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Reduced feedback for selective fading MIMO broadcast channels
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 45 (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.
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
In wireless communication, the multipath approximation is often used to describe the wireless channel model. In this case, the received signal is the sum of all multipath components and the line of sight. Each path induces a variation of the signal strength due to the Doppler spread ν caused by the movement of the transmitters, receivers, and scatterers and a delay shift τ at the receiver side. In general, the number of paths is very high, which makes logical to model the multipath effect by a continuous linear timevarying (LTV) system. The LTV system is described by a linear operator that maps an input signal into an output signal r(t), related by the following noisefree relationship,
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
In digital communications, the linear operator is random and LTV channel models are studied under the widesense stationary and uncorrelated scattering (WSSUS) assumption. This property consists in assuming that the random channel is widesense stationary in time t and uncorrelated in scattering (delay) τ, which means that
The WSSUS property implies that the timevarying transfer function is widesense stationary in both time and frequency, and the spreading function is uncorrelated in delay τ and in Doppler ν, i.e.,
The scattering function is the 2D Fourier transform of the timefrequency correlation function such that
C. Underspread LTV operator
As a consequence of the limited velocity of transmitter, receiver, and scatterers in the propagation environment, the maximum Doppler shift is limited to ν_{0}. We also assume that the maximum delay is bounded by 0 and +τ_{0}. The scattering function is therefore supported on a rectangle of area known as channel spread, such that
The underspread assumption is of a practical interest as the time taken by channel to change significantly (1/ν_{0}) is usually much longer than the delay spread τ_{0}, i.e.,
Channels satisfying these characteristics are called underspread selective fading channels and are shown by Kozek [16] to satisfy the two following properties.

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 as
(5)where 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.

For grid parameters chosen such that and and hence
(6)it has been shown in [16], [15] that the eigenvalues of can be well approximated by,
where is the timevarying transfer function.
D. Signaling scheme: equivalent parallel model
In the following, we base our results on the strategy used by Dirusi et al. [15] to derive the discrete timefrequency input output relation. This strategy consists of transmitting and receiving on the orthonormal channel eigenfunctions g_{ m, l } (t). The transmitted signal can be therefore written using the channel WH set as,
where D = MT is the approximate time duration of s(t) and W = N_{ c }F is its approximate bandwidth.
The projection of the noisy received signal onto the channel WH set is given by,
which imply that,
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}.
It can be easily seen from (8) that using the WH set, the timefrequency parallel channel is converted into N parallel statistically dependent timefrequency channels where the correlation matrix between the scalar subchannels is denoted by . As shown in Section 2B, the timevarying transfer function is widesense stationary in time and in frequency, and the covariance matrix is a twolevel Hermitian Toeplitz matrix. Using the same notations as in [15], the covariance matrix can be written as,
where is a Toeplitz matrix with coefficients given in (2) and defined in (9).
E. Multiuser BC model
In the rest of this article, we consider a MIMO BC depicted in Figure 1 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 } . This condition can be of a practical interest if we consider that the K users are selected randomly among a large number of users and have an aggregate number of receive antennas that does not exceed the number of transmit antennas.
The fading process considered in this paper are characterized by nondisjoint scattering function such that
By choosing the sampling period T, and sampling frequency F such that T ≤ 1/ν_{0} and F ≤ 1/τ_{0}, channels can be diagonalized using the same WH sets. At each destination k, the received signal y _{k}(n) is given by
where is the channel matrix at the timefrequency slot n, and n = 0 ... N  1 with N being the total number of timefrequency slots. The vector is the transmitted signal, and n^{[1]}(n),..., n^{[K]}(n) are independent complex Gaussian noise terms with unit variance. The transmitter is subject to an average power constraint P, such that
We assume that channels are spatially uncorrelated, that for a given timefrequency slot n, H^{[k]}(n) has i.i.d entries. The channels corresponding to different destinations are assumed to be statistically independent. However, channels are correlated across n for a given destination k, according to
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.
