n e l , t h eOFDM symbol itself will not be distort-ed. Thus, by discarding the gua rd inter-val, the effects of intersymbol interfer-ence are thrown a wa y a s well.

Intrasymbol interferenceTh e g u a r d i n t e r va l i s n o t u s e d i n

p r ac t i c al sy st e m s b e c au se i t d oe s n otp r eve n t a n O F D M sy m b ol f r om i n t e r-

fering with itself. This type of interfer-e n ce i s c a l l ed i n t r a s y m b ol i n t e r f er -e n c e . T h e sol u t i on t o t h e p r ob l e m ofi n t r a s y m b o l i n t e r f e r e n c e i n v o l v e s adiscrete-time property. Recall that incontinuous-time, a convolution in timeis equivalent to a multiplication in thef r e q u e n c y - d o m a i n . T h i s p r o p e r t y i strue in discrete-time only if the signals

ar e of i n f in i t e l en g t h or i f a t l east on eo f t h e s i g n a l s i s p e r i od i c o v e r t h eran ge of the convolution. I t is not pra c-tical to have an infinite-length OFDMsymbol, however, it is possible to maket h e O F D M s y m b o l a p p e a r p e r io d ic .Th i s p e r i o d i c f o r m i s a c h i e v e d b yr e p l a c i n g t h e g u a r d i n t e r v a l w i t hsomething known as a cyclic prefix oflength L P samples. The cyclic prefix isa r e p l i c a of t h e l ast L P sa m p le s of t h eOFDM symbol where L P > L C. Since i tc o n t a i n s r e d u n d a n t i n f o r m a t i o n , t h e

cyclic prefix is discarded at the receiv-er. Like the case of the guard interval ,this step removes the effects of inter-s y m b o l i n t e r f e r e n c e. B e ca u s e o f t h ew a y i n w h i c h t h e c y cl i c p r e f ix w a sformed, the cyclically-extended OFDMs y m b o l n o w a p p e a r s p e r i o d i c w h e nconvolved w ith the channel. An impor-t a n t r e s ul t i s t h a t t h e e f fe ct o f t h echannel becomes multiplicative.

In a digital communications system,the symbols that arrive at the receiverh a v e b e e n c o n v o l v e d w i t h t h e t i m e -d om ai n c h an n e l i m p u l se r e sp on se ofl en g t h L C samples. Thus, the effect ofthe channel is convolutional . In order

t o u n d o t h e e f f e ct s o f t h e c h a n n e l ,a n o t h e r c on v o l u t i o n m u s t b e p e r -f or m e d a t t h e r e c ei v er u s i n g a t i m e -d om a i n f il t er k n ow n a s a n e q u a l iz e r .The length of the equalizer needs to beo n t h e o r d e r o f t h e t i m e s p a n o f t h echannel. The equalizer processes sym-b ol s i n or d e r t o ad ap t i t s r e sp on se i nan attempt to remove the effects of the

c h a n n e l . S u c h a n e q u a l i z e r c a ne x pe n s iv e t o i m pl em e n t i n h a r d wan d of t e n r e q u i r e s a l ar g e n u m b esymbols in order to adapt i ts respoto a good setting.

In OFDM, the t ime-domain signs t i l l c on v o l v e d w i t h t h e c h a nr e spon se. H ow e ve r , t h e d at a w i l l m a t e l y b e t r a n s f o r m e d b a c k i n t of r e q u en c y -d o m a i n b y t h e F F T i nreceiver. Because of th e periodic naof the cyclically-extended OFDM sbol, this t ime-domain convolution

result in the multiplication of the strum of the OFDM signal (i .e. , theq u e n c y -d o m a i n c on s t e l l a t i o n p oiw i t h t h e f r e q u e n c y r e s p o n s e o fchannel. The result is that each subr i e r ’ s sy m b ol w i l l b e m u l t i p l i e d bcomplex number equal to the channfrequency response at that subcarrf r e q u e n c y . E a c h r e c ei v e d s u b c a rexperiences a complex gain (ampliand phase distortion) due to the cnel. In order to undo t hese effects, aquency-domain equalizer is emploSuch an equalizer is much simpler a time-domain equalizer. The frequcy-domain equalizer consists of a

g l e c o m p l e x m u l t i p l i c a t i o n f o r esu b c ar r i e r . F or t h e si m p l e c ase onoise, th e ideal va lue of the equa lir e s p o n s e i s t h e i n v e r s e o f t h e c hnel’s frequency response. An examis shown in Figure 7. With such at i n g , t h e f r e q u e n c y - d om ai n e q u alw o u l d c a n c e l o u t t h e m u l t i p l i c aeffect of the channel.

