Broadband MIMO-OFDM Wireless
`Communications
`
`GORDON L. STÜBER, FELLOW, IEEE, JOHN R. BARRY, MEMBER, IEEE,
`STEVE W. MCLAUGHLIN, SENIOR MEMBER, IEEE, YE (GEOFFREY) LI, SENIOR MEMBER, IEEE,
`MARY ANN INGRAM, SENIOR MEMBER, IEEE, AND THOMAS G. PRATT, MEMBER, IEEE
`
`Invited Paper
`
`frequency division multiplexing (OFDM) is a
`Orthogonal
`popular method for high data rate wireless transmission. OFDM
`may be combined with antenna arrays at the transmitter and
`receiver to increase the diversity gain and/or to enhance the
`system capacity on time-variant and frequency-selective channels,
`resulting in a multiple-input multiple-output (MIMO) config-
`uration. This paper explores various physical
`layer research
`challenges in MIMO-OFDM system design, including physical
`channel measurements and modeling, analog beam forming tech-
`niques using adaptive antenna arrays, space–time techniques for
`MIMO-OFDM, error control coding techniques, OFDM preamble
`and packet design, and signal processing algorithms used for per-
`forming time and frequency synchronization, channel estimation,
`and channel tracking in MIMO-OFDM systems. Finally, the paper
`considers a software radio implementation of MIMO-OFDM.
`Keywords—Adaptive antennas, broadband wireless, mul-
`tiple-input multiple-output (MIMO), orthogonal frequency division
`multiplexing (OFDM), software radio, space–time coding, syn-
`chronization.
`
`I. INTRODUCTION
`
`Orthogonal frequency division multiplexing (OFDM)
`has become a popular technique for transmission of signals
`over wireless channels. OFDM has been adopted in several
`wireless standards such as digital audio broadcasting (DAB),
`digital video broadcasting (DVB-T), the IEEE 802.11a [1]
`local area network (LAN) standard and the IEEE 802.16a [2]
`metropolitan area network (MAN) standard. OFDM is also
`being pursued for dedicated short-range communications
`(DSRC) for road side to vehicle communications and as
`a potential candidate for fourth-generation (4G) mobile
`wireless systems.
`
`Manuscript received June 23, 2003; revised November 3, 2003. This work
`was supported in part by the Yamacraw Mission (http://www.yamacraw.org)
`and in part by the National Science Foundation under Grant CCR-0121565.
`The authors are with the School of Electrical and Computer Engineering,
`Georgia Institute of Technology, Atlanta, GA 30332 USA.
`Digital Object Identifier 10.1109/JPROC.2003.821912
`
`into a
`OFDM converts a frequency-selective channel
`parallel collection of frequency flat subchannels. The sub-
`carriers have the minimum frequency separation required
`to maintain orthogonality of
`their corresponding time
`domain waveforms, yet the signal spectra corresponding to
`the different subcarriers overlap in frequency. Hence, the
`available bandwidth is used very efficiently. If knowledge of
`the channel is available at the transmitter, then the OFDM
`transmitter can adapt its signaling strategy to match the
`channel. Due to the fact that OFDM uses a large collection
`of narrowly spaced subchannels, these adaptive strategies
`can approach the ideal water pouring capacity of a fre-
`quency-selective channel. In practice this is achieved by
`using adaptive bit loading techniques, where different sized
`signal constellations are transmitted on the subcarriers.
`OFDM is a block modulation scheme where a block of
`information symbols is transmitted in parallel on
`sub-
`carriers. The time duration of an OFDM symbol is
`times
`larger than that of a single-carrier system. An OFDM modu-
`lator can be implemented as an inverse discrete Fourier trans-
`form (IDFT) on a block of
`information symbols followed
`by an analog-to-digital converter (ADC). To mitigate the ef-
`fects of intersymbol interference (ISI) caused by channel
`time spread, each block of
`IDFT coefficients is typically
`preceded by a cyclic prefix (CP) or a guard interval consisting
`of
`samples, such that the length of the CP is at least equal
`to the channel length. Under this condition, a linear convo-
`lution of the transmitted sequence and the channel is con-
`verted to a circular convolution. As a result, the effects of
`the ISI are easily and completely eliminated. Moreover, the
`approach enables the receiver to use fast signal processing
`transforms such as a fast Fourier transform (FFT) for OFDM
`implementation [3]. Similar techniques can be employed in
`single-carrier systems as well, by preceding each transmitted
`data block of length
`by a CP of length
`, while using fre-
`quency-domain equalization at the receiver.
