# Space-Time Interference Alignment and Degrees of Freedom Regions for the
MISO Broadcast Channel with Periodic CSI Feedback ††thanks: N. Lee and R.
Heath were funded in part by the Army Research Labs, Grant W911NF1010420. This
paper was presented in part [19] at IEEE 50th Annual Allerton Conference on
Communication, Control, and Computing, Monticello, Oct. 2012.

Namyoon Lee and Robert W. Heath Jr.

Wireless Networking and Communications Group
Department of Electrical and Computer Engineering
The University of Texas at Austin, Austin, TX 78712 USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper characterizes the degrees of freedom (DoF) regions for the multi-
user vector broadcast channel with periodic channel state information (CSI)
feedback. As a part of the characterization, a new transmission method called
_space-time interference alignment_ is proposed, which exploits both the
current and past CSI jointly. Using the proposed alignment technique, an inner
bound of the sum-DoF region is characterized as a function of a normalized CSI
feedback frequency, which measures CSI feedback speed compared to the speed of
user’s channel variations. One consequence of the result is that the
achievable sum-DoF gain is improved significantly when a user sends back both
current and outdated CSI compared to the case where the user sends back
current CSI only. Then, a trade-off between CSI feedback delay and the sum-DoF
gain is characterized for the multi-user vector broadcast channel in terms of
a normalized CSI feedback delay that measures CSI obsoleteness compared to
channel coherence time. A crucial insight is that it is possible to achieve
the optimal DoF gain if the feedback delay is less than a derived fraction of
the channel coherence time. This precisely characterizes the intuition that a
small delay should be negligible.

##  I Introduction

Channel state information at the transmitter (CSIT) is important for
optimizing wireless system performance. In the multiple-input-single-output
(MISO) broadcast channel, CSIT allows the transmitter to send multiple data
symbols to different receivers simultaneously without creating mutual
interference by using interference suppression techniques [1]-[2]. Prior work
on the MISO broadcast channel focused on the CSIT uncertainty caused by
limited rate feedback [5]-[9]. This prior work showed no degrees of freedom
(DoF) are lost compared to the perfect CSIT case when the CSI feedback rate
per user linearly increases with signal to noise ratio (SNR) in dB scale.
Meanwhile, there is little work on the effect of CSIT uncertainty due to
feedback delay. In particular, if a transmitter has past CSI that is
independent of the current CSI, i.e., outdated CSIT, it was commonly believed
that only unity DoF gain could be achieved, which is the same DoF gain in the
absence of channel knowledge at the transmitter [10]. Recently, it was shown
in [11] that outdated CSIT is helpful to obtain the DoF gains beyond that
achieved by time-division-multiple-access (TDMA). The key idea from [11] was
to exploit the perfect outdated CSIT as side-information so that the
transmitter aligns inter-user interference between the past and the currently
received signals. Motivated by the work in [11], extensions have been
developed for other networks [12]-[15]. The DoF gain was studied for the
interference channel and the X channel when only outdated CSI is used at the
transmitter in [12]. The DoF was characterized for the two-user MIMO
interference channel with delayed CSI according to different transmit-and-
receive-antenna configurations in [13]. The outdated CSI feedback framework
was also considered for the two-hop interference channel in [14] and [15],
where the DoF gain increased beyond that achieved by the TDMA method.

The common assumption of previous work [11]-[15] is that the transmitter has
delayed CSI only. Depending on a feedback protocol and relative feedback delay
compared to channel variations, however, it may be possible for the
transmitter to have knowledge of outdated as well as current CSI. Therefore,
one of the fundamental questions is how much gain can the system provide if
the transmitter can have both current and outdated CSI? An answer for this
question is addressed in the periodic CSI feedback framework in this paper.

###  I-A Two Different Models of Periodic CSI Feedback

Unlike aperiodic CSI feedback where each user opportunistically sends back CSI
whenever the channel condition is changed [21], periodic CSI feedback allows
the transmitter to receive CSI periodically from mobile users through
dedicated feedback channels. This is suitable for current cellular standards
such as 3GPP LTE [22] because it is implementable using a control signal with
lower overhead. In such a periodic CSI feedback protocol, two different models
can be considered depending on the dominant factor that causes delayed CSIT.

