signal processing
`
`The principles
`of OFDM
`
`Multicarrier modulation techniques
`are rapidly moving from the textbook
`to the real world of modern
`communication systems
`
`By Louis Litwin and
`Michael Pugel
`
`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-
`
`digital 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 system
`Frequency 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 OFDM
`If the FDM system above had been able to use a
`set of subcarriers that were orthogonal to each other,
`niques, and also discuss common impairments and
`a higher level of spectral efficiency could have been
`how, in some cases, OFDM mitigates their effect.
`achieved. The guardbands that were necessary to
`Where applicable, the impairment effects and tech-
`allow individual demodulation of subcarriers in an
`niques will be compared to those in a single carrier
`FDM system would no longer be necessary. The use
`system. A brief overview of some modern applica-
`of orthogonal subcarriers would allow the subcarri-
`tions will conclude the article.
`ers’ spectra to overlap, thus increasing the spectral
`The single-carrier
`efficiency. As long as orthogonality is maintained, it
`modulation system
`is still possible to recover the individual subcarriers’
`A typical single-carrier modulation spectrum is
`signals despite their overlapping spectrums.
`shown in Figure 1. A single carrier system modu-
`If the dot product of two deterministic signals is
`lates information onto one carrier using frequency,
`equal to zero, these signals are said to be orthogo-
`phase, or amplitude adjustment of the carrier. For
`nal to each other. Orthogonality can also be
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`The principles of orthogonal frequency division
`
`Figure 1. Single carrier spectrum example.
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`son that the OFDM subcarriers’ spec-
`trums can overlap without causing
`interference.
`A simple OFDM example
`Figure 3 shows 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 4 shows
`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 system
`The idea behind the analog imple-
`mentation of OFDM can be extended to
`
`Figure 2. FDM signal spectrum example.
`
`viewed 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 OFDM. Later 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-
`
`Figure 3. A Simple OFDM generator. N subcarri-
`ers transmitting 1 bit of information each, by
`turning on and off at time intervals T.
`
`Figure 4. Overall spectrum of the simple OFDM signal shown with four subcarriers within. Note
`that the zero crossings all correspond to peaks of adjacent subcarriers.
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`Multipath channels and
`the use of cyclic prefix
`A major problem in most wireless
`systems is the presence of a multipath
`channel. In a multipath environment,
`the transmitted signal reflects off of sev-
`eral objects. As a result, multiple
`delayed versions of the transmitted sig-
`nal arrive at the receiver. The multiple
`versions of the signal cause the received
`signal to be distorted. Many wired sys-
`tems also have a similar problem where
`reflections occur due to impedance mis-
`matches in the transmission line.
`A multipath channel will cause two
`problems for an OFDM system. The
`first problem is intersymbol interfer-
`ence. This problem occurs when the
`received OFDM symbol is distorted by
`the previously transmitted OFDM sym-
`bol. The effect is similar to the inter-
`symbol interference that occurs in a sin-
`gle-carrier system. However, in such
`systems, the interference is typically
`due to several other symbols instead of
`just the previous symbol; the symbol
`period in single carrier systems is typi-
`cally much shorter than the time span
`of the channel, whereas the typical
`OFDM symbol period is much longer
`than the time span of the channel. The
`second problem is unique to multicarri-
`er systems and is called Intrasymbol
`Interference. It is the result of interfer-
`ence amongst a given OFDM symbol’s
`own subcarriers. The next sections illus-
`trate how OFDM deals with these two
`types of interference.
`Intersymbol interference
`Assume that the time span of the
`channel is LC samples long. Instead of a
`single carrier with a data rate of R sym-
`bols/second, an OFDM system has N
`subcarriers, each with a data rate of
`R/N symbols/second. Because the data
`rate is reduced by a factor of N, the
`OFDM symbol period is increased by a
`factor of N. By choosing an appropriate
`
`tude and phase of the sinusoid for
`that subcarrier. The IFFT output is
`the summation of all N sinusoids.
`Thus, the IFFT block provides a sim-
`ple way to modulate data onto N
`orthogonal subcarriers. The block of N
`output samples from the IFFT make
`up a single OFDM symbol. The length
`of the OFDM symbol is NT where T is
`the IFFT input symbol period men-
`tioned above.
