Exhibit 2031
`
`Exhibit 2031
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`

`

`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 4, APRIL 2007
`
`717
`
`Interference Analysis of Filtered Multitone
`Modulation Over Time-Varying Frequency-
`Selective Fading Channels
`
`Tiejun (Ronald)Wang, John G. Proakis, Life Fellow, IEEE, and James R. Zeidler, Fellow, IEEE
`
`Abstract—We consider in this paper filtered multitone (FMT)
`modulation over frequency-selective time-varying fading channels.
`Due to the phase and amplitude distortion introduced by the fading
`channel, not only is the orthogonality among different subcarriers
`destroyed, but also the perfect Nyquist sampling condition of the
`baseband matched filters is no longer valid. Consequently, inter-
`channel, as well as intersymbol, interference will cause distortions
`to the transmitted signals. In this paper, the interference caused by
`the channel frequency selectivity and time variance is quantified
`by analyzing the demodulated signals at the receiver under several
`different fading-channel conditions. An analysis of the average car-
`rier-to-interference (C/I) ratio of the FMT system is provided in
`order to demonstrate the underlying tradeoff between spectral ef-
`ficiency and system performance. For comparison purposes with
`other multicarrier communication systems (or modulation tech-
`niques), the C/I ratio of the conventional orthogonal frequency-di-
`vision multiplexing system is also provided and compared with that
`of the FMT system under the same channel conditions and spectral
`efficiency. Finally, numerical and simulation results are given that
`confirm the C/I ratio results obtained.
`Index Terms—Carrier-to-interference (C/I) ratio, filtered multi-
`tone (FMT), interchannel interference (ICI), intersymbol interfer-
`ence (ISI), orthogonal frequency-division multiplexing (OFDM).
`
`I. INTRODUCTION
`
`F ILTERED MULTITONE (FMT) modulation is a form of
`
`multicarrier modulation that satisfies the perfect recon-
`struction conditions, in a sense that the sampling signals from
`the matched filter at the receiver side are free from intersymbol
`interference (ISI) as well as interchannel interference (ICI).
`FMT modulation avoids spectral overlapping between subcar-
`riers by resorting to a noncritical sampling technique [1], where
`the transmitter upsampling factor
`is larger than the number
`of subcarriers
`. FMT has been proposed for data transmission
`on very-high-speed digital subscriber lines (VDSL) [1], [2] and
`wireless channels [3] to provide high-data-rate communica-
`tions with high spectral efficiency, and convenience in spectrum
`management [1]–[3].
`
`Paper approved by S. N. Batalama, the Editor for Spread Spectrum and Es-
`timation of the IEEE Communications Society. Manuscript received February
`7, 2005; revised November 28, 2005. This work was supported by the Center
`for Wireless Communications under the CoRe Research Grants 00-10071 and
`03-10148. This paper was presented in part at the IEEE Global Telecommuni-
`cations Conference, St. Louis, MO, November/December 2005.
`The authors are with the Center for Wireless Communications, Department
`of Electrical and Computer Engineering, University of California, San Diego,
`La Jolla, CA 92093-0407 USA (e-mail: ronald@cwc.ucsd.edu).
`Digital Object Identifier 10.1109/TCOMM.2007.892455
`
`In contrast, conventional orthogonal frequency-division mul-
`tiplexing (OFDM) [4] has overlapping spectra and rectangular
`impulse responses. Consequently, each OFDM subchannel
`exhibits a sinc-shape frequency response. Therefore, the time
`variations of the channel during one OFDM symbol duration
`destroy the orthogonality of different subcarriers, and result
`in power leakage among subcarriers, known as intercarrier
`interference, which causes degradation in system performance
`[5]–[7]. To be specific, the orthogonality of the channel re-
`sponses in different OFDM subchannels is destroyed at the
`receiver by the amplitude and phase distortion caused by the
`fading channel impulse response (CIR) under certain channel
`conditions. The fading-channel impairments on the orthogo-
`nality conditions generally fall into three categories depending
`on the fading characteristics: quasi-static frequency-flat fading
`channels do not introduce ISI or ICI to a multicarrier system;
`frequency-selective fading channels introduce ISI, but pre-
`serve the orthogonality among different subchannel signals;
`time-varying frequency-selective channels introduce both ISI
`and ICI.