Figure 1: Motivation of the proposed periodic CSI feedback model.

  1. 1.

Feedback frequency limited model (Model 1): Consider a block fading channel
where the channel values are constant for the channel coherence time
$T_{\textrm{c}}$ and the channels change independently between blocks.
Further, assume that feedback delay $T_{\textrm{fb}}$ is much less than the
channel coherence time $T_{\textrm{c}}$. In such a channel, consider the case
where a transmitter cannot track all channel changes of users because each
user has a limited CSI feedback opportunity given by the periodic feedback
protocol. In other words, the CSI feedback frequency is not high enough to
continually track the user’s channel variations. Consider the example
illustrated in Fig. 1-(a). If a user is able to use the feedback channel over
the feedback duration $T_{\textrm{f}}=2T_{\textrm{c}}$ among the overall
feedback period $T=3T_{\textrm{c}}$, i.e.,
$\omega=\frac{T_{\textrm{f}}}{T}=\frac{2}{3}$, the transmitter may have
current CSI for two-thirds of the channel variations, assuming that the CSI
feedback delay is much smaller than the channel coherence time. Instead of
sending back currently estimated CSI only, however, the transmitter can
acquire more channel knowledge, provided that each user sends back a set of
CSI including outdated and current CSI during the feedback durations. For
example, each user may send back one outdated and two current channel state
estimates to the transmitter during feedback durations as shown in Fig. 1-(b).
In this model, CSIT is outdated due to the lower feedback frequency rather
than the feedback link delay. Hence, it is possible to consider an equivalent
CSI feedback model in a fast fading channel without considering feedback delay
to focus on the effect of CSI feedback frequency as shown in Fig. 2-(a). The
equivalence comes from the fact that both models allow for the transmitter to
use two outdated and one current observation of CSI periodically. In this
model, we establish how the CSI feedback frequency affects the sum-DoF gain in
this fast-fading channel configuration.

Figure 2: Periodic CSI feedback models for both fast and block fading
channels. In Model 1, outdated CSIT is due to the lower CSI feedback frequency
than channel variations. Meanwhile, outdated CSIT comes from the CSI feedback
delay mainly in Model 2.

  2. 2.

Feedback link delay limited model (Model 2): We also consider a different CSI
feedback scenario in which the feedback link delay is the dominant factor that
causes outdated CSIT. Consider a block fading channel where the transmitter
can track all channel variations for each user, assuming that the feedback
interval $T$ is equal to channel coherence time $T_{\textrm{c}}$ for
simplicity. Further, the feedback delay $T_{\textrm{fb}}$ is not negligible
compared to the channel coherence time. In this scenario, outdated CSIT is no
longer affected by the feedback frequency because the transmitter can track
the channel variations over all channel blocks. Rather, the relative
difference between the CSI feedback delay and the channel coherence time
causes the transmitter have the delayed CSI during a fraction of channel
coherence time. For instance, when the feedback delay is one-third of channel
coherence time, only delayed CSIT is available during one-third of channel
coherence time, but both delayed and current CSIT is available in the
remaining time as illustrated in Fig 2-(b).

###  I-B Contribution

In this paper, we consider the MISO broadcast channel where a transmitter with
$N_{\textrm{t}}=K-1$ antennas supports $K$ users with a single antenna. The
main contribution of this paper is to show that joint use of outdated and
current CSIT improves significantly the sum-DoF of the vector broadcast
channel in both periodic CSI feedback models. For the feedback frequency
limited model, we provide a characterization of the achievable sum-DoF region
as a function of CSI feedback frequency
$\omega=\frac{T_{\textrm{f}}}{T_{\textrm{f}}+T_{\textrm{n}}}$. As illustrated
in Fig. 3, we show that it is possible to increase the achievable sum-DoF
region substantially from the additional outdated CSI feedback over all the
range $\omega\in(0,1)$ compared to the case where each user sends back current
CSI only at the feedback instant. One notable point is that when
$\omega\geq\frac{K-1}{K}$, it is possible to achieve the optimal
$\min\\{N_{\textrm{t}},K\\}=K-1$ DoF gain (cut-set bound) without knowledge of
all current CSIT, i.e., $\omega<1$. This result is appealing because it was
believed that the cut-set bound can be achieved when the transmitter only has
current CSIT over all channel uses. Our result, however, reveals the fact that
current CSIT over all channel uses is not necessarily required, and joint use
of outdated and current CSIT allows the optimal DoF gain to be achieved.