`After some additional processing,
`the time-domain signal that results
`from the IFFT is transmitted across
`the channel. At the receiver, an FFT
`block is used to process the received
`signal and bring it into the frequency-
`domain. Ideally, the FFT output will
`be the original symbols that were
`sent to the IFFT at the transmitter.
`When plotted in the complex plane,
`the FFT output samples will form a
`constellation, such as 16-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 the
`concept of a constellation (such as
`symbol slicing) must occur in the fre-
`quency-domain. The block diagram
`in Figure 5 illustrates the switch
`between frequency-domain and time-
`domain in an OFDM system.
`
`Figure 6. Example of intersymbol interference. The green symbol was transmitted first, followed by the
`blue symbol.
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`Figure 5. Block diagram of a simple OFDM system.
`
`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-
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`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 channel null.
`
`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
`LC 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 interference
`The 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 signals
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`are 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 LP samples. The cyclic prefix is
`a replica of the last LP samples of the
`OFDM symbol where LP > LC. 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 LC 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.
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`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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`COFDM: Coded OFDM
`Coded 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”
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`and those bits would not be weighted as
`heavily as bits from a strong subcarrier.
`COFDM systems are able to achieve
`excellent performance on frequency-
`selective channels because of the com-
`bined benefits of multicarrier modula-
`tion and coding.
`Non-ideal effects in an
`OFDM system
`This section will examine the effects
`of non-idealities in an OFDM system.
`These effects will include impairments
`and receiver offsets. Because the fourier
`transform is a fundamental operation in
`OFDM, the effects of several offsets can
`be intuitively understood by applying
`fourier transform theory.
`Local oscillator
`frequency offset
`At start-up, the local oscillator (LO)
`frequency at the receiver is typically dif-
`ferent from the LO frequency at the
`transmitter. A carrier tracking loop is
`used 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 Fourier
`Transform theory. The LO offset can be
`expressed mathematically by multiply-
`ing the received time-domain signal by
`a complex exponential whose frequency
`is equal to the LO offset amount. Recall
`from Fourier Transform theory that
`multiplication by a complex exponential
`in 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 condition
`called “loss of orthogonality” to occur.
`The frequency shift causes the OFDM
`subcarriers to no longer be orthogonal.
`The orthogonality of the subcarriers is
`lost because the bins of the FFT will no
`longer line up with the peaks of the
`received signal’s since pulses. The
`result is a distortion called inter-bin
`interference or IBI. IBI occurs when
`energy from one bin spills over into
`adjacent bins and this energy distorts
`the affected subcarriers. In Fourier
`Transform theory this effect is called
`DFT leakage.
`The left plot of Figure 8 shows the
`spectrum of a received OFDM signal
`with no LO offset. For the purpose of
`clarity, only one non-zero subcarrier
`was transmitted. Note that this subcar-
`rier is not interfering with its adjacent
`subcarriers. The spectrum of the non-
`zero subcarrier actually extends over
`the entire range of the FFT, however,
`due to the orthogonal nature of the sig-
`nal, the zero-crossings of the spectrum
`exactly line up with the other FFT bins.
`The right plot of Figure 8 shows the
`received spectrum of the same signal
`with one non-zero subcarrier, however,
`in this case there is an LO offset. This
`offset has resulted in a loss of orthogo-
`nality, and the zero-crossings of the
`non-zero subcarrier’s spectrum no
`longer line up with the FFT bins. The
`result is that energy from the non-zero
`subcarrier is spread out among all of
`the other subcarriers, with those sub-
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`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 offset
`It 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 filter
`coefficient in a frequency-domain equal-
`izer multiplies its corresponding subcar-
`rier by a complex gain (i.e., amplitude
`scaling and phase rotation). The equal-
`izer’s coefficients can be used to correct
`for a small phase rotation as long as the
`rotation doesn’t cause the constellation
`points to rotate beyond the symbol deci-
`sion regions. Larger phase rotations are
`corrected by a carrier tracking loop.
