On Optimizing Feedback Interval for Temporally Correlated MIMO Channels With Transmit Beamforming And Finite-Rate Feedback

TL;DR

This paper analyzes the optimal feedback interval for MIMO channels with temporal correlation, demonstrating that proper timing of feedback significantly improves transmission rates with limited feedback.

Contribution

It introduces a method to determine the optimal feedback interval for correlated MIMO channels, enhancing rate performance over existing schemes.

Findings

Optimal feedback interval maximizes average received power.

Large system approximation accurately predicts finite system performance.

Quantized beamforming with optimal interval outperforms Kalman-filter scheme and frequent feedback.

Abstract

A receiver with perfect channel state information (CSI) in a point-to-point multiple-input multiple-output (MIMO) channel can compute the transmit beamforming vector that maximizes the transmission rate. For frequency-division duplex, a transmitter is not able to estimate CSI directly and has to obtain a quantized transmit beamforming vector from the receiver via a rate-limited feedback channel. We assume that time evolution of MIMO channels is modeled as a Gauss-Markov process parameterized by a temporal-correlation coefficient. Since feedback rate is usually low, we assume rank-one transmit beamforming or transmission with single data stream. For given feedback rate, we analyze the optimal feedback interval that maximizes the average received power of the systems with two transmit or two receive antennas. For other system sizes, the optimal feedback interval is approximated by maximizing the rate difference in a large system limit. Numerical results show that the large system approximation can predict the optimal interval for finite-size system quite accurately. Numerical results also show that quantizing transmit beamforming with the optimal feedback interval gives larger rate than the existing Kalman-filter scheme does by as much as 10% and than feeding back for every block does by 44% when the number of feedback bits is small.

Equations

$$\mathbf{r}[kL+l]=\mathbf{H}(k)\mathbf{v}(k)x_s[kL+l]+\mathbf{n}[kL+l],1\le l\le L$$

$$\mathbf{H}(k)=\alpha \mathbf{H}(k-1)+\sqrt{1-{\alpha }^2}\mathbf{W}(k)$$

$$R=E\left[\log \left(1+\rho \mathbf{v}(k{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\mathbf{v}(k)\right)\right]$$

$$\mathcal{V}=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\cdots ,{\mathbf{v}}_n\}$$

$$\hat{\mathbf{v}}(k)$$

$$=\arg \underset{{\mathbf{v}}_j\in \mathcal{V}}{\max }{\mathbf{v}}_j^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k){\mathbf{v}}_j.$$

$$R=E\left[\log \left(1+\rho \hat{\mathbf{v}}(k{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(k)\right)\right].$$

$$\bar{R}$$

$$\le \frac{1}{K}\sum _{k=1}^K\log \left(1+\rho E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(1)\right]\right)$$

$$\le \log \left(1+\rho \frac{1}{K}\sum _{k=1}^{K}E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(1)\right]\right)$$

$$K^{\ast }=\arg \underset{K\in {\mathbb{Z}}^{+}}{\max }\frac{1}{K}\sum _{k=1}^{K}E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(1)\right],$$

$$\begin{gathered} E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(1)\right] \\ ={\alpha }^{2k-2}\left({\gamma }_{2\times N_r}(BK)-N_r\right)+N_r \end{gathered}$$

$${\gamma }_{2\times N_r}(BK)$$

$$\displaystyle=\frac{1}{(N_{r}-1)!(N_{r}-2)!}\Big{[}\phi(N_{r}+2,N_{r}-1)$$

$$-2\varphi (N_r+1,N_r)+\varphi (N_r,N_r+1)$$

$$\displaystyle\quad-\frac{1}{2^{BK}+1}\big{(}\phi(N_{r}+2,N_{r}-1)$$

$$-3\varphi (N_r+1,N_r)+3\varphi (N_r,N_r+1)$$

$$\displaystyle\quad-\phi(N_{r}-1,N_{r}+2)\big{)}\Big{]}$$

$$\begin{gathered} \varphi (m,n)=mn\varphi (m-1,n-1)-\frac{(m-n)(m+n-1)!}{2^{m+n+1}}, \\ \forall m,n\ge 1 \end{gathered}$$

$$\begin{gathered} \frac{1}{K}\sum _{k=1}^{K}E\left[{\hat{\mathbf{v}}}^{†}(1){\mathbf{H}}^{†}(k)\mathbf{H}(k)\hat{\mathbf{v}}(1)\right] \\ =N_r+\frac{1}{K}\left(\frac{1-{\alpha }^{2K}}{1-{\alpha }^2}\right)({\gamma }_{2\times N_r}(BK)-N_r). \end{gathered}$$

$$\underset{\alpha \to 1}{\lim }\frac{1}{K}\sum _{k=1}^{K}E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(1)\right]={\gamma }_{2\times N_r}(BK).$$

$$N_r+\frac{1}{K}\left(\frac{1-{\alpha }^{2K}}{1-{\alpha }^2}\right)({\gamma }_{2\times N_r}(BK)-N_r)>{\gamma }_{2\times N_r}(B)$$

$$K$$

$$<\frac{1}{1-{\alpha }^2}\left(\frac{{\gamma }_{2\times N_r}(BK)-N_r}{{\gamma }_{2\times N_r}(B)-N_r}\right).$$

$$K<\frac{1}{1-{\alpha }^2}.$$

$$\begin{gathered} \frac{1}{K}\sum _{k=1}^{K}E\left[{\hat{\mathbf{v}}}^{†}(1){\mathbf{H}}^{†}(k)\mathbf{H}(k)\hat{\mathbf{v}}(1)\right] \\ =2+\frac{1}{K}\left(\frac{1-{\alpha }^{2K}}{1-{\alpha }^2}\right)({\gamma }_{N_t\times 2}(BK)-2) \end{gathered}$$

$${\gamma }_{N_t\times 2}(BK)=E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(1{)}^{†}\mathbf{H}(1)\hat{\mathbf{v}}(1)\right]$$

$$\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(1{)}^{†}\mathbf{H}(1)\hat{\mathbf{v}}(1)=\underset{1\le j\le 2^{BK}}{\max }{\mathbf{v}}_j^{†}\mathbf{H}(1{)}^{†}\mathbf{H}(1){\mathbf{v}}_j.$$

$$F_{\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}|\lambda_{1},\lambda_{2}}(x)\\ =\left\{\begin{array}[]{ll}1-\frac{\lambda_{1}}{\lambda_{1}-\lambda_{2}}\left(1-\frac{x}{\lambda_{1}}\right)^{N_{t}-1}&\\ \quad+\frac{\lambda_{2}}{\lambda_{1}-\lambda_{2}}\left(1-\frac{x}{\lambda_{2}}\right)^{N_{t}-1}&:\,0\leq x\leq\lambda_{2}\\ 1-\frac{(\lambda_{1}-x)^{N_{t}-1}}{(\lambda_{1}-\lambda_{2})\lambda_{1}^{N_{t}-2}}&:\,\lambda_{2}\leq x\leq\lambda_{1}.\end{array}\right.$$

