Exhibit 2030
`
`Exhibit 2030
`
`

`

`Techniques for Suppression of Intercarrier
`Interference in OFDM Systems
`∗
`
`Tiejun (Ronald) Wang, John G. Proakis, and James R. Zeidler
`Center for Wireless Communications
`University of California, San Diego
`La Jolla, CA 92093-04047
`
`Abstract— This paper considers an orthogonal frequency divi-
`sion multiplexing (OFDM) system over frequency selective time-
`varying fading channels. The time variations of the channel
`during one OFDM frame destroy the orthogonality of different
`subcarriers and results in power leakage among the subcarriers,
`known as Intercarrier Interference (ICI), which results in a
`degradation of system performance.
`In this paper, channel state information is used to minimize the
`performance degradation caused by ICI. A simple and efficient
`polynomial surface channel estimation technique is proposed to
`obtain the necessary channel state information. Based on the
`estimated channel
`information, we describe a minimal mean
`square error (MMSE) based OFDM detection technique that
`reduces the performance degradation caused by ICI distortion.
`Performance comparisons between conventional OFDM and the
`proposed MMSE-based OFDM receiver structures under the
`same channel conditions are provided in this article. Simulation
`results of the system performance further confirm the effective-
`ness of the new technique over the conventional OFDM receiver
`in suppressing ICI in OFDM systems.
`
`I. INTRODUCTION
`
`Orthogonal Frequency Division Multiplexing (OFDM) is
`a widely known modulation scheme in which a serial data
`stream is split into parallel streams that modulate a group
`of orthogonal subcarriers [1]. OFDM is widely used and
`considered a promising technique for high speed data trans-
`mission in digital broadcasting, wireless LANs, HDTV, and
`next generation mobile communications.
`OFDM symbols are designed to have a relatively long time
`duration, but a narrow bandwidth. Hence OFDM is robust
`to channel multipath dispersion and results in a decrease in
`the complexity of equalizers for high delay spread channels
`or high data rates. However, the increased symbol duration
`makes an OFDM system more sensitive to the time varia-
`tions of mobile radio channels. In particular, the effect of
`Doppler spreading destroys the orthogonality of the subcar-
`riers, resulting in intercarrier interference (ICI) due to power
`leakage among OFDM subcarriers. In paper [2], the carrier to
`interference (C/I) ratio has been introduced to demonstrate the
`effect of the ICI under different maximum Doppler spreads and
`different Doppler spectra. Performance degradation of OFDM
`systems due to Dopper spreading is also analyzed in [3].
`
`This work was supported by the Center for Wireless Communications under
`the CoRe research grant core 00-10071.
`
`In this paper, the channel state information is assumed to be
`unavailable at the receiver and has to be estimated in the first
`place. In [4], a time-frequency polynomial model for channel
`estimation in OFDM systems is proposed, which does not have
`to estimate channel statistics such as the channel correlation
`matrix and average SNR per bit. However, in practice such
`knowledge is usually not available and the channel statistics
`may vary by time. But a large polynomial order is required
`in order to represent the 2-D frequency channel response with
`sufficient accuracy, since the frequency selectivity makes the
`channel changes relatively fast over the frequency domain.
`Therefore, we need to design a channel estimation method
`under the frequency selective and time-varying fading channel
`with low complexity. In this paper, we propose a new modified
`polynomial channel estimator with better performance. In
`contrast to the estimator which estimates the channel response
`in the frequency domain, the modified estimation algorithm
`directly estimates the time domain response (has relatively
`slower variation and requires lower order to polynomial func-
`tion representations), and hence achieves better estimation
`quality. Another point is that in [4], the fading channel is
`modeled as constant within one OFDM frame, but changes
`from frame to frame, which is inaccurate for most cases,
`especially when ICI is to be shown and analyzed. The modified
`polynomial channel estimator in our paper not only estimates
`the variations frame by frame, but also within one OFDM
`frame.
`We also propose in this paper a minimum mean square error
`(MMSE) criterion-based OFDM receiver structure that takes
`into account both additive noise and the ICI disturbance. The
`numerical simulation results of the system performance that
`are provided under various channel conditions confirm the
`superior performance of the MMSE-OFDM receiver over the
`conventional OFDM receiver.
`The rest of the paper is organized as follows: In Section II
`we describe the OFDM system model as well as the frequency
`selective time varying fading channel model considered in
`this paper. In Section III, the polynomial model is described
`for the channel and an estimation algorithm is provided and
`applied to perform the OFDM channel estimation. In Section
`IV, two different OFDM receiver structures, the conventional
`OFDM receiver and the MMSE-based receiver are descibed. In
`Section V, the numerical system performance of these different
`detection techniques are presented and compared. Finally, our
`conclusions are contained in Section VI.
