 Research
 Open Access
Reduced feedback for selective fading MIMO broadcast channels
 Lina Mroueh^{1}Email author,
 Stéphanie RouquetteLéveil^{2} and
 JeanClaude Belfiore^{3}
https://doi.org/10.1186/16871499201145
© Mroueh et al; licensee Springer. 2011
 Received: 5 October 2010
 Accepted: 27 July 2011
 Published: 27 July 2011
Abstract
In this article, we analyze the selective multipleinput multipleoutput broadcast channel, where links are assumed to be selective in both time and frequency. The assumption of full channel knowledge at the transmitter side requires a large amount of feedback, and it is therefore not practical to be implemented in real systems. A more feasible solution with finite rate feedback originally proposed by Jindal in IEEE Trans. Inf. Theory is applied here to the selective fading case, where the minimal number of feedback bits required to achieve the full multiplexing is derived. We show that the correlation between time frequency channels can be used in order to minimize the number of feedback bits to the transmitter side while conserving the maximal multiplexing gain. Finally, the practical implementation of a timefrequency channel quantization scheme is addressed, and a lowcomplexity scheme that also achieves the multiplexing gain is proposed.
Keywords
 timefrequency selective channel
 underspread channel
 broadcast channel
 random vector quantization
 Grassmann manifold
 multiplexing gain
1. Introduction and motivation
The challenge of the next generation of wireless communication is to offer at the receiver side a high data rate with a high quality of service. The multipleinput multipleoutput (MIMO) transmission and the multiuser communication have been recently introduced in almost all new standards. These two techniques of transmission have been extensively studied in the literature over the last few years aiming to boost the quality of service of wireless systems close to the one of wireline systems.
In this article, we consider the broadcast channel (BC), where a common source transmits data simultaneously to different receivers that do not cooperate. We assume that communication occurs on channels that exhibits memory in both time and frequency. Our objective here is to propose a transmission strategy for the high data rate communications when the channel is known partially at the transmitter side.
When full CSIT is assumed at the transmitter side, the capacity region of the BC has been characterized in [1]. It has been shown that the Dirty Paper coding technique achieves the maximal capacity region. Despite of its optimality, this technique is not feasible to be implemented in practical system as it brings high complexity at the transmitter side. Many more practical downlink transmission techniques including linear precoding schemes (e.g., channel inversion [2] and block diagonalization (BD) in [3]) and nonlinear precoding schemes (e.g., vector perturbation technique [4]) have been proposed. Although the vector perturbation technique improves the error performance compared to linear precoding schemes, this comes at the expense of an increased complexity caused by the use of a sphere encoder at the transmitter side. Linear precoding schemes considered in this paper are less complex to operate than other precoding schemes and are shown to achieve the full multiplexing gain at the high SNR regime [5].
The full CSIT assumption is not generally of a practical interest as it requires a large amount of feedback. The quantization schemes of flat fading or frequency selective MIMO BC has been extensively addressed in literature [6][13]. A limited feedback solution with finite rate feedback for the flat fading channel has been studied by Jindal for the case of single antenna users [6] and later extended to the MIMO case in [7]. More realistic feedback schemes with noisy feedback scheme and delay were considered in [8]. For the frequency selective fading MIMO BC, most of stateoftheart techniques proposed in literature, e.g., [9][12] deal with the quantization of the frequency response with a focus on the quantization codebook design. A complete tutorial on these limited feedback strategies with their applications in standards can be found in [13]. The majority of these works pointed out that using an adequate number of feedback bits that scales as SNR, the full multiplexing gain can be also achieved using limited feedback.
While most of the above results address the case where the channels between source and destination are assumed to be flat fading or frequency selective, real communications occur on channel that exhibits memory in time and frequency [14]. The timefrequency selective channel gives an accurate model of the wireless channel, and especially for the case of applications that exhibit duration and bandwidth that exceed the coherence time and bandwidth of the channel. A complete description of the considered channel model can be found in [14, 15].
A. Contributions
In this paper, we analyze the selective MIMO BC, where links are selective in both time and frequency. Based on the fact that the timefrequency selective channel model can be well approximated by the parallel correlated (timefrequency) channels as in [15], we show how the correlation between these timefrequency channels can be used in a selective MIMO BC to reduce the number of feedback bits while conserving the full multiplexing gain. The two main contributions of this paper can be summarized as following: (i) We show that the timefrequency selective channel can be characterized by a finite number of Gaussian invariants parameters, and we propose strategies with a limited number of feedback bits to quantize these invariants parameters. (ii) A practical approach to achieve the full multiplexing gain with a low computational complexity scheme is proposed to quantize the timefrequency selective channel.