`As compared with an OFDM system using a cyclic prefix
`(sacrificing spectral efficiency) to remove ISI, an FMT system
`uses a noncritical sampling rate technique to mitigate the in-
`terference caused by the fading channel. The interference (in-
`cluding ISI and ICI) in an FMT system may be suppressed by
`choosing a larger noncritical sampling factor (the ratio of the
`upsampling factor over the number of subcarriers), and hence,
`sacrificing spectral efficiency. The tradeoff between possible
`ICI (and ISI) and the corresponding spectral efficiency (or data
`rate) of the FMT system is, therefore, an important issue for the
`system design and comparison with conventional OFDM. Re-
`cently, several interesting papers have appeared, addressing this
`problem from different perspectives in a time-variant and pos-
`sibly frequency-selective fading environment [8]–[11]. Assalini
`et al. investigated in [8] and [9] the effects of frequency offsets
`and phase noise in FMT and OFDM systems. By evaluating the
`achievable bit rates of the two systems over different types of
`channels, they found that FMT has a higher spectral efficiency
`and is more robust to frequency offset than OFDM. Tonello [10],
`[11] calculated the exact matched-filter performance bound for
`multitone-modulated signals in time-varying and frequency-se-
`lective fading channels when optimal maximum-likelihood de-
`tection is employed. In [12]–[14], practical FMT systems in-
`cluding the filter-bank design were considered, with the objec-
`tive of minimizing the ISI and ICI, while at the same time max-
`imizing the spectral efficiency.
`
`0090-6778/$25.00 © 2007 IEEE
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`VIS EXHIBIT 2031
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`Fig. 1. System model of the FMT modulation system.
`
`In this paper, a system performance analysis is provided
`for the FMT system over time-frequency dispersive channels.
`It is known that when the baseband filter at the transmitter
`and the matched filter at the receiver maintain orthogonality
`among the subcarriers and also satisfy the Nyquist sampling
`criterion, there is no ICI or ISI in the system as long as the
`fading CIR is flat and stationary. However, this is not the case
`for most practical wireless environments. Instead of simulating
`the achievable system information rate or calculating the
`bit-error rate bounds, we focus our attention on investigating
`the average system carrier-to-interference (C/I) ratio of the
`FMT system over time-varying frequency-selective fading
`channels. Through the analysis of the average C/I ratio of the
`FMT system, an understanding of the tradeoff between spectral
`efficiency and system performance degradation is provided.
`Furthermore, we provide in this paper comparisons (both in
`system C/I ratio and average symbol-error rate) between FMT
`and OFDM systems under the same channel conditions and
`spectral efficiency. Numerical and simulation results of the
`system C/I ratio, as well as the average symbol-error rate,
`further confirm and support the obtained analytical results.
`The paper is organized as follows. In Section II, we describe
`the FMT system model, as well as the frequency-selective
`time-varying fading channel model considered in this paper. In
`Section III, the demodulated signal at the receiver is analyzed
`in detail under several different channel conditions. The av-
`erage C/I ratio is also determined to demonstrate the tradeoff
`between performance and spectral efficiency. In Section IV,
`numerical results of FMT system performance are presented
`and compared with an OFDM system under several different
`channel conditions. Section V concludes the paper.
`
`The transmitted signal
`quency-shifted versions of the
`at the transmission rate of
`
`is obtained by adding the fre-
`filtered outputs from the filters
`, which is given by
`
`(1)
`
`is the total number of subchannels in the FMT system
`where
`and
`is the frequency spacing between adjacent sub-
`channels. The obtained signal
`is transmitted over a
`time-varying, frequency-selective channel. At the receiver end,
`the received signal after sampling may be expressed as
`
`(2)
`
`where
`
`represents the CIR of the th path at time
`represents the total number of paths of the frequency-
`,
`selective fading channel, and
`represents the additive
`Gaussian noise with zero mean and variance
`.