$$\begin{gathered} E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(1{)}^{†}\mathbf{H}(1)\hat{\mathbf{v}}(1)|{\lambda }_1,{\lambda }_2\right] \\ ={\lambda }_1-{\int }_0^{{\lambda }_1}{\left(F_{{\mathbf{v}}^{†}\mathbf{\Lambda }\mathbf{v}}(x)\right)}^{2^{BK}}\mathrm{d}x. \end{gathered}$$

$$\begin{gathered} E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(1{)}^{†}\mathbf{H}(1)\hat{\mathbf{v}}(1)\right] \\ ={\int }_0^{\infty }{\int }_0^{{\lambda }_1}E\left[\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(1{)}^{†}\mathbf{H}(1)\hat{\mathbf{v}}(1)|{\lambda }_1,{\lambda }_2\right] \\ \times f_{\mathbf{\Lambda }}({\lambda }_1,{\lambda }_2)\mathrm{d}{\lambda }_1\mathrm{d}{\lambda }_2 \end{gathered}$$

$$\begin{gathered} {\gamma }_{N_t\times 2}(BK)=\frac{\varphi (N_t+2,N_t-1)}{(N_t-1)!(N_t-2)!} \\ -{\int }_0^{\infty }{\int }_0^{{\lambda }_1}{\int }_0^{{\lambda }_1}{\left(F_{{\mathbf{v}}^{†}\mathbf{\Lambda }\mathbf{v}}(x)\right)}^{2^{BK}}{f}_{\mathbf{\Lambda }}({\lambda }_1,{\lambda }_2)\mathrm{d}x\mathrm{d}{\lambda }_1\mathrm{d}{\lambda }_2. \end{gathered}$$

$$R_{△}$$

$$=E\left[\log \left(\frac{1}{\rho N_t}+\frac{1}{N_t}\hat{\mathbf{v}}(k{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(k)\right)\right].$$

$${\bar{R}}_{△}=\frac{1}{K}\sum _{k=1}^{K}E\left[\log \left(\frac{1}{\rho N_t}+\frac{1}{N_t}\hat{\mathbf{v}}(1{)}^{†}\mathbf{H}(k{)}^{†}\mathbf{H}(k)\hat{\mathbf{v}}(1)\right)\right]$$

Full text

I Introduction

Employing multiple antennas at transmitters and/or receivers has been shown to increase spatial diversity and spectral efficiency [2, 3]. To achieve higher potential of multiple antennas, some channel state information (CSI) at both the transmitter and receiver is required. At a receiver, CSI can be estimated from pilot signals. However, estimating the channel at a transmitter is not possible for frequency-division duplex (FDD) where forward and backward channels are in different frequency bands. Consequently, a transmitter in FDD must obtain CSI from a receiver via a low-rate feedback channel. Many researchers have proposed schemes to quantize and feed back CSI and analyze the associated performance (see [4] and references therein). With finite feedback rate, the beamforming vector is selected from a quantization set or a codebook, which is known a priori at the transmitter and the receiver. The codebook index of the selected vector is then fed back to the transmitter, which subsequently adjusts its beamforming coefficients accordingly. Different codebooks have been proposed and analyzed in [5, 6, 7, 8]. The optimal Grassmannian codebook that maximizes the minimum chordal distance between any two codebook entries was proposed in [5]. In [6], a random vector quantization (RVQ) codebook whose entries are independent isotropically distributed, is analyzed. RVQ codebook is simpler to construct than Grassmannian codebook and performs close to the optimum. To reduce search complexity of RVQ, the codebook entries are organized in a tree structure in [7]. In [8], PSK and QAM codebooks were proposed with low-complexity search based on noncoherent detection algorithm. If CSI at the receiver is also not perfect due to limited channel training, the rate performance will degrade further. Imperfect CSI at the receiver in conjunction with limited feedback has been considered in our previous work [9].

Feeding back quantized beamforming coefficients may not be useful in a fast fading channel since they are quickly outdated [10]. If the channel fades slowly, the beamforming coefficients may not need to be updated frequently. Thus, the feedback scheme should be adapted to temporal correlation of the channel [11, 12, 13, 14, 15, 16, 17]. Switched codebook quantization was proposed in [11] where the codebook selection was based on channel spatial and temporal correlations. In [12], quantized CSI was modeled as a first-order finite-state Markov chain and beamforming feedback is based on the channel dynamics. An adaptive feedback period (AFP) scheme in which the receiver feeds back to the transmitter periodically was considered in [13]. However, the authors were only concerned with MISO channels in which the number of receive antennas is fixed to 1. The optimal feedback period for coordinated multi-point (COMP) systems was considered in [14] where channels are also modeled as a first-order Gauss-Markov process. In [15], the minimum feedback rate of a differential feedback scheme was analyzed. The authors in [16] have proposed a differential codebook, which is rotated according to channel correlation, feedback rate, and the previous transmit beamforming. In [18, 17], a differential precoder, which depends on temporal correlation of the channel, adjusts the quantized transmit precoder to be closer to the optimal precoder.

Another line of work [19, 20, 21] applied Kalman filter (KF) to predict the current transmission channel based on previous estimates and channel correlation. References [19, 20] proposed quantizing and feeding back an innovation term, which is the difference between the received signal and its estimate, to the transmitter. The current channel estimate then can be computed by the transmitter using KF with a sequence of the previous quantized innovations. In [20], only 2 bits per update were required to send back innovations and were used to compute the beamforming vector by the transmitter. CSI at the receiver was obtained via a pilot signal and was not perfect. Reference [21] improved the training phase of KF beamforming in massive MIMO systems by reducing the amount of pilot.

For this work, we consider block Rayleigh-fading MIMO channels with time evolution modeled by a first-order Gauss-Markov process. (An uncorrelated block-fading model was considered in our previous work [6, 9]). Antennas are assumed to be sufficiently far apart that they are independent. We analyze the performance of quantized beamforming (rank-one precoding) in the AFP scheme first proposed by [13], which considered only MISO channels. In our previous work [22], we also considered quantizing transmit beamforming in MISO channels, but in conjunction with orthogonal frequency-division multiplexing (OFDM), and optimize the size of subcarrier cluster. To quantize transmit beamforming, we apply random vector quantization (RVQ) codebook, which has been shown to perform close to the optimum codebook [6, 23]. Furthermore, RVQ can be analyzed to obtain some insights into the limited feedback performance. Although transmission with beamforming or rank-one precoding does not achieve full spatial multiplexing gain in MIMO channels, the amount of CSI feedback required for beamforming is substantially smaller than that with full-rank precoding [6]. As subsequent results will show, the AFP scheme with our proposed feedback interval outperforms other schemes in low-feedback regimes. Also, when feedback rate is low, the optimal rank of the precoding matrix that maximizes achievable rate is also low and thus, transmit beamforming can be optimal or close to optimal [6]. Hence, our contribution, which is stemmed from quantizing transmit beamforming, will be most beneficial for systems with very limited feedback.