B. Outline
The rest of the article is organized as following. In Section 2, we present the channel and the signal model, and we propose a general representation of the selective channel. Then, using this channel representation, we show in Section 3 how the correlation between these timefrequency channels can be used in a selective MIMO BC to reduce the number of feedback bits while conserving the full multiplexing gain. Moreover, a practical feedback scheme with a low computational complexity is also addressed in this section. The optimality of the reduced feedback channel quantization is also illustrated using numerical results. Finally, Section 4 concludes the article.
C. Notation
The notation used in this paper is as follows. Boldface lower case letters v denote vectors, boldface capital letters M denote matrices. M^{†} denotes conjugate transposition. I_{ N } stands for the N × N identity matrix. represents the complex Gaussian random variable. is the mathematical expectation w.r.t. to the random variable X. The pulse distribution is denoted by δ_{ n } = 1 if n = 0 and 0 otherwise. ⌊x⌋ denotes the floor of x and 〈x, y〉 is the scalar product between two vectors x and y.
2. System and channel model
In this article, we consider a multipleantenna BC where a source S with n_{ t } transmit antennas wants to communicate simultaneously with K destinations D_{ i } having n_{ r } receive antennas each, with n_{ t } ≥ Kn_{ r } . We assume that all communications occur on timefrequency selective fading channels. In the following, we start by briefly recalling from [15] the approximate decomposition of timefrequency selective channels into statistical correlated parallel channels for the pointtopoint case. Then, the corresponding inputoutput relation at each destination for the BC is provided.
A. Timefrequency selective SISO channel model
where s(t) is the transmitted signal and h(t, τ) is the timevarying impulse response.
The LTV system is also characterized by two other functions. The delayDoppler spreading function defined as Fourier transform (t → ν) of h(t, τ) and the timevarying transfer function defined as the Fourier transform (τ → f ) of h(t, τ).
B. WSSUS assumption and statistical channel description
C. Underspread LTV operator

The underspread LTV operator admits structured sets of orthonormal eigenfunctions {g_{ m, l } (t)} that are independent of the channel operator, well localized in time and frequency and known as WeylHeisenberg (WH) set. This set is obtained by translating in time and modulating in frequency a prototype g(t). In the following, this set is denoted aswhere m, n ∈ ℤ, T and F are the grid parameter of WH set. The triple g(t), T, F are chosen such that g(t) has unit energy and that g_{ m, n } (t) form an orthonormal base, i.e.,
Finally, the grid parameters T and F should satisfy TF > 1 to guarantee that g_{ m, n } (t) form an orthonormal basis and are well localized in time and frequency^{a} (more details about the choice of grid parameters can be found in [15] and references therein). Heuristically, the optimal choice of TF that minimizes the intersymbol interference (ISI) and intercarrier interference (ICI) and maximizes the number of degrees of freedom is TF ≈ 1.25.

where is the timevarying transfer function.
D. Signaling scheme: equivalent parallel model
where D = MT is the approximate time duration of s(t) and W = N_{ c }F is its approximate bandwidth.
Note that due to the orthonormal WH set, z[m, l] are i.i.d for all (m, l) ∈ {0 ... M  1} × {0 ... N_{ c }  1}, such that and . In the rest of the paper, we let n denote the timefrequency slot (m, l), with n = 0 ... N  1 and N = MN_{ c } is the total number of timefrequency slots. We finally denote by h the N × 1 vector containing parameters with (m, n) ∈ {0 ... M  1} × {0 ... N_{ c }  1}.
E. Multiuser BC model
where is the N × N correlation between the scalar subchannels with rank equal to ρ, is an n_{ r } × Nn_{ t } matrix with i.i.d. entries. For simplicity of notations, we assume that all scalar subchannels have the same correlation function.
In the following, we assume that the transmitter does not know the instantaneous value of the channel but knows the probabilistic channels' law^{b} including the knowledge of .
F. Impact of the correlation on the channel model
In this section, we propose a general representation of the timefrequency selective channel. We show that the MIMO channel between the source and each destination at each timefrequency slot can be written as given in Lemma 1.
where are the eigenvalues of the covariance matrix , and w_{ i, p } is the i th entry of the eigenvector w_{ p } of corresponding to .
Proof: Please refer to Appendix A for the proof of this lemma. ■
G. Physical interpretation of Lemma 1
The channel model in Lemma 1 gives a general representation of any selective fading channel and models the cases where the channel is selective either in time, frequency or in time and frequency.
