OFDM原理介绍

The principles of orthogonal frequency division multiplexing (OFDM)modulation have been in existence for several decades.However,in recent years these techniques have quickly moved out of textbooks and research laboratories and into prac-tice in modern communications systems.The tech-niques are employed in data delivery systems over the phone line,digital radio and television,and wireless networking systems.What is OFDM?And why has it recently become so popular?This article will review the fundamentals behind OFDM tech-niques,and also discuss common impairments and how,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 applica-tions will conclude the article.The single-carrier modulation systemA typical single-carrier modulation spectrum is shown in Figure 1.A single carrier system modu-lates information onto one carrier using frequency,phase,or amplitude adjustment of the carrier.Fordigital signals,the information is in the form of bits,or collections of bits called symbols,that are modu-lated onto the carrier.As higher bandwidths (data rates)are used,the duration of one bit or symbol of information becomes smaller.The system becomes more susceptible to loss of information from impulse noise,signal reflections and other impairments.These impairments can impede the ability to recover the information sent.In addition,as the bandwidth used by a single carrier system increases,the sus-ceptibility to interference from other continuous sig-nal sources becomes greater.This type of interfer-ence is commonly labeled as carrier wave (CW)or frequency interference.Frequency division multiplexing modulation systemFrequency division multiplexing (FDM)extends the concept of single carrier modulation by using multiple subcarriers within the same single channel.The total data rate to be sent in the channel is divided between the various subcarriers.The data do not have to be divided evenly nor do they have to originate from the same information source.Advantages include using separate modula-tion/demodulation customized to a particular type of data,or sending out banks of dissimilar data that can be best sent using multiple,and possibly dif-ferent,modulation schemes.Current national television systems committee (NTSC)television and FM stereo multiplex are good examples of FDM.FDM offers an advantage over single-carrier modulation in terms of narrowband frequency interference since this interference will only affect one of the frequency subbands.The other subcarriers will not be affected by the interference.Since each subcarrier has a lower information rate,the data symbol periods in a digital system will be longer,adding some additional immunity to impulse noise and reflections.FDM systems usually require a guard band be-tween modulated subcarriers to prevent the spec-trum of one subcarrier from interfering with an-other.These guard bands lower the system’s effective information rate when compared to a single carrier system with similar modulation.Orthogonality and OFDMIf the FDM system above had been able to use a set of subcarriers that were orthogonal to each other,a higher level of spectral efficiency could have been achieved.The guardbands that were necessary to allow individual demodulation of subcarriers in an FDM system would no longer be necessary.The use of orthogonal subcarriers would allow the subcarri-ers’spectra to overlap,thus increasing the spectral efficiency.As long as orthogonality is maintained,it is still possible to recover the individual subcarriers’signals despite their overlapping spectrums.If the dot product of two deterministic signals is equal to zero,these signals are said to be orthogo-nal to each other.Orthogonality can alsobeFigure 1.Single carrier spectrum example.signal processingThe principlesof OFDMMulticarrier modulation techniques are rapidly moving from the textbookto the real world of modern communication systemsBy Louis Litwin andMichael Pugelviewed from the standpoint of stochas-tic processes.If two random processes are uncorrelated,then they are orthogonal.Given the random nature of signals in a communications sys-tem,this probabilistic view of orthogo-nality provides an intuitive under-standing of the implications of orthog-onality in ter in this article,we will discuss how OFDM is imple-mented in practice using the discrete fourier transform (DFT).Recall from signals and systems theory that the sinusoids of the DFT form an orthogo-nal basis set,and a signal in the vec-tor space of the DFT can be represent-ed as a linear combination of the orthogonal sinusoids.One view of the DFT is that the transform essentially correlates its input signal with each of the sinusoidal basis functions.If the input signal has some energy at a cer-tain frequency,there will be a peak in the correlation of the input signal and the basis sinusoid that is at that corre-sponding frequency.This transform is used at the OFDM transmitter to map an input