Figure 7. Left plot shows the frequency response of a channel, and the right plot shows the corresponding frequency-domain equalizer response. Note thequalizer response is large when the channel response is small in order to counteract the effect of a channel null.


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COFDM: Coded OFDMCoded OFDM, or COFDM, is a term

u s ed f o r a s y s t e m i n w h i c h t h e e r r o rcontrol coding and OFDM modulationp r o ce s s e s w o r k c l o s el y t o g e t h e r . A nimportant step in a COFDM system isto interleave and code the bits prior tothe IFFT. This step serves the purposeo f t a k i n g a d j a c e n t b i t s i n t h e s o u r c ed a t a a n d s p r e a d i n g t h e m o u t a c r o s smultiple subcarriers. One or more sub-carriers may be lost or impaired due toa f r e q u e n c y n u l l , an d t h i s l oss w ou l dcause a contiguous stream of bit errors.

Such a burst of errors would typicallybe ha rd to correct. The interleaving a tthe transmitter spreads out the contigu-ous bits such that the bit errors becomespaced far apart in t ime. This spacingmakes i t easier for the decoder to cor-rect the errors. Another important stepi n a C O F D M sy st em i s t o u se ch an n e linforma tion from the equa lizer t o deter-mine th e reliability of the received bits.T h e val u e s of t h e e q u al i ze r r e sp on sea r e u s e d t o i n f e r t h e s t r e n g t h o f t h ereceived subcarriers. For example, if theequalizer response had a large value ata certain frequency, it would correspondto a frequency null at that point in the

channel. The equalizer response wouldhave a large value at tha t point becauseit is t rying to compensate for the w eakreceived signal. This reliability informa-tion is passed on to t he decoding blockss o t h a t t h e y c a n p r o p e r l y w e i g h t t h ebits when making decoding decisions.In the case of a frequency null , the bitsw ou l d b e m ar k e d as “ l ow c on f i d e n c e ”

and those bits would not be weighted asheavily as bits from a strong subcarrier.C O F D M s y s t e m s a r e a b l e t o a c h i e v ee xc e l l e n t p e r f or m an c e on f r e q u e n c y -selective channels because of the com-bined benefits of multicar rier modula-tion and coding.

Non-ideal effects in anOFDM system

This section will examine the effectsof non-ideali t ies in an OFDM system.These effects will include impairmentsan d receiver offsets. B ecause the fourier

transform is a fundamental operation inOFDM, t he effects of several offsets canbe intuitively understood by applyingfourier tra nsform theory.

Local oscillatorfrequency offset

At start-up, the local oscillator (LO)frequency at the receiver is typically dif-f e r e n t f r o m t h e L O f r e q u e n c y a t t h etransmitter. A carrier tracking loop isu s e d t o a d j us t t h e r e ce iv e r ’s L O f r e -quency in order to match the transmit-ter’s LO frequency as closely as possi-ble. The effect of having an LO frequen-c y of f set c an b e e xp lai n e d b y F ou r ie r

Tra nsform th eory. The LO offset can beexpressed mathematically by multiply-ing the received time-domain signal bya complex exponential whose frequencyis equal to the LO offset amount. Recallf r o m F o u r i e r T r a n s f o r m t h e o r y t h a tmultiplication by a complex exponentiali n t i m e i s e q u i val e n t t o a sh i f t i n f r e -quency. The LO offset results in a fre-

quency shift of the received signa l st r u m . Th i s s h i f t c a u s e s a c on d icalled “loss of orthogonali ty” to ocThe frequency shift causes the OFsubcarriers to no longer be orthogoThe orthogonality of the subcarrielost because the bins of the FFT will o n g e r l i n e u p w i t h t h e p e a k s o fr e c e i v e d s i g n a l ’ s s i n c e p u l s e s . r e su l t i s a d i st or t i on c al l e d i n t e r -i n t e r f er e n ce o r I B I . I B I o cc u r s we n e r g y f r o m o n e b i n s p i l l s o v e r ad j ac en t b i n s an d t h i s e n e r gy d i stt h e a f f e ct e d s u b ca r r i e r s . I n F o u

Transform theory this effect is caDFTleakage.