`
`IEEE Communications Society / WCNC 2005
`
`39
`
`0-7803-8966-2/05/$20.00 © 2005 IEEE
`VIS EXHIBIT 2030
`
`Page 1 of 6
`
`

`

`(cid:2)(cid:4)(cid:4)hl
`
`t[n]
`
`(cid:4)(cid:4)2(cid:3)
`
`statistically independent. We also assume that they have an
`exponential power delay profile, which is given by
`1 − exp(−β)
`= α· exp(−β · l), α =
`1 − exp(−L · β) . (3)
`The number of fading taps L is given by τmax/Ts, where τmax
`is the maximum multipath delay, and Ts = 1/W , where W
`is the channel (OFDM signal) bandwidth. In the time domain,
`the fading coefficients hl
`t[n] are correlated and have a Doppler
`power spectrum density modeled as in Jakes [6], given by
`(cid:8)
`|f| ≤F d
`1
`1−
`0
`
`E
`
`II. CHANNEL MODEL
`
`d [k]
`t
`
`ts [n]
`
`Binary
`
`Source
`
`M−ary QAM
`
`S/P
`
`Modulator
`
`Converter
`
`IFFT
`
`Guard
`Interval
`Insertion
`
`P/S
`
`Converter
`
`l
`th [n]
`
`Channel
`
`AWGN
`
`Noise
`
`Channel Estimation
`
`tR [k]
`
`Data
`
`Output
`
`M−ary QAM
`
`Symbol
`
`Detection
`
`r [n]
`t
`
`FFT
`
`Guard
`Interval
`Removal
`
`S/P
`
`Converter
`
`Fig. 1. Baseband model of the OFDM system
`
`,
`
`(4)
`
`otherwise
`
`(cid:10)2
`
`f F
`
`d
`
`(cid:9)
`
`
`
`
`πFd·
`
`D(f) =
`
`(cid:1)t
`
`,
`
`(7)
`
`(cid:14)(cid:4)(cid:4)(cid:4)(cid:4)shif t
`
`[n],··· , h0
`t [n]
`
`Ht =
`
`(cid:13)
`
`h[n] =
`
`t
`
`t
`
`st[n] =
`
`where Fd is the Doppler bandwidth. Hence hl
`t[n] has an
`(cid:12)
`(cid:11)
`autocorrelation function given by
`2π(n−m)FdTs
`
`t[n]·hlt[m](cid:2)] =α ·exp(−β·l)·J0
`E[hl
`. (5)
`where J0(·) is the first kind Bessel function of zero order.
`Written in a concise matrix form, we can represent (2) as,
`rt = Ht · st + n
`(6)
`,
`t are vectors of size N × 1, and the channel
`where rt and n(cid:1)
`(cid:14)H
`(cid:13)
`matrix Ht is given by
`h[0]H , h[1]H ,··· , h[N − 1]H
`where h[n]H is the right cyclic shift by n + 1 positions of a
`zero padded vector given by
`(cid:15)
`(cid:16)(cid:17)
`(cid:18)
`0, 0,··· , 0
`, hL−1
`[n], hL−2
`N−L
`
`n+1
`
`. (
`
`(9)
`
`,
`
`(cid:1)t
`
`An OFDM system with N subcarriers is considered in this
`paper. A block of N · log2 M bits of data is first mapped
`into a sequence {d[k]} of M-ary complex symbols of length
`N, each modulating an orthonormal exponential function
`N ), k = 0,··· , N − 1. Each data symbol d[k] is
`exp(j 2πkn
`normalized to have unit average signal power E[|d[k]|2] =
`1. As demonstrated in Fig.1, information bearing sequences
`{d[k]} is first serial to parallel converted and processed by an
`N−1(cid:1)
`IFFT operation, given by
`1√
`dt[k] · exp(j
`N
`
`2πkn
`N
`
`k=0
`
`), 0 ≤ n, k ≤ N − 1 ,
`(1)
`where the subscript t represents the tth OFDM frame. A
`cyclic prefix is inserted into the transmitted signal to prevent
`possible intersymbol
`interference (ISI) between successive
`OFDM frames. After parallel to serial conversion, the signals
`are transmitted through a frequency selective time varying
`fading channel. At the receiver end, after removing the cyclic
`guard interval, the sampled received signal is characterized in
`the following format, by applying the tapped-delay-line model
`[5]
`
`[n], 0 ≤ n ≤ N −1 , (2)
`
`(cid:1)t
`
`+ n
`
`L−1(cid:1)
`
`rt[n] =
`
`t[n]· st
`hl
`
`(cid:2)|n− l|N
`
`(cid:3)
`
`l=0
`where hlt[n] represents the channel response of the lth path
`
`during the tth OFDM frame, L represents the total number
`[n]
`of paths of the frequency-selective fading channel, n
`represents the additive Gaussian noise with zero mean and
`[n]|2] =σ 2 = N0/Es (Es/N0 represents the
`variance E[|n
`system signal to noise ratio), and |·|N represents the modular
`N operation.