For the feedback link delay limited model, we also characterize a trade-off
region between CSI feedback delay and the sum-DoF gain for the multi-user
vector broadcast channel by the proposed STIA algorithm and leveraging results
in [11]. From the proposed region, we show that the joint use of outdated and
current CSIT improves significantly the sum-DoF gain compared to that achieved
by the separate exploitation of outdated and current CSIT. One consequence of
the derived region is that not too delayed CSIT achieves the optimal DoF for
the system. This is because the proposed trade-off region achieves the cut-set
bound for the range of $0\leq\gamma\leq\frac{1}{K}$. Further, we demonstrate
that the proposed trade-off region is optimal for the three-user $2\times 1$
vector broadcast channel using a recently derived converse result [30].

To show these results, we propose a new transmission method called space-time
interference alignment (STIA). The key idea of STIA is to design the transmit
beamforming matrix by using both outdated and current CSI jointly so that
multiple inter-user interference signals between the past observed and the
currently observed are aligned in both space and time domains. Therefore, STIA
is a generalized version of the transmission method developed in [11] for
using both outdated and current CSI.

In terms of related work, new transmission methods combining MAT in [11] and
partial zero-forcing (ZF) using both current and outdated CSI were developed
for the two-user vector broadcast channel in [16] and [17]. The main
difference lies in the system model. In [16] and [17], the imperfect current
CSI estimated from the temporal channel correlations is used in the
transmission algorithms. Meanwhile, our transmission algorithm exploits
perfect current CSI. The problem of exploiting perfect outdated and current
CSIT has been studied for the two-user vector broadcast channel [29] to obtain
the DoF gain, which is more closely related to our work. The limitation of
work in [29] is that it considers a simple two-user vector broadcast channel.

Figure 3: An achievable sum-DoF region for the $K$-user $(K-1)\times 1$ MISO
broadcast channel with CSI feedback frequency limited model. The achievability
of point B and C in the region is derived by the proposed STIA. Figure 4: A
CSI feedback delay-DoF gain trade-off region for the $K$-user $(K-1)\times 1$
MISO broadcast channel with CSI feedback link delay limited model.

This paper is organized as follows. In Section II, the system model and
periodic CSI feedback models are described. The key concept of the proposed
transmission method is explained by giving an illustrative example in Section
III. In Section IV, the achievable sum-DoF region for the $K$-user MISO
broadcast channel with the feedback frequency limited model (Model 1) is
characterized. In Section V, a CSI feedback delay and DoF gain trade-off is
studied with the feedback link delay limited model (Model 2). In Section VI,
the proposed algorithm is compared with the prior work. The paper is concluded
in Section VII.

Throughout this paper, transpose, conjugate transpose, inverse, and trace of a
vector ${\bf x}$ are represented by ${\bf x}^{T}$ , ${\bf x}^{*}$, ${\bf
x}^{-1}$ and $\textmd{Tr}\left({\bf x}\right)$, respectively. In addition,
$\mathbb{C}$, $\mathbb{Z}^{+}$, and $\mathcal{CN}(0,1)$ a complex value, a
positive integer value and a complex gaussian random variable with zero mean
and unit variance.

##  II System Model

###  II-A Singal Model

Consider a $K$-user MISO broadcast channel where a transmitter with
$N_{\textrm{t}}=K-1$ multiple antennas sends independent messages to a
receiver with a single antenna. The input-output relationship at the $n$-th
channel use is given by