Lemma 1 (Timefrequency MIMO channel matrix): The user k channel matrix at a timefrequency slot n can be written as
where is a Gaussian matrix with i.i.d entries such that
and Γ(n) is a ρn_{ t } × n_{ t } deterministic matrix that depends only on the channel statistics, such that,
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. ■
In the following, we set
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
The time selective channel models the case when the channel remains constant during a block n and changes in a statistical independent manner across block, such that
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
For the frequency selective channel, it is well known that the channel can be decomposed into N parallel statistically dependent subchannels when using a cyclic prefix OFDM system. In this case, n stands for the frequency slot, N the total number of subcarriers, and the correlation matrix is a circulant matrix with rank ρ equal to the number of channels taps L. The eigenvectors of are the columns of an N × N FFT matrix. Consequently, Γ(n) in Lemma 1 is such that
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.
It can be deduced from Lemma 1 that when statistical channel knowledge is available at the receiver side, the knowledge of the timevarying channel at any time frequency slot n = (m, l) depends only on the knowledge of the Gaussian vector h_{ ω, i, j } = [h_{ ω, i, j } [0] ... h_{ ω, i, j } [ρ  1]], where
Another different way to identify the LTV channel has also been established by Kailath [17] where a channel reconstruction strategy based on the 2D Shannon sampling theorem was proposed. A complete description on the LTV channel identification can be found in Chapters 1 and 2 of [14]. It should be emphasized here that the vector containing the finite number of Kailath parameters' required to identify the channel is not Gaussian with i.i.d. entries and cannot be easily quantized using classical quantization technique. The statistical distribution of these parameters is not easy to be derived. The Kailath channel identification strategy states also that the LTV channel can be identified by a finite number of parameters, which is equal to (⌊ν_{0}D⌋ + 1)(⌊τ_{0}W⌋ + 1). This number of free parameters can give a rough estimate of the rank ρ of the covariance matrix , which is nothing but the number of independent Gaussian parameters required to identify the channel. This implies that ρ is approximatively equal to
Remark 1: From the choice of the grid parameters (ν_{0}T ≤ 1, τ_{0}F ≤ 1 and TF = 1.25), it can be easily shown that in this case, that the ratio is bounded by
For wireless channel, the channel spread is generally . Moreover, the total number of timefrequency slots is generally very large. This implies that scales as
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
When linear precoding is used, the transmitted signal vector x(n) is a linear function of the destinations' data symbols . Let V^{[k]}(n) denote the precoding matrix of user k at a timefrequency slot, such as
The received signal for user k is given by,
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)
The zero forcing (ZF) precoder eliminates the multiuser and the interantenna interference. In this case, the l th columns of the precoding matrices V^{[j]}(n) are chosen such that,
The received signal is given by
2) Block diagonalization
When the BD precoding schemes is used, the precoding matrices are chosen in order to eliminate the multiuser interference only, such that
The received signal at each user side is given by
3) Achieving the full multiplexing gain
When full CSIT is assumed, it is known from [5] that both schemes achieve the full multiplexing gain of min(n_{ t } , Kn_{ r } ) = Kn_{ r } , which means that the sum capacity C_{sum}(P) of the BC scales as,
More recently, it has been shown in [6, 7] that the full CSIT assumption can be reduced to a partial CSIT when performing digital quantization of the flat MIMO channel using a number of feedback bits N_{f, flat}that scales as
B. A first approach: straightforward approach (SA)
The first approach to quantize the timefrequency selective channel can be straightforwardly deduced from the quantization of the flat fading channel. Each timefrequency channel H^{[k]}(n) can be simply seen a flat fading channel and is quantized by its own using
feedback bits in order to achieve the full multiplexing gain. Over the N timefrequency slots, the total number of feedback bits is such that
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.
The training phase in Figure 2 aims to estimate and quantize these channel invariant parameters and includes: (1) The estimation of the LTV invariants parameters at each receiver side. (2) The estimated invariant parameters of the channel are quantized and feedback to the transmitter side.
1) Perfect channel estimation at the receiver side
For the MIMO BC, each receiver needs to estimate the ρn_{ t } invariants parameters^{d} at each receive antennas. For this, the transmitter sends a common pilot common n_{ t } × 1 vector signal of length n_{ t }ρ to all receivers such that,
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}.