Th e l e f t p l ot of F i g u r e 8 sh ow ssp e c t r u m of a r e c e i ve d O F D M siw i t h n o L O off set . F or t h e p u r p osc l ar i t y , on l y on e n on - ze r o su b c arwas transmitted. Note that this subrier is not interfering with i ts a djasubcarriers. The spectrum of the nze r o su b c ar r i e r ac t u al l y e xt e n d s othe entire range of the FFT, howedue to the orthogonal nature of thenal, the zero-crossings of the spectexactly line up with t he other FFT bTh e r i g h t p lo t o f F i g u r e 8 s h o w sr e c e i ve d sp e c t r u m of t h e sam e si

with one non-zero subcarrier, howein th is case there is an LO offset . offset has resulted in a loss of orthn a l i t y , a n d t h e z e r o - c r o s s i n g s o fn o n - z er o s u b c a r r i e r ’ s s p e c t r u mlonger l ine up with the FFT bins. result is that energy from the non-su b c ar r i e r i s sp r e ad ou t am on g at h e ot h e r su b car r i e r s , w i t h t h ose

Figure 8. Received spectrum with one non-zero subcarrier. The left plot is for the case of no LO offset, and the right plot is for the presence of an LO offset.


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carriers closest to the non-zero subcarri-er receiving the most interference. Thissimple example was for the case of onlyone non-zero subcarrier. In a practicals y s t e m , a l m o s t a l l o f t h e s u b c a r r i er swould be actively used for transmitting

data . A given subcarrier would experi-e n ce I B I d u e t o e n e r gy f r om al l of t h eother active subcarriers in the system.T h e c e n t r al l i m i t t h e or e m st at e s t h att h e su m of a l ar g e n u m b e r of r an d omprocesses will result in a signal that hasa Gaussian distribution. Because of thisproperty, t he IB I wil l manifest i tself asadditive Ga ussian noise, thus loweringthe effective SNR of the system.

The effect of an LO frequency offsetcan be corrected by multiplying t he sig-nal by a correction factor. The correc-tion factor would be a sinusoid with af r e q u e n c y t h at i s i d e al l y e q u al t o t h ea m o u n t o f t h e L O f r e q u e n c y o f f s e t .

V a r i o u s c a r r i e r t r a c k i n g a l g o r i t h m sexist tha t can a daptively determine thefrequency that will correct for the offset.

LO phase offsetI t i s a l s o p o s s i b l e t o h a v e a n L O

phase offset , separate from an LO fre-quency offset. The two offsets can occurin conjunction or one or the other can bepresent by itself. As the name suggests,an LO pha se offset occurs when t here isa d i ff e r en c e b et w e e n t h e p h ase of t h eLO output a nd t he phase of the receivedsignal . This effect can be representedm a t h e m a t i c a l l y b y m u l t i p l y i n g t h etime-domain signal by a complex expo-

n e n t i a l w i t h a c on s t a n t p h a s e . Th er e su l t i s a c on st an t p h ase r ot at i on f orall of t he subcarriers in t he frequency-d om ai n . T h e c on st e l l at i on p oi n t s f ore ac h su b c ar r i e r e xp e r i e n c e t h e sam edegree of rotation. If the phase rotation

is small, the frequency-domain equaliz-e r c an c or r e c t t h i s e f f e c t . E ac h f i l t e rcoefficient in a frequency-domain equal-izer multiplies its corresponding subcar-rier by a complex gain (i .e. , a mplitudescaling and phase rotation). The equal-

izer’s coefficients can be used to correctfor a small phase rotat ion a s long a s therotation doesn’t cause the constellationpoints t o rotat e beyond the sym bol deci-sion regions. Larger phase rotations arecorrected by a carrier tracking loop.