`
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`Fig. 1. Q  L MIMO-OFDM system, where Q and L are the numbers of inputs and outputs,
`respectively.
`
`Multiple antennas can be used at the transmitter and
`receiver, an arrangement called a multiple-input mul-
`tiple-output
`(MIMO) system. A MIMO system takes
`advantage of the spatial diversity that is obtained by spa-
`tially separated antennas in a dense multipath scattering
`environment. MIMO systems may be implemented in a
`number of different ways to obtain either a diversity gain
`to combat signal fading or to obtain a capacity gain. Gen-
`erally, there are three categories of MIMO techniques. The
`first aims to improve the power efficiency by maximizing
`spatial diversity. Such techniques include delay diversity,
`space–time block codes (STBC) [4], [5] and space–time
`trellis codes (STTC) [6]. The second class uses a layered
`approach to increase capacity. One popular example of
`such a system is V-BLAST suggested by Foschini et al. [7]
`where full spatial diversity is usually not achieved. Finally,
`the third type exploits the knowledge of channel at the
`transmitter. It decomposes the channel coefficient matrix
`using singular value decomposition (SVD) and uses these
`decomposed unitary matrices as pre- and post-filters at the
`transmitter and the receiver to achieve near capacity [8].
`OFDM has been adopted in the IEEE802.11a LAN and
`IEEE802.16a LAN/MAN standards. OFDM is also being
`considered in IEEE802.20a, a standard in the making for
`maintaining high-bandwidth connections to users moving at
`speeds up to 60 mph. The IEEE802.11a LAN standard op-
`erates at raw data rates up to 54 Mb/s (channel conditions
`permitting) with a 20-MHz channel spacing, thus yielding a
`bandwidth efficiency of 2.7 b/s/Hz. The actual throughput is
`highly dependent on the medium access control (MAC) pro-
`tocol. Likewise, IEEE802.16a operates in many modes de-
`pending on channel conditions with a data rate ranging from
`4.20 to 22.91 Mb/s in a typical bandwidth of 6 MHz, trans-
`lating into a bandwidth efficiency of 0.7 to 3.82 bits/s/Hz.
`Recent developments in MIMO techniques promise a signif-
`icant boost in performance for OFDM systems. Broadband
`MIMO-OFDM systems with bandwidth efficiencies on the
`order of 10 b/s/Hz are feasible for LAN/MAN environments.
`The physical (PHY) layer techniques described in this paper
`are intended to approach 10 b/s/Hz bandwidth efficiency.
`This paper discuss several PHY layer aspects broadband
`MIMO-OFDM systems. Section II describes the basic
`
`MIMO-OFDM system model. All MIMO-OFDM receivers
`must perform time synchronization, frequency offset esti-
`mation, and correction and parameter estimation. This is
`generally carried out using a preamble consisting of one or
`more training sequences. Once the acquisition phase is over,
`receiver goes into the tracking mode. Section III provides an
`overview of the signal acquisition process and investigates
`sampling frequency offset estimation and correction in
`Section IV. The issue of channel estimation is treated in Sec-
`tion V. Section VI considers space–time coding techniques
`for MIMO-OFDM, while Section VII discusses coding
`approaches. Adaptive analog beam forming approaches
`can be used to provide the best possible MIMO link.
`Section VIII discusses various strategies for beamforming.
`Section IX very briefly considers medium access control
`issues. Section X discusses a software radio implementation
`for MIMO-OFDM. Finally, Section XI wraps up with some
`open issues concluding remarks.