`The fading channel coefficients hl
`t[n] are modeled as zero
`mean complex Gaussian random variables. Based on the Wide
`Sense Stationary Uncorrelated Scattering (WSSUS) assump-
`tion, the fading channel coefficients in different delay taps are
`
`(cid:1)t
`
`(cid:1)t
`
`2πnm
`N
`From another point of view, the received signal can be
`expressed in terms of the equivalent frequency domain channel
`model as
`
`(11)
`
`[n],
`
`(cid:1)t
`
`+ n
`
`(cid:10)
`
`(cid:9)
`
`dt[k] · ˜ht
`k[n] exp
`
`2πkn
`N
`
`j
`
`N−1(cid:1)
`
`k=0
`
`rt[n] =
`
`1√
`N
`
`where ˜ht
`k[n] is the frequency domain channel response for the
`kth subcarrier during the tth OFDM frame. We incorporate
`the result given by (2) into (11), then we have the frequency
`domain channel response ˜ht
`k[n] as
`
`L−1(cid:1)
`
`l=0
`
`˜ht
`k[n] =
`
`hl
`t[n] exp
`
`(cid:9) − j
`
`(cid:10)
`
`2πlk
`N
`
`, n = 0, 1,··· , N − 1 .
`(12)
`
`IEEE Communications Society / WCNC 2005
`
`40
`
`0-7803-8966-2/05/$20.00 © 2005 IEEE
`
`8)
`Hence the received signal rt is related to the data vector dt
`as,
`rt = HtW · dt + n
`where W is the inverse Fourier transformation matrix given
`by
`√
`
`(cid:2)
`
`wn,m
`
`W =
`
`(cid:3)
`
`N×N , wn,m = exp(j
`
`)/
`
`N .
`
`(10)
`
`Page 2 of 6
`
`

`

`channel frequency response, and I is the length of the guard
`interval inserted into each OFDM frame to avoid intersymbol
`interference between consecutive frames. The objective is to
`estimate the polynomial coefficients {ai,j}.
`Estimation may be performed every T OFDM frames,
`which means that the assumed model region is a N×T (N +I)
`(f requency × T ime) 2-D plane. Pilots are inserted every pf
`subcarriers, and every pt OFDM frames. Thus, the overhead
`1
`pf pt .
`ratio attributed to the pilots is
`In order to estimate the coefficients {ai,j} of the channel
`frequency response, let us first rewrite (18) in the matrix form
`as
`k[n] = aH · qtk,n ,
`˜ht
`(cid:2)
`where vector a given by
`ap,0, ap−1,1,··· , ai,j ··· , a0,0
`is the polynomial channel coefficients to be estimated, and
`(cid:10)j
`qt
`k,n is a vector given by
`
`qt
`k,n =
`
`t(N + I) +n
`
`(cid:3)H
`
`.
`
`(21)
`
`
`
`(cid:3)
`
`,
`
`(19)
`
`(20)
`
`a =
`
`
`(cid:2)kp,··· , ki ·(cid:9)
`(cid:4)(cid:4)(cid:4)Rt[k] −
`
`(22)
`
`(cid:4)(cid:4)(cid:4)2
`
`,
`
`(cid:19)g
`
`tk
`
`(cid:1)
`
`i∈Pf
`
`(cid:1)
`
`(t,k)∈P
`
`,··· , 1
`Since the additive noise in equation (14) is spherically sym-
`metric and zero mean Gaussian,
`the ML estimate of the
`coefficients vector a is chosen such that
`,i · dt[i]
`
`At the receiver side, the FFT operation is performed on each
`block of N received signal samples rt[n]. Thus, we obtain
`1√
`rt[n] · exp
`N
`
`(cid:10)
`
`,
`
`(13)
`
`2πkn
`N
`
`N−1(cid:1)
`
`(cid:9) − j
`
`Rt[k] =
`
`(cid:1)t
`
`n=0
`where Rt[k] represent the decision symbol at the kth subcar-
`rier of the tth OFDM frame. It also can be written in a concise
`matrix form as
`= Gt · dt + nt ,
`Rt = WH · rt = WHHtW · dt + WH · n
`(14)
`where Rt, dt and nt are column vectors of size N × 1,
`vector nt is i.i.d complex AWGN noise due to the orthonormal
`t of the original noise n(cid:1)t, and the matrix
`transformation WH
`(cid:14)
`(cid:13)
`
`Gt is related to the channel frequency response as follow:
`
`Gt = WHHtW =
`
`gt
`i,j
`
`,
`
`N×N
`
`(15)
`
`(cid:10)
`
`.