Assuming a perfect estimation at the receiver side^{e} and using the above pilot sequence, the value of the timefrequency channel between transmit antenna i and receiver antenna j is available at each antenna j for a timefrequency slot equal to n_{ t }q + (i  1) with q = 0 ... ρ  1. The ρ invariant parameters h_{ ω, i, j } [s] (s = 0 ... ρ  1) of the channel between transmit antenna i and receive antenna j can be therefore evaluated as a solution of a linear system with ρ independent equations and ρ unknown variables, given by
with q = 0 ... ρ  1. The duration of this estimation phase is given by,
2) Quantization and feedback of the Gaussian vector
The estimated invariant parameters of the channel at each receiver are then quantized using N_{ f } bits. This quantization is based on grouped or partitioned feedback that will be detailed in Sections 3D and 3E. The duration of this phase depends on the uplink reliable rate R_{ u } on which the feedback bits are conveyed to the transmitter side and is such that
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.
Generally, the exact derivation of the effective maximal rate should take into account the training phase, meaning that the effective duration of transmission is,
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)
At each time frequency slot n, the channel seen by each receive antenna j for a user k can be written using Lemma 1 as
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.
Theorem 1: For the Kselective MIMO BC with n_{ t } transmit antennas at the source and n_{ r } receive antennas at the destinations (n_{ t } ≥ Kn_{ r } ) when a zero forcing scheme is used, the total spatial multiplexing gain of Kn_{ r } can be achieved using the above RVQ scheme if the number of feedback bits N_{ f } sent by each user scales as
where ρ is the rank of the selective fading channel covariance matrix. At high SNR, the rate loss incurred by the above quantization scheme is upper bounded by,
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.
As the timefrequency channel matrices H^{[k]}(n) are correlated, it is not necessary that the receiver feeds back the channel at each timefrequency slot. It can be easily deduced from Lemma 1 that the knowledge of is sufficient to know the channel at each timefrequency slot. Usually, quantized matrices are chosen in a Grassmannian manifold , where T > M. That's why, the quantization problem of selective fading channel consists in finding a quantization for (with^{f}ρn_{ t } ≥ n_{ r } ). The quantization codebook is supposed to be known at the transmitter side as well at the receivers side. Each receiver uses a different codebook C_{ k } of 2 ^{B} unitary matrices in , such that
Each user (to say user k) feeds back the index of the W matrix that is closest in term of its chordal distance to the channel matrix , i.e.,
where d is the chordal distance between two matrices. Each of the 2 ^{B} unitary matrices are chosen independently and are uniformly distributed over a Grassmannian . As shown in [7] (and references therein), the distortion associated with a given codebook C_{ k } for the quantization of is such that
where is equivalent when the number of bits B goes to infinity to
and C is a constant independent of B given by
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.
Theorem 2: For the Kselective MIMO BC with n_{ t } transmit antennas at the source and n_{ r } receive antennas at the destinations(n_{ t } ≥ Kn_{ r } ) when a BD scheme is used, the total spatial multiplexing gain of Kn_{ r } can be achieved using the above grouped Grassmannian quantization scheme if the number of feedback bits N_{ f } broadcast by each user scales as,
where ρ is the rank of the selective fading channel covariance matrix. At high SNR, the rate loss incurred by the above quantization scheme is upper bounded by
where C is a constant defined as
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.
This requires to write as the concatenation of ρ vectors such that
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.
The received signal at the base station is,
This implies that the unbiased MMSE estimated norm is,
where e_{ i } is the feedback Gaussian noise such that .
Its direction by finding the optimal orthonormal vector in a random codebook of size 2 ^{B} that is closest in term of its angle to h _{ω, j}[i] using a RVQ technique, such that,
Finally, the normalized corresponding ρn_{ t } × 1 quantized vector is constructed as following
In the following, let
and
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 .
Lemma 2: The quantization error of the PRF scheme operating at high SNR is upper bounded by,
The expected quantization error is upper bounded by,
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.