`
`II. MIMO-OFDM SYSTEM MODEL
`
`A multicarrier system can be efficiently implemented in
`discrete time using an inverse FFT (IFFT) to act as a modu-
`lator and an FFT to act as a demodulator. The transmitted data
`are the “frequency” domain coefficients and the samples at
`the output of the IFFT stage are “time” domain samples of the
`transmitted waveform. Fig. 1 shows a typical MIMO-OFDM
`implementation.
`Let
`denote the length-
`data symbol block. The IDFT of the date block
`yields the
`time domain sequence
`, i.e.,
`
`IFFT
`
`(1)
`
`To mitigate the effects of channel delay spread, a guard in-
`terval comprised of either a CP or suffix is appended to the
`sequence
`. In case of a CP, the transmitted sequence with
`guard interval is
`
`(2)
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`the received samples
`demodulated using an
`
`are repeated
`-point FFT as
`
`times and
`
`FFT
`
`(3)
`
`(4)
`
`where
`. The demodulated OFDM sample
`matrix
`of dimension (
`) for the th subcarrier can
`be expressed in terms of the transmitted sample matrix
`of
`of dimension (
`), the channel coefficient matrix
`dimension (
`) and the additive white Gaussian noise
`matrix
`of dimension (
`) [24] as
`
`(5)
`
`can viewed as either a collection of
`, and
`,
`where
`matrices of dimension
`or as a collection of
`vectors of length
`.
`
`A. Preamble Design for MIMO-OFDM Systems
`Least square channel estimation schemes require that all
`training symbol matrices
`,
`,
`be unitary so that only
`OFDM symbols
`are needed for channel estimation [25]. A straightforward so-
`lution is to make each
`a diagonal matrix. However, the
`power of the preamble needs to be boosted by
`dB
`in order to achieve a performance similar to the case when
`the preamble signal is transmitted from all the antennas. This
`has the undesirable effect of increasing the dynamic range re-
`quirements of the power amplifiers. Hence, methods are re-
`quired so that sequences can be transmitted from all the an-
`tennas while still having unitary
`matrices. One approach
`adapts the work by Tarokh et al. on space–time block codes
`[5], [26]. For
`and
`orthogonal designs exist.
`For example, for
`and , we can choose the preamble
`structures of the form
`
`(6)
`
`(7)
`
`. This
`vector
`is the length-
`where
`results in unitary
`matrices. As it turns out, transmitting
`the same sequence from all the antennas in this fashion is
`advantageous when performing synchronization. A similar
`structure for
`exists. For other values of
`, a least
`squares (LS) solution for the channel estimates can be ob-
`tained by either transmitting more than
`training sequences
`or by making the training symbol matrices unitary by using
`a Gram–Schmidt orthonormalization procedure as described
`in [24].
`
`B. Pilot Insertion
`Channel coefficients require constant tracking. This is
`aided by inserting known pilot symbols at fixed or vari-
`able subcarrier positions. For example, the IEEE 802.16a
`
`Fig. 2. Frame structure for the Q  L OFDM system.
`
`is the guard interval length in samples, and
`where
`is the residue of modulo
`. The OFDM complex enve-
`lope is obtained by passing the sequence
`through a pair
`of ADCs (to generate the real and imaginary components)
`with sample rate
`s, and the analog
`and
`signals are
`upconverted to an RF carrier frequency. To avoid ISI, the CP
`length must equal or exceed the length of the discrete-time
`channel impulse response
`. The time required to transmit
`one OFDM symbol
`is called the OFDM
`symbol time. The OFDM signal is transmitted over the pass-
`band RF channel, received, and downconverted to base band.
`Due to the CP, the discrete linear convolution of the trans-
`mitted sequence with the channel impulse response becomes
`a circular convolution. Hence, at the receiver the initial
`samples from each received block are removed, followed by
`an
`-point discrete Fourier transform (DFT) on the resulting
`sequence.
`The frame structure of a typical MIMO-OFDM system is
`shown in Fig. 2. The OFDM preamble consists of
`training
`symbols of length
`, where
`,
`and an integer that divides
`. Often the length of the guard
`interval in the training period is doubled; for example, in
`IEEE802.16a [1], to aid in synchronization, frequency offset
`estimation and equalization for channel shortening in cases
`where the length of the channel exceeds the length of the
`guard interval.
