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  • Jean-Paul Linnartz is Senior Scientist with the Natuurkundig Laboratorium of Philips Research in Eindhoven, The Netherlands. He leads of team of researchers working on conditional access, copy control, security and electronic watermarking. He joined Philips Research in 1995. From 1994-1998, he also was Assistant Adjunct Professor at the University of California at Berkeley. 1988-1991 and in 1994, he worked at T.U. Delft as Assistant and Associate Professor, respectively. In 1992-1994, he was Assistant Professor with the University of California at Berkeley. In 1993, he was the first to use the term Multi-Carrier CDMA (MC-CDMA) for a spread spectrum transmission method that combines Direct-Sequence CDMA with Orthogonal Frequency Division Multiplexing (OFDM). During his M.Sc. and Ph.D project he worked on mathematical methods to evaluate the performance of wireless data and cellular phone networks. At Philips Research, his research mainly is on Electronic Watermarks, Conditional Access and Information Security.
  • Code Division Multiple Access (CDMA) is the simultaneous use of the radio spectrum by multiple users, whereby different users make temporally and spectrally overlapping transmissions. Nonetheless their signals can be separated by code signatures. That is, the transmit spectrum has intentionally been widened (‘spread spectrum’) by using some well defined signal processing operation. In the case of transmission of multiple signals from a common base base station, signals are Code Division Multiplexed, so one should preferably speak about Code Division Multiplexing. Nonetheless, many authors use the term CDMA (Code Division Multiple Access ) in stead. The term multiple access is more appropriate for the uplink (many mobiles to a commo base station). This presentation addresses various aspects of MC-CDM(A). After a general introduction, it covers transmitter and receiver design, performance and capacity. Comparisons are made to DS-CDMA and OFDM.
  • Orthogonal Frequency Division Multiplexing (OFDM) is special form of multi-carrier modulation, patented in 1970. It is particularly suited for transmission over a dispersive channel. In a multipath channel, most conventional modulation techniques are sensitive to intersymbol interference unless the channel symbol rate is small compared to the delay spread of the channel. OFDM is significantly less sensitive to intersymbol interference than conventional modulation such as BPSK or QAM, because a special set of signals is used to build the composite transmitted signal. The basic idea is that each bit occupies a frequency-time window which ensures little or no distortion of the waveform. In practice, it means that bits are transmitted in parallel over a number of frequency-nonselective channels.
  • Multipath reception and user mobility lead to channel behavior that is time and frequency dependent. In the following frequency-time diagrams, we depict where in frequency and when in time most of the bit energy is located. It is important to select a modulation scheme that is appropriate for the Doppler Spread and the Delay Spread of the channel. In narrowband communication (NB), the occupied bandwidth is small and the bit duration is long. The bit energy footprint is stretched in vertical direction. Intersymbol interference is neglectable is the bit duration is, say, ten times or more longer than the delay spread. However, due to multipath the entire signal (bandwidth) may vanish in a deep fade. Is fading is very fast, the channel may change during a bit transmission. This can occur if the Doppler spread is large compared to the transmit bandwidth. In wideband transmission (WB), i.e., transmission at high bit rate, the signal is unlikely to vanish completely in a deep fade, because its bandwidth is so wide that some components will be present at frequencies where the channel is good. Such frequency selective fading, however, leads to intersymbol interference, which must be handled, for instance by an adaptive equalizer. In MultiCarrier or OFDM systems, multiple bits are transmitted in parallel over multiple subcarriers. If one subcarrier is in a fade, the other may not. Error correction coding can be used to correct bit errors on faded subcarriers. Rapid fading (Doppler) may erode the orthogonality of closely spaced subcarriers.