Theorem 3: For a selective fading MIMO BC with n_{ t } ≥ Kn_{ r } , the total spatial multiplexing gain of Kn_{ r } can be achieved using the above partitioned vector quantization scheme if

The direction of each vector is quantized using

The norm of each vector is quantized using
The total number of feedback bits N_{ f } sent by each user is,
At high SNR, the upper bound on the rate loss incurred by the PRF quantization scheme is
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
Although the number of feedback bits required by the PRF scheme is slightly increased compared to the GRF scheme, the PRF strategy reduces significantly the complexity of the system. The quantization problem of the whole vector is decomposed into ρ subquantization problems where each subquantization deals with a smaller size of vector. Instead of performing one search in one codebook of size as shown in Subsection 3D, where B_{ g } = (ρn_{ t }  1) log_{2}P, the PRF schemes PRF perform ρ searches in ρ different codebooks of size B_{ p } each with B_{ p } = ρ(n_{ t }  1) log_{2}P. Thus,
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
In order to illustrate the proposed reduced quantized schemes, we consider the MIMO BC with K = 3 destinations having n_{ r } = 2 receive antennas each and a source with n_{ t } = 6 transmit antennas. We assume that the communication occurs over a radio channel characterized by the parameters in Table 1 (Table 2.1 in [19]). These parameters correspond to the context of the standard IEEE 802.16 (or WIMAX).
In this case, it can be easily verified that the signal duration and bandwidth are much larger than the coherence bandwidth and the coherence time of the channel. The channel is therefore selective in time and frequency. This channel can be approximately decomposed into parallel timefrequency channel using WH sequences as explained in Section 2. We assume in the following that the grids parameters are chosen such that TF = 1.25, where T = 0.1 ms and F = 12.5 KHz. The number of timefrequency slots is therefore,
and the sufficient number of parameters required to identify the channel is ρ ≈ (⌊ν_{0}D⌋ + 1)(⌊τ_{0}W⌋ + 1) = 12 ≪ N. The required estimation period can be deduced from (20) and is equal to,
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].
Figure 3 compares the number of feedback bits required by each strategy. We can notice that the number of feedback bits using reduced feedback strategies is significantly reduced compared to straightforward strategies and that the number of feedback bits required by the PRF strategy is slightly increased compared to the GRF  ZF. The total duration of the feedback period in (21) depends on this number of feedback bits and the quantization strategy. For instance, the feedback time required by the PRF scheme operating at an SNR = 30 dB is . However, for the SABD, the feedback time is equal to 12 ms considered as a large delay as it constitutes nearly 1/4 of the total duration of the communication.
At the end of the training phase, the source reconstructs the channel using one of the above strategies and adapts itself to the quantized channel. Figure 4 compares the sum capacity of the MIMO BC when the above quantization strategies are used. When the number of feedback bits of each strategy scales with SNR as shown in Theorems 1 and 3, we can observe that the full multiplexing gain of min(n_{ t } , Kn_{ r } ) = 6 can be achieved. Moreover, the performances of the straightforward strategies are the same as the reduced ones. As stated before, the PRF scheme reduces significantly the complexity of the system as it deals with the quantization of vector of 1 × 6 vector instead of 1 × 72 as it is the case for the GRF scheme. We can also see in Figure 4 that the full multiplexing is achieved and there is a negligible rate loss between the PRF and the GRFZF of nearly ~0.1 dB. It can be deduced that the same performance of the GRFZF strategy can be achieved but with a much lower computational complexity.
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.
Finally, we can observe in Figure 5 that when the number of feedback bits N_{f} = 200 does not scale with SNR that the GRF scheme gives a better reconstruction of the channel at the transmitter side and outperforms all the other strategies. This result is not surprising as the 200 bits will be divided for the SAZF between all the timefrequency slots, and each vector is quantized using 4 bits only. For the PRF scheme, a part of the 200 feedback bits is used for the analog feedback of subvectors norm and the remaining feedback bits are used to quantize n_{ r }ρ subvectors. Each subvector is therefore quantized using only a few number of feedback bits that are not sufficient for channel reconstruction.
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 .