`First consider the preamble portion of the OFDM frame.
`The length-
`preamble sequences are obtained by
`exciting every th coefficient of a length-
`frequency-do-
`main vector with a nonzero training symbol from a chosen
`alphabet (the remainder are set to zero). The frequency-do-
`main training sequences transmitted from the th antenna are
`, where
`and
`The individual length-
`time domain training sequences
`are obtained by taking an
`-point IDFT of the sequence
`, keeping the first
`time-domain coefficients
`and discarding the rest. A CP is appended to each length-
`time-domain sequence. Let
`be the vector of subchannel
`coefficients between the th transmit and the th receive an-
`be the received sample sequence at
`tenna and let
`the th receiver antenna. After removing the guard interval,
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`is the optimum coarse timing
`where
`acquisition instant and
`. The fre-
`quency offset can then be removed from the
`received sample sequence by multiplying it with
`during the preamble and
`during the data portion. Note
`that by reducing the length of the training symbol
`by a factor of
`, the range of the frequency offset
`estimate in the time domain can be increased by
`a factor of
`.
`Step 3) Residual Frequency Offset Correction—Should
`the range of the time domain frequency offset es-
`timation be insufficient, frequency-domain pro-
`cessing can be used. Suppose that the same fre-
`quency-domain training sequence
`is
`transmitted from all the antennas. The residual
`frequency offset, that is, an integer multiple of
`the subcarrier spacing, can be estimated by com-
`puting a cyclic cross-correlation of
`with the received, frequency corrected (from Step
`II), demodulated symbol sequence, viz.,
`
`where
`
`(10)
`
`(11)
`
`The residual frequency offset is estimated as
`. Note that
`the fractional part of the relative frequency offset
`is estimated in the time domain in Step II while
`the integer part is estimated in the frequency do-
`main in Step III.
`Step 4) Fine Time Synchronization—Fine time acquisi-
`tion locates the start of the useful portion of the
`OFDM frame to within a few samples. Once the
`frequency offset is removed, fine time synchro-
`nization can be performed by cross correlating the
`frequency corrected samples with the transmitted
`preamble sequences. The fine time synchroniza-
`tion metric is
`
`(12)
`
`.
`where
`For systems using two and four and eight transmit
`antennas using the orthogonal designs discussed
`in Section II-A, only one cross correlator is
`needed per receiver antenna. Once again the
`threshold is set at 10% of the energy contained
`in
`received samples. Since fine time synchro-
`nization is computationally expensive process, it
`is carried out for a small window centered around
`the coarse time synch. instant
`.
`Finally, the net time synchronization instant for the en-
`tire receiver is selected to be
`.
`An added negative offset of a few samples is applied to the
`
`Fig. 3. Pilot tone generation.
`
`standard recommends the insertion of eight pilot tones at
`fixed positions on subcarriers [12, 36, 60, 84, 172, 196, 220,
`244] (assuming
`). Fig. 3 shows the method for
`generating the pilot sequences used in the IEEE 802.16a
`standard. In the downlink (DL) and the uplink (UL), the
`shift register is initialized with sequences as shown. A 0 at
`the output
`is mapped to
`and a 1 is mapped to
`.
`For a MIMO system with
`and
`antennas, the pilot
`sequences
`can be coded over space and time to form
`structures in (6) and (7), respectively, thereby admitting a
`simple LS channel estimate. For more information on the
`pilot sequence construction, readers can refer to [27].
`
`III. SYNCHRONIZATION IN THE ACQUISITION MODE
`
`Time and frequency synchronization can be performed se-
`quentially in the following steps [28].
`Step 1) Coarse Time Synchronization and Signal Detec-
`tion—Coarse time acquisition and signal detec-
`tion locates the start of an OFDM frame over an
`approximate range of sample values. Due to the
`presence of the CP (or suffix), coarse time ac-
`quisition during the preamble can be performed
`by correlating the received samples that are at a
`distance of
`from each other over a length-
`window ([25], [29]), viz.