`
`(16)
`
`2π(j − i)n
`N
`
`N−1(cid:1)
`
`(cid:9)
`
`j
`
`i[n] · exp
`˜ht
`
`1 N
`
`where
`
`gt
`i,j =
`
`n=0
`The decision symbol Rt[k] at the receiver end is unfortunately
`distorted not only by additive Gaussian noise, but also by ICI
`as well. We expand (14), and represent Rt[k] in the following
`form:
`
`(cid:11)
`
`Rt[k] =
`
`(cid:12)
`
`N−1(cid:1)
`
`(cid:11)
`
`N−1(cid:1)
`
`(cid:19)g
`
`tk
`
`(cid:10)
`
`is minimized, where the noisy estimated channel response
`,i
`is provided in the following form by applying the polynomial
`channel model in (18),
`
`(cid:9)
`
`j
`
`(cid:20)˜ht
`k[n] · exp
`N−1(cid:1)
`
`n=0
`
`N−1(cid:1)
`(cid:21)
`
`n=0
`
`1 N
`
`1 N
`
`(cid:19)g
`
`,i =
`
`tk
`
`2π(i − k)n
`(cid:10)(cid:22)
`(cid:9)
`N
`2π(i − k)n
`k,n · exp
`qt
`= aH
`. (23)
`j
`N
`and the set P in (22) contains all the pilot locations in the
`detection region, while Pf contains the frequency locations of
`(cid:4)(cid:4)(cid:4)m = 0, pf , 2pf ,··· , (N/pf − 1)pf ;
`(cid:23)
`the pilots,
`(cid:24)
`P =
`(cid:4)(cid:4)(cid:4)m = 0, pf , 2pf ,··· , (N/pf − 1)pf
`(cid:24)
`(cid:23)
`(cid:4)(cid:4)(cid:4)2
`(cid:4)(cid:4)(cid:4)Rt[k] − aH · ut,k
`(cid:1)
`(cid:9)
`
`(m, n)
`n = 0, pt, 2pt,··· , (T /pt − 1)pt
`
`,
`
`m
`
`Pf =
`Hence, we can restate the problem of channel estimation,
`which is equivalent to finding the ML solution of (22), as
`solving
`
`. (24)
`
`,
`
`(25)
`
`(cid:10) · qt
`
`k,n .
`
`(26)
`
`min
`a
`
`(t,k)∈P
`
`dt[i] exp
`
`j
`
`2π(i − k)n
`N
`
`(cid:1)
`
`N−1(cid:1)
`
`i∈Pf
`
`n=0
`
`1 N
`
`ut,k =
`
`where ut,k is given by
`
`N−1(cid:1)
`(cid:9)
`
`n=0
`
`j
`
`1 N
`
`k[n] ·
`˜ht
`
`n=0
`
`1 N
`
`(17)
`
`· dt[k] +
`˜ht
`k[n]
`(cid:10)(cid:12)
`i=0
`i(cid:2)=k
`2π(i − k)n
`· dt[i] + nt[k],
`× exp
`N
`The first term in (17) is the desired signal, the second term
`represents the ICI from the other subcarriers, and finally the
`third term is the additive noise. Our objective is to reduce the
`effect of the ICI term by using knowledge of the channel state
`information.
`
`III. CHANNEL ESTIMATION
`In this paper, we assume that perfect channel state infor-
`mation is not available at the receiver. The channel frequency
`response ˜ht
`k[n] and time domain impulse response hl
`t[n] are
`estimated at the receiver end by inserting pilots at some of the
`subcarriers and, thus, estimating the frequency response of the
`channel at selected frequencies.
`
`A. A 2-D Polynomial Surface Channel Estimator
`We assume that the frequency-selective time-varying chan-
`nel response is a mathematically smooth function with respect
`to time and frequency. Hence, we may model the continuous
`channel frequency response ˜ht
`k[n] as a 2-D polynomial surface
`function within a certain time-frequency region as in [4]. That
`is,
`
`(cid:1)
`˜ht
`k[n] =
`t(N + I) +n
`(18)
`,
`i+j≤p
`(cid:10)
`(cid:9)
`where {ai,j} are the polynomial coefficients up to order p, k
`t(N + I) +n
`and
`are the frequency and time indexes of the
`
`ai,j · ki ·(cid:9)
`
`(cid:10)j
`
`IEEE Communications Society / WCNC 2005
`
`41
`
`0-7803-8966-2/05/$20.00 © 2005 IEEE
`
`Page 3 of 6
`
`

`

`(cid:9)
`(cid:9)
`
`j
`
`(cid:19)g
`
`tk
`
`(cid:10) ·
`
`−2πil
`N
`2π(i − k)n
`N
`
`j
`
`estimated channel response
`according to (14),
`
`L−1(cid:1)
`(cid:11)
`
`l=0
`
`×
`
`a
`
`1 N
`
`(cid:19)g
`
`,i =
`
`tk
`
`(cid:1)lH · exp
`N−1(cid:1)
`
`exp
`
`,i is given by the following form
`
`n=0
`Hence the problem of channel estimation is reduced to finding
`the optimal a(cid:1)
`which minimizes the following:
`(cid:1)H · vt,k
`
`(cid:12)
`
`(cid:10) · qt
`
`n
`
`.