In this study, we can summarize our contribution as follows

• •

We derive a closed-form expression of the averaged received power for channels with two transmit antennas and arbitrary number of receive antennas, which is based on the eigenvalue distribution of the channel matrix [24]. For channels with arbitrary number of transmit and two receive antennas, the expression for the averaged received power is also derived, but needs to be evaluated numerically. We formulate the problems that find the optimal feedback interval and compare the rate performance of AFP scheme and the minimum feedback-period (MFP) scheme, which updates feedback for every fading block. Similar study has been performed in [13] for MISO channels and in [14] for COMP system with a single-antenna receiver. However, our results, which apply to MIMO models as well, are different and not simple extension of [13] or [14]. We find that the maximum feedback interval where the AFP scheme outperforms the MFP one, depends more on the number of receive antennas especially when feedback rate is low.

• •

For channels with an arbitrary number of transmit and receive antennas, we derive the averaged rate difference in a large system limit in which the numbers of transmit and receive antennas and the number of feedback bits tends to infinity with fixed ratios. Numerical examples show that the large system results can be used to approximate the optimal feedback interval of finite-size systems. Some of the large system results were presented in part in [1].

• •

Our numerical results show that the AFP scheme with the optimal feedback interval outperforms KF beamforming with quantized innovation in all feedback-rate regimes and the performance gain can be significant in MIMO channels. We also find that with very low feedback rate, the AFP scheme achieves larger averaged received power than the differential codebook proposed by [16], which is adapted with the channel. Although the optimal feedback interval is analyzed for RVQ codebook, the numerical results show that the optimal feedback interval for RVQ is close to that for Grassmannian codebook, which achieves optimal rate for channels with finite number of antennas.

The paper is organized as follows. Section II introduces the channel model and feedback schemes. In Section III, we analyze the optimal feedback interval for systems with two transmit and/or two receive antennas. Large system analysis is shown in Section IV. The numerical results and conclusions are in Sections V and VI, respectively.

II System Model

We consider a point-to-point discrete-time multiple-antenna channel with $N_{t}$ transmit and $N_{r}$ receive antennas. We assume block fading in which the channel gains remain static for $L$ symbols and change in the next block of symbols. To allow meaningful feedback of CSI from a receiver, the block length $L$, which is also a coherence period, is assumed to be sufficiently long. During the $k$th fading block, an $N_{r}\times 1$ receive vector during symbol index $kL+l$ is given by

[TABLE]

where we use square brackets and parentheses to indicate symbol index and block index, respectively. In (1), $x_{s}[i]$ is the $i$th transmitted symbol with zero mean and unit variance, $\bm{n}[i]$ is an $N_{r}\times 1$ additive white Gaussian noise (AWGN) vector during symbol index $i$ with zero mean and covariance $\sigma_{n}^{2}\bm{I}$ where $\bm{I}$ is an identity matrix, $\bm{v}(k)$ is an $N_{t}\times 1$ unit-norm beamforming vector for the $k$th fading block, and $\bm{H}(k)=[h_{ij}(k)]$ is an $N_{r}\times N_{t}$ channel matrix whose element $h_{ij}(k)$ is the channel gain between the $i$th receive and the $j$th transmit antennas during the $k$th fading block. Here, we consider rank-one transmit precoding or beamforming. Arbitrary-rank transmit precoding with multiple independent data streams in temporally uncorrelated MIMO channels was considered in [6]. Assuming an ideal scattering environment, $h_{ij}(k)$ is modeled as a complex Gaussian random variable with zero mean and unit variance. Also, we assume that adjacent antennas in antenna arrays at both the transmitter and receiver are placed sufficiently far apart that elements of $\bm{H}(k)$ are independent.

To model a time evolution of the channel considered, we adopt the first-order Gauss-Markov process, which has been widely used for its tractability [25, 11, 26, 13]. Thus, the channel matrix of the $k$th fading block relates to that of the previous block as follows

[TABLE]

where $\bm{W}(k)$ is an $N_{r}\times N_{t}$ innovation matrix with independent zero-mean unit-variance complex Gaussian entries, and $\alpha\in[0,1)$ denotes a temporal correlation coefficient between adjacent blocks. Note that $\alpha\rightarrow 1$ produces a time-invariant channel. On the other hand, $\alpha=0$ indicates a channel with no temporal correlation and thus, the channel fades independently from one coherence block to the next. For the Jakes/Clarke fading model [27], $\alpha=\mathbb{J}_{0}(2\pi D_{s}T_{s})$ where $\mathbb{J}_{0}(\cdot)$ is the zeroth-order Bessel function, $D_{s}$ is the Doppler spread, and $T_{s}$ is the time duration of a block. For example, for a channel with 900-MHz carrier frequency and 5-ms average fading block, $\alpha$ ranges from 0.5 to 0.9999 as mobile’s velocity varies from 60 km/h to 1 km/h.

The associated ergodic achievable rate of this channel is given by

[TABLE]

where $\rho=E[|x_{s}|^{2}]/\sigma_{n}^{2}=1/\sigma_{n}^{2}$ denotes the background signal-to-noise ratio (SNR), $[\cdot]^{{\dagger}}$ denotes the Hermitian transpose, and $E[\cdot]$ denotes the expectation operator. We note that the expectation in (3) is over channel matrix. To achieve the desired rate, the transmitter encodes the transmitted symbols across many different fading blocks with equal power per symbol. In addition to SNR, the achievable rate also depends on the beamforming vector $\bm{v}(k)$. If the transmitter can track the channel perfectly (perfect CSI), the optimal $\bm{v}(k)$ is the eigenvector of $\bm{H}(k)^{{\dagger}}\bm{H}(k)$ corresponding to the maximum eigenvalue. In other words, the optimal beamforming vector is in the direction of the strongest channel mode.

With FDD, the transmitter is not able to estimate the channel directly and has to rely on CSI fed back from the receiver via a rate-limited channel. The receiver can estimate the channel from pilot signals, which is known a priori at the transmitter and receiver. Assuming perfect CSI, the receiver selects the optimal beamforming vector and sends it back via a feedback channel to the transmitter. Since the feedback channel is rate-limited, the selected beamforming vector needs to be quantized. Here, we quantize the transmit beamforming vector with an RVQ codebook

[TABLE]

where entries $\bm{v}_{j}$ are independent isotropically distributed and $n$ denotes the number of entries in the RVQ codebook. For given $\log_{2}n$ quantization bits, RVQ performs close to the optimal codebook [23, 6] for channels with finite number of transmit and receive antennas. In a large system limit to be defined, RVQ is optimal (i.e., maximizes achievable rate) [6, 28].