`
`(8)
`
`. In ad-
`where
`dition to maximizing
`, it should also exceed a
`certain threshold to reduce the probability of false
`alarm (
`). We chose the threshold to be 10%
`of the incoming signal energy of the correlation
`window.
`Step 2) Frequency Offset Estimation in the Time Do-
`main—Any frequency offset between
`the
`transmitter and the receiver local oscillators
`is reflected in the time domain sequence as a
`progressive phase shift
`, where
`is the frequency offset and is defined as the ratio
`of the actual frequency offset to the intercarrier
`spacing. A frequency offset estimate of up to
`subcarrier spacings can be obtained based
`on the phase of the autocorrelation function in
`(8) as follows:
`
`(9)
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`Table 1
`SUI-4 Channel Model
`
`fine time synchronization instant in order to ensure that the
`OFDM windows for all the receivers falls into an ISI-free
`zone.
`
`A. Example
`Consider a 2
`4 broadband MIMO-OFDM
`2 and a 4
`system [2] operating at a carrier frequency of 5.8 GHz on the
`SUI-4 channel shown in Table 1. The OFDM signal occupies
`a bandwidth of 4.0 MHz. The uncorrected frequency offset
`(
`) is 1.25 subcarrier spacings. The OFDM blocksize is
`, and the guard interval is kept at
`. Out of
`256 tones, the dc tone and 55 other tones at the band edges
`are set to zero. Hence, the number of used tones
`.
`The length of the sequences used in the preamble is varied
`from
`to
`to
`. The preamble insertion period
`is chosen to be ten. STBCs are used to encode the data. For
`a 2
`2 system, the Alamouti STBC is used with code rate
`1, whereas for a 4
`4 system, code rate is 3/4 [26]. In the
`data mode, each of the tones is modulated using a 16-QAM
`constellation and no channel coding is employed. LS channel
`estimates obtained using the preamble are used to process the
`entire frame [28]. For training sequences of length
`,
`frequency-domain linear interpolation and extrapolation are
`used. Afterwards, frequency-domain smoothing is used, such
`that channel estimates at the band-edges are kept as they are,
`whereas all the other channel estimates are averaged using
`
`(13)
`
`Fig. 4 shows the coarse and fine time synchronization per-
`formance for a 4
`4 MIMO-OFDM system with
`,
`, and signal-to-noise ratio (SNR) of 10 dB.
`Fig. 5 shows the overall bit error rate (BER) performance
`of a 2
`2 MIMO-OFDM system using the suggested algo-
`rithms.
`
`IV. SAMPLE FREQUENCY OFFSET CORRECTION AND
`TRACKING
`
`MIMO-OFDM schemes that use coherent detection need
`accurate channel estimates. Consequently, the channel coef-
`ficients must be tracked in a system with high Doppler. In
`the broadband fixed wireless access (BFWA) system IEEE
`
`802.16a, the channel is nearly static. However, channel vari-
`ations are still expected due to the presence of sampling fre-
`quency offset between transmitter and the receiver RF oscil-
`lators. Generally, the components in the customer premises
`equipment (CPE) have a low tolerance with typical drift of
`20 parts per million (ppm). This means a signal with a BW
`of 4 MHz produces an offset of 80 samples for every 1 s of
`transmission. Sample frequency offset causes phase rotation,
`amplitude distortion and loss in synchronization.
`Even after successful signal acquisition and synchroniza-
`tion, the OFDM system must guard against sample frequency
`offset (SFO) and phase offset. It must also guard against drift
`in the RF local oscillator and sampling clock frequency with
`time [30].
`Let
`
`be the sampling time at the receiver and let
`be the normalized offset in the sampling
`time. The received and demodulated OFDM symbol with the
`sampling time offset can be approximated by (14), at the
`bottom of the page. where
`,
`and
`is the running index of the OFDM
`symbol in time, and
`.
`Due to the sampling frequency offset
`, the received
`demodulated symbol suffers phase rotation as well as
`amplitude distortion. In general, the value of
`is very
`small. For example, for a sampling clock tolerance of 20
`ppm and sampling frequency
`MHz,
`.