signal onto a set of orthogo-nal subcarriers,i.e.,the orthogonal basis functions of the DFT.Similarly,the transform is used again at the OFDM receiver to process the received subcarriers.The signals from the sub-carriers are then combined to form an estimate of the source signal from the transmitter.The orthogonal and uncorrelated nature of the subcarriers is exploited in OFDM with powerful results.Since the basis functions of the DFT are uncorrelated,the correla-tion performed in the DFT for a given subcarrier only sees energy for that corresponding subcarrier.The energy from other subcarriers does not con-tribute because it is uncorrelated.This separation of signal energy is the rea-son that the OFDM subcarriers’spec-trums can overlap without causing interference.A simple OFDM exampleFigure 3shows a simple representa-tion of an OFDM system.These types of systems have been built but the practicality of such construction quick-ly diminishes as the number of subcar-riers increases.Each subcarrier car-ries one bit of information (N bits total)by its presence or absence in the output spectrum.The frequency of each subcarrier is selected to form an orthogonal signal set,and these fre-quencies are known at the receiver.Note that the output is updated at a periodic interval T that forms the sym-bol period as well as the time bound-ary for orthogonality.Figure 4shows the resultant frequency spectrum.In the frequency domain,the resulting sin function side lobes produce over-lapping spectra.The individual peaks of subbands all line up with the zero crossings of the other subbands.This overlap of spectral energy does not interfere with the system’s ability to recover the original signal.The receiv-er multiplies (i.e.,correlates)the incoming signal by the known set of sinusoids to produce the original set of bits sent.The digital implementation of an OFDM system will enhance these simple principles and permit more complex modulation.Implementation of an OFDM systemThe idea behind the analog imple-mentation of OFDM can be extendedtoFigure 3.A Simple OFDM generator.N subcarri-ers transmitting 1bit of information each,by turning on and off at time intervalsT.Figure 4.Overall spectrum of the simple OFDM signal shown with four subcarriers within.Note that the zero crossings all correspond to peaks of adjacentsubcarriers.Figure 2.FDM signal spectrum example.the digital domain by using the discrete Fourier Transform(DFT)and its coun-terpart,the inverse discrete Fourier Transform(IDFT).These mathematical operations are widely used for trans-forming data between the time-domain and frequency-domain.These trans-forms are interesting from the OFDM perspective because they can be viewed as mapping data onto orthogonal sub-carriers.For example,the IDFT is used to take in frequency-domain data and convert it to time-domain data.In order to perform that operation,the IDFT cor-relates the frequency-domain input data with its orthogonal basis functions, which are sinusoids at certain frequen-cies.This correlation is equivalent to mapping the input data onto the sinu-soidal basis functions.In practice,OFDM systems are implemented using a combination of fast Fourier Transform(FFT)and inverse fast Fourier Transform(IFFT) blocks that are mathematically equiv-alent versions of the DFT and IDFT, respectively,but more efficient to implement.An OFDM system treats the source symbols(e.g.,the QPSK or QAM symbols that would be present in a single carrier system)at the transmitter as though they are in the frequency-domain.These symbols are used as the inputs to an IFFT block that brings the signal into the time-domain.The IFFT takes in N symbols at a time where N is the number of subcarriers in the system.Each of these N input symbols has a symbol period of T seconds.Recall that the basis functions for an IFFT are N orthogonal sinusoids.These sinusoids each have a different frequency and the lowest frequency is DC.Each input symbol acts like a complex weight for the corresponding sinu-soidal basis function.Since the input symbols are complex,the value of the symbol determines both the ampli-tude and phase of the sinusoid forthat subcarrier.The IFFT output isthe summation of all N sinusoids.Thus,the IFFT block provides a sim-ple way to modulate data onto Northogonal subcarriers.The block of Noutput samples from the IFFT makeup a single OFDM symbol.The lengthof the OFDM symbol is NT where T isthe IFFT input symbol period men-tioned above.After some additional processing,the time-domain signal that resultsfrom the IFFT is transmitted acrossthe channel.At the receiver,an FFTblock is used to process the receivedsignal and bring it into the frequency-domain.Ideally,the FFT output willbe the original symbols that weresent to the IFFT at the transmitter.When plotted in the complex plane,the FFT output samples will form aconstellation,such as16-QAM.However,there is no notion of a con-stellation for the time-domain