Given $\log_{2}n$ bits and channel matrix $\bm{H}(k)$, the receiver selects from the RVQ codebook

[TABLE]

The index of the selected beamforming vector is then fed back to the transmitter, which adjusts its beamforming vector accordingly. We assume that the time duration to feed back the selected index is negligible when compared to one fading block and that the feedback channel is error-free. The associated achievable rate with a quantized transmit beamformer is given by

[TABLE]

Since the channel is time-varying, the transmit beamforming needs to be quantized and fed back for every fading block. This may not be practical due to the limited feedback rate. However, the system can take advantage of temporal correlation of the channel in order to reduce the number of bits needed. In this paper, we consider feedback schemes that reduce the number of feedback bits while maintaining performance.

III On Optimizing Feedback Interval

Suppose that there are $B$ feedback bits available per fading block. Since the overhead must be kept small, $B$ bits per fading block may not be sufficient to meaningfully quantize a beamforming vector $\bm{v}$. In the AFP scheme proposed by [13], $\bm{v}$ is quantized and fed back at the beginning of every interval of $K$ fading blocks with $BK$ bits instead of every block with $B$ bits. However, the transmit beamforming vector quantized to the first fading block with more feedback bits will gradually be outdated as time passes. Thus, the feedback interval $K$ should be adjusted to the temporal correlation of the channel. In this section, we analyze the optimal feedback interval for MIMO channels in the AFP scheme. Note that the feedback interval was analyzed for MISO channels by [13]. Here we analyze the achievable rate for MIMO channels with either two transmit or two receive antennas. The analysis involves the eigenvalue distribution of the channel matrix and the distribution of the received power with RVQ codebook conditioned on the channel [24], which becomes more complex as the system size increases. Thus, our results are not simple extension of those in [13].

First, we determine an average achievable rate over $K$ fading blocks given by

[TABLE]

where $\hat{\bm{v}}(1)$ is the quantized transmit beamformer for the channel $\bm{H}(1)$ in the first fading block and we apply Jensen’s inequality to obtain the upper bound (10). From (8), we see that for the AFP scheme, the quantized beamformer of the first block is used for all $K$ consecutive blocks. Since the expression of the average rate in (8) is not tractable, we choose to instead maximize the rate upper bound in (10) and obtain the feedback interval as follows

[TABLE]

which is an integer optimization problem. The problem in (11) is to maximize the average received power over $K$ blocks. If $K$ is not too large, an exhaustive search can be performed to find the optimal feedback interval $K^{*}$. We expect $K^{*}$ to be a good estimate of the feedback interval that maximizes the average rate (8) in a low-SNR regime since in that regime, logarithm increases approximately linearly with the received power.

III-A $2\times N_{r}$* Channels*

For a point-to-point channel with 2 transmit antennas and $N_{r}>1$ receive antennas, the following lemma gives the expected received power during the $k$th fading block when the quantized transmit beamforming for the first block is used.

Lemma 1

The received power for the $k$th block of a $2\times N_{r}$ channel with $BK$ bits to quantize $\bm{v}(1)$, is given by

[TABLE]

where

[TABLE]

and $\phi(m,n)$ is a recursive function given by

[TABLE]

with the following initial conditions: $\phi(4,1)=\frac{45}{8}$, $\phi(3,2)=\frac{11}{8}$, $\phi(2,3)=\frac{5}{8}$, and $\phi(1,4)=\frac{3}{8}$.

The proof is in Appendix -A.

From (12), we see that as $k$ increases, the received power decreases since the channel becomes less matched to the transmit beamformer $\hat{\bm{v}}(1)$. However, if the channel is highly correlated ($\alpha$ close to 1), the received power will gradually decrease with time. Averaged over the whole feedback interval, the received power for a $2\times N_{r}$ channel is given by

[TABLE]

We note that the average received power increases with $B$. To determine $K^{*}$ that maximizes the average received power, we substitute (16) into (11) and solve the problem. To obtain some insight on $K^{*}$, we can consider the two extreme regimes. When channels are less correlated ($\alpha\to 0$) and $B$ is large, $K^{*}$ will be close to 1. This is due to the diminishing return of $\gamma_{2\times N_{r}}(x)$. $K^{*}\approx 1$ implies that feedback must occur as frequently as possible when the channel is fast changing and feedback rate is high. When channels are highly correlated ($\alpha\to 1$), we can show with L’Hôpital’s rule that

[TABLE]

Thus, the optimal interval $K^{*}\to\infty$ since $\gamma_{2\times N_{r}}(x)$ is increasing with $K$. In other words, if the channel is relatively static, the feedback interval should be large. For other values of $\alpha$ (e.g., $\alpha=0.8$), our numerical results in Fig. 1 show that $K^{*}$ does not depend much on $N_{r}$ since increasing the number of receive antennas seems to increase the received signal power uniformly for all $K$.

In [13], the performance of the AFP scheme is compared with that of the minimum feedback period (MFP) scheme in which transmit beamforming is quantized and fed back to the transmitter for every fading block ($K=1$). However, [13] only considers MISO channels. In MIMO channels with a given feedback rate of $B$ bits per fading block, we find that the AFP scheme (with $K>1$) outperforms the MFP scheme (with $K=1$) if

[TABLE]

where the right-hand side of (18) is the average received power in (16) with $K=1$. With some algebraic manipulation, we obtain

[TABLE]

Thus, (20) gives the range of $K$ in which the performance of AFP exceeds that of MFP and the maximum $K$ with that property. If we consider a large $B$ regime or $B\to\infty$, the inequality (20) becomes

[TABLE]

Thus, we can conclude that when the feedback rate is large, the maximum feedback interval of the AFP scheme that outperforms the MFP scheme depends largely on the temporal correlation $\alpha$. Thus, the feedback interval for the AFP scheme can be set larger when channels are highly correlated and should be shortened when channels are less correlated.

III-B $N_{t}\times 2$* Channels*

Next, we consider channels with $N_{t}>2$ transmit antennas and two receive antennas. We can follow the derivation of the averaged received power for $2\times N_{r}$ channels in Section III-A to obtain the averaged received power for $N_{t}\times 2$ channels,

[TABLE]

where the above expression follows (16) with $N_{r}=2$, and

[TABLE]

is the received power of the first block. Recall that

[TABLE]

Since the RVQ codebook is employed, the probability density function (pdf) of $\bm{v}_{j}^{{\dagger}}\bm{H}(1)^{{\dagger}}\bm{H}(1)\bm{v}_{j}$ is identical for all $j$ and is equal to $\bm{v}_{j}^{{\dagger}}\bm{\Lambda}\bm{v}_{j}$ [24] where $\bm{\Lambda}$ is an $N_{t}\times N_{t}$ diagonal matrix whose main diagonal entries are the ordered eigenvalues of $\bm{H}(1)^{{\dagger}}\bm{H}(1)$. For this channel, there are only two nonzero eigenvalues, which are denoted by $\lambda_{1}$ and $\lambda_{2}$ and $\lambda_{1}\geq\lambda_{2}>0$. We derive the distribution of $\bm{v}_{j}^{{\dagger}}\bm{\Lambda}\bm{v}_{j}$ and obtain the following lemma.