1) Time selective channel (or block fading channel
In this case, the correlation matrix and consequently , where the elements e_{ n, j } of the N × 1 vector e_{ n } are such that e_{ n, j } = δ_{ n  j }.
2) Frequency selective channel
where and σ^{2} is the variance of the channel taps.
3) Time frequency selective channel
As stated before, due to the delay timevarying channel, the considered LTV channel induces ISI and ICI at each receiver side. As mentioned in Section 2D, by projecting the transmitted signal and the received signal on the channel eigenfunctions, the ISI and ICI interferences are canceled. The LTV channel is then decomposed into parallel timefrequency channel. The timefrequency channels change at each timefrequency slot. However, for an LTV channel that it is characterized by a scattering function that is compactly supported in a rectangle as in (3), it is well known from [17] that this variation depends only on a finite number of parameters that are invariant during all the duration of the transmission.
and, therefore, the rank of the covariance matrix ρ is much lower than the total number of timefrequency slots.
Remark 2: As stated in Section 2D, the covariance matrix is a twolevel Hermitian Toeplitz matrix. The eigenvectors of such matrix are not generally well structured as it is the case for circulant matrix unless its dimensions are sufficiently large, i.e., M → ∞ and N → ∞ as shown in [18]. The eigenvectors can be well approximated in this case by he eigenvectors of a twolevel circulant matrix. For the twolevel circulant matrix, the eigenvectors correspond to the kronecker product between all the columns of the fast Fourier transform matrix with dimensions M × M and N_{ c } × N_{ c } . In the following, no restriction on the values of M and N is considered. However, we assume that the correlation matrix is known at the transmitter side^{c} and the channel matrix Γ(n) can be deduced straightforwardly from the knowledge of as shown in Lemma 1.
3. Reduced feedback for the selective MIMO BC
The main objective of this section is to show how to achieve the total multiplexing gain in a selective fading MIMO BC as illustrated in Figure 1 when a limited feedback bits are used to quantize the channel. For this, we start first by giving some basic preliminaries on the linear precoding over the MIMO BC in Sections 3A and 3B. Then, we give in Section 3C a global overview on the general concept of the proposed quantization schemes of LTV selective fading channel. In Section 3D, we propose quantization schemes that take advantage of the correlation between timefrequency to reduce the number of feedback bits when a zero forcing or a BD scheme are used. A practical approach to achieve the full multiplexing while keeping a low computational complexity is proposed in Section 3E. Numerical illustrations are provided in Section 3F.
A. Basic preliminaries
where the second term represents the multiuser interference from every other user's signal. In the rest of this section, let denote the channel seen by the receive antenna i of user k at a timefrequency slot n.
1) Zero forcing (ZF)
2) Block diagonalization
3) Achieving the full multiplexing gain
B. A first approach: straightforward approach (SA)
Although, the straightforward strategy achieves the full multiplexing gain, it is not optimal in the sense that the number of feedback bits sent to the source is very large. Moreover, this feedback contains redundant information about the channels. In the next two subsections, we will show how the correlation between the timefrequency channels can be used in order to reduce the number of feedback bits.
C. Quantizing the selective fading channel: general concepts
In this section, we give a global overview on the general concept of the proposed quantization of LTV selective fading channel.
1) Perfect channel estimation at the receiver side
where p = (n mod n_{ t } ) + 1 and e_{ j } is the jth vector of the n_{ t } × n_{ t } identity matrix with entries e_{ j, n } = δ_{n  j}.
2) Quantization and feedback of the Gaussian vector
3) Channel reconstruction and precoder design
After this training phase, the transmitter should be able to reconstruct the channel using the quantized channel invariant parameters and the statistical channel knowledge as shown in Lemma 1. At each timefrequency slot, the linear precoder is adapted to the quantized timefrequency channel, and data are transmitted to the different users as shown in Figure 2.
However, to make a fair comparison with the perfect CSIT (where the training phase is often omitted), the rate gap between the quantized rate and the perfect CSIT that will be considered in the following is derived considering only the effective timefrequency slot where data information is transmitted using timefrequency slots.
D. Grouped reduced feedback (GRF) for selective fading BC
In this section, the estimated elements are grouped into one 1 × ρn_{ t } vector (respectively into one ρn_{ t } × n_{ r } matrix) and quantized using a random vector quantization (RVQ) when zero forcing precoder is used (respectively using a Grassmannian quantization with BD).