  • Direct Sequence intentionally broadens the transmit spectrum by multiplying user bits with a fast random sequence. The wideband signal is unlikely to fade completely. If frequency selective fading occurs, the receiver sees a series of time-shifted versions of the chipped bit. A ‘rake’ receiver can separate the individual resolvable paths. Frequency Hopping : The carrier frequency is shifted frequently, to avoid fades or narrowband interference. While the signal may vanish during one hop (at a particular frequency), most likely other hops jump to frequencies where the channel is good. Error correction is applied to correct bits lost at faded frequencies. Slow and fast hopping methods differ in the relative duration spent at one frequency, compared to the user bit duration. Orthogonal MC-CDMA is a spectrum-spreading method which exploits advantages of OFDM and DS-CDMA. Each user bit is transmitted simultaneously (in parallel) over multiple subcarriers. This corresponds to a 90 degree rotation of a bit energy map in the frequency-time plane, when compared to DS-CDMA. While the signal may be lost some subcarriers, reliable communication is ensured through appropriate combining of energy from various subcarriers, taking into account the signal-to-interference ratios. In an interference-free environment, Maximum Ratio Diversity combining is the preferred receive strategy. In a noise free environment, equalization (i.e. making all subcarrier amplitudes equal) is preferred.
  • Multi Carrier Code Division Multiple Access (MC-CDMA) is a relatively new concept. Its development aimed at improved performance over multipath links. MC-CDMA is a modulation method that uses multi-carrier transmission (more precisely OFDM) of DS-CDMA-type signals. This scheme was first proposed at PIMRC '93 in Yokohama by Linnartz, Yee (U. of California at Berkeley) and Fettweis (Teknekron, Berkeley, currently at U. of Dresden, Germany). Independently, Fazal and Papke proposed a similar system. Chouly presented a similar idea at ICC ‘93. Since 1993, MC-CDMA rapidly has become a hot topic of research.
  • What is orthogonal MC-CDMA? There are many equivalent ways to describe MC-CDMA: 1.MC-CDMA is a form of CDMA or spread spectrum, but we apply the spreading in the frequency domain (rather than in the time domain as in Direct Sequence CDMA). 2.MC-CDMA is a form of Direct Sequence CDMA, but after spreading, a Fourier Transform (FFT) is performed. 3.MC-CDMA is a form of Orthogonal Frequency Division Multiplexing (OFDM), but we first apply an orthogonal matrix operation to the user bits. Therefor, MC-CDMA is sometimes also called "CDMA-OFDM". 4.MC-CDMA is a form of Direct Sequence CDMA, but our code sequence is the Fourier Transform of a Walsh Hadamard sequence. 5.MC-CDMA is a form of frequency diversity. Each bit is transmitted simultaneously (in parallel) on many different subcarriers. Each subcarrier has a (constant) phase offset. The set of frequency offsets form a code to distinguish different users. The MC-CDMA method described here is NOT the same as DS-CDMA using multiple carriers. In the latter system the spread factor per subcarrier can be smaller than with conventional DS-CDMA. Such a scheme is sometimes called MC-DS-CDMA. This does not use the special OFDM-like waveforms to ensure dense spacing of overlapping, yet orthogonal subcarriers. MC-DS-CDMA has advantages over DS-CDMA as it is easier to synchronize to this type of signals.
  • The downlink is the link from base to mobile It is also called the forward or outbound link. Here all signal originate at the same transmitter. Thus it is fairly simple to reduce mutual interference from users within the same cell, by assigning orthogonal (e.g. Walsh-Hadamard) codes. The downlink can use synchronous transmission. This model applies to the case of a single broadcaster simultaneously sending data symbols over one MC-CDMA link, as well as to the case of symbols from multiple users which are multiplexed onto a common multi-carrier signal. We will use the term multi-user interference (MUI) for any mutual interference between different symbols due to frequency dispersion of the channel, eventhough in the broadcast scenario all signals belong to the same user.
  • In MC-CDMA, after recovery of the subcarriers, the signals at the output of the FFT have to be ‘unspread’, by applying the inverse code matrix. However some weighing is needed to optimize performance and to mitigate the effects of the channel. At this point we restrict ourselves to the class of (linear) receivers which make decisions based on linear combinations of all subcarrier signals. Mathematically the receiver is equivalent to a single matrix operation. Nonetheless, w e explicitly introduce the FFT, the inverse code matrix C -1 and a generic weigh matrix W . This allows us to address a simple implementation for the receiver, where the weighing reduces to a simplified adaptive diagonal matrix, while the FFT and C -1 are nonadaptive, and can be implemented efficiently using standard butterfly topologies.