FFT window location offsetA n o t h e r n o n - i d e a l e f f e c t t h a t c a n

occur in a real-world OFDM system isan FFT window location offset . An N-point FF T at the receiver processes dat ai n b l oc k s o f N s a m p l e s a t a t i m e .Ideally, the N samples taken in by theFFT will correspond t o the N sa mples ofa single transmitted OFDM symbol. In

practice, a correlation is often used w itha known preamble sequence locat ed a tthe beginning of the transmission. Thiscorrelation operation aids the receiverin synchronizing itself with the receiveds i g n a l ’s O F D M s y m b o l b o u n d a r i e s .However, inaccuracies still remain, andthey manifest themselves as an offset inthe FFT window location. The result isthat the N sa mples sent to the FFT willnot line up exactly w ith t he correspond-ing OFDM symbol. I f the offset is verylarge, part of the N sa mples w ill be fromone OFDM symbol, and the rest of sam-ples wil l be from another OFDM sym-bol. Such a si tuation would result in a

severe distortion of the received subcar-rier’s constellat ions. F ortuna tely, such alarge offset does not typically occur if ar o b u s t s y n c h r o n i z a t i o n a l g o r i t h m i sused. More likely, a n FFT window loca-t i on o f f s et o f ju s t a f ew s a m p l e s w i l l

occur. The presence of the cyclic prg i v es e n o u g h h e a d r o om t o e n a bsmall offset to be present without ing samples from more than one OFsymbol. However, even an offset of one sample wil l cause some degre

d i s t o r t i o n . A g a i n , t h e e f f e c t c a nu n d e r s t o o d f r o m F o u r i e r T r a n s ft h e or y . Th e of f se t c an b e vi e w e d ash i f t i n t i m e . As l on g as t h e F F T wd o w l oc a t i o n o f f s et d o es n o tb e y o n d a n O F D M s y m b o l b o u n dt h i s s h i f t i n t i m e i s e q u iv a l e n t l i n e ar l y - i n c r e asi n g p h ase r ot at i ot h e f r e q u e n c y - d om ai n c on st e l l at iC o n s t e l l a t i o n s o n s u b c a r r i e r s c os p o n d i n g t o l o w f r e q u e n c i e s w i lrotated slightly, whereas constellaton h i g h e r - f r e q u e n c y su b c ar r i e r s e x p e r i e n c e a l a r g e r r o t a t i o n . amount of rotation increases linearlt h e s u b c a r r i e r ’ s F F T b i n l o ca

increases. Examples of the effects offerent degrees of FFT window locaoffsets a re shown in Figure 9.

FFT window locat ion offsets a re ocorrected by performing a time-domc o r r e l a t i o n w i t h a k n o w n t r a i nsequence embedded in the tra nsmisignal . The location of the peak ofcorrelation al lows the receiver to chronize itself with the incoming sig

Sampling frequency offsetA n ot h e r p ot e n t i al l y h ar m f u l s i

t ion is the presence of a sampling q u e n c y of f se t . T h i s c on d i t i on oc cwhen the A/D converter output is s

pled either too fast or too slow. Ret h a t F S /2 is th e highest a vaila blequency in discrete-time w here F S issampling frequency. Sampling too essential ly increases the value of a n d t h e r e s u l t i s a c on t r a c t e d (

Figure 9. Effect of different FFT window offsets.


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s q u a s h e d ) s p e ct r u m . S i m i la r l y , s a m -p l i n g t oo sl ow d e c r e ase s t h e val u e ofF S /2 a nd results in an expan ded spec-

t r u m . I f t h e s p ec t r u m e x p a n d s t o om u c h , a l i a s i n g o f t h e s p e c t r u m c a noccur. Either type of sampling frequen-

cy offset results in IBI since the expan-sion or contra ction of the spectrum pre-vents the received subcarriers from lin-ing up with the FFT bin locations. Theeffect of sampling too fast is illustratedin Figure 10 and simulation results tod e m on s t r a t e t h i s e ff ec t a r e s h o w n i nFigure 11. A sampling frequency offsetcan be corrected by generating an errort e r m t h at i s u se d t o d r i ve a sam p l i n grat e converter.