`The fading channel coefficients
`are modeled as
`zero-mean complex Gaussian random variables. Based on the
`wide-sense stationary and uncorrelated scattering (WSSUS) as-
`sumption, the fading channel coefficients in different delay taps
`are statistically independent. We also assume that they have an
`exponential power delay profile, which is given by
`
`II. SYSTEM MODEL FOR FMT MODULATION
`
`As illustrated in Fig. 1, the complex-valued quadrature ampli-
`tude modulation (QAM) symbols
`,
`, are provided at the symbol rate of
`. After upsampling by
`a factor of
`, each symbol stream is filtered by a baseband filter
`with frequency response
`and impulse response
`.
`
`(3)
`The number of fading taps
`, where
`is given by
`is the en-
`is the maximum multipath delay, and
`tire channel bandwidth of the FMT system. The parameter
`in
`(3) controls the coherence bandwidth of the channel. The 3 dB
`channel coherence bandwidth is given by
`.
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`WANG et al.: INTERFERENCE ANALYSIS OF FILTERED MULTITONE MODULATION
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`
`Fig. 2. Frequency spectrum of an FMT system with M subcarriers under upsampling (or downsampling) factor of K.
`
`are cor-
`In the time domain, the fading coefficients
`related and have a Doppler power spectrum density modeled as
`in Jakes [16], given by
`
`factor of
`at time
`
`. Therefore, the th output of the substream
`is given by
`
`(4)
`
`otherwise
`
`where
`
`is the maximum Doppler bandwidth. Hence,
`has an autocorrelation function given by
`
`(5)
`
`is the first kind Bessel function of zero order.
`where
`As demonstrated in Fig. 2, the baseband filter
`is de-
`signed to have the following two important properties. First, it
`has a limited bandwidth of
`, and hence, adjacent sub-
`channels have nonoverlapping frequency responses, which can
`be represented as
`
`(6)
`and which is equivalent to the following condition in the time
`domain:
`
`are different positive integers.
`and
`where
`It also satisfies the Nyquist perfect sampling condition, which
`can be expressed in the following way:
`
`(7)
`
`(8)
`
`Note that both conditions given by (7) and (8) hold only ap-
`proximately in real systems, where the practical filters have fi-
`nite-length time-domain impulse responses. However, it is as-
`sumed in this paper that the filter-impulse response
`is suf-
`ficiently long, and hence, conditions (7) and (8) are valid.
`The sampled signal
`at the receiver is frequency-
`shifted, which generates
`substreams. Each stream is then fil-
`tered by a matched filter
`, followed by subsampling by a
`
`(9)
`
`III. INTERFERENCE ANALYSIS
`In this section, an analysis of the interference generated under
`different channel conditions is provided, and further used to de-
`fine the tradeoffs between spectrum efficiency and system per-
`formance degradation.
`
`A. FMT Transmitter and Receiver Structures
`Similar to the derivations given in [1], we first briefly describe
`in this subsection the FMT transmitter and receiver structure for
`the purpose of interference analysis. With the change of vari-
`ables by
`, we can rewrite (1) into the following
`form:
`
`where
`signal
`
`is the inverse Fourier transformation of the input
`, and is given by
`
`(10)
`
`into the following sum of integer
`We further decompose
`index and fractional index as:
`
`(11)
`
`Then, the transmitted signal (10) can be simplified as
`
`(12)
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`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 4, APRIL 2007
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`(13)
`
`(18)
`
`where
`
`belongs to a filter set
`, whose elements are the
`polyphase components (with respect to
`) of the prototype
`filter with impulse response
`. The filter index
`of
`provides the address of the polyphase component
`that needs to be applied to the th
`output of
`the inverse discrete Fourier transform (IDFT) to generate the
`transmitted signal
`. Therefore, the transmitted signal
`can actually be viewed as an OFDM signal
`subcarriers passing through
`baseband filters with
`with
`periodically shifting filter index of period
`.