  • Because of delay spread and frequency dispersion due to multipath fading, subcarriers are received with different amplitudes. An importance aspect of the receiver design is how to treat the individual subcarriers, depending on their amplitude  n,n . Options are Linear combining, by weighting the i th subcarrier by a factor w i : Maximum Ratio Combining: w n =  n,n * . This optimally combats noise, but does not exploit interference nulling. the system acts as a maximum ratio combining diversity receiver. This receiver combines the subcarrier energy to minimize the signal-to-noise ratio before the slicer. However, the multi-user interference is filtered out sub-optimally. Only in a channel that is non-selective over the entire OFDM bandwidth, signals from other user bits remain orthogonal. If subcariers have different attenuation, the orthogonality of user signals is damaged, so multi-user interference (MUI) occurs. This setting further worsens this effect. Equal Gain: | w n | = 1, arg( w n ) = arg(  n,n ). The simplest solution. (See also EGC diversity) Equalization: w n = 1/  n,n . This perfectly restores orthogonality and nulls interference, but excessively boosts noise. MMSE: w n =  n,n /(|  n,n | 2 + c). This gives the best post-combiner signal-to-noise-plus-interference ratio. Maximum likelihood detection (non-linear!) Linnartz and Yee showed that MC-CDMA signals can also be detected with fairly simple receiver structures, using an FFT and a variable gain diversity combiner, in which the gain of each branch is controlled only by the channel attenuation at that subcarrier. At PIMRC '94 in The Hague, optimum gain control functions were presented. Results showed that a fully loaded MC-CDMA system, i.e., one in which the number of users equals the spread factor, can operate in a highly time dispersive channel with satisfactory bit error rate. These results appeared in contrast to the behaviour of a fully loaded DS-CDMA link that typically does not work satisfactorily with large time dispersion.
  • AD MMSE: As a compromise between MUI rejection and noise suppression, a joint optimization can be derived from the following model. The Minimum Mean-Square Error Estimate of the user data is the conditional expectation E B | Y . We can rewrite this as E B | Y = E C -1 A | Y = C -1 E A | Y . This shows that one can, without loss of performance, estimate the modulation of each subcarrier, and then perform an inverse of the code matrix. Let the estimate A of the user data be a linear combination of Y , namely A = WY . The optimum choice of matrix W follows from the orthogonality principle that the estimation error is uncorrelated with the received data, viz., E( A - A ) Y H = 0 N with 0 N an all-zero matrix of size N by N. Thus we arrive at W = E[ AY H ] R YY -1 , for the optimum estimation matrix. Here Y = HA + N , where H (diagonal) channel matrix, describing the subcarrier amplitudes. We normalize the modulation as E AiAi * = 1. Then, E[ AY H ] is the diagonal matrix of the complex conjugates of the subcarrier amplitudes, and R YY is the covariance matrix of Y , with R YY = E YY H = HH H . In the more general case of time-varying channels with ICI implementation of this MMSE solution is quite involved because W does not reduce to a diagonal matrix. This implies that the optimum filter requires a (channel-adaptive) matrix inversion. Techniques have been studied to (blindly) estimate channel parameters in real-time. Mostly, such studies assume a limited number of (dominant) propagation paths, so in this respect these differ from our Rayleigh model. In practice, it may not always be feasible or economic to estimate all  m,n accurately, invert the covariance matrix in real-time, while adapting fast enough for the time-variations of the channel.
  • Various definitions of BER’s are relevant to a system designer: the instantaneous BER B 0 of an individual subcarrier with a given amplitude, and the local-mean BER B 1 , thus averaged over all channels. We compute B 0 as the BER for a given  n,n , but otherwise averaged over all channels, i.e., averaged over  m,n ( m  n ). That is, B 0 can be interpreted as the expected value of the BER if only the subcarrier amplitude is known (or estimated) from measurements, but without any knowledge about the instantaneous value of the ICI. A typical OFDM receiver would forward such side information to the error correction decoder. We consider a quasi-stationary radio link, in which channel variations cause ICI, but the power P 0 =  n,n  n,n */2 for each subcarier is reasonably constant during an OFDM frame. Formally these two assumptions conflict, but for small Doppler shifts they may be reasonably accurate.