$\displaystyle{y}^{(k)}[n]={\bf h}^{(k)^{T}}[n]{\bf x}[n]+z^{(k)}[n],$ (1)

where ${\bf x}[n]\in\mathbb{C}^{N_{\textrm{t}}\times 1}$ denotes the signal
sent by the transmitter, ${{\bf
h}^{(k)}}[n]=\left[{h}^{(k)}_{1}[n],\ldots,{h}^{(k)}_{N_{\textrm{t}}}[n]\right]^{T}$
represents the channel vector from the transmitter to user $k$; and
$z^{(k)}[n]\sim\mathcal{CN}(0,1)$ for $k\in\\{1,2,\ldots,K\\}$. We assume that
the transmit power at the transmitter satisfies an average constraint
$\mathbb{E}\left[\textmd{Tr}\left({\bf x}[n]{{\bf x}}^{*}[n]\right)\right]\leq
P$. We assume that all the entries of channel vector ${\bf h}^{(k)}[n]$ are
drawn from a continuous distribution and the absolute value of all the channel
coefficients is bounded between a nonzero minimum value and a finite maximum
value to avoid the degenerate cases where channel values are equal to zero or
infinity. Each receiver has a perfect estimate of its CSI, i.e. has perfect
CSIR.

###  II-B Periodic CSI Feedback Model

####  II-B1 Model 1 (CSI feedback frequency limited model)

A feedback cycle ($T\in\mathbb{Z}^{+}$ time slots) consists of two sub
durations: a non-feedback duration ($T_{\textrm{n}}$ time slots) and a CSI
feedback duration ($T_{\textrm{f}}$ time slot). During the non-feedback
duration, each user cannot send back CSI to the transmitter at all. Starting
with the $T_{\textrm{n}}+1$ time slot, each user sends it back to the
transmitter until $T_{\textrm{n}}+T_{\textrm{f}}$ time slots. In the first
time slot in the feedback duration, the user sends back both current CSI and
the estimated CSI obtained from the previous non-feedback durations. From time
slot $n=T_{\textrm{n}}+2$ to $n=T_{\textrm{n}}+T_{\textrm{f}}$, only current
CSI is provided to the transmitter $\left\\{{\bf h}^{(k)}[n]\right\\}$. For
example, user $k$ sends back $T_{\textrm{n}}-1$ outdated CSI $\left\\{{\bf
h}^{(k)}[1],{\bf h}^{(k)}[2],\ldots,{\bf h}^{(k)}[T_{\textrm{n}}]\right\\}$
and one current CSI, $\left\\{{\bf h}^{(k)}[T_{\textrm{n}}\\!+\\!1]\right\\}$
through the feedback link at $T_{\textrm{n}}\\!+\\!1$ time slot. From time
slot $t=T_{\textrm{n}}\\!+\\!2$ to $t=T_{\textrm{n}}\\!+\\!T_{\textrm{f}}$,
the only current CSI is sent back to the transmitter $\left\\{{\bf
h}^{(k)}[t]\right\\}$. Since the channel values are independently changed at
every time slot, the transmitter is able to obtain $T_{\textrm{n}}-1$ outdated
or obsolete CSI $\left\\{{\bf h}^{(k)}[1],{\bf h}^{(k)}[2],\ldots,{\bf
h}^{(k)}[T_{\textrm{n}}]\right\\}$ and $T_{\textrm{f}}$ current CSI
$\left\\{{\bf h}^{(k)}[T_{\textrm{n}}\\!+\\!1],\ldots,{\bf
h}^{(k)}[T_{\textrm{n}}\\!+\\!T_{\textrm{f}}]\right\\}$ within a feedback
cycle $T=T_{\textrm{n}}+T_{\textrm{f}}$, where $k\in\\{1,2,\ldots,K\\}$. The
CSI is provided through a noiseless and delay-free feedback link at regularly-
spaced time intervals.

In this model, we define a parameter that measures the normalized feedback
speed, i.e., feedback frequency, which is

$\displaystyle\omega=\frac{T_{\textrm{f}}}{T_{\textrm{n}}+T_{\textrm{f}}}.$
(2)

For example, if $\omega=1$, then the transmitter obtains instantaneous CSI
over all time slots because the feedback is frequent enough to track all
channel variations, i.e., the perfect CSIT case. Alternatively, if $\omega=0$,
then the transmitter cannot obtain any CSI from the receivers, i.e., no CSIT
case.