`For similar reasons, the polyphase components (with respect
`) of the matched filter at the receiver can be denoted as
`
`to
`
`(14)
`is the matched prototype filter, and the transforma-
`where
`tion of the indexes is illustrated as follows:
`
`B. FMT Over Quasi-Static Frequency-Flat Fading Channel
`
`In order to see how time variations and frequency selectivity
`affect the ICI (or ISI), it is insightful to first investigate an FMT
`system over frequency-flat and quasi-static fading channels,
`where the channel response remains static within the data burst.
`The performance of the FMT system over such channel condi-
`tions was analyzed and characterized in [1]. In this subsection,
`we briefly describe the main results.
`First, by substituting (2) (for the case of quasi-static flat-
`fading channel) into (18), the signal
`can be repre-
`sented as
`
`where
`is the channel gain or impulse response of the fre-
`quency-flat fading channel. According to the property (8), the
`following results can be obtained:
`
`(19)
`
`Correspondingly, the received signal
`resented in the polyphase form given by
`
`(15)
`can also be rep-
`
`(16)
`
`Following similar derivations given by (13) and by employing
`the change of variable
`, the th output signal of
`the demodulator at time
`at the receiver can be represented
`in the following form:
`
`Therefore, (19) can be simplified to
`
`(20)
`
`(21)
`
`where
`variance
`
`(17)
`
`is the equivalent additive Gaussian noise with
`
`(22)
`
`where
`
`is given by
`
`Hence, the output signal from the demodulator is given by
`
`(23)
`where
`is the equivalent (frequency-domain) additive
`Gaussian noise of the th subcarrier at time instant
`and has
`the same variance as
`.
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`C. FMT Over Frequency-Selective Channel
`When the FMT system is transmitting over a frequency-selec-
`tive fading environment, the spectrum of the received signal is
`shaped by the channel frequency response. Hence, the Nyquist
`perfect sampling condition of the baseband filters of the FMT
`system is no longer satisfied, and consequently, we have ISI. In
`this case, by substituting (2) and (1) into (9), the output signals
`from the demodulator are given by
`
`(27)
`where the second term is the ISI caused by frequency selectivity,
`and
`is the time-domain autocorrelation function of the
`prototype filter impulse response
`defined as
`
`as the Fourier transformation of the
`If we further define
`weighted autocorrelation function, in the following form:
`
`(24)
`
`(28)
`
`It is evident that the first term of (24) can be reorganized as the
`following form:
`
`then (24) can be simplified to be
`
`(29)
`
`(30)
`
`It is evident that the output signal from the demodulator is cor-
`rupted by both additive noise and ISI, which is represented by
`the second term of (30). After some manipulations, the variance
`of the ISI disturbance conditioned on the instantaneous channel
`realizations can be expressed by the following:
`
`Due to the property of the filter impulse response given by (7),
`we can obtain the following equality:
`
`(25)
`
`where
`
`is given by
`
`(31)
`
`(32)
`
`Therefore, given a certain channel realization, the signal-to-in-
`terference-plus-noise ratio (SINR) is given by
`
`(26)
`
`(33)
`
`By substituting (25) and (26) into (24), the output signals from
`the demodulator can be simplified into the following form:
`
`given by (32) comes only
`Notice that the interference
`from the same subchannel and there is no ICI. This is due
`to the fact that the baseband filter
`is bandlimited to be
`within a frequency band of width
`, and hence, maintains
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`
`orthogonality among different subcarriers. When the channel
`is frequency-selective, the frequency response of the received
`signal after the matched filter no longer satisfies the Nyquist
`perfect sampling condition, but still maintains nonoverlapping
`frequency bands. Therefore, different subcarriers remain or-
`thogonal to each other and the received signal only contains
`ISI, but no ICI.