signal.When plotted on the complex plane,the time-domain signal forms a scat-ter plot with no regular shape.Thus,any receiver processing that uses theconcept of a constellation(such assymbol slicing)must occur in the fre-quency-domain.The block diagramin Figure5illustrates the switchbetween frequency-domain and time-domain in an OFDM system.Multipath channels andthe use of cyclic prefixA major problem in most wirelesssystems is the presence of a multipathchannel.In a multipath environment,the transmitted signal reflects off of sev-eral objects.As a result,multipledelayed versions of the transmitted sig-nal arrive at the receiver.The multipleversions of the signal cause the receivedsignal to be distorted.Many wired sys-tems also have a similar problem wherereflections occur due to impedance mis-matches in the transmission line.A multipath channel will cause twoproblems for an OFDM system.Thefirst problem is intersymbol interfer-ence.This problem occurs when thereceived OFDM symbol is distorted bythe previously transmitted OFDM sym-bol.The effect is similar to the inter-symbol interference that occurs in a sin-gle-carrier system.However,in suchsystems,the interference is typicallydue to several other symbols instead ofjust the previous symbol;the symbolperiod in single carrier systems is typi-cally much shorter than the time spanof the channel,whereas the typicalOFDM symbol period is much longerthan the time span of the channel.Thesecond problem is unique to multicarri-er systems and is called IntrasymbolInterference.It is the result of interfer-ence amongst a given OFDM symbol’sown subcarriers.The next sections illus-trate how OFDM deals with these twotypes of interference.Intersymbol interferenceAssume that the time span of thechannel is LCsamples long.Instead of asingle carrier with a data rate of R sym-bols/second,an OFDM system has Nsubcarriers,each with a data rate ofR/N symbols/second.Because the datarate is reduced by a factor of N,theOFDM symbol period is increased by afactor of N.By choosing anappropriateFigure6.Example of intersymbol interference.The green symbol was transmitted first,followed by the blue symbol.value for N,the length of the OFDM symbol becomes longer than the time span of the channel.Because of this configuration,the effect of intersymbol interference is the distortion of the first L C samples of the received OFDM sym-bol.An example of this effect is shown in Figure 6.By noting that only the first few samples of the symbol are distorted,one can consider the use of a guard interval to remove the effect of inter-symbol interference.The guard interval could be a section of all zero samples transmitted in front of each OFDM symbol.Since it does not contain any useful information,the guard interval would be discarded at the receiver.If the length of the guard interval is prop-erly chosen such that it is longer than the time span of the channel,the OFDM symbol itself will not be distort-ed.Thus,by discarding the guard inter-val,the effects of intersymbol interfer-ence are thrown away as well.Intrasymbol interferenceThe guard interval is not used in practical systems because it does not prevent an OFDM symbol from inter-fering with itself.This type of interfer-ence is called intrasymbol interfer-ence.The solution to the problem of intrasymbol interference involves a discrete-time property.Recall that in continuous-time,a convolution in time is equivalent to a multiplication in the frequency-domain.This property is true in discrete-time only if the signalsare of infinite length or if at least one of the signals is periodic over the range of the convolution.It is not prac-tical to have an infinite-length OFDM symbol,however,it is possible to make the OFDM symbol appear periodic.This periodic form is achieved by replacing the guard interval with something known as a cyclic prefix of length L P samples.The cyclic prefix is a replica of the last L P samples of the OFDM symbol where L P >L C .Since it contains redundant information,the cyclic prefix is discarded at the receiv-er.Like the case of the guard interval,this step removes the effects of inter-symbol interference.Because of the way in which the cyclic prefix was formed,the cyclically-extended OFDM symbol now appears periodic when convolved with the channel.An impor-tant result is that the effect of the channel becomes multiplicative.In a digital communications system,the symbols that arrive at the receiver have been convolved with the time-domain channel impulse response of length L C samples.Thus,the effect of the channel is convolutional.In order to undo the effects of the channel,another convolution must be per-formed at the receiver using a time-domain filter known as an equalizer.The length of the equalizer needs to be on the order of the time span of the channel.The equalizer processes sym-bols in order to adapt its response in an attempt to remove the effects of the channel.Such an equalizer can