  • In the receiver, each subcarrier (with index n ) is weighed by a factor w n which depends only on the signal strength  n,n in that subcarrier and the noise variance. The decision variable for user bit 0, after combining all subcarrier signals consists of x 0 + x MUI + x ICI where x 0 is the wanted signal, x MUI is the multi-user interference (due to imperfect restoration of the subcarrier amplitudes), x ICI is the intercarrier interference (due to crosstalk  m,n between a n and y m ), and x noise is the noise. We propose a receiver that estimates only  n,n , but no off-diagonal (ICI) terms  m,n with m  n. We take the weight settings of W to be as in (20), except that the noise N 0 / T s is replaced by the joint contribution from noise and ICI. In the next section we calculate the BER. In the appendix, we study the statistical behavior of  n,n and w n,n for Rayleigh channels with Doppler, in particular M ij defined as M ij = E ch |  n,n | i | w n,n | j . In contrast to most previous expressions in this section, the expectation is taken over all channels, and denoted as E ch . We exploit that |  n,n | is Rayleigh with mean square value P 0 .
  • The value of x MUI highly depends on the choice of C and on the channel. For orthogonal spreading codes (  n c 0 [ n ] c k [ n ] = 0) and a non-dispersive channel (  n,n is constant with subcarrier frequency n ), x MUI can be made zero by taking w n,m =  n,m . For a dispersive channel, the orthogonality of spreading codes is eroded but the MUI level can remain low if the weight factors w n,m are appropriately chosen, as shown in a previous section. In such case, the variance of the MUI can be evaluated by observing that for any two orthogonal codes c j [ n ] and c k [ n ] with j  k , one can partition the set of subcarrier indixes n with n = 0, 1, N -1 into two sets, both with exactly N /2 elements, such that A - = { n : c j [ n ] c k [ n ] = -1/ N } and A + = { n : c j [ n ] c k [ n ] = +1/ N } [24]. Here A +  A - = A ensures that  A +  A - c j [ n ] c k [ n ] = 0. The ICI contribution stems from crosstalk between subcarriers. Signal components which are present in a n =  k c k [ n ] b k are spilled into y m = y n +  , with strength  n+  ,n . In the receiver, these are weighted by w n +  ,n +  and unspread by c 0 [ n +  ]. In the analysis we assume statistical independence of the data and the channel. This simplifies the calculation of second moments (variances) We attribute no specific ICI-reducing properties to the spreading code, i.e., we take, E[ c 0 [ n ] c k [ n - m ]] 2 = N -2. Here the question arises whether a system designer can chose the spreading matrix C such that the ICI is mitigated. This poses requirements on the cross-correlation  n c 0 [ n ] c k [ n-m ]. It is the time-frequency dual of the well known problem of finding good codes for asynchronous DS-CDMA with good cross and autocorrelation properties to combat delay spread. Walsh-Hadamard codes have no particular properties to achieve good autocorrelation properties, and their autocorrelation behaviour can be approximated by the behavior of randomly chosen codes. Here the situation here more involved, because each term c 0 [ n ] c k [ n - m ] is multiplied by  m,n and w n,n , which are complex valued with random mutually independent arguments. Thus, even if the code had good autocorrelation properties, the channel delay spread erodes the attenuation of the ICI hoped for.