The vector can be written in function of its covariance matrix such that
where is an i.i.d. vector with the same dimension as . Using the eigenvalue decomposition of ,
where in (36) is also a random Gaussian vector , since W^{†} is a unitary matrix. It follows from (36) that
Note here that the number of independent parameters is nothing but the rank of the covariance matrix . As channels between transmit antennas and receive antennas are not correlated, this implies that H^{[k]}(n) can be written as shown in (38).
Equivalently,
By using some simple matricial manipulations, we get,
where
which completes the proof.
Appendix B
Proof of Theorem 1
A. Relationship between the matrix and its quantification
Let a be the quantization error between the normalized vector and its quantified vector such that,
As shown in Lemma 2 in [6], the vector can be written as the sum of two vectors, one, in the direction of the quantized vector and the second, s_{ j } is isotropically distributed in the nullspace of , independent of a as shown in [6], such that
This implies that,
with .
B. Throughput analysis
Let ΔR_{ k } = R_{Quant}  R_{Full}_{CSIT} be the rate loss incurred by the quantization. Due to the isotropic nature of the channel matrices, the rate loss can be written such that,
Using the relation between the channel vector and its quantized channel vector in (40), and from the ZF constraint,
it follows that
It should be emphasized here that for the selective fading channel and as a consequence of the zero forcing constraint in (41), the vector is isotropically distributed in the nullspace of . Moreover, it can be easily shown by using the value of matrix Γ(n) in (15) that the norm of this vector is such that,
It can be deduced therefore that the two vectors s_{ j } and are two unit vector isotropically distributed in the nullspace of . Then by following the same reasoning as in [6], these two vectors are distributed in the ρn_{ t }  1 nullspace of , and the angle between these two vectors is beta distributed with parameters β(1, ρn_{ t }  2). Hence,
It is well known from [6] that the quantization error a corresponds the minimum angle between the channel vector and the 2 ^{B} codebooks vector, and therefore it is distributed as the minimum between 2 ^{B} beta variables, and
Then, the rate loss is therefore upper bounded by,
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 scale as (ρn_{ t }  1) log_{2}P, then ΔR_{ k } is constant and independent of P, such that
where . Consequently,
and therefore the maximal multiplexing gain can be achieved, but with a constant capacity gap.
Appendix C
Proof of Theorem 2
The relationship between the quantized channel matrix can be deduced using Lemma 1 in [7]. By doing similar manipulations as in Theorem 1, it can be shown that in this case the rate loss incurred by the quantization can be bounded such that,
where D_{ s } is the distortion error rate and is such that,
where is equivalent for large B to
to and C is a constant independent of B defined as in (29). It follows that,
It can be easily deduced that, ΔR is independent of P, if and only if B scales as n_{ r } (ρn_{ t }  n_{ r } ) log_{2}P, then
where c is a constant independent of P. Consequently,
The maximal multiplexing gain can be therefore achieved, but with a constant capacity gap.
Appendix D
Proof of Lemma 2
Let denote the quantization error between the channel vector and its quantized version. Then,
In order to find an upper bound on a_{ p } , we propose to find separately upper bounds on and . Using (33), the scalar product between this two vectors can be related to the partitioned vectors as
where
As , then
Let and then,
The CauchySchwartz inequality,
implies that,
Next, by noticing that , we get the upper bound on
where,
and
At high SNR, the analog estimation in (32) is approximately equal to . This implies that,
The expected quantization error is therefore,
The random variable is beta distributed with parameters a = n_{ t } and b = (ρ  1)n_{ t } . The expectation of this variable is a/(a + b) = 1/ρ. The expectation of quantization error incurred by the RVQ is upper bounded by . Then, at high SNR, the quantization error is upper bounded by,
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 upper bound can be derived following the same steps as Appendix B. It turns out that at high SNR, the gap rate is upper bounded by,
Using Lemma 2, the expected gap rate is upper bounded by,
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].
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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.
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Mroueh, L., RouquetteLéveil, S. & Belfiore, J. Reduced feedback for selective fading MIMO broadcast channels. J Wireless Com Network 2011, 45 (2011) doi:10.1186/16871499201145
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Keywords
 timefrequency selective channel
 underspread channel
 broadcast channel
 random vector quantization
 Grassmann manifold
 multiplexing gain