be expensive to implement in hardware and often requires a large number of symbols in order to adapt its response to a good setting.In OFDM,the time-domain signal is still convolved with the channel response.However,the data will ulti-mately be transformed back into the frequency-domain by the FFT in the receiver.Because of the periodic nature of the cyclically-extended OFDM sym-bol,this time-domain convolution will result in the multiplication of the spec-trum of the OFDM signal (i.e.,the fre-quency-domain constellation points)with the frequency response of the channel.The result is that each subcar-rier’s symbol will be multiplied by a complex number equal to the channel’s frequency response at that subcarrier’s frequency.Each received subcarrier experiences a complex gain (amplitude and phase distortion)due to the chan-nel.In order to undo these effects,a fre-quency-domain equalizer is employed.Such an equalizer is much simpler than a time-domain equalizer.The frequen-cy-domain equalizer consists of a sin-gle complex multiplication for each subcarrier.For the simple case of no noise,the ideal value of the equalizer’s response is the inverse of the chan-nel’s frequency response.An example is shown in Figure 7.With such a set-ting,the frequency-domain equalizer would cancel out the multiplicative effect 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 that the equalizer response is large when the channel response is small in order to counteract the effect of a channelnull.COFDM:Coded OFDMCoded OFDM,or COFDM,is a term used for a system in which the error control coding and OFDM modulation processes work closely together.An important step in a COFDM system is to interleave and code the bits prior to the IFFT.This step serves the purpose of taking adjacent bits in the source data and spreading them out across multiple subcarriers.One or more sub-carriers may be lost or impaired due to a frequency null,and this loss would cause a contiguous stream of bit errors. Such a burst of errors would typically be hard to correct.The interleaving at the transmitter spreads out the contigu-ous bits such that the bit errors become spaced far apart in time.This spacing makes it easier for the decoder to cor-rect the errors.Another important step in a COFDM system is to use channel information from the equalizer to deter-mine the reliability of the received bits. The values of the equalizer response are used to infer the strength of the received subcarriers.For example,if the equalizer response had a large value at a certain frequency,it would correspond to a frequency null at that point in the channel.The equalizer response would have a large value at that point because it is trying to compensate for the weak received signal.This reliability informa-tion is passed on to the decoding blocks so that they can properly weight the bits when making decoding decisions. In the case of a frequency null,the bits would be marked as“low confidence”and those bits would not be weighted asheavily as bits from a strong subcarrier.COFDM systems are able to achieveexcellent performance on frequency-selective channels because of the com-bined benefits of multicarrier modula-tion and coding.Non-ideal effects in anOFDM systemThis section will examine the effectsof non-idealities in an OFDM system.These effects will include impairmentsand receiver offsets.Because the fouriertransform is a fundamental operation inOFDM,the effects of several offsets canbe intuitively understood by applyingfourier transform theory.Local oscillatorfrequency offsetAt start-up,the local oscillator(LO)frequency at the receiver is typically dif-ferent from the LO frequency at thetransmitter.A carrier tracking loop isused to adjust the receiver’s LO fre-quency in order to match the transmit-ter’s LO frequency as closely as possi-ble.The effect of having an LO frequen-cy offset can be explained by FourierTransform theory.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.Recallfrom Fourier Transform theory thatmultiplication by a complex exponentialin time is equivalent to a shift in fre-quency.The LO offset results in a fre-quency shift of the received signal spec-trum.This shift causes a conditioncalled“loss of orthogonality”to occur.The frequency shift causes the OFDMsubcarriers to no longer be orthogonal.The orthogonality of the subcarriers islost because the bins of the FFT will nolonger line up with the peaks of thereceived signal’s since pulses.Theresult is a distortion called inter-bininterference or IBI.IBI occurs whenenergy from one bin spills over intoadjacent bins and this energy distortsthe affected subcarriers.In FourierTransform theory this effect is calledDFT leakage.The left plot of Figure8shows thespectrum of a received OFDM signalwith no LO offset.For the purpose ofclarity,only one non-zero subcarrierwas transmitted.Note that this subcar-rier is not interfering with its adjacentsubcarriers.The spectrum of the non-zero subcarrier actually extends overthe entire range of the FFT,however,due to the orthogonal nature of the sig-nal,the zero-crossings of the spectrumexactly line up with the other FFT bins.The right plot of Figure8shows thereceived spectrum of the same signalwith one non-zero subcarrier,however,in this case there is an LO offset.Thisoffset has resulted in a loss of orthogo-nality,and the zero-crossings of thenon-zero subcarrier’s spectrum nolonger line up with the FFT bins.Theresult is that energy from the non-zerosubcarrier is spread out among all ofthe other subcarriers,with thosesub-Figure8.