####  II-B2 Model 2 (CSI feedback link delay limited model)

We consider an ideal block fading channel where the channel values are
constant for the channel coherence time $T_{\textrm{c}}$ and change
independently between blocks. Each user sends backs CSI to the transmitter
every $T_{\textrm{c}}$ time slots periodically where $T_{\textrm{c}}$ denotes
channel coherence time. If we assume that the feedback delay time
$T_{\textrm{fb}}$ is less than the channel coherence time, i.e.,
$T_{\textrm{fb}}<T_{\textrm{c}}$, then the transmitter learns the CSI
$T_{\textrm{fb}}$ time slot after it was sent by the user. For example, if a
user sends back CSI at time slot $n$, the transmitter has CSI at time slot
$n+T_{\textrm{fb}}$ in our model.

Let us define a parameter for the ratio between the CSI feedback delay and the
channel coherence time. We call this the normalized CSI feedback delay:

$\displaystyle\gamma=\frac{T_{\textrm{fb}}}{T_{\textrm{c}}}.$ (3)

We refer to the case where $\gamma\geq 1$ as the completely outdated CSI
regime as considered in [11]. In this case, only completely outdated CSI is
available at the transmitter. We refer to the case where $\gamma=0$ as the
current CSI point. Since there is no CSI feedback delay, the transmitter can
use the current CSI in each slot. As depicted in Fig. 1-(b), if
$\gamma=\frac{1}{3}$, the transmitter is able to exploit instantaneous CSI
over two-thirds of the channel coherence time and outdated CSI for the
previous channel blocks.

###  II-C Sum-DoF Region with Periodic CSI Feedabck

Since the achievable data rate of the users depends on the parameters $\omega$
or $\gamma$ and SNR, it can be expressed as a function of
$\lambda\in\\{\omega,\gamma\\}$ and SNR. Using this notion, for codewords
spanning $n$ channel uses, a rate of user $k$
$R^{(k)}(\lambda,\textrm{SNR})=\frac{\log|m^{(k)}(\lambda,\textrm{SNR})|}{n}$
is achievable if the probability of error for the message $m^{(k)}$ approaches
zero as $n\rightarrow\infty$. The sum-DoF characterizing the high SNR behavior
of the achievable rate region is defined as

$\displaystyle
d(\lambda)=\sum_{k=1}^{K}d^{(k)}(\lambda)=\lim_{\textrm{SNR}\rightarrow\infty}\frac{\sum_{k=1}^{K}R^{(k)}(\lambda,\textrm{SNR})}{\log(\textrm{SNR})}.$
(4)

##  III Space-Time Interference Alignment

Before proving our main results, we explain the space-time interference
alignment (STIA) signaling structure. The periodic CSI feedback protocol
motivates us to develop the STIA because it exploits current and outdated CSI
jointly. In this section, we focus on the special case of $K=3$,
$N_{\textrm{t}}=2$, and $\omega=\frac{2}{3}$ to provide an intuition why joint
use of both CSI is useful. We generalize this construction to a more general
set of parameters in the sequel.

###  III-A Precoding and Decoding Structure

In this section, we will show that $2$ of DoF (outer bound) is achieved for
the $3$-user $2\times 1$ vector broadcast channel when $\omega=\frac{2}{3}$
($T_{\textrm{n}}=1$ and $T_{\textrm{f}}=2$) under the assumptions of Model 1
as depicted Fig. 2-(a). The proposed STIA algorithm consists of two phases.
The first phase involves sending multiple symbols per user to create an
interference pattern without CSI knowledge at the transmitter. The second
phase involves sending linear combinations of the same symbols to recreate the
same interference patterns using both current and outdated CSI. Using the fact
that each user receives the same interference patterns during the two phases,
it cancels the interference signals to obtain the desired symbols in the
decoding procedure.