Uniform noiseA d d i t i v e w h i t e G a u s s i a n n o i s e

(AWGN) is the most common impair-ment encountered in a communications

system. In a wireless medium, the noisesource is typically considered to be ther-mal noise that is Gaussian a nd uniformacross the frequency ra nge. Additionaln o i s e s o u r c e s i n c l u d e a t m o s p h e r i csou r ce s an d solar r ad i at i on . I n a c on -t ai n e d m e d i a, su c h as a c oaxi al c ab l esystem, thermal noise wil l be present,

b u t t h e s y s t e m m a y a l s o h a v e ot h e rsources that can increase the noise inthe system. The effect of AWGN on an

OFDM system is similar to its effect ona single carrier syst em. The signa l-to-n oi se r at i o (S N R ) i s a f u n c t i on of t h e

total signal power over the total noisepower across t he r eceived channel. Theuniform n oise contributes t o the S NR ofe ac h su b c ar r i e r i n t h e O F D M sy st e man d t h e n e t r e su l t i s e q u i val en t t o t h eeffect on single channel systems.

Non-uniform noiseNoise in a communications channel

c an of t e n b e sh ap e d , or “ c ol or e d ” , b yv a r i o u s e f f e ct s . Th e s e e f f e ct s c a ninclude transmit signal imperfections,transmission channel characteristics, orreceiver frequency shaping. The impli-c at i on s of t h e se e f f e c t s f or an O F D Msystem can be different compared to its

single-carrier counterpart. The modula-t i on o f t h e O F D M s y s t e m c a n b e t a i -lored for the noise characteristics. Onemethod previously mentioned involvesl ow e r i n g t h e m o d u la t i on (n u m b e r o fb i t s/sy m b ol ) on su b c a r r i e r s i n a l ow S N R e n v i r o n m e n t a s i l l u s t r a t e d i nF i g u r e 12. A n ot h e r m e t h od i n vol ve s

sending the same data on several c a r r i e r s , o r s e n d i n g d a t a t h a t c a nconsidered lower priority. In extrc ases, t h e su b car r i er s c an t r an sm idata, essentially turning them off.

Impulse noiseImpulse noise is a common imp

ment in a communications system ing from motors or l ightning. Impn o i s e i s t y p i c a l l y c h a r a c t e r i z e d short time-domain burst of energy. burst may be repetitive or may be a gle event. In either case, the freques p e c t r u m f r o m t h i s e n e r g y b u r sw i d e b an d , t y p i c al l y m u c h w i d e r tt h e c h an n e l , b u t i s p r e se n t f or onshort time period.

One of the most important conct o u n d er s t a n d a b ou t O F D M a n dproperties related t o the FFT algorii s h o w t h e a l g or i t h m c h a n g e s

nature of the signal. In a single-cars y s t e m , t h e s y m b ol c a n b e v i ew eoccupying all of the available frequesp ec t r u m f or t h e t i m e d u r at i on ofsymbol. A group of symbols then opies a ll of the spectrum for the duraof the whole group, but in a t ime dsion arrangement.

OFDM, using t he FFT, ta kes syman d c r eat e s t h e se g r ou ps d i r e ct l y t h e n t r a n s f o r m s t h e m . T h e y a r elonger t ime-domain multiplexed, ar e n ow frequency-domain multipleThe OFDM symbol is now a collectiothese source symbols, and this OFs y m b ol n o w h a s a m u ch l on g er d

tion. Each original symbol occupies a small frequency region, but now op i e s t h at r e g i on f or t h e e n t i r e O Fsymbol duration. Figure 13 i llustrt h i s c o n c e p t . F o r i m p u l s e s t h a tsh or t i n d u r at i on , t h e i m pu l se e nmasks a smaller percentage of t ime a c h O F D M s y m b ol c om p a r e d t osingle carrier case. Impulse noisetherefore have less of an effect on sdura tion noise.

Carrier interferenceS i n g l e - c a r r i e r i n t e r f e r e n c e a r

from other sources that may co-exithe frequency range of interest . Th

can be generated by nearby circuitother transmission sources. The sicarrier system must handle this inference as a noise source for a ll infortion sent. The OFDM system can athe frequency region of interferencdisabling or turning off the a ffected c a r r i e r s . N a r r o w b a n d m o d u l asources of interference can be con

Figure 10. Illustration of the effect of a sampling frequency offset.