`
`(33)
`
`,
`
`(34)
`
`H
`
`H
`
`t,k
`
`j
`
`exp
`
`j
`
`,
`
`(cid:12)
`
`.
`
`(35)
`
`n
`
`(cid:1)
`(cid:11) N−1(cid:1)
`
`i∈Pf
`
`1 N
`
`×
`
`H
`
`v0
`t,k
`
`vt,k =
`
`vl
`t,k =
`
`,··· , vL−1
`(cid:9)
`, v1
`t,k
`−2πil
`dt[i] · exp
`(cid:9)
`N
`2π(i − k)n
`N
`
`n=0
`Finally, the modified polynomial channel estimator results in
`the solution
`
`(cid:1)
`(cid:1)
`
`a
`V
`
`By taking the complex derivative of (25) with respect to aH
`and, after straightforward manipulations, we obtain
`(cid:1)
`−1 · b ,
`a = U
`ut,k · uH
`(cid:1)
`U =
`t,k ,
`t [k] · ut,k .
`RH
`
`b =
`
`(t,k)∈P
`
`(t,k)∈P
`
`(27)
`
`Once the coefficients vector aH is obtained, the estimated
`channel response within the considered region can be obtained
`
`by the following equation,(cid:20)˜ht
`k[n] = aH · qt
`k,n ,
`where a and qt
`k,n are given by (27) and (21).
`
`(cid:1)
`
`(cid:4)(cid:4)(cid:4)Rt[k] − a
`
`(t,k)∈P
`
`min
`a(cid:1)
`
`(28)
`
`where vt,k is given by
`
`(cid:13)
`
`(cid:4)(cid:4)(cid:4)2
`(cid:14)H
`(cid:10) ·
`(cid:10) · qt
`
`(cid:20)˜ht
`
`B. A Modified Polynomial Channel Estimator
`In the estimator described above, a large polynomial order
`p may be required in order to represent the 2-D frequency
`channel response ˜ht
`k[n] with sufficient accuracy. If so, it is
`quite possible that some numerical problems may result from
`the large condition numbers of the matrix U in (27).
`However, if we carefully study equation (12), we will ob-
`serve that the variation of ˜ht
`k[n] with respect to the frequency
`index k is actually caused by the Fourier transformation of
`the time domain channel impulse response hl
`t[n]. Therefore,
`it is much easier to estimate hl
`t[n] directly using the same
`polynomial channel model. Thus, the time domain channel
`response hl
`t[n] requires a small order polynomial function to
`describe when the channel fading is slow.
`Hence we consider the modified polynomial channel esti-
`mator based on the model
`(cid:1)l
`a
`
`i ·(cid:9)
`
`(cid:1) i
`
`≤p
`
`hl
`t[n] =
`
`(cid:10)i
`
`,
`
`t(N + I) +n
`
`0 ≤ l ≤ L − 1 , (29)
`
`where hl
`t[n] represent the time-varying fading channel re-
`sponse for the lth path, and {a
`i} is the corresponding
`(cid:1)l
`polynomial coefficient up to order p. Written in a matrix form,
`we have the following:
`
`hl
`t[n] = a
`
`(30)
`
`(cid:1)lH · qt
`n ,
`where a(cid:1)l and qt
`n are column vectors of length (p + 1) given
`l(cid:3)H
`(cid:2)
`by,
`
`,··· , a
`(cid:1)l =
`(cid:10)p−1
`(cid:10)p
`(cid:2)(cid:9)
`(cid:9)
`a
`,··· , 1
`qt
`t(N + I) + n
`t(N + I) + n
`n =
`,
`(cid:1)L−1H(cid:3)H
`(cid:2)
`By stacking the coefficients a(cid:1)l as a column vector,
`,··· , a
`(cid:1)
`(cid:1)0H
`(cid:1)1H
`, a
`and applying the same criterion as in the 2-D channel esti-
`mator, the optimal polynomial channel coefficient vector a(cid:1)
`is
`chosen such that equation (22) is minimized, where now the
`
`(cid:3)H
`
`. (31)
`
`,
`
`(cid:1)0
`
`l
`
`(cid:1)p
`
`, a
`
`−1
`
`l
`
`(cid:1)p
`
`a
`
`a
`
`=
`
`a
`
`,
`
`(32)
`
`k[n] is the estimated frequency-domain channel re-
`where
`sponse obtained by the estimation algorithm described in the
`previous section.