Lemma 2

Let $\bm{v}$ be an $N_{t}\times 1$ isotropically distributed vector with $N_{t}>2$ and $\bm{\Lambda}=\text{diag}([\lambda_{1},\lambda_{2},\underbrace{0,0,...,0}_{N_{t}-2}])$ with $\lambda_{1}\geq\lambda_{2}>0$. The cumulative distribution function (cdf) of $\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}$ conditioned on $\lambda_{1}$ and $\lambda_{2}$ is given by

[TABLE]

We remark that the expression of the cdf for $\lambda_{2}\leq x\leq\lambda_{1}$ is obtained from [24] and is shown in Lemma 2 for completeness. However, the expression of the cdf for $0\leq x\leq\lambda_{2}$ is not derived in [24] and is not a simple extension of the earlier case. The proof of Lemma 2 is shown in Appendix -B.

With (24) and (25), it is straightforward to show that

[TABLE]

Thus,

[TABLE]

where $f_{\bm{\Lambda}}(\lambda_{1},\lambda_{2})$ is the joint pdf of the two ordered eigenvalues of $\bm{H}(1)^{{\dagger}}\bm{H}(1)$ and is stated in (59) where $N_{t}$ replaces $N_{r}$. Substitute (26) into (27) and evaluate the first integral to obtain

[TABLE]

The recursive function $\phi$ is defined in (15). The integral in (28) can be evaluated by any numerical method. We remark that the expression for the average received power in (28) does not apply for $N_{t}=2$ since the cdf derived in Lemma 2 only applies when $N_{t}>2$. We find the optimal feedback interval $K^{*}$ by maximizing the average received power in (22), which is determined by (28). The same conclusion made for the previous channel model on the maximum feedback interval of the AFP scheme still applies for this channel model. However, [6] has shown that in order to maintain $\gamma_{N_{t}\times 2}$, $B$ needs to scale with $N_{t}$ as $N_{t}$ becomes large. Otherwise, if $B/N_{t}\to 0$, then the quantization error of transmit beamforming vector will be large and, hence the received power $\gamma_{N_{t}\times 2}$ will be close to that with no CSI. Thus, for a fixed feedback rate, the maximum feedback interval of the AFP scheme must increase as $N_{t}$ increases.

From the analysis, we see that optimizing the feedback interval requires the temporal correlation coefficient $\alpha$, which in practice, has to be estimated. For instance, a least-square estimator [29] can be applied to determine $\alpha$. Since channel statistics does not change as often as channel realization does, $\alpha$ may not need to be estimated frequently.

In this section, our analytical results only apply to channels with either two transmit or two receiver antennas. For channels with arbitrary $N_{t}$ and $N_{r}$, the expression for the received power is not tractable due to the pdf of $\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}$ and the joint pdf of the ordered eigenvalues of $\bm{H}(1)^{\dagger}\bm{H}(1)$. However, the performance of the system with an arbitrary number of antennas can be well approximated by its performance in a large system regime to be defined in the next section.

IV Large System Analysis

The large system limit refers to one of which $N_{t},N_{r},B$ tend to infinity with fixed $\bar{N}_{r}\triangleq N_{r}/N_{t}$ and $\bar{B}\triangleq B/N_{t}$. In a large system limit, the pdf of the ordered eigenvalues converges to a deterministic function [30] and hence, performance analysis of systems with arbitrary size becomes accessible. It is shown by [6] that with some feedback ($\bar{B}>0$) and fixed $\bar{N}_{r}$, the achievable rate defined in (7) increases with $\log(\rho N_{t})$. Thus, we define an achievable rate difference as follows

[TABLE]

Therefore, $R_{\triangle}$ is a rate difference between an actual rate and $\log(\rho N_{t})$ and the difference increases with $\bar{B}$ [6]. With feedback rate $\bar{B}$ per fading block, we apply the AFP scheme described in Section III and compute the average rate difference over an interval of $K$ fading blocks given by

[TABLE]

where the quantized beamforming of the first block is used for the whole interval of $K$ blocks. We note that in the previous section, we chose to evaluate the upper bound on the rate via the average received power due to the intractability of the rate analysis. However, in this section, we evaluate the rate difference.

IV-A Large System With $\bar{N}_{r}>0$

First we consider the large system with $\bar{N}_{r}>0$. In other words, the numbers of transmit and receive antennas are increasing at the same rate. Similar to the analysis of the system with a finite number of antennas, we determine the received power per transmit antenna $\frac{1}{N_{t}}\hat{\bm{v}}^{{\dagger}}(1)\bm{H}^{{\dagger}}(k)\bm{H}(k)\hat{\bm{v}}(1)$ by applying the Gauss-Markov equation in (2) and evaluate each term after substitution. The first of the two nonzero terms is shown by [6, 28] to converge in a large system limit

[TABLE]

where $\bar{B}K$ is the normalized feedback bits used for quantizing $\hat{\bm{v}}(1)$ and the expression for the function $\gamma_{\infty}(x)$ is as follows [6]. Suppose

[TABLE]

For $0\leq x\leq\beta$, $\gamma_{\infty}$ satisfies

[TABLE]

and for $x\geq\beta$,

[TABLE]

The second nonzero term can be shown to converge to

[TABLE]

Applying (32) and (36), we obtain

[TABLE]

Consequently, the expression for the asymptotic rate difference is given by

[TABLE]

We would like to maximize the asymptotic achievable rate difference averaged over the feedback interval $K$. For a given feedback rate of $\bar{B}$ and $\bar{N}_{r}>0$, the optimal feedback interval that maximizes the asymptotic achievable rate difference is therefore given by

[TABLE]

Similar to a finite-size system, exhaustive search over some range of $K$ can be used to obtain a suboptimal feedback interval. We note that the optimal feedback interval in (40) will depend on the temporal correlation coefficient, feedback rate, and the number of transmit and receiver antennas. Next we consider two extreme regimes for which $\alpha\to 0$ and $\alpha\to 1$. When the channel does not change ($\alpha\to 1$), the optimal feedback interval $K^{*}$ can be shown to be infinite from (40). This implies that only one feedback update at the start with all available feedback bits giving the maximum rate difference.

When the channel fades independently from a current block to the next block ($\alpha\to 0$), the rate difference in (39) becomes

[TABLE]

Maximizing the rate-difference expression in (41), the optimal feedback interval is given by

[TABLE]

which depends on $\bar{N}_{r}$ and $\bar{B}$. We remark that for moderate to large $\bar{B}$, $K^{*}=1$. Hence, if the channel is temporally uncorrelated, the feedback update must occur as frequent as possible. In other words, the MFP scheme will outperform the AFP scheme.