Figure 11. Simulation results showing the effect of a sampling frequency that is too high. Note that thesample that was originally at bin 15 is now at bin 8.


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ered similar to carrier interference intheir impairment.

Phase noise

Noise can also be added to the signalthrough a frequency-conversion stage.The local oscillat or used in t he converterwill inherently have some phase noise( u n c er t a i n t y o f a c t u a l f r e q u e n c y o rp h ase of t h e si g n al ) t h at w i ll b e t r an s-ferred to the desired signal . Figure 14shows the effect of phase noise on a localoscillator. Phase noise is shaped and isprimarily concentrated near the carrier(or center frequency) of the s igna l.

An OFDM signal set contains multi-p l e s u b c a r r i e r s , e a c h o f w h i c h i s as m a l l e r p e r c e n t a g e o f t h e t o t a l f r e -q u e n c y b a n d w i d t h t h a n i n a s i n g l ec a r r i e r s y s t e m . As a r e s u l t , p h a s e

n o i s e i s a s m a l l e r p e r c e n t a g e o f t h eb an d w i d t h i n a s i n g le -c ar r i er sy st e m .

F or t h i s r e ason , p h ase n oise d e g r ad e st h e p e r f or m an c e of an O F D M sy st e mm or e t h an i n a s i n g l e c ar r i e r sy st e m .P h a s e n o is e e f fe ct s i n a n O F D M s y s -

t e m c a n b e s e p a r a t e d i n t o t w o c a t e -gories: phase noise maintained withino n e s u b c a r r i e r s p a c i n g , a n d p h a s en o i s e t h a t e x t e n d s a c r o s s s u b c a r r i e rs p a c i n g s . P h a s e n o i s e t h a t e x t e n d sac r oss su b c ar r i e r sp ac i n g s i s c on si d -ered extreme an d results in demodula-t i o n e r r o r s . P h a s e n o is e w i t h i n o n es u b c a r r i e r s p a c i n g e s s e n t i a l l y h a s asimilar but scaled effect as for the sin-g l e c a r r i e r s y s t e m . T h e p h a s e n o i s er e su l t s i n p h a s e u n c e rt a i n t y i n t h ec on st e l l at i on p oi n t p r od u c i n g an ar c -sh ap e d n oi se p a t t e r n i n t h e c on st e ll a-tion of each subcarrier.

I n or d e r t o h e l p t h e O F D M sy st e m

h an d l e p h ase n oi se , p i l ot su b c ar r i e r sare often used. These pilot subcarriers

ar e g e n e r at e d b y t h e I F F T an d c au s e d t o p r ov i de a s t a b l e p h a s e r eence for the receiver circuitry. Adt h e se p i l ot s l ow e r s t h e avai l ab l e dr at e of t h e sy st e m b e c au se t h e se c a r r i e r s a r e n o l o n g e r a v a i l a b l

t r a n s m it d a t a .

Non-linear circuits in thetransmitter and receiver

Al l t r a n s m i t t e r s a n d r e c e iv e rc om m u n i c a t i o n s s y s t e m s c on tdevices such as amplifiers and mithat have non-linear transfer functiTh e se n on -l i n ear i t i e s c r e at e an at i o n a l p e r f o r m a n c e l i m i t a t i o n . receiver performance is typically limb y d i s t o r t i o n g e n e r a t e d i n t h e i nam p l if i er or m i xe r i n t h e p r e sen cstrong undesired signals. The transter performance is limited primarilp ow e r am p l i f i e r l i n e ar i t y . A n O F

signal is made up of multiple simuneous signals that , for a given avepower, have a higher peak signal leOFDM signals result in an increasthe peak-to-average rat io (PAR) ofsi g n al . F or m u l t i -c ar r i er sy st e m s,PAR value is often expressed in teof statistics because the probability a l l s u b c a r r i e r s w i l l s i m u l t a n e o ur e a c h p e a k a m p l i t u d e i s l o w , et h ou g h t h e si m u l t an e ou s p e ak amtude va lue is la rge. These higher pamplitude levels will create more sed i s t o r t i o n t h a n a s i n g l e c a r r i e r ceven if the a verage power levels of ear e the sa me. The higher distortion