`(cid:20)˜ht
`By substituting for Rt[k] from (17) into (37), we obtain
`˜ht
`k[n]
`· dt[k]
`(cid:10)(cid:12)
`2π(i − k)n
`N
`
`n=0
`
`n=0
`
`(cid:9)(cid:26)N−1
`(cid:10)(cid:9)(cid:26)N−1
`(cid:25)dt[k] =
`(cid:4)(cid:4)(cid:4)(cid:26)N−1
`(cid:4)(cid:4)(cid:4)2
`(cid:20)˜ht
`(cid:11) N−1(cid:1)
`N−1(cid:1)
`(cid:9)
`k[n]
`k[n] · exp
`˜ht
`
`n=0
`
`j
`
`+
`
`i=0
`i(cid:2)=k
`
`n=0
`
`(cid:10)H
`
`k[n]
`
`· dt[i]
`
`= V
`=
`
`(cid:1)
`(cid:1)−1 · b
`(cid:1)
`,
`vt,k · vH
`(cid:1)
`t,k ,
`t [k] · vt,k .
`RH
`
`(t,k)∈P
`
`(t,k)∈P
`
`(36)
`
`(cid:1)
`b
`
`=
`
`IV. OFDM DATA DETECTION TECHNIQUES
`In this section, we describe a MMSE-based data detection
`technique and the conventional OFDM signal detection tech-
`nique. Both detection techniques rely on the channel estimates
`that are obtained as described in section III.
`
`A. Conventional OFDM Signal Detection
`According to equation (17), we observe that the simplest
`and most commonly used OFDM detection technique performs
`detection based on the following decision symbol,
`· Rt[k] ,
`
`(cid:25)dt[k] =
`
`N
`
`n=0
`
`(cid:9)(cid:26)N−1
`(cid:20)˜ht
`(cid:4)(cid:4)(cid:4)(cid:26)N−1
`(cid:20)˜ht
`
`n=0
`
`k[n]
`
`k[n]
`
`(cid:10)H
`(cid:4)(cid:4)(cid:4)2
`
`(37)
`
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`
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`
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`
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`
`

`

`(cid:9)(cid:26)N−1
`(cid:4)(cid:4)(cid:4)(cid:26)N−1
`
`n=0
`
`n=0
`
`×
`
`(cid:20)˜ht
`(cid:20)˜ht
`
`k[n]
`
`k[n]
`
`(cid:10)H
`(cid:4)(cid:4)(cid:4)2 · +
`
`(cid:9)(cid:26)N−1
`(cid:20)˜ht
`(cid:4)(cid:4)(cid:4)(cid:26)N−1
`(cid:20)˜ht
`
`N
`
`n=0
`
`n=0
`
`k[n]
`
`k[n]
`
`(cid:10)H
`(cid:4)(cid:4)(cid:4)2
`
`· nt[k].
`
`(38)
`
`E
`
`As expected, the decision symbol is distorted not only by the
`additive Gaussian noise, but also the intercarrier interference
`(ICI) from other subcarriers. Hence the system performance
`is affected by the time variation of the channel through the
`introduction of the ICI. As shown in Section V, there exists
`an error floor resulting from the ICI, even when the signal to
`additive noise ratio is sufficient high.
`
`B. MMSE Based Detection
`Motivated by the similarities between ICI distortion in
`OFDM systems and ISI distortion in single carrier systems,
`we consider an MMSE based detection technique to suppress
`ICI. By taking both ICI and additive noise into account, the
`MMSE-based OFDM detection technique is superior to the
`conventional OFDM detection scheme described above.
`Starting from equation (14), suppose the best linear receiver
`structure of the data dt[k], which minimizes the mean square
`error, is given by
`
`(cid:25)dt[k] = ct
`H · Rt ,
`k
`where the optimal coefficient vector ct
`(cid:14)
`(cid:13)(cid:4)(cid:4)dt[k] − (cid:25)dt[k]
`k is a column vector of
`(cid:4)(cid:4)2
`size N × 1, and minimizes the following mean square error,
`min
`ct
`(cid:14)
`k
`Rt · (dt[k] − (cid:25)dt[k])H
`(cid:14)
`(cid:13)
`
`(39)
`
`(40)
`
`(41)
`
`E
`
`.
`
`By applying the orthogonality principle, we have,
`
`(cid:13)
`
`E
`
`(cid:13)
`
`= 0 .