  • Because of the mathematical structure of result for the BER, we can introduce the figure of merit  , and rewrite the SNR after the MC-CDM subcarrier weighting and despreading as E N / N 0 =  P 0 T s / N 0 , thus a fixed non-fading value. Here  is a system parameter, which gives the improvement of MC-CDMA in a Rayleigh fading channel, over narrowband transmission in a non-fading channel. For typical SNR’s (say E n / N 0 of about 6 .. 20 dB),  is about 4.. 6 dB.. For very poor local-mean signal to noise ratios (large P 0 T s / N 0 ), the noise largely dominates over the MUI , and the MC-CDMA MMSE receiver acts mainly as a maximum ratio combiner and  tends to unity (0 dB). The curve plots the local-mean BER for BPSK versus the signal-to-noise ratio (SNR) in a very-slowly changing Rayleigh-fading channel, without Doppler spread and ICI ( v = 0). Curves (AWGN), (OFDM) and (3 - MC-CDMA) are theoretical results. Curve (AWGN) depicts the BER of BPSK in a channel without fading, using erfc  ( E N / N 0 ). Curve (OFDM) gives the BER for a narrowband Rayleigh fading channel ( N = 1) which is the same as the local-mean BER for OFDM, before any error correction. Moreover, it models MC-CDMA with N = 1. Curve (3) is the local-mean BER for MC-CDMA for N  . Curve (1) and (2) are Monte Carlo simulations for a system with N is 8 and 64 subcarriers, respectively. Curves (4) and (5) for correlated Rayleigh fading have been simulated in the frequency domain.
  • Formally, the Shannon capacity of OFDM and MC-CDMA are identical because the weighting operation W and the inverse code matrix C -1 are invertable operations. This is understood from the Data Processing Theorem. However, in order to achieve the full capacity of the link, the receiver must jointly detect MC-CDMA symbols and address the fact that the noise for the various user symbols become correlated if W uses different weights per subcarrier. However, a practical receiver would not take advantage of this. A loss of performance occurs (relative to ideally coded OFDM) in a system that extracts N MC-CDMA symbols and processes these as if they were transmitted over an AWGN, Linear Time Invariant, dispersion-free channel. It is reasonable to estimate the capacity per dimension of such MC-CDMA system as ½ log 2 (1 +  P 0 T s / N 0 ). It is easy to understand that OFDM can achieve the same capacity as this Rayleigh fading channel. For large SNR, C OFDM   /(2 ln2) + ½ log 2 (2 P 0 T s / N 0 ), thus asymptotically, OFDM on a Rayleigh fading channel with local-mean SNR P 0 T s / N 0 has approximately 0.4 bit less capacity per dimension than a non-fading channel with the SNR fixed to P 0 T s / N 0 .
  • In a practical multi-user system with intermittent transmissions, inbound messages are sent via a multiple-access channel, whereas in outbound channel, signals destined for different users can be multiplexed. In the latter case, the receiver in a mobile station can maintain carrier and bit synchronisation to the continuous incoming bit stream from the base station, whereas the receiver in the base station has to acquire synchronisation for each user slot. Moreover, in packet-switched data networks, the inbound channel has to accept randomly occurring transmissions by the terminals in the service area. Random-access protocols are required to organise the data traffic flow in the inbound channel, and access conflicts ('contention') may occur. In cellular networks with large traffic loads per base station, spread-spectrum modulation can be exploited in the downlink to combat multipath fading, whereas in the uplink, the signal powers from the various mobile subscribers may differ too much to effectively apply spread-spectrum multiple access unless sophisticated adaptive power control techniques are employed.
  • The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of I w reflected waves. Each wave has its particular Doppler frequency offset  i , path delay T i and amplitude D i , each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset  i = 2  f i lies within the Doppler spread -2  f    i  2  f  , with f  = vf c / c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2  f c =  c . The received signal r ( t ) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m -th subcarrier frequency, thus with exp{- j  c t-jm  s t +  m } during an appropriately chosen interval T s . We take phase compensation  m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [ y 0 , y 1 , .., y N -1 ] T . We assuming rectangular pulses of duration T s . We write y m =  n a n  m,n T s where  m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m . This result can be interpreted as sampling in frequency domain: the I w multipath channel contributions appear weighed according to their individual Doppler offset  i . It confirms that due to the Doppler shifts, the detected signal y m contains contributions from all n subcarrier signals, not only from m = n . All  m,n ‘s with m  n lead to ICI, with amplitudes weighed by sinc(  +  i /  s ).