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.carriers closest to the non-zero subcarri-er receiving the most interference.This simple example was for the case of only one non-zero subcarrier.In a practical system,almost all of the subcarriers would be actively used for transmitting data.A given subcarrier would experi-ence IBI due to energy from all of the other active subcarriers in the system. The central limit theorem states that the sum of a large number of random processes will result in a signal that has a Gaussian distribution.Because of this property,the IBI will manifest itself as additive Gaussian noise,thus lowering the effective SNR of the system.The effect of an LO frequency offset can be corrected by multiplying the sig-nal by a correction factor.The correc-tion factor would be a sinusoid with a frequency that is ideally equal to the amount of the LO frequency offset. Various carrier tracking algorithms exist that can adaptively determine the frequency that will correct for the offset. LO phase offsetIt is also possible to have an LO phase offset,separate from an LO fre-quency offset.The two offsets can occur in conjunction or one or the other can be present by itself.As the name suggests, an LO phase offset occurs when there is a difference between the phase of the LO output and the phase of the received signal.This effect can be represented mathematically by multiplying the time-domain signal by a complex expo-nential with a constant phase.The result is a constant phase rotation for all of the subcarriers in the frequency-domain.The constellation points for each subcarrier experience the same degree of rotation.If the phase rotation is small,the frequency-domain equaliz-er can correct this effect.Each filtercoefficient in a frequency-domain equal-izer multiplies its corresponding subcar-rier by a complex gain(i.e.,amplitudescaling and phase rotation).The equal-izer’s coefficients can be used to correctfor a small phase rotation as long as therotation doesn’t cause the constellationpoints to rotate beyond the symbol deci-sion rger phase rotations arecorrected by a carrier tracking loop.FFT window location offsetAnother non-ideal effect that canoccur in a real-world OFDM system isan FFT window location offset.An N-point FFT at the receiver processes datain blocks of N samples at a time.Ideally,the N samples taken in by theFFT will correspond to the N samples ofa single transmitted OFDM symbol.Inpractice,a correlation is often used witha known preamble sequence located atthe beginning of the transmission.Thiscorrelation operation aids the receiverin synchronizing itself with the receivedsignal’s OFDM symbol boundaries.However,inaccuracies still remain,andthey manifest themselves as an offset inthe FFT window location.The result isthat the N samples sent to the FFT willnot line up exactly with the correspond-ing OFDM symbol.If the offset is verylarge,part of the N samples will be fromone OFDM symbol,and the rest of sam-ples will be from another OFDM sym-bol.Such a situation would result in asevere distortion of the received subcar-rier’s constellations.Fortunately,such alarge offset does not typically occur if arobust synchronization algorithm isused.More likely,an FFT window loca-tion offset of just a few samples willoccur.The presence of the cyclic prefixgives enough headroom to enable asmall offset to be present without tak-ing samples from more than one OFDMsymbol.However,even an offset of justone sample will cause some degree ofdistortion.Again,the effect can beunderstood from Fourier Transformtheory.The offset can be viewed as ashift in time.As long as the FFT win-dow location offset does not gobeyond an OFDM symbol boundary,this shift in time is equivalent to alinearly-increasing phase rotation inthe frequency-domain constellations.Constellations on subcarriers corre-sponding to low frequencies will berotated slightly,whereas constellationson higher-frequency subcarriers willexperience a larger rotation.Theamount of rotation increases linearly asthe subcarrier’s FFT bin locationincreases.Examples of the effects of dif-ferent degrees of FFT window locationoffsets are shown in Figure9.FFT window location offsets are oftencorrected by performing a time-domaincorrelation with a known trainingsequence embedded in the transmittedsignal.The location of