For general $\bar{N}_{r}$ and $\alpha$, to find the range of $K$ in which the AFP scheme performs better than the MFP scheme, we solve for $K$

[TABLE]

where $\bar{R}_{\triangle}^{\infty}$ is stated in (39).

IV-B Large System With $\bar{N}_{r}\to 0$

Next we examine the system in which $\bar{N}_{r}\to 0$ in a large system limit. The results will apply to the system in which the receiver is equipped with only single antenna (MISO channel) or a fixed number of antennas while the transmitter is equipped with much larger number of antennas. First we evaluate the large system limit of $\frac{1}{N_{t}}\hat{\bm{v}}(1)^{{\dagger}}\bm{H}(k)^{{\dagger}}\bm{H}(k)\hat{\bm{v}}(1)$. For $\bar{N}_{r}=0$, [6] shows that

[TABLE]

while

[TABLE]

Thus, the asymptotic achievable rate difference is given by

[TABLE]

for $0<\alpha\leq 1$.

Maximizing the asymptotic achievable rate difference in (48) gives the optimal feedback interval as follows

[TABLE]

If the integer constraint is removed, we can find $K^{*}$ from the first derivative of $\bar{R}_{\triangle}^{\infty}$ in (48) and obtain the following approximation

[TABLE]

where $0<\alpha<1$. The asymptotic $K^{*}$ obtained from (50) is close to that for a finite-size system. We note that for large feedback $\bar{B}$, $K^{*}$ is small. The solution implies that the feedback update should occur often when a large number of feedback bits is available. For a small-feedback regime ($\bar{B}\to 0$), $K^{*}$ is approximated as follows

[TABLE]

We note that $K^{*}$ is increasing with $\alpha$. Thus, we can conclude that with a low feedback rate and a highly correlated channel, the feedback interval should be large or the feedback update should occur less frequently.

Comparing the rates obtained from the AFP and MFP schemes, we find that the feedback interval $K$ for the AFP scheme must be larger than

[TABLE]

Hence, as channels become less correlated (small $\alpha$), $K$ can be large. This bound is obtained by solving

[TABLE]

where $\bar{R}_{\triangle}^{\infty}(K)$ is stated in (48).

V Numerical Results

To illustrate the performance of the considered schemes, Monte Carlo simulation is performed with 3,000 channel realizations. First, we compare the analytical results derived in Section III with the simulation results. Fig. 1 shows the average received power normalized by the average received power with perfect feedback, over the feedback interval of the AFP scheme with the feedback interval $K$. The feedback rate $B=1$ bit per block and correlation coefficient $\alpha=0.8$. We have two sets of system sizes. For the first set, $N_{t}$ is fixed at 2 with various $N_{r}$ ($2\times 2$, $2\times 3$, and $2\times 4$). We see that the analytical result in (16), which is shown with a solid line, perfectly matches with the simulation result, which is shown with circles. For all $N_{r}$, the optimal feedback interval $K^{*}$ is 3. Adding more receive antennas will increase the received power since the receiver can capture more transmitted signal. With 4 receive antennas, the system with $K^{*}$ achieves closer to 85% of the performance with infinite feedback. The AFP scheme with the optimal $K$ ($K=3$) can outperform the MFP scheme ($K=1$) by close to 11%.

For the second set of system sizes in which $N_{r}=2$ and $N_{t}$ varies ($2\times 2$, $3\times 2$, $4\times 2$, and $5\times 2$), the analytical result comes from (22), and (28). We see that the optimal interval $K^{*}$ increases with $N_{t}$ since the number of bits ($BK$) required to quantize the beamforming vector increases with $N_{t}$. For a larger system ($5\times 2$), the AFP with $K^{*}=5$ can outperform the MFP by as much as 27%.

For $2\times 2$ channels, we see that the AFP scheme with $2\leq K\leq 7$ gives larger averaged received power than the MFP scheme. The range of $K$ is accurately predicted by (20). For the $2\times 4$ channel, the range of $K$ for which the AFP performs better is $2\leq K\leq 8$, which can also be obtained by (20).

In Fig. 2, we compare the performance of RVQ codebook with that of the Grassmannian codebook [5], which maximizes the minimum chordal distance between any two codebook entries. The Grassmannian codebook is optimal for channels with finite number of antennas and hence, is shown in the figure to outperform RVQ codebook. However, the Grassmannian codebook is more complex to construct than RVQ codebook especially when the number of entries is large. Thus, in the figure, we do not have results of the Grassmannian codebook beyond $K=8$. We note that the performance shown in Fig. 2 is the averaged received power normalized by the received power with infinite feedback. We see a larger performance gap between the two codebooks when $BK$ is small or when the number of quantization bits is small. For all 3 cases shown, $K^{*}$ for RVQ codebook and the optimal $K$ that maximizes the received power for Grassmannian codebook only differs by 1. This implies that the optimal feedback interval derived for RVQ codebook in this study can be applied to the Grassmannian codebook with some small degradation. For the $3\times 2$ channel, we see that the gain of AFP with the optimal $K$ over MFP ($K=1$) increases when the channel becomes more correlated ($\alpha$ closer to 1). For the 3 x 2 channel with $\alpha=0.95$, the Grassmannian codebook with $K^{*}=7$ ($K^{*}$ is derived with RVQ codebook) achieves approximately 82% of the rate with perfect feedback while the Grassmannian codebook with $K=1$ or the MFP scheme achieves only 57% of the rate with perfect feedback. Thus, the performance gain of the AFP scheme over the MFP scheme in this instance is about 44%.

Fig. 3 shows the optimal feedback interval $K^{*}$ for a $2\times 2$ channel with different values of correlation coefficient $\alpha$ and the number of feedback bits per fading block per transmit antenna $B/N_{t}$. We consider a mobile system operating at 900 MHz with 5-ms average fading block for which $\alpha$ varies from 0.5 to 0.9999 as the speed of mobile decreases from 60 km/h to 1 km/h [27]. We see that for a slow fading channel $(\alpha\rightarrow 1)$, feedback update can be less frequent and thus, the feedback interval is large. On the other hand, fast fading channels (smaller $\alpha$) require frequent feedback updates. If the feedback rate per transmit antenna ($B/N_{t}$) is increased from 0.5 to 1, we see that $K^{*}$ decreases.

In Fig. 3, we also show the optimal interval $K^{*}$ of a large system with $\bar{N}_{r}=1$ obtained by solving (40). We remark that $K^{*}$ for a large system is obtained by maximizing the rate difference while $K^{*}$ for a $2\times 2$ channel is obtained by maximizing the averaged received power. However, we see that the large system results can give a good approximation of those of a very small system.