1) Zero forcing with grouped reduced feedback (GRFZF)
where is a Gaussian vector with i.i.d entries. As it can be noticed from (23), it is sufficient to know to determine the channel at each time frequency slot and at each antenna j = 1 ... n_{ r } . For the selective fading BC when a zero forcing precoder is used, we prove in Theorem 1 that it is sufficient to quantize n_{ r } unit norm vectors, at each user k to achieve the full multiplexing gain using a RVQ technique. We assume that each destination uses n_{ r } different codebooks to quantize each vector in order to prevent quantizing two different vectors by the same vector. The quantization codebook containing 2 ^{ B } unit norm Gaussian 1 × ρn_{ t } vectors is assumed to be known at the transmitter and receiver side. At each user k, each antenna j feeds the index F^{[k, j]}of the ω vectors that is closest (in term of its angle) to its channel vector . The minimal number of feedback bits required to achieve the full multiplexing gain is summarized in Theorem 1 as following.
Proof: The proof of this theorem is mainly based on the previous quantization result of the flat fading channel in [6] and is detailed in Appendix B.
2) BD with grouped reduced digital feedback (GRFBD)
In this section, we propose a quantization scheme for the BD when a timefrequency selective channel is considered. Based on the observation that timefrequency selective channel slots are correlated, we compute the minimal number of feedback bits required to achieve the full multiplexing gain.
with g = n_{ r } (ρn_{ t }  n_{ r } ) is the dimensionality of the Grassmannian manifold. The minimal number of feedback bits required to achieve the full multiplexing gain is summarized in Theorem 2 as following.
with g = n_{ r } (ρn_{ t }  n_{ r } ) is the dimensionality of the Grassmannian manifold.
Proof: Please refer to Appendix C for the proof of this theorem. ■
E. Partitionedreduced feedback (PRF): a practical approach to achieve the full multiplexing gain
It can be noticed from Section 3D that when grouping at each receiver all the invariants parameters of the channel into one 1 × ρn_{ t } vector (respectively one ρn_{ t } × n_{ r } matrix), the size of the codebook required to achieve full multiplexing gain is very large, and consequently the search complexity of the optimal vector (respectively matrix) in the codebook becomes very high.
For instance, if we consider a MIMO BC with a ZF precoder with n_{ t } = 6 antennas, a covariance matrix rank ρ = 12 and n_{ r } = 2, the number of feedback bits required to quantize a 1 × ρn_{ t } vector at an SNR = 30 dB is 707 bits. In order to quantize this vector, one needs to search the optimal vector in a codebook of size 2^{707}, which is not always feasible to be implemented in a practical system.
Motivated by this issue, we propose in this subsection a practical approach to quantize the timefrequency selective channel, which will be called in the following PRF. The proposed strategy guarantees to exploit all the available degrees of freedom in the MIMO BC when a zero forcing precoder is used while keeping a low computational complexity.
1) PRF strategy
The main objective of the proposed PRF scheme with ZF precoding is to quantize the channel vector containing the invariants Gaussian parameters seen at the receive antenna j of user k in a partitioned way.
The proposed PRF scheme consists to find for each partitioned channel vector h _{ω, j}[i]:
Its quantized norm using a noisy analog feedback scheme. We assume that these coefficients are sent β times on an unfaded uplink AWGN channel with the same power as the downlink scheme.
where e_{ i } is the feedback Gaussian noise such that .
In order to derive the gap rate with the full CSIT, we characterize in Lemma 2 the angle between the normalized vector vector and its quantized vector .
Proof: Please refer to Appendix D for the proof of this lemma.
2) Achieving the full multiplexing gain
For a selective fading MIMO BC with n_{ t } ≥ Kn_{ r } , we show in Theorem 3 that when using the above PRF scheme with a sufficient number of feedback bits and a zero forcing precoder, the total multiplexing gain can be also achieved. The following result is summarized in the following theorem.
Proof: Please refer to Appendix E for the proof of this theorem. ■
Remark 3: It should be emphasized here that the quantization of the directions of the subvectors is not sufficient alone to achieve the full multiplexing gain if it is not coupled with the feedback of these vectors' norms. This is also illustrated in the numerical results in Section 3F and can be analytically proved following the same reasoning as above.
3) Reducing the computational search complexity
It can deduced therefore that the total search complexity in the PRF scheme is reduced by a factor of compared to the GRF scheme. This factor becomes very significant for high SNR ranges.
We finally note that partitioning the vectors is not restricted to ρ vectors with n_{ t } elements. Increasing the vectors partitions comes at the expense of an increased feedback, but a significant reduced computational complexity of the system. A tradeoff between the complexity of the system and the number of feedback bits should be considered.