`If we use binary phase-shift keying (BPSK) modulation and
`further approximate the interference as a Gaussian random vari-
`able [6], [7], then the average bit-error probability (BEP) of the
`FMT system over the frequency-selective channel is given by
`
`(34)
`
`It is evident from (34) that the average BEP does not have an
`analytical closed-form expression, and hence, it is very difficult
`to provide useful insight into the tradeoff between performance
`degradation and spectral efficiency. Therefore, we evaluate the
`C/I ratio of the FMT system, which is given by
`
`Moreover, by substituting the expression of
`into (38), it can be further simplified to the form
`
`given by (36)
`
`Following similar derivations, we also obtain the following ex-
`pectation:
`
`(39)
`
`(40)
`
`Substituting (39) and (40) into (35), we obtain the C/I ratio
`in terms of the number of subcarriers
`, upsampling factor
`, and the baseband filter
`. Hence, we have provided a
`closed-form expression of the tradeoff between performance
`degradation and spectral efficiency
`.
`
`(35)
`
`In order to simplify the above equation, let us denote
`as
`the Fourier transformation of
`, which is given by
`
`D. FMT Over Frequency-Selective Time-Varying Channel
`
`In this section, we consider the time-varying fading channel,
`where the orthogonality among different subchannels is no
`longer valid. Therefore, both ICI and ISI are going to disturb
`the transmitted signal. In this case, the output signal from the
`FMT demodulator is given by
`
`where
`given by
`
`is the frequency response of the baseband filter,
`
`(36)
`
`(37)
`
`By substituting the power delay profile given by (3) into the
`Fourier transformation of the weighted autocorrelation function
`given by (29), we obtain the following result:
`
`where
`
`is defined as
`
`(38)
`
`(41)
`
`(42)
`
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`WANG et al.: INTERFERENCE ANALYSIS OF FILTERED MULTITONE MODULATION
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`If we further denote
`
`as the following:
`
`Given the fact that
`
`then (41) can be easily represented as
`
`we can simplify the following expectation:
`
`(43)
`
`(44)
`It is evident from the above equation that the second term repre-
`sents the ISI, the third term represents the ICI, and the last term
`represents the additive Gaussian noise. Similarly, the SINR of
`the th subcarrier at time instant
`can be expressed as the fol-
`lowing:
`
`Therefore, the summation of the average signal power
`average interference power
`is given by
`
`723
`
`(49)
`
`(50)
`and the
`
`(51)
`
`(45)
`The average BEP of the FMT system over the fading channel
`can also be expressed by (34) when BPSK modulation is used.
`Due to the fact that (34) is intractably complicated, we use
`the C/I ratio to analyze the tradeoffs between the spectral effi-
`ciency and performance degradation of the FMT system over
`time-varying fading channels. First, according to the uncorre-
`lated assumption between different fading-path responses, we
`have the following result:
`
`where the right-hand side of the above equation can further be
`simplified to be
`
`(46)
`
`where
`
`is the discrete Doppler spectrum, given by
`
`(47)
`
`otherwise.
`(48)
`
`is the power spectrum density of the interfer-
`where
`ences plus signals, which is given by
`
`Following similar derivations, the average signal power
`can
`also be obtained as the following form after some manipula-
`tions:
`
`(52)
`
`(53)
`
`given by (53) and the
`Finally, by substituting signal power
`interference power
`from (52) into the C/I
`by subtracting
`ratio definition, the C/I ratio of the FMT system over frequency-
`selective time-varying channel can be readily obtained.
`
`E. Performance Comparison With OFDM System
`We know that both FMT and OFDM are different forms of
`multicarrier modulation. Therefore, it is insightful to compare
`the performance of these two modulations under the same
`channel conditions and with the same spectral efficiency.