`
`(cid:14)
`
`which is equivalent to solving the linear equation,
`· ct
`Rt · RH
`Rt · dt[k]H
`k = E
`(cid:13)
`(cid:14)
`It is straightforward to show that,
`(cid:14)
`Rt · RH
`E
`Rt · dt[k]H
`where gtk is the kth column of matrix Gt, and σ2 is the noise-
`
`to-signal ratio N0/Es. Hence the optimal linear weighting
`vector ct
`k is given by
`
`E
`
`(cid:13)
`
`E
`
`t
`
`.
`
`(42)
`
`t
`
`
`
`= GtGHt + σ2IN ,
`
`= gt
`k ,
`
`(43)
`
`(44)
`
`(cid:9)
`
`ct
`k =
`
`
`
`GtGHt + σ2IN
`
`k .
`
`(45)
`
`(cid:10)−1 · gt
`(cid:10)−1 · Rt .
`(cid:9)
`(cid:25)dt = GH
`When we replace Gt in equation (46) by (cid:25)Gt, we obtain
`(cid:9)(cid:25)Gt
`(cid:10)−1 · Rt .
`(cid:25)dt = (cid:25)GH
`(cid:25)GH
`
`If written in a concise matrix format, the MMSE-based
`OFDM estimated symbol vector is given by,
`GtGHt + σ2IN
`
`
`
`t
`
`t
`
`t + σ2IN
`
`(46)
`
`(47)
`
`(cid:14)
`
`(cid:10)−1gt
`
`k . (48)
`
`
`
`GtGHt + σ2IN
`
`k
`
`(cid:14)
`
`(cid:14)(cid:12)
`
`(cid:10)−1Gt
`
`. (49)
`
`H(cid:9)
`(cid:13)(cid:4)(cid:4)dt[k] − (cid:25)dt[k]
`(cid:4)(cid:4)2
`(cid:11)
`(cid:13)
`(cid:9)
`
`(cid:13)(cid:4)(cid:4)dt[k]−(cid:25)dt[k]
`(cid:4)(cid:4)2
`N−1(cid:1)
`
`1 N
`
`By substituting (45) into (40), we obtain the minimum mean
`square error, which is given as
`= 1− gt
`Thus, the average mean square error for the tth OFDM frame
`is
`
`E
`
`k=0
`
`tr
`
`GH
`t
`
`
`
`GtGHt + σ2IN
`
`= 1 − 1
`N
`V. NUMERICAL RESULTS
`In this section, we provide numerical and simulation results
`for the channel estimation algorithm as well as the perfor-
`mance of the different OFDM detection techniques described
`in the previous sections.
`
`A. Channel Estimation
`In general, a multivariate polynomial function of infinite
`order p → ∞ is required to describe a 2-D continuous
`frequency selective and time-varying channel ˜ht
`k[n]. However,
`since the channels in which practical OFDM systems operate
`are slowly time varying and have limited multipath delays,
`they can be represented by a small order polynomial.
`In this paper, we consider an uncoded 16-QAM OFDM
`system that has 16 subcarriers with carrier frequency of 5
`GHz. This system is transmitting over a 4-path frequency
`selective fading channel having an exponential power delay
`profile, with a bandwidth W = 1/T = 1M Hz and a Doppler
`spread Fd = 312.5Hz, corresponding to a mobile speed
`of 67.5 Km/hr. In the system, one entire OFDM channel
`estimation block is composed of T = 16 OFDM frames,
`where each frame has N = 16 orthonormal subcarriers. Pilots
`are inserted every pf = 4 subcarriers, and every pt = 4
`OFDM frames, with pilots overhead ratio 1/(pt·pf ) = 6.25%.
`The actually channel as well as the estimated channel are
`shown in Fig. 2 and Fig. 3 with system signal to noise ratio
`Es/N0 = 4dB, where the polynomial channel model is of
`order up to p = 2. As we can expect, the channel estimation
`error will become even smaller with higher order p.
`
`Channel Frequency Response |H(f,t)|
`
`Channel Frequency Response |H(f,t)|
`
`Fig. 2. Actual Channel
`
`Fig. 3. Estimated Channel
`
`As we can see from the above plot when the normalized
`Doppler spread is small, the channel experiences slow varia-
`tions over the time index and hence can be represented by a
`low order polynomial function. However, frequency-selectivity
`makes the channel changes relatively fast over the frequency
`
`as the final decision symbol for the new MMSE-based detec-
`tion technique.
`
`IEEE Communications Society / WCNC 2005
`
`43
`
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`
`Page 5 of 6
`
`

`

`domain (index). Therefore, the modified polynomial channel
`estimator can provide better estimation results with a low
`order p because it only estimates the time domain polynomial
`coefficients. The frequency response of the actual channel as
`well as the estimated channel (generated from the estimated
`coefficients a(cid:1)
`of the time domain channel impulse response)
`are shown in Fig. 4 and Fig. 5, where the polynomial model
`is of order up to p = 2.