F. Numerical results
Channel and signal parameters
Key channel and signal parameters  Values 

Carrier frequency f_{c}  2.5 GHz 
Communication bandwidth W  1 MHz 
Delay requirement D  50 ms 
Doppler spread ν_{0}  100 Hz 
Coherence time T_{ c }  2.5 ms 
Delay spread τ_{0}  1 μ s 
Coherence bandwidth W_{c}  500 KHz 
Channel spread  10^{4} 
Reliable uplink rate  1 Mbps 
For this channel and signal model, we compare^{g} the proposed strategies: the SA with ZF (SA  ZF), the SA with ZF (SA  BD), the GRF scheme^{h} with zero forcing (GRF  ZF) and the PRF. The classical comparisons of ZF and BD are extensively addressed in [6, 7] and the same behavior as for the flat fading channel can be observed for the TF selective channel. In the rest of this subsection, we focus mainly on the performance of the practical proposed PRF scheme. We note here that the PRF scheme is not compatible with the BD construction as it is based on a RVQ and not on a quantization over a Grassmann manifold [7].
When no feedback information on the subvectors norms is provided to the transmitter side, we can observe in Figure 4 that the full multiplexing gain cannot be achieved. The directional knowledge should be coupled with the subvectors' norms quantification to achieve the full multiplexing gain.
4. Conclusions and perspectives
In this article, we studied the selective MIMO BC with limited feedback. We showed that as timefrequency channels are correlated it is not necessary to quantize each timefrequency channel. However, it is sufficient to reconstruct the channel based on a finite number of parameters by making use of the correlation in time and frequency while conversing the full spatial multiplexing gain. The optimal number of feedback bits required to achieve the full multiplexing gain is computed. Moreover, the practical implementation of a TF channel quantization scheme is addressed and a lowcomplexity scheme that also achieves the multiplexing gain is proposed. The design of a PRF matricial scheme compatible with the BD scheme will be addressed in our forthcoming works.
Appendix A
Proof of Lemma 1
Let be the N × 1 stacked channel vector that contains the N timefrequency channel's components, and its N × N Hermitian covariance matrix such that . The covariance channel matrix coefficients can be deduced from (2) and is supposed to be known at both the transmitter and the receiver side. In the following, we set the rank of and its eigenvalue decomposition where .
which completes the proof.
Appendix B
Proof of Theorem 1
A. Relationship between the matrix and its quantification
with .
B. Throughput analysis
and therefore the maximal multiplexing gain can be achieved, but with a constant capacity gap.
Appendix C
Proof of Theorem 2
The maximal multiplexing gain can be therefore achieved, but with a constant capacity gap.
Appendix D
Proof of Lemma 2
This completes the proof.
Appendix E
Proof of Theorem 3
For each receive vector, the number of feedback bits required to estimated the partitioned vectors norms is at most equal to the capacity of the uplink AWGN channel N_{ f, a } = βρ log_{2}P.
The maximal multiplexing gain can be achieved, if the gap capacity between the full CSIT and the quantized capacity are independent of P. This occurs if the number of bits required to quantize each part of the vector scales as (n_{ t }  1) log_{2}P. In total there is ρ parts in each vector, and the total number of feedback bits is therefore ρ(n_{ t }  1) log_{2}P + βρ log_{2}P.
Endnotes
^{a}Please note that the considered WH set with parameters T and F is a Riez sequence and is constructed as a dual of a WH frame characterized by grid parameters , and such that . ^{b}This assumption is commonly used for when considering noncoherent setting as defined in [15] and references therein. ^{c}In practical system, this assumption can be feasible as it requires only the feedback of N different values of . The twolevel Toeplitz matrix can be constructed according to (9). ^{d}This corresponds to parameters h_{ ω, i, j } [s] in (16). ^{e}For the perfect estimation, we assume that the I/O relationship in (11) is noisyfree. ^{f}This inequality can be easily verified by noticing that ρ ≥ 1, K ≥ 1 and n_{ t } ≥ Kn_{ r } . This implies that ρn_{ t } ≥ n_{ t } ≥ Kn_{ r } ≥ n_{ r } . ^{g}For simplicity, we only consider the data transmission over the first 100 TF slots that follow the training phase to plot the numerical results. ^{h}We note here that the implementation of the GRF with BD is not possible with Matlab when dealing with a 72 × 2 channel using the classical numerical generation as in [7].
Declarations
Acknowledgements
The authors would like to thank Professor Helmut Bölcskei for helpful discussions about the LTV channel model, and the anonymous reviewers for their valuable comments and suggestions that improved significantly the quality of this paper.
Authors’ Affiliations
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