####  III-A1 Phase One (Obtain the Interference Pattern)

This phase consists of one time slot. During phase one, the transmitter does
not have any CSI knowledge according to Model I because each user cannot
feedback CSI during the non-feedback duration, i.e., $T_{\textrm{n}}=1$. In
time slot 1, the transmitter sends six independent symbols where $s^{(1)}_{1}$
and $s^{(1)}_{2}$ intended for user 1, $s^{(2)}_{1}$ and $s^{(2)}_{2}$
intended for user 2, and $s^{(3)}_{1}$ and $s^{(3)}_{2}$ intended for user 3
without precoding as

$\displaystyle{\bf x}[1]=\sum_{k=1}^{3}{\bf s}^{(k)},$ (5)

where ${\bf s}^{(k)}=\left[s^{(k)}_{1},s^{(k)}_{2}\right]^{T}$. Neglecting
noise at the receiver, the observation at each receiver is a function of three
linear combinations:

$\displaystyle y^{(1)}[1]$
$\displaystyle=L^{(1,1)}[1]+L^{(1,2)}[1]+L^{(1,3)}[1],$ (6) $\displaystyle
y^{(2)}[1]$ $\displaystyle=L^{(2,1)}[1]+L^{(2,2)}[1]+L^{(2,3)}[1],$ (7)
$\displaystyle y^{(3)}[1]$
$\displaystyle=L^{(3,1)}[1]+L^{(3,2)}[1]+L^{(3,3)}[1],$ (8)

where $L^{(k,i)}[n]$ denotes a linear combination seen by user $k$ for the
transmitted symbols for user $i$ at the $n$-th time slot:

$\displaystyle L^{(1,1)}[1]$
$\displaystyle=h^{(1)}_{1}[1]s^{(1)}_{1}+h^{(1)}_{2}[1]s^{(1)}_{2},$
$\displaystyle L^{(1,2)}[1]$
$\displaystyle=h^{(1)}_{1}[1]s^{(2)}_{1}+h^{(1)}_{2}[1]s^{(2)}_{2},$
$\displaystyle L^{(1,3)}[1]$
$\displaystyle=h^{(1)}_{1}[1]s^{(3)}_{1}+h^{(1)}_{2}[1]s^{(3)}_{2},$
$\displaystyle L^{(2,1)}[1]$
$\displaystyle=h^{(2)}_{1}[1]s^{(1)}_{1}+h^{(2)}_{2}[1]s^{(1)}_{2},$
$\displaystyle L^{(2,2)}[1]$
$\displaystyle=h^{(2)}_{1}[1]s^{(2)}_{1}+h^{(2)}_{2}[1]s^{(2)}_{2},$
$\displaystyle L^{(2,3)}[1]$
$\displaystyle=h^{(2)}_{1}[1]s^{(3)}_{1}+h^{(2)}_{2}[1]s^{(3)}_{2},$
$\displaystyle L^{(3,1)}[1]$
$\displaystyle=h^{(3)}_{1}[1]s^{(1)}_{1}+h^{(3)}_{2}[1]s^{(1)}_{2},$
$\displaystyle L^{(3,2)}[1]$
$\displaystyle=h^{(3)}_{1}[1]s^{(2)}_{1}+h^{(3)}_{2}[1]s^{(2)}_{2},$
$\displaystyle\textrm{and}\quad L^{(3,3)}[1]$
$\displaystyle=h^{(3)}_{1}[1]s^{(3)}_{1}+h^{(3)}_{2}[1]s^{(3)}_{2}.$

In summary, during phase one, each receiver obtains a linear combination of
one desired equation $L^{(i,i)}[1]$ and two interference equations
$L^{(i,j)}[1]$ for $i\neq j$.

####  III-A2 Phase Two (Generate the Same Interference Pattern)

The second phase uses two time slots, i.e. $n\in\\{2,3\\}$. In this phase, the
transmitter has knowledge of both current and outdated CSI thanks to feedback.
For example, during time slot 2, the transmitter has outdated CSI for the
first time slot and current CSI for the second time slot. Using this
information, in time slot 2 and time slot 3, the transmitter sends
simultaneously two symbols for the dedicated users by using linear beamforming
as