In Fig. 4, we compare the achievable rate difference of a large system derived in (39) with that of a finite-size system for various feedback rates per transmit antenna $\bar{B}$. The feedback interval $K$ is fixed at 8 blocks and SNR $\rho$ is at 10 dB. The averaged rate gain of finite-size and large systems is obtained from (31) and (39), respectively. We see that as the system size increases from $N_{t}=4$ to 8, 16, and 24, the simulation results approach the large system results. However, we note that the convergence to the asymptotic results is slow. Thus, unless the system size is very large, the gap between the actual and the asymptotic rate difference might be significant. We also note that the rate difference increases with $\bar{B}$ as expected, but rate of increase is different for different values of $\alpha$. When the channel is less correlated ($\alpha=0.5$), the quantized beamforming vector of the first block is not a good substitute for that of the next blocks. Consequently, we do not see much increase in that case although the feedback rate is increased. On the contrary, we see a large increase when the channel is more correlated ($\alpha=0.9$) since the quantized beamforming vector of the first block performs well for all subsequent blocks in the same interval. Since we quantize beamforming vectors with the RVQ codebook, which requires an exhaustive search to find the quantized vector, the search complexity can be too large for large $B$. Thus, some of the plots in Fig. 4 do not extend to a larger feedback rate.

In Fig. 5, we set $\bar{B}=0.25$ and vary $K$ for channels with different temporal correlation. We compare the rate difference of $4\times 4$ channels and that of a large system with $\bar{N}_{r}=1$. For a $4\times 4$ channel with $\alpha=0.9$, the AFP scheme with $K=5$ performs almost twice as much as the MFP scheme does (the green line with pluses). For time-invariant channels ($\alpha=1$), the optimal $K$ is large. Although the difference between the results of small-size and large systems can be large as shown in Fig. 4, the optimal feedback interval $K^{*}$ obtained from the two results is close (either off by 1 or identical). We also compare the optimal $K$ from the simulation and analytical results with different system sizes, feedback rates, and channel correlation coefficients in Fig. 6. The results reinforce that the optimal feedback interval that maximizes the rate difference of a finite-size system, can be predicted quite accurately by the large-system analysis.

We plot the optimal feedback interval $K^{*}$ with the temporal correlation $\alpha$ for large-system channels with different $\bar{N}_{r}$ and $\bar{B}$ in Fig. 7. The same trend as shown in Fig. 3 is also observed in this figure. $K^{*}$ increases with $\alpha$. However, we note that $K^{*}$ is mostly unchanged across different values of $\bar{N}_{r}$, except when $\bar{B}$ is extremely low. Similar observation regarding to different number of receive antennas was also noted for a finite-size system.

Fig. 8 shows how the optimal feedback interval $K^{*}$ increases with the number of transmit antennas $N_{t}$, but decreases with the number of feedback bits per fading block $B$. We note that $K^{*}$ is obtained by first substituting (22) into (11) and then solving (11) numerically. We set $N_{r}=2$ and $\alpha=0.8$. For larger $N_{t}$, the number of bits to quantize the beamforming vector needs to increase to maintain the rate performance and hence, the feedback interval has to increase as well. Similar to the results in Fig. 3, as $B$ increases, the feedback interval can be reduced.

In Fig. 9, we compare the AFP and MFP schemes with existing Kalman-filter scheme and differential-feedback scheme in the literature for $3\times 1$ and $3\times 3$ channels. In [20], KF scheme is applied to construct the channel vector (or channel matrix) at the transmitter, which then can compute the optimal transmit beamforming. For a fair comparison, we assume that the channel estimation at the receiver is perfect. The receiver quantizes an innovation term, which is the difference between the received signal and its estimate based on channel estimates from the previous blocks. The innovation can be straightforwardly shown to be zero-mean Gaussian with some finite variance. Thus, for quantization, we apply a generalized Lloyd algorithm [31], which minimizes the mean square error. The quantized innovation is fed back to the transmitter for every fading block. To construct the channel vector, we follow the steps in [20, 19]. The performance of KF scheme is shown in Fig. 9.

For the performance of differential feedback, we apply method 1 in [16]. The codebook that quantizes transmit beamforming vector is not fixed, but is gradually updated by the rotation matrix selected from a rotation codebook and the normalized radius, which is a function of $N_{t}$, $B$, $\alpha$, and block index $k$. The rotation codebook consists of unitary matrices. For the optimal rotation codebook, the minimum distance defined by [16] between two entries is maximized. For the results in this figure, we generate 10000 random codebooks with the desired number of entries and find the codebook with the largest minimum distance between any two codebook entries. Thus, our rotation codebook is suboptimal, but should be close to the optimum due to a larger number of trials.

In Fig. 9, we $\alpha=0.9$ and SNR = 10 dB. We note that some feedback schemes may require some initial feedback bits and thus, their performance does not extend to $B=0$ or small $B$. For example, the KF scheme needs at least $2N_{r}$ bits to quantize an innovation, which is an $N_{r}$-dimensional complex vector. From the figure, we see that, Grassmannian codebook with $K^{*}$, which is obtained from our analysis, performs the best for low to moderate feedback rates and is followed closely by RVQ codebook with $K^{*}$. For the $3\times 1$ channel with $B=2$, the Grassmannian codebook with $K^{*}$ outperforms KF scheme by about 10%. The differential feedback scheme by [16] performs better than other schemes when feedback rate is larger and performs worse when feedback rate is small. As mentioned in [16], the scheme requires some sufficient feedback to compensate for cumulative quantization error. We see that codebooks with $K^{*}$ outperform the KF scheme for all feedback rates for both $3\times 1$ and $3\times 3$ channels. Performance degradation is quite significant for the KF scheme when applied to MIMO channels. If feedback is not sufficient, the KF scheme does not track channel matrix well and hence, produces transmit beamforming, which is not aligned with the strongest channel mode.

VI Conclusions

We have analyzed the feedback interval that maximizes either the average received power or the rate difference for MIMO channels. For the channel model with either two transmit or two receive antennas, the optimal interval depends more on the channel correlation, the number of transmit antennas, and the feedback rate, and less on the number of receive antennas. For that model, we formulated the received-power maximizing problem in which the exact feedback interval can be found. For systems with arbitrary number of transmit and receive antennas, large system analysis can be used to predict the optimal interval accurately as shown by the numerical examples. The optimal feedback interval is a function of the channel correlation, the number of feedback bits per antenna, and the ratio between the number of transmit and receive antennas. However, the optimal feedback interval also is less sensitive to the change in the number of receive antennas.

When the feedback rate is low, the AFP scheme with the optimal feedback interval outperforms the other schemes including the KF scheme and differential feedback scheme. The performance gain of the AFP scheme over the other schemes can be as much as 10%. Thus, the feedback interval should be adapted according to channel condition. However, when the feedback rate is high, the performance difference among the different schemes may not be significant. We also note that the optimal feedback interval derived for RVQ codebook and be applied with Grassmannian codebook, which is optimal for finite-size channels, with small degradation.

In this work, we assume that training of the channel is sufficient and thus, CSI at the receiver is perfect. For a system with limited training, the actual performance of the AFP scheme will be lower than that obtained in the paper and the KF scheme may perform better. Since we only consider a point-to-point channel in the present work, broadcast or multiple-access channels are also of interest and can be considered in future work.