i n c r e ase t h e S N R n e e d e d t o m ai na d e q u a t e p e r f o r m a n c e . L i n e arequirements in both the receivert r a n s m i t t e r m u s t b e a d ju s t e d“backed off” to account for this incrin PAR value. The PAR value, and the amount of l inearity compensatwill depend on a number of paramei n c lu d i n g t h e n u m b e r o f s u b ca r ra n d t h e l ev e l o f S N R t h a t m u smaintained.

Modern applicationsO F D M h as b e e n c h ose n f or se v

current an d future communicat ions tems a ll over t he w orld. It is well-su

for systems in which the channel cacteristics make it difficult to mainadequate communications l ink perma nce. Asynchronous digital subscrline (ADSL) provides a method of de r i n g h i g h sp e e d d at a ove r t h e p hl i n e . T h e s y s t e m u s e s O F D M t eniques, call ing their variation discmulti-tone (DMT). DMT includes


Figure 13. Comparison of single carrier versus OFDM spectrum.

Figure 12. Uniform and Non-uniform noise and SNR. OFDM can tailor its modulation to the shape of thenoise spectrum.


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t u r e s f o r a l l o w i n g t h eremoval of subcarriers andf or a d j u s t i n g m o d u la t i onf o r m a t ( f r o m 1 t o 1 5 b i t sp e r sy m b ol ) on a p e r su b -carrier basis to best suit the

tran smission channel char-acteristics. The system alsop er m i t s “ d y n a m i c a l l oc a -tion” of these para meters.

E u r op ean d i g it a l t e le vi-sion is based on the DVB-T( d i g i t a l v i d e o b r o a d c a s t -t e r r es t r ia l ) s t a n d a r d t h a tu s e s e i t h e r 2 0 4 8 ( 2 K ) o r8192 (8K) subcarriers with-i n a s t a n d a r d 8 M H z TVchannel. The system speci-f i ca t i on s a n d c od i n g w e r es p e c i f i c a l l y d e s i g n e d t oal l ow m u l t i p oi n t r e p e at e rs i g n a l i n g t h a t c r e a t e s c o -

c h a n n e l s i g n a l s . D i s cu s -s i o n s a r e o n g o i n g i n t h eU . S . t o l o ok a t a s i m i l a rsystem and J apan is close to adopting asimilar sta ndard for t heir future digitalTV broadcast system.

The next generation of radio broad-cast may also make use of OFDM tech-n i q u e s. I n t h e U. S . , t h e sy st e m u n d e rconsideration will initially “co-exist” inthe same frequency slot as the currentanalog broadcast. OFDM allows the sys-tem designers to shape the digital spec-trum by disabling the subcarriers thatcorrespond t o th e current ana log spec-t r u m d u r i n g t h e c o- e xi st e n c e p e r i od .

After th e co-existence period the s ubcar-riers can be enabled and t he subsequentdata rate increased.

Various high-speed wireless network-i n g st an d ar d s i n t h e 5 G H z f r e q u e n c yregion employ OFD M modulation. TheU.S. IEEE 802.11a and European ETSIH i p e r la n /2 s t a n d a r d s u t i l i z e s i m i la rphysical layer structures with 64-carri-er OFDM and modulation ranging fromBPSK to 64-QAM per subcarrier. Vari-ou s d at a r at e s f r om 6 t o 54 M b p s ar ep os s i b le . O F D M w o r k s w e ll i n h o m ean d of f i c e e n vi r on m e n t s f or h an d l i n gwall reflections and movement withinthe structure.

ConclusionsOFDM techniques are quickly becom-

ing a popular method for adva nced com-m u n i ca t i on s n e t w o r k s . A d v a n c es i nVLSI technology have made it possibleto efficiently implement an FFT block inh a r d w a r e . D e s p i t e t h e a d v a n t a g e sOFDM can offer, the hardw are t o imple-

ment it can still make up a sizeable andexpensive portion of the design. OFDMshould not be considered for every com-m u n i c a t i o n s y s t e m b e c a u s e o f i t sincreased complexity and higher trans-mitter and receiver demands. However,for certain systems, modern digital sig-nal processing techniques now make itpossible to use this modulation systemto improve the reliability of the commu-nications link.