the peak of thecorrelation allows the receiver to syn-chronize itself with the incoming signal.Sampling frequency offsetAnother potentially harmful situa-tion is the presence of a sampling fre-quency offset.This condition occurswhen the A/D converter output is sam-pled either too fast or too slow.Recallthat FS/2is the highest available fre-quency in discrete-time where FSis thesampling frequency.Sampling too fastessentially increases the value of FS/2and the result is a contracted(i.e.,squashed)spectrum.Similarly,sam-pling too slow decreases the value of F S /2and results in an expanded spec-trum.If the spectrum expands too much,aliasing of the spectrum can occur.Either type of sampling frequen-cy offset results in IBI since the expan-sion or contraction of the spectrum pre-vents the received subcarriers from lin-ing up with the FFT bin locations.The effect of sampling too fast is illustrated in Figure 10and simulation results to demonstrate this effect are shown in Figure 11.A sampling frequency offset can be corrected by generating an error term that is used to drive a sampling rate converter.Uniform noiseAdditive white Gaussian noise (AWGN)is the most common impair-ment encountered in a communicationssystem.In a wireless medium,the noise source is typically considered to be ther-mal noise that is Gaussian and uniform across the frequency range.Additional noise sources include atmospheric sources and solar radiation.In a con-tained media,such as a coaxial cable system,thermal noise will be present,but the system may also have other sources that can increase the noise in the system.The effect of AWGN on anOFDM system is similar to its effect on a single carrier system.The signal-to-noise ratio (SNR)is a function of the total signal power over the total noise power across the received channel.The uniform noise contributes to the SNR of each subcarrier in the OFDM system and the net result is equivalent to the effect on single channel systems.Non-uniform noiseNoise in a communications channel can often be shaped,or “colored”,by various effects.These effects can include transmit signal imperfections,transmission channel characteristics,or receiver frequency shaping.The impli-cations of these effects for an OFDM system can be different compared to itssingle-carrier counterpart.The modula-tion of the OFDM system can be tai-lored for the noise characteristics.One method previously mentioned involves lowering the modulation (number of bits/symbol)on subcarriers in a low SNR environment as illustrated in Figure 12.Another method involvessending the same data on several sub-carriers,or sending data that can be considered lower priority.In extreme cases,the subcarriers can transmit no data,essentially turning them off.Impulse noiseImpulse noise is a common impair-ment in a communications system aris-ing from motors or lightning.Impulse noise is typically characterized as a short time-domain burst of energy.The burst may be repetitive or may be a sin-gle event.In either case,the frequency spectrum from this energy burst is wideband,typically much wider than the channel,but is present for only a short time period.One of the most important concepts to understand about OFDM and its properties related to the FFT algorithm is how the algorithm changes the nature of the signal.In a single-carrier system,the symbol can be viewed as occupying all of the available frequency spectrum for the time duration of the symbol.A group of symbols then occu-pies all of the spectrum for the duration of the whole group,but in a time divi-sion arrangement.OFDM,using the FFT,takes symbols and creates these groups directly and then transforms them.They are no longer time-domain multiplexed,they are now frequency-domain multiplexed.The OFDM symbol is now a collection of these source symbols,and this OFDM symbol now has a much longer dura-tion.Each original symbol occupies only a small frequency region,but now occu-pies that region for the entire OFDM symbol duration.Figure 13illustrates this concept.For impulses that are short in duration,the impulse energy masks a smaller percentage of time of each OFDM symbol compared to the single carrier case.Impulse noise can therefore have less of an effect on short duration noise.Carrier interferenceSingle-carrier interference arises from other sources that may co-exist in the frequency range of interest.These can be generated by nearby circuits or other transmission sources.The single carrier system must handle this inter-ference as a noise source for all informa-tion sent.The OFDM system can avoid the frequency region of interference by disabling or turning off the affected sub-carriers.Narrowband modulated sources of interference can beconsid-Figure 10.Illustration of the effect of a sampling frequencyoffset.Figure 11.Simulation results showing the effect of a sampling frequency that is too high.Note that the sample that was originally at bin 15is now at bin 8.。