`
`is applied to obtain the frequency channel response. Fig. 7
`demonstrates the system performance curve in terms of av-
`erage symbol errors versus the signal to noise ratio of con-
`ventional OFDM and MMSE-based OFDM receiver structure
`using the estimated channel state information. The pilots
`pattern described in Section V-B is used under the same
`OFDM system as well as the fading channel parameters.
`
`Performance Comparison among different Receiver Structures, Estimated Channel State Information
`
`Conventional−OFDM, 16−Subcarriers
`Conventional−OFDM, 256−Subcarriers
`MMSE−OFDM, 16−Subcarriers
`MMSE−OFDM, 256−Subcarriers
`IDEAL: NO−ICI
`
`100
`
`10−1
`
`10−2
`
`10−3
`
`10−4
`
`10−5
`
`Symbol−Error Probability, Ps
`
`10−6
`−10
`
`0
`
`10
`
`30
`20
`Signal−to−Noise Ratio, Eb/N0
`
`40
`
`50
`
`60
`
`Fig. 7. Error rate performance using modified polynomial channel estimation
`
`As expected, we observe from Fig. 7 that MMSE-based
`OFDM receiver structure outperforms the conventional OFDM
`receiver over the entire signal to noise range.
`
`VI. CONCLUSIONS
`In this paper, in order to combat the ICI distortion caused
`by Doppler-spread of the time-varying fading channels, the
`MMSE-based OFDM receiver structure is proposed as a
`detection technique. System performance is compared with
`the conventional OFDM receivers under the same channel
`conditions.
`By applying the a polynomial surface channel estimator,
`we provide in this paper the simulation results of the overall
`system performance, which further confirms that
`the new
`MMSE-based OFDM receiver can reduce the symbol error
`rate more than conventional OFDM receivers under the same
`Doppler-spread channel environments.
`
`REFERENCES
`[1] S. Weinstein and P. Ebert, “Data transmission by frequency-division
`multiplexing using the discrete Fourier transform,” IEEE Trans. Commun.,
`vol. 19, pp. 628-634, Oct. 1971.
`[2] P. Robertson, and S. Kaiser,“Analysis of the loss of orthogonality through
`Doppler spread in OFDM system”, in Proc. Globecom’99, pp. 701-706,
`Dec., 1999.
`[3] T. Wang, J. Proakis, J. Zeidler, “Performance Analysis of High QAM
`OFDM System Over Frequency Selective Time-Varying Fading Channel,”
`in Proc. 14th IEEE PIMRC, vol. 1, pp. 793-798, Sept., 2003.
`[4] X. Wang and K. J. R. Liu, “OFDM channel estimation based on time-
`frequency polynomial model of fading multipath channel”, Vehicular
`Technology Conference 2001, Vol. 1, pp. 460-464, 2001.
`[5] J. G. Proakis,Digital Communications, 2nd ed., New York: McGraw-Hill,
`1989.
`[6] W. C. Jakes, Microwave Mobile Communications, IEEE Press, Reprinted,
`1994.
`
`Channel Frequency Response |H(f,t)|
`
`Fig. 5. Estimated Channel
`Fig. 4. Actual Channel
`B. Performance Comparison under Perfectly Known Channel
`Before applying the channel estimation technique proposed
`in Section III, a comparison among different detection schemes
`with perfect channel state information is insightful. Fig. 6
`demonstrates the system performance curves in terms of
`the average number of symbol errors versus the signal to
`noise ratio of conventional OFDM and MMSE-based OFDM
`receiver structures. The simulation is performed on the same
`OFDM system with 16 and 256 subcarriers over the frequency
`selective time varying fading channel with the same parame-
`ters as is described in Section V-A.
`
`Performance Comparison among different Receiver Structures, Ideal Known Channel
`
`Conventional−OFDM, 16−Subcarriers
`Conventional−OFDM, 256−Subcarriers
`MMSE−OFDM, 16−Subcarriers
`MMSE−OFDM, 256−Subcarriers
`IDEAL: NO−ICI
`
`0
`
`10
`
`30
`20
`Signal−to−Noise Ratio, Eb/N0
`
`40
`
`50
`
`60
`
`100
`
`10−1
`
`10−2
`
`10−3
`
`10−4
`
`10−5
`
`Symbol−Error Probability, Ps
`
`10−6
`−10
`
`Fig. 6. Error rate performance for a known channel
`
`As we observe from Fig. 6, the conventional OFDM receiver
`structure has an error floor and its performance is limited
`by the ICI. The proposed MMSE-based receiver structure
`performs not only slightly better in low signal to noise ratio
`range, but is also able to remove the error floor completely in
`high signal to noise range.
`C. Performance Comparison under Channel Estimation
`When the channel state information is not perfectly known
`at the receiver, the modified polynomial channel estimator
`
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`
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`
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`
`Page 6 of 6
`
`