`In OFDM systems, a cyclic prefix is used to combat the mul-
`tipath spread and to mitigate the ISI. In practical situations,
`we select the length of the cyclic prefix to be longer than the
`channel maximum multipath delay spread, and thus, the system
`
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`
`is ISI-free. According to the ICI analysis of an OFDM system
`in a time-varying fading channel as provided in [6] and [7], the
`carrier over interference (ICI only) ratio of the th subcarrier in
`an OFDM system can be represented as
`
`(54)
`
`is the OFDM symbol interval.
`where
`In contrast to the OFDM system, both ICI and ISI exist in
`the FMT system in a frequency-selective time-varying fading
`channel. As demonstrated in (44), the second term of the right-
`hand side represents the ISI, and the third term represents the
`ICI. In order to understand the effects of these two interferences
`on FMT system performance, the average power of the carrier
`signal (first term), ISI (second term), and ICI (third term) of (44)
`are analyzed separately. After similar manipulations as given in
`Section III-D, the average ICI and ISI powers are given by the
`following form:
`
`(55)
`
`(56)
`
`is given by (53), and
`where
`lowing form:
`
`,
`
`are given by the fol-
`
`(57)
`
`(58)
`
`, which is equal to
`We know that the total interference power
`the sum of both interference
`and
`, is sometimes a coarse
`performance measure and can not represent the individual
`effects of each interference, especially when compared with
`the OFDM system which has only ICI. Therefore, it would be
`reasonable to compare both the carrier-over-interference ratio
`and the carrier-over-ICI ratio
`with the C/I
`ratio of an OFDM system, in order to understand the overall
`interference as well as the ICI on FMT system performance.
`As can be expected, the performance degradation of both sys-
`tems depends heavily on the channel conditions (frequency se-
`lectivity and time variance), as well as the system spectral ef-
`ficiencies. First, in the case when the channel is slowly fading
`(low Doppler frequency), an OFDM system with cyclic guard
`
`intervals is slightly distorted only by the ICI, and has a per-
`formance close to the interference-free situation in the limit
`when
`. However, for FMT systems, the ISI always ex-
`ists due to the loss of the Nyquist sampling condition caused
`by the channel frequency-selectivity, even for channels with
`small Doppler spreading and systems with low spectral effi-
`ciency
`. Therefore, OFDM outperforms FMT in a
`slowly fading environment. Second, it is reasonable to expect
`the FMT system to have a smaller ICI than an OFDM system,
`since the baseband signal in FMT modulation is pulse-shaped
`by a filter as contrast to the simple rectangular baseband filter in
`OFDM modulation. Hence, the
`ratio of an FMT system
`is expected to be larger than that of an OFDM system. Further-
`more, it is apparent from (54) that the ICI (power) does not
`depend on channel frequency-selectivity and grows inversely
`quadratically with respect to the maximum Doppler frequency
`. In this sense, the FMT system is expected to outperform
`OFDM in a fast-fading (with large Doppler spread) environment
`as ICI dominates the overall interference. Finally, we observe
`that the spectral efficiency of the OFDM system depends on
`the channel frequency-selectivity in the sense that the length of
`the cyclic prefix should be larger than the maximum multipath
`delay. Thus, an ISI-free OFDM system has a maximum spectral
`efficiency given by
`
`(59)
`
`is the maximum multipath delay. Therefore, under
`where
`the same spectral efficiency, an FMT system is likely to outper-
`form an OFDM system over highly frequency-selective fading
`channels with long channel delay spread.
`
`F. Intuitive Discussion on Per-Channel Equalization
`Although considering the problem of equalization for FMT
`systems over time-varying frequency-selective fading channels
`is outside the scope of this paper, it would be interesting and in-
`sightful to provide some intuitive discussion on the performance
`and implementation complexity of the per-channel equalization
`techniques based on the interference analysis provided in pre-
`vious sections.