$\displaystyle{\bf x}[n]=\sum_{k=1}^{3}{\bf V}^{(k)}[n]{\bf s}^{(k)},$ (9)

where ${\bf V}^{(k)}[n]\in\mathbb{C}^{2\times 2}$ denotes the beamforming
matrix used for carrying the same symbol vector ${\bf
s}^{(k)}=\left[s^{(k)}_{1},s^{(k)}_{2}\right]^{T}$ at time slot $n$, where
$n\in\\{2,3\\}$ and $k\in\\{1,2,3\\}$. The main idea for designing precoding
matrix ${\bf V}^{(k)}[n]$ is to allow all the receivers observe the same
linear combination for interference signals received during time slot 1 by
exploiting current and outdated CSI. For instance, user 2 and user 3 received
the interference signals in the form of
$L^{(2,1)}[1]=h^{(2)}_{1}[1]s^{(1)}_{1}+h^{(2)}_{2}[1]s^{(1)}_{2}$ and
$L^{(3,1)}[1]=h^{(3)}_{1}[1]s^{(1)}_{1}+h^{(3)}_{2}[1]s^{(1)}_{2}$. Therefore,
to deliver the same linear combination for the undesired symbols to user 2 and
user 3 repeatedly during $n\in\\{2,3\\}$, the transmitter constructs the
beamforming matrix carrying symbols, $s^{(1)}_{1}$ and $s^{(1)}_{2}$ to
satisfy

$\displaystyle\left[\begin{array}[]{c}{{\bf h}^{(2)T}}[n]\\\ {{\bf
h}^{(3)T}}[n]\\\ \end{array}\right]{\bf
V}^{(1)}[n]=\left[\begin{array}[]{c}{{\bf h}^{(2)T}}[1]\\\ {{\bf
h}^{(3)T}}[1]\\\ \end{array}\right].$ (14)

Similarly, to ensure the interfering users receive the same linear combination
of the undesired symbols, which is linearly dependent (aligned) with the
previously overheard equation during time slot 1, the beamforming matrices
carrying data symbols for user 2 and user 3 are constructed to satisfy the
space-time inter-user interference alignment conditions:

$\displaystyle\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[n]\\\ {{\bf
h}^{(3)T}}[n]\\\ \end{array}\right]{\bf
V}^{(2)}[n]=\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[1]\\\ {{\bf
h}^{(3)T}}[1]\\\ \end{array}\right],$ (19)

and

$\displaystyle\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[n]\\\ {{\bf
h}^{(2)T}}[n]\\\ \end{array}\right]{\bf
V}^{(3)}[n]=\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[1]\\\ {{\bf
h}^{(2)T}}[1]\\\ \end{array}\right].$ (24)

Since we assume that the channel coefficients are drawn from a continuous
distribution, matrix inversion is guaranteed with high probability. Therefore,
it is possible to construct transmit beamforming matrices ${\bf V}^{(1)}[n]$,
${\bf V}^{(2)}[n]$ and ${\bf V}^{(3)}[n]$ as

$\displaystyle{\bf V}^{(1)}[n]$ $\displaystyle=\left[\begin{array}[]{c}{{\bf
h}^{(2)T}}[n]\\\ {{\bf h}^{(3)T}}[n]\\\
\end{array}\right]^{-1}\left[\begin{array}[]{c}{{\bf h}^{(2)T}}[1]\\\ {{\bf
h}^{(3)T}}[1]\\\ \end{array}\right],$ (29) $\displaystyle{\bf V}^{(2)}[n]$
$\displaystyle=\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[n]\\\ {{\bf
h}^{(3)T}}[n]\\\ \end{array}\right]^{-1}\left[\begin{array}[]{c}{{\bf
h}^{(1)T}}[1]\\\ {{\bf h}^{(3)T}}[1]\\\ \end{array}\right],$ (34)

and

$\displaystyle{\bf V}^{(3)}[n]$ $\displaystyle=\left[\begin{array}[]{c}{{\bf
h}^{(1)T}}[n]\\\ {{\bf h}^{(2)T}}[n]\\\
\end{array}\right]^{-1}\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[1]\\\ {{\bf
h}^{(2)T}}[1]\\\ \end{array}\right].$ (39)

If we denote ${{\bf\tilde{h}}^{(1)T}}[n]={{\bf h}^{(1)T}}[n]{\bf V}^{(1)}[n]$
and $L^{(1,1)}[n]={{\bf\tilde{h}}^{(1)T}}[n]{\bf s}^{(1)}$ for $n=2,3$, at
time slot 2 and time slot 3, the received signals at user 1 are given by