  • Clarke and Aulin studied a uniform probability density of the angle  i at which multipath waves arrive at the mobile, thus f  (  ) = 1/(2  ). The Doppler shift per wave equals f i = ( v / c ) f c cos(  i ). This can be used to derive a U-shaped Doppler spectrum. for an omni-directional antenna. We denote the local mean received power, per subcarrier as p T . The variance of the ICI signal leaking from transmit subcarrier n into received subcarrier m = n +  equals P  . It can be expressed in terms of the ratio of the Doppler spread over the subcarrier spacing, defined as  = f  /f s ,  and p T. . The figure plots the received power P 0 , and the ICI powers P 1 , P 2 , and P 3 versus the normalized Doppler spread  for p T = 1.
  • The effect of Doppler spreading at 4 GHz is introduced here. We consider a system with (infinitely) many subcarriers. We inserted typical values for DTTB but consider MC-CDMA instead of the standardized OFDM. The frame duration is T s = 896 microseconds, with an FFT size of N = 8192. This corresponds to a subcarrier spacing of f s = 1.17 kHz and a data rate of 9.14 Msymbols/s. The local-mean BER is plotted versus antenna speeds v for E b / N 0 of 10, 20 and 30 dB. MC-CDMA appears to largely outperform uncoded OFDM.
  • The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of I w reflected waves. Each wave has its particular Doppler frequency offset  i , path delay T i and amplitude D i , each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset  i = 2  f i lies within the Doppler spread -2  f    i  2  f  , with f  = vf c / c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2  f c =  c . The received signal r ( t ) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m -th subcarrier frequency, thus with exp{- j  c t-jm  s t +  m } during an appropriately chosen interval T s . We take phase compensation  m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [ y 0 , y 1 , .., y N -1 ] T . We assuming rectangular pulses of duration T s . We write y m =  n a n  m,n T s where  m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m . This result can be interpreted as sampling in frequency domain: the I w multipath channel contributions appear weighed according to their individual Doppler offset  i . It confirms that due to the Doppler shifts, the detected signal y m contains contributions from all n subcarrier signals, not only from m = n . All  m,n ‘s with m  n lead to ICI, with amplitudes weighed by sinc(  +  i /  s ).
  • The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of I w reflected waves. Each wave has its particular Doppler frequency offset  i , path delay T i and amplitude D i , each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset  i = 2  f i lies within the Doppler spread -2  f    i  2  f  , with f  = vf c / c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2  f c =  c . The received signal r ( t ) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m -th subcarrier frequency, thus with exp{- j  c t-jm  s t +  m } during an appropriately chosen interval T s . We take phase compensation  m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [ y 0 , y 1 , .., y N -1 ] T . We assuming rectangular pulses of duration T s . We write y m =  n a n  m,n T s where  m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m . This result can be interpreted as sampling in frequency domain: the I w multipath channel contributions appear weighed according to their individual Doppler offset  i . It confirms that due to the Doppler shifts, the detected signal y m contains contributions from all n subcarrier signals, not only from m = n . All  m,n ‘s with m  n lead to ICI, with amplitudes weighed by sinc(  +  i /  s ).
  • The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of I w reflected waves. Each wave has its particular Doppler frequency offset  i , path delay T i and amplitude D i , each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset  i = 2  f i lies within the Doppler spread -2  f    i  2  f  , with f  = vf c / c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2  f c =  c . The received signal r ( t ) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m -th subcarrier frequency, thus with exp{- j  c t-jm  s t +  m } during an appropriately chosen interval T s . We take phase compensation  m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [ y 0 , y 1 , .., y N -1 ] T . We assuming rectangular pulses of duration T s . We write y m =  n a n  m,n T s where  m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m . This result can be interpreted as sampling in frequency domain: the I w multipath channel contributions appear weighed according to their individual Doppler offset  i . It confirms that due to the Doppler shifts, the detected signal y m contains contributions from all n subcarrier signals, not only from m = n . All  m,n ‘s with m  n lead to ICI, with amplitudes weighed by sinc(  +  i /  s ).

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