`Hence,
`and its effect is negligible. With this
`assumption, the demodulated OFDM sample matrix
`in
`(5) becomes
`
`(15)
`
`is diagonal matrix representing the phase ro-
`where
`tations of the received demodulated samples due to the pres-
`ence of sampling frequency offset.
`
`A. Sample Frequency Offset Estimation
`If the MIMO-OFDM transmission is being carried out in
`blocks of
`OFDM symbols, then phase rotation between
`consecutive blocks of OFDM symbols increases in a linear
`fashion. Hence let the received sample matrix corresponding
`to the preamble be given by
`
`(16)
`
`The received sample matrix for the next block of OFDM
`symbols corresponding to the pilot tones is then given by
`
`(17)
`If the channel does not change much for
`consecutive
`blocks of OFDM symbols as is the case for wireless
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`Fig. 4. Coarse and fine time synchronization for a 4  4 system with N = 128, SNR = 10 dB,
`freq. off. + = 1 + 0:25. Steps IB, IIIB.
`
`Fig. 5. Uncoded BER as a function of SNR for a 2  2 system using 16-QAM modulation, P = 10.
`
`LAN/MAN applications, then we can correlate
`and
`to obtain an initial estimate of
`per subcarrier as
`
`trace
`
`B. Channel Estimation
`
`Once initial estimates of
`are obtained, channel estima-
`tion can be carried out using the LS technique as
`
`(19)
`
`This estimate of the sampling frequency offset estimate is
`then averaged over all the subcarriers.
`
`(18)
`
`where
`. This ensures that the initial effect of
`the sampling frequency offset is taken into account when
`the channel is estimated. More elaborate channel estimation
`schemes are considered in Section V.
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`Fig. 6. BER performance for a 4  4 system with

= 0 and 10 ppm, with frame length of 80
`OFDM symbols.
`
`C. Sampling Frequency Offset Tracking
`are obtained, open loop sam-
`Once initial estimates of
`pling frequency offset estimation is obtained by minimizing
`the metric
`
`trace
`
`(20)
`
`. This results in the LS solution of the
`
`where
`type
`
`BER results. The bottom curve is the ideal results with per-
`fect synchronization and channel estimation and zero sam-
`pling frequency offset. The next curve up shows the per-
`formance with synchronization and channel estimation, but
`without any sampling frequency offset present in the system.
`The next curve up shows the performance with synchroniza-
`tion, channel estimation, and sample frequency offset correc-
`tion. The top curve shows the performance when an uncor-
`rected sample frequency offset is present.
`
`(21)
`
`V. MIMO-OFDM CHANNEL ESTIMATION
`
`introduced
`where is a small number of the order of 1 10
`to guard against ill-conditioned matrices and is the identity
`matrix. If the variance of the noise at the receiver is known
`then this factor can be applied instead of
`. From
`, the
`new value of sampling frequency offset may be extracted by
`correlating the diagonal elements of the
`matrix as
`
`(22)
`
`is then passed through a first-order
`The new value of
`low-pass filter and the output of the filter is used to obtain
`the filtered estimate of
`. This then is used to form the new
`. The sampling frequency offset in the tracking
`estimate
`.
`mode is then compensated for as
`
`Channel state information is required in MIMO-OFDM
`for space–time coding at the transmitter and signal detection
`at receiver. Its accuracy directly affects the overall perfor-
`mance of MIMO-OFDM systems. In this section, we present
`several approaches for MIMO-OFDM channel estimation.
`
`A. Basic Channel Estimation
`time and
`Channel estimation for OFDM can exploit
`frequency correlation of the channel parameters. A basic
`channel estimator has been introduced in [31].
`As discussed before, for a MIMO system with
`transmit
`antennas, the signal from each receive antenna at the th sub-
`channel of the th OFDM block can be expressed as1
`
`D. Example
`Simulations are carried out for the same 4
`4 broadband
`fixed wireless access system described in Section III. The
`sampling frequency offset
`is 10 parts per million (ppm)
`and is allowed to vary around that value in a random walk.