-A Proof of Lemma 1

Apply the Gauss-Markov model in (2) and some algebraic manipulation to obtain

[TABLE]

where the expectation of the cross term that consists of $\bm{W}$ is equal to zero since $\bm{W}$ has zero mean and is independent of $\hat{\bm{v}}(1)$ and all channel matrices $\bm{H}(k),\forall k$.

We proceed to analyze the first expectation in (54). Following the same argument pertaining to the received power in (24), $\bm{v}^{{\dagger}}_{i}\bm{H}^{{\dagger}}(1)\bm{H}(1)\bm{v}_{i}$ is independent and has the same distribution as $\bm{v}^{{\dagger}}_{i}\bm{\Lambda}\bm{v}_{i}$ [24] where $\bm{\Lambda}=\text{diag}([\lambda_{1},\lambda_{2}])$, and $\lambda_{1}$ and $\lambda_{2}$ are the ordered eigenvalues of $\bm{H}(1)^{{\dagger}}\bm{H}(1)$ with $\lambda_{1}\geq\lambda_{2}\geq 0$. The cdf for $\bm{v}^{{\dagger}}_{i}\bm{\Lambda}\bm{v}_{i}$ conditioned on $\lambda_{1}$ and $\lambda_{2}$ is given by [24]

[TABLE]

We then apply integration by parts to obtain the expression for the conditional expectation as follows

[TABLE]

Averaging over the two eigenvalues gives

[TABLE]

where $f_{\bm{\Lambda}}(\lambda_{1},\lambda_{2})$ is a joint pdf for the two ordered eigenvalues of a Wishart matrix $\bm{H}(1)^{{\dagger}}\bm{H}(1)$ given by [32]

[TABLE]

Suppose

[TABLE]

By substituting (59) into (58) and rearranging the terms, we can write $E\left[\hat{\bm{v}}(1)^{{\dagger}}\bm{H}(1)^{{\dagger}}\bm{H}(1)\hat{\bm{v}}(1)\right]$ in (58) in terms of $\phi(\cdot,\cdot)$ as shown in (14).

To evaluate (60), we apply integration by parts to the inner integral to obtain

[TABLE]

where in (61), we switch the order of integration for the first integral and evaluate the second integral. We again evaluate the inner integral in (62) and switch the order of integration to obtain

[TABLE]

Adding the last two terms in (63) gives (15). The initial conditions are obtained by evaluating the double integral in (60).

Since $\bm{W}(k-i)$ and $\hat{\bm{v}}(1)$ are independent and $E[\bm{W}(k-i)^{{\dagger}}\bm{W}(k-i)]=N_{r}\bm{I}$, we have that

[TABLE]

where $\operatorname{tr}\{\cdot\}$ denotes the trace operator.

Finally, we substitute (63) and (67) into (54) and simplify to obtain (12).

-B Proof of Lemma 2

Since the considered matrix $\bm{H}(1)^{{\dagger}}\bm{H}(1)$ has rank 2,

[TABLE]

Applying the results from [24, eq. (18)], we obtain in Lemma 2 the expression for the cdf $F_{\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}}(x)$ for $\lambda_{2}\leq x\leq\lambda_{1}$ only. Next we derive the expression of the cdf when $0\leq x\leq\lambda_{2}$. The derivation is inspired by [33] where evaluating the cdf $F_{\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}}(x)$ was formulated as finding the surface area of an $N_{t}$-dimensional spherical cap. The results in [33] apply when $\bm{\Lambda}$ is full rank. In our case, $\bm{\Lambda}$ has rank 2 with nonzero diagonal entries $\lambda_{1}$ and $\lambda_{2}$.

Recall that $\bm{v}=[v_{1}\ v_{2}\ \cdots\ v_{N_{t}}]^{T}$ is an $N_{t}\times 1$ isotropically distributed vector with unit norm. Therefore, we have

[TABLE]

In $N_{t}$-dimensional space, we can view the set of vectors $\{\bm{v}\in\mathbb{C}^{N_{t}}|\sum_{i=1}^{N_{t}}|v_{i}|^{2}=1\}$ as a surface of an $N_{t}$-dimensional unit ball centered at the origin. We can rearrange $\lambda_{1}|v_{1}|^{2}+\lambda_{2}|v_{2}|^{2}\geq x$ as follows

[TABLE]

The above inequality describes the region outside a two-dimensional ellipse centered at the origin. Because $\lambda_{1}\geq\lambda_{2}$, the widest part of the ellipse is determined by $\frac{x}{\lambda_{2}}$. Since we consider the regime where $0\leq x\leq\lambda_{2}$ or $0\leq\frac{x}{\lambda_{2}}\leq 1$, geometrically, the ellipse is completely contained in the $N_{t}$-dimensional unit ball.

We take the same analytical approach as the one in [33] by first finding the volume of the $N_{t}$-dimensional object prescribed by $\lambda_{1}|v_{1}|^{2}+\lambda_{2}|v_{2}|^{2}\geq x$ and $\sum_{i=1}^{N_{t}}|v_{i}|^{2}\leq r^{2}$ where $r\geq 1$. (In the final steps, we will set $r=1$.) Then, we compute its surface area, which is shown to be proportional to the desired cdf $F_{\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}}(x)$ [33].

The volume of the region $\{\bm{v}\in\mathbb{C}^{N_{t}}|\lambda_{1}|v_{1}|^{2}+\lambda_{2}|v_{2}|^{2}\geq x,\|\bm{v}\|^{2}\leq r^{2}\}$ is denoted by

[TABLE]

where the volume of an $N_{t}$-dimensional ball with radius $r$ is given by [33]

[TABLE]

and $\textsf{Vol}(\lambda_{1}|v_{1}|^{2}+\lambda_{2}|v_{2}|^{2}\leq x,\|\bm{v}\|^{2}\leq r^{2})$ is the volume of the ellipsoid that is completely contained in the hyperball with radius $r$.

To compute the volume of the ellipsoid, we first apply the following transformation

[TABLE]

where $r_{i}$ and $\theta_{i}$ are the magnitude and phase of $v_{i}$, respectively. Using spherical coordinates, the volume of the ellipsoid is given by

[TABLE]

We note that the multiple integral in the brackets in (74) is the volume of an $(N_{t}-2)$-dimensional ball with radius $r^{2}-r_{1}^{2}-r_{2}^{2}$. Applying (72), we have

[TABLE]

To compute the surface area of the volume, we differentiate the volume as follows

[TABLE]

The surface area of the $N_{t}$-dimensional unit ball is given by

[TABLE]

The pdf of $\bm{v}^{{\dagger}}\bm{\Lambda}\bm{v}$ is given by [24, eq. (115)]

[TABLE]

Finally, the expression of the cdf for $0\leq x\leq\lambda_{2}$ in (25) can be obtained by integrating the pdf in (82).