`FFT window location offset
`Another non-ideal effect that can
`occur in a real-world OFDM system is
`an FFT window location offset. An N-
`point FFT at the receiver processes data
`in blocks of N samples at a time.
`Ideally, the N samples taken in by the
`FFT will correspond to the N samples of
`a single transmitted OFDM symbol. In
`practice, a correlation is often used with
`a known preamble sequence located at
`the beginning of the transmission. This
`correlation operation aids the receiver
`in synchronizing itself with the received
`signal’s OFDM symbol boundaries.
`However, inaccuracies still remain, and
`they manifest themselves as an offset in
`the FFT window location. The result is
`that the N samples sent to the FFT will
`not line up exactly with the correspond-
`ing OFDM symbol. If the offset is very
`large, part of the N samples will be from
`one OFDM symbol, and the rest of sam-
`ples will be from another OFDM sym-
`bol. Such a situation would result in a
`severe distortion of the received subcar-
`rier’s constellations. Fortunately, such a
`large offset does not typically occur if a
`robust synchronization algorithm is
`used. More likely, an FFT window loca-
`tion offset of just a few samples will
`
`occur. The presence of the cyclic prefix
`gives enough headroom to enable a
`small offset to be present without tak-
`ing samples from more than one OFDM
`symbol. However, even an offset of just
`one sample will cause some degree of
`distortion. Again, the effect can be
`understood from Fourier Transform
`theory. The offset can be viewed as a
`shift in time. As long as the FFT win-
`dow location offset does not go
`beyond an OFDM symbol boundary,
`this shift in time is equivalent to a
`linearly-increasing phase rotation in
`the frequency-domain constellations.
`Constellations on subcarriers corre-
`sponding to low frequencies will be
`rotated slightly, whereas constellations
`on higher-frequency subcarriers will
`experience a larger rotation. The
`amount of rotation increases linearly as
`the subcarrier’s FFT bin location
`increases. Examples of the effects of dif-
`ferent degrees of FFT window location
`offsets are shown in Figure 9.
`FFT window location offsets are often
`corrected by performing a time-domain
`correlation with a known training
`sequence embedded in the transmitted
`signal. The location of the peak of the
`correlation allows the receiver to syn-
`chronize itself with the incoming signal.
`Sampling frequency offset
`Another potentially harmful situa-
`tion is the presence of a sampling fre-
`quency offset. This condition occurs
`when the A/D converter output is sam-
`pled either too fast or too slow. Recall
`that FS/2 is the highest available fre-
`quency in discrete-time where FS is the
`sampling frequency. Sampling too fast
`essentially increases the value of FS/2
`and the result is a contracted (i.e.,
`
`Figure 9. Effect of different FFT window offsets.
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`squashed) spectrum. Similarly, sam-
`pling too slow decreases the value of
`FS/2 and results in an expanded spec-
`
`but the system may also have other
`sources that can increase the noise in
`the system. The effect of AWGN on an
`
`Figure 10. Illustration of the effect of a sampling frequency offset.
`
`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 10 and 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 noise
`Additive white Gaussian noise
`(AWGN) is the most common impair-
`ment encountered in a communications
`
`OFDM 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 noise
`Noise 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 its
`
`Figure 11. Simulation results showing the effect of a sampling frequency that is too high. Note that the
`sample that was originally at bin 15 is now at bin 8.
`
`system. 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,
`
`single-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 involves
`
`sending 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 noise
`Impulse 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 13 illustrates
`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 interference
`Single-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 be consid-
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`are generated by the IFFT and can be
`used to provide a stable phase refer-
`ence for the receiver circuitry. Adding
`these pilots lowers the available data
`rate of the system because these sub-
`carriers are no longer available to
`transmit data.
`Non-linear circuits in the
`transmitter and receiver
`All transmitters and receivers in
`communications systems contain
`devices such as amplifiers and mixers
`that have non-linear transfer functions.
`These non-linearities create an addi-
`tional performance limitation. The
`receiver performance is typically limited
`by distortion generated in the input
`amplifier or mixer in the presence of
`strong undesired signals. The transmit-
`ter performance is limited primarily by
`power amplifier linearity. An OFDM
`signal is made up of multiple simulta-
`neous signals that, for a given average
`power, hav