`Hence
`ppm
`ppm . Fig. 6 shows the
`
`th
`is the channel frequency response at the
`where
`subchannel of the th OFDM block corresponding to the th
`transmit antenna, and
`is the additive white Gaussian
`noise. The challenge with MIMO channel estimation is
`
`1We omit the index for receive antenna here, since channel estimation for
`each receive antenna is performed independently.
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`Fig. 7. Basic channel parameter estimator for MIMO-OFDM with two transmit antennas.
`
`that each received signal corresponds to several channel
`parameters.
`Since the channel response at different frequencies is cor-
`related, channel parameters at different subcarriers can be ex-
`pressed as
`
`is the temporal estimation of channel parameter
`where
`vector, defined as
`
`and
`
`,
`
`,
`
`, and
`
`are definded as
`
`(23)
`
`. The pa-
`and
`for
`rameter
`depends on the ratio of the delay span of
`wireless channels and the OFDM symbol duration, and
`. Hence, to obtain
`, we only
`
`.
`need to estimate
`from the th transmit an-
`If the transmitted signals
`2, then
`, a temporal es-
`tenna are known for
`, can be found by minimizing the following
`timation of
`cost function:
`
`Direct calculation in [31] yields
`
`or
`
`...
`
`...
`
`...
`
`...
`
`...
`
`...
`
`(24)
`
`(25)
`
`(26)
`
`...
`
`...
`
`2During the training period, transmitted signals are know to the receiver.
`In the data transmission mode, a decision-directed approach can be used
`
`(27)
`
`(28)
`
`and
`
`respectively.
`From the temporal estimation of channel parameters, ro-
`bust estimation can be obtained using the approach devel-
`oped in [32], which exploits the time correlation of channel
`parameters. Robust estimation of channel parameter vectors
`at the th OFDM block can be obtained by
`
`) are the coefficients for the robust channel
`’s (
`where
`estimator [31], [32].
`Fig. 7 illustrates the block diagram of the basic channel
`estimator for a MIMO-OFDM system with two transmit
`antennas. To calculate temporal estimation in the figure, a
`matrix inversion is need to get the temporal
`and
`. In general, a
`estimation of
`matrix inversion is required for a MIMO-OFDM system
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`

`Fig. 8.
`(a) WER and (b) MSE of a 2  2 MIMO-OFDM system when a wireless channel with
`40-Hz Doppler frequency and the two-ray and the COST207 HT delay profiles, respectively.
`
`transmit antennas, which is computationally in-
`with
`tensive. To reduce the computational complexity,
`the
`significant-tap-catching estimator has been proposed in
`[31].
`To study the impact of channel estimation error on
`MIMO-OFDM performance, a 2
`2 MIMO-OFDM system
`with space–time coding is simulated. The parameters of the
`simulated OFDM system are similar to those in [31] and
`[32]. The OFDM signal consists of 128 tones, including
`eight guard tones on each side, and with 160- s symbol
`duration. A 40- s guard interval is used, resulting in a total
`block length
`s and a subchannel symbol rate
`
`kbaud. A 16-state 4-PSK space–time code is used.
`In brief, the overall system can transmit data at a rate of
`1.18 Mb/s over an 800-kHz channel, i.e., the bandwdith
`efficiency is 1.475 b/s/Hz.
`Fig. 8 compares the performance between channels with
`the two-ray and the COST207 HT delay profiles with
`Hz. From the figure, the system has the same perfor-
`mance when the ideal parameters of the previous OFDM
`block are used for decoding. However, when estimated pa-
`rameters are used, the system has better performance for the
`two-ray delay profile than for the HT profile, since the es-
`timator has lower MSE for the two-ray delay profile as we
`
`STÜBER et al.: BROADBAND MIMO-OFDM WIRELESS COMMUNICATIONS
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`

`can see from Fig. 8(b). When the seven-tap or nine-tap sig-
`nificant-tap-catching technique in [31] is used, the required
`SNR for a 10% WER is 8 dB for the two-ray delay profile
`and and about 8.6 dB for the COST207 HT delay profile, re-
`spectively.