`To be specific, we first consider the per-channel equalizer
`in the FMT system. Since it is shown in Section III-D that
`both ISI and ICI exist in the unequalized FMT system over
`frequency-selective and time-varying channels, the expected
`performance gain by using the per-channel equalizer to mitigate
`the ISI caused by frequency-selective channel distortion would
`be significant. On the other hand, OFDM systems usually
`rely on the presence of the cyclic prefix to combat the ISI.
`As an intuitive comparison, the system performance of the
`per-equalized FMT system should outperform the conventional
`OFDM system, especially over highly frequency-selective
`fading channels and with fast time-variations.
`On the other hand, if viewed from the implementation com-
`plexity perspective, equalizers of FMT systems are more com-
`plicated than those of OFDM systems, since two different types
`of interferences are to be equalized. Consequently, compared
`with OFDM systems, the coefficients of the equalizers in FMT
`
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`725
`
`Fig. 3. C/I ratio graphs of an FMT system with M = 64 subcarriers under
`different upsampling factors K = 71; 80; 91; 107; 128.
`
`Fig. 4. Carrier-over-ICI ratio (C=I
`) graphs of an FMT system with M = 64
`subcarriers under different upsampling factors K = 68; 71; 80; 91; 107; 128.
`
`systems are more dependent on the channel state information. In
`practical communication systems, the channel state information
`is subject to channel estimation errors, and hence, FMT equal-
`izers are more sensitive to the quality of the channel estimates
`than equalizers in OFDM systems. Therefore, a per-channel
`equalized OFDM system is more likely to outperform the equal-
`ized FMT system over fast-fading channels where channel esti-
`mation is not very accurate.
`
`IV. NUMERICAL AND SIMULATION RESULTS
`
`In this section, we assume that the baseband filter
`has a root-raised cosine frequency response, which is given by
`
`(60)
`
`where the roll-off factor
`.
`is given by
`We first demonstrate in Fig. 3 the C/I ratio graphs of an
`FMT system having 64 subcarriers and several different up-
`sampling factors
`. Simulation results
`are shown on the graphs as discrete points. The spectral ef-
`ficiency
`, defined as the ratio of the subcarrier
`number over the upsampling factor, varies from
`to
`, which parameterizes the C/I ratio curves. The FMT
`system has a total bandwidth
`0.5 MHz, where adjacent
`subcarriers have frequency spacing 7.81 kHz. By setting the
`system under the above conditions, we fix the total bandwidth
`and vary the spectral efficiency by changing the data rate.
`The FMT system is transmitting over a frequency-selective
`taps fading channel with 3 dB coherence bandwidth
`0.125 MHz. For comparison purposes, the C/I ratio
`graph of an OFDM system having 64 subcarriers with the same
`subcarrier frequency spacing is also shown in the plot. For
`
`sake of fair comparisons, we only consider ISI-free OFDM
`systems, where the length of the cyclic prefix is longer than the
`. Therefore,
`maximum multipath number (tap number)
`the OFDM system considered has maximum spectral efficiency
`.
`From Fig. 3, we observe that when the FMT system has mod-
`erate-to-low spectral efficiencies (
`in this case), it out-
`performs the OFDM systems at the cost of losing spectral ef-
`ficiency
`. However, at high spectral efficiency
`regimes (
`in this case), OFDM has a better C/I ratio than
`the regular FMT system. Therefore, we conclude from these re-
`sults that OFDM is superior to an FMT system under the same
`spectral efficiency. We also observe from the left part of the
`plot that an OFDM system outperforms FMT (even with smaller
`spectral efficiency) over slowly time-varying fading channels
`with low Doppler frequencies (
`100 Hz).