$\displaystyle{y}^{(1)}[2]$ $\displaystyle=\sum_{k=1}^{3}{{\bf
h}^{(1)T}}[2]{\bf V}^{(k)}[2]{\bf s}^{(k)}$ $\displaystyle={{\bf
h}^{(1)T}}\\![2]{\bf V}^{(1)}[2]{\bf s}^{(1)}+{{\bf h}^{(1)T}}[2]{\bf
V}^{(2)}\\![2]{\bf s}^{(2)}$ $\displaystyle+{{\bf h}^{(1)T}}[2]{\bf
V}^{(3)}[2]{\bf s}^{(3)}$ $\displaystyle={{\bf\tilde{h}}^{(1)T}}[2]{\bf
s}^{(1)}+{{\bf{h}}^{(1)T}}[1]{\bf s}^{(2)}+{{\bf{h}}^{(1)T}}[1]{\bf s}^{(3)}$
$\displaystyle=L^{(1,1)}[2]+L^{(1,2)}[1]+L^{(1,3)}[1],$ (40)
$\displaystyle{y}^{(1)}[3]$ $\displaystyle=\sum_{k=1}^{3}{{\bf
h}^{(1)T}}[3]{\bf V}^{(k)}[3]{\bf s}^{(k)}$ $\displaystyle={{\bf
h}^{(1)T}}[3]{\bf V}^{(1)}[3]{\bf s}^{(1)}+{{\bf h}^{(1)T}}\\![3]{\bf
V}^{(2)}\\![3]{\bf s}^{(2)}$ $\displaystyle+{{\bf h}^{(1)T}}[3]{\bf
V}^{(3)}\\![3]{\bf s}^{(3)}$ $\displaystyle={{\bf\tilde{h}}^{(1)T}}[3]{\bf
s}^{(1)}+{{\bf{h}}^{(1)T}}[1]{\bf s}^{(2)}+{{\bf{h}}^{(1)T}}[1]{\bf s}^{(3)}$
$\displaystyle=L^{(1,1)}[3]+L^{(1,2)}[1]+L^{(1,3)}[1].$ (41)

If we denote ${{\bf\tilde{h}}^{(2)T}}[n]={{\bf h}^{(2)T}}[n]{\bf V}^{(2)}[n]$
and $L^{(2,2)}[n]={{\bf\tilde{h}}^{(2)T}}[n]{\bf s}^{(2)}$ for $n=2,3$, the
received signals at user 2 during time slot 2 and 3 are given by

$\displaystyle{y}^{(2)}[2]$ $\displaystyle=\sum_{k=1}^{3}{{\bf
h}^{(2)T}}[2]{\bf V}^{(k)}[2]{\bf s}^{(k)}$ $\displaystyle={{\bf
h}^{(2)T}}[2]{\bf V}^{(1)}[2]{\bf s}^{(1)}+{{\bf h}^{(2)T}}[2]{\bf
V}^{(2)}[2]{\bf s}^{(2)}$ $\displaystyle+{{\bf h}^{(2)T}}[2]{\bf
V}^{(3)}[2]{\bf s}^{(3)}$ $\displaystyle={{\bf{h}}^{(2)T}}[1]{\bf
s}^{(1)}+{{\bf{\tilde{h}}}^{(2)}}[2]{\bf s}^{(2)}+{{\bf{h}}^{(2)T}}[1]{\bf
s}^{(3)}$ $\displaystyle=L^{(2,1)}[1]+L^{(2,2)}[2]+L^{(2,3)}[1],$ (42)
$\displaystyle{y}^{(2)}[3]$ $\displaystyle=\sum_{k=1}^{3}{{\bf
h}^{(2)T}}[3]{\bf V}^{(k)}[3]{\bf s}^{(k)}$ $\displaystyle={{\bf
h}^{(2)T}}[3]{\bf V}^{(1)}\\![3]{\bf s}^{(1)}+{{\bf h}^{(2)T}}[3]{\bf
V}^{(2)}[3]{\bf s}^{(2)}$ $\displaystyle+{{\bf h}^{(2)T}}[3]{\bf
V}^{(3)}[3]{\bf s}^{(3)}$ $\displaystyle={{\bf{h}}^{(2)T}}[1]{\bf
s}^{(1)}+{{\bf{\tilde{h}}}^{(2)T}}[3]{\bf s}^{(2)}+{{\bf