`
`B. Optimum Training Sequences for Channel Estimation
`In this section, we describe optimum training that can
`simplify initial channel estimation and optimize estimation
`performance.
`For simplicity, we assume that modulation results in con-
`stant-modulus signals, that is,
`. From (27)
`
`the performance of temporal channel parameter estimation
`during training period.
`
`C. Simplified Channel Estimation
`In the above section, we have introduced optimum se-
`quences for channel estimation, which not only improve the
`initial channel estimation during the training period but also
`simplify channel estimation. During the data transmission
`period (
`), transmitted symbols are random; therefore,
`. Here, we introduce an approach
`we cannot control
`that simplifies channel estimation during data transmission
`mode.
`From (25), for the th OFDM block, we have
`
`where
`
`since
`
`’s for
`
`for
`,
`, where
`
`denotes the unit impulse function. Consequently,
`, where
`is a
`identity matrix. If the
`’s are chosen such that
`training sequences
`for
`, then, from (26),
`, and no matrix
`inversion is required for channel estimation.
`To find
`for
`with
`and
`, it is sufficient to find
`for
`only consists of
`if
`.
`To construct training sequences such that
`, let
`be any se-
`the training sequence for the first antenna
`quence that is good for time and frequency synchronization
`and other properties, such as low PAPR. For a MIMO-OFDM
`system with the number of transmit antennas,
`, less than or
`equal to
`, let
`
`and
`, where
`for
`denotes the largest integer no larger than . Then for any
`
`(29)
`
`Note that
`
`; therefore
`
`and
`
`(30)
`
`(31)
`
`(32)
`
`Consequently,
`
`or
`), which results
`.
`
`,
`
`for
`(equivalent to
`for all
`. If
`in
`.
`for all
`Hence,
`It should be indicated that the above optimum training
`sequence design approach is not applicable to those
`MIMO-OFDM systems with more than
`transmit
`antennas.
`It is proved in [31] that the MSE of the basic temporal
`channel estimation reaches the low bound when
`for
`. Therefore, optimum training sequence can not only
`reduce the complexity of channel estimation but also improve
`
`(33)
`
`for
`. In the above expression, the subscript has
`been added to indicate that those vectors and matrices are re-
`lated to the th OFDM block. From the discussion in the pre-
`vious section, for an OFDM system with constant modulus
`for
`, and, therefore
`modulation,
`
`(34)
`
`for
`
`. From the above equations, if
`are known, then
`estimated without any matrix inversion.
`If robust estimation of channel parameter vectors at pre-
`vious OFDM block,
`’s for
`are used to
`substitute
`on the right side of (34), then
`
`’s for
`can be
`
`(35)
`
`, and the matrix inversion in (26) can be
`
`for
`avoided.
`The simplified channel estimation described above signif-
`icantly reduces the computational complexity of channel es-
`timation; it may also cause some performance degradation.
`However, it is demonstrated by theoretical analysis and com-
`puter simulation in [33] that the performance degradation is
`negligible.
`
`D. Enhanced Channel Estimation
`In [31] and [33] (Sections V-A–V-C), we have intro-
`duced channel parameter estimators and optimum training
`sequences for OFDM with multiple transmit antennas.
`Furthermore, for a MIMO-OFDM system where many inde-
`pendent channels with the same delay profile are involved,
`the channel delay profile can be more accurately estimated.
`By exploiting the estimated channel delay profile, channel
`parameter estimation can be further improved.
`From the above discussion, for the
`th OFDM block,
`channel parameters corresponding to the
`th transmit and
`in (23), can be estimated
`the th receive antenna pairs,
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`

`Fig. 9. MSE comparison of the basic and the enhanced channel estimation techniques for
`a 4  4 MIMO-OFDM system.
`
`using the correlation of channel parameters at different times
`, the estimated
`, the channel
`and frequencies. With
`frequency response at the th tone of the
`th OFDM block
`can be reconstructed by
`
`(36)
`
`with
`If
`
`.
`is,
`that
`the channel’s delay profile,
`for
`’s
`, is known, and it can be used to
`, the
`reconstruct the channel frequency response from
`can be significa