`It might seem counterintuitive that the simple OFDM system
`without any baseband pulse-shaping filters outperforms an FMT
`system with a baseband filter design. However, it can be ex-
`plained by the different interference types of both systems. As
`stated in Section III-E, an OFDM system with a cyclic prefix
`does not have any ISI, while an FMT system is distorted by
`both ISI and ICI. Therefore, due to the additional ISI impair-
`ment, it is reasonable to see that OFDM outperforms FMT with
`the same spectral efficiency. In order to compare purely the ICI
`effects on the system performance, we demonstrate in Fig. 4 the
`carrier-over-ICI ratios of the same FMT system over the same
`fading channel conditions as in Fig. 3. We observe from the plot
`that an FMT system has less ICI distortion (hence, larger
`ratio) compared with an OFDM system having the same spec-
`tral efficiency. Furthermore, the group of C/I and
`curves
`provided in Figs. 3 and 4 demonstrate the tradeoff between spec-
`tral efficiency and system performance (or performance degra-
`dation).
`In order to compare the system performance over channels
`with high frequency-selectivity, we demonstrate in Fig. 5 the
`CIR curves of the same FMT system with 64 subcarriers and
`
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`726
`
`IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 4, APRIL 2007
`
`Fig. 5. C/I ratio graphs of an FMT system with M = 64 subcarriers under
`different upsampling factors K = 80; 84; 91; 107; 128.
`
`Fig. 7. Comparison of symbol-error probability of an FMT system with M =
`64 subcarriers under different spectral efficiency over highly frequency-selec-
`tive and fast-fading channel.
`
`0.125
`
`delay profile with the 3 dB coherence bandwidth
`MHz, and the maximum Doppler spread is fixed to be
`100 Hz; while the channel in Fig. 7 has a coherence bandwidth
`31.25 kHz with
`and Doppler spread
`600 Hz. For comparison purposes, the average symbol-error
`probability of an OFDM system with 64 subcarries over the
`same fading channels is also included in both plots. From the
`above two plots, we observe the tradeoff between spectral ef-
`ficiency and performance degradation. Furthermore, consistent
`with the results of the C/I ratio curves provided in this sec-
`tion, an OFDM system outperforms (in average symbol-error
`rate sense) an FMT system with the same spectral efficiency
`in slowly fading and low frequency-selective fading channels,
`but is outperformed by FMT modulation in fast-fading chan-
`nels with high frequency selectivity. Therefore, FMT modula-
`tion is a promising alternative to OFDM in the multicarrier com-
`munications family in certain wireless environments, and pro-
`vides better performance in combating frequency-selective and
`time-varying fading channels.
`
`V. CONCLUSION
`In this paper, we considered an FMT modulation system in
`frequency-selective time-varying fading channels. The ICI and
`ISI of the FMT system, which is caused by the time variations
`and frequency selectivity of the fading channel, has been an-
`alyzed in detail. By analyzing the demodulated signals at the
`receiver under different fading channels, we showed analyti-
`cally that quasi-static frequency-flat fading channels do not in-
`troduce ISI or ICI to a multicarrier system, frequency-selective
`fading channels introduce ISI, but preserve the orthogonality be-
`tween different subchannel signals, whereas time-varying fre-
`quency-selective channels introduce both ISI and ICI in an FMT
`system. We provided in this paper the analysis of the effects of
`ISI and ICI in an FMT system separately under different channel
`conditions. The obtained C/I ratio analysis provided insight into
`the tradeoff between spectral efficiency and system performance
`
`Fig. 6. Comparison of symbol-error probability of an FMT system with M =
`64 subcarriers under different spectral efficiency over low frequency-selective
`and slowly time-varying fading channel.
`
`, with
`different upsampling factors
`. The system is
`to
`ranging from
`transmitting over a highly frequency-selective fading channel
`with 3 dB coherence bandwidth
`31.25 kHz and the
`number of independent taps
`with an exponential delay
`profile. We observe from the above plot that the FMT modula-
`tion outperforms an OFDM system with the same spectral ef-
`ficiency over highly frequency-selective fading channels with
`high Doppler frequency (moderate-to-fast fading channels).
`We also demonstrate in Figs. 6 and 7 the average symbol-error
`probability of the same FMT system with 16-QAM under sev-
`eral different spec