R E F E R E N C E S

[ 1] B i n g h a m , J . A . C . , M u l t i car r i er Modul at i on f or Dat a Transmi ssi on: A n

i d e a w h o s e t i m e h a s c o m e , I E E ECommunications Ma gaz ine, Vol. 28, no.5, pp. 5-14, May 1990.

[2] J .M. Cioffi, A M u l t i c a r r i e r P r i m e r ,i n AN S I T1 E 1. 4 C om m i t t eeContribution, No. 91-157, Boca Raton,FL, Nov. 1991.

[3] Weinstein, S.B ., E bert , P .M., D a t a T r a n s m i s si o n b y F r eq u en c y -D i v i s i on

M ul t i pl exi ng Usi ng t he Di scret e Fouri er T r an sf or m , I E E E Tr a n s a c t i o n s o nCommunication Technology, Vol. COM-19, no. 5, pp. 628-634, October 1971.

[4] J . Stott , The Effects of Phase Noise i n C O F D M , EB U Technical Review,Summer1998

[5 ] P . S h e l s w e ll T h e C O FM o d u l a t i o n S y s t e m , T h e H ea r

D i g i t a l A u d i o B r o a d sca s t i n g , BR e s e a r c h a n d D e v e lo pm e n t R e pB B C RD 1996/8

About the authors

Michael P ugel is a principal meber of the technical staff at ThomsMultimedia, Indianapolis. He is cr e n t l y w or ki n g i n c or p or at e i n n ot i o n a n d R e s e a r c h f o r t h e F u t uC o m m u n i c a t i o n s S y s t e m s G r oi n vest i g at i n g h om e n e t w or ki n g ccepts a nd a dvan ced front-end t echno g y . P r e v i ou s l y , h e h a s w o r k e danalog and digital television, digsatellite receiver, RF remote contand cable modem front-end prod

development. He currently holdsU.S. pat ents and has received numous internal Thomson awards. He c o-a u t h or e d se ve r a l p a p e r s a n dt u t o r i a l p r e v i o u s l y p r e s e n t e dI C C E . H e g r a d u a t e d f r om P u r dU n i v e r si t y w i t h h i s B S E E i n 1 9an d M S E E i n 1991. H e c an b e cta cted a t: 317-587-4027; fax 317-56898; e-ma il: PugelM @tce.com.

L o u i s L i t w i n i s a m e m b e r o f t e ch n i ca l s t a f f w i t h Th o m sM u l t i m e d i a C o r p o r a t e R e s e a rw h e r e h e i s w o r k i n g o n a w i r e lOFDM-based m odem for digita l honetworking applications. Mr. Litw

received his M.S. degree in electrengineering from Purdue Universin 1999 an d his B .S. degree in eleccal engineering (summa cum laufrom Drexel University in 1997. w a s n a m ed b y E t a K a p pa N u a s Alton B. Zerby and Carl T. Koerno u t s t a n d i n g e l ec t r i ca l e n g i n e erst u d e n t f or 1997. H e h as p u b l i shove r a d oze n t e c h n i c al ar t i c l e s aconferences papers on various topr e l a t e d t o d i g i t a l c om m u n i ca t ia n d a l s o h a s f i v e p a t e n t s p e n d ir e l a t e d t o O F D M . H i s p r o fe s s iointerests include digital communitions w ith a particular focus on a d

t i v e e q u a l i z a t i o n a n d e r r or -c on tcoding. He can be contacted at : 3587-4745; fax 317-587-6898; e-mli tw in [email protected]

Th e a u t h o r s w o u l d l i k e t o t h aM a x B e l o t s e r k o v s k y ( Th o m sM u l t i m e d i a ) f o r e x p a n d i n g t hOFDM h orizons .

Figure 14. Phase noise on a LO. The upper picture shows a signal

with very little phase noise, and the lower picture shows the samesignal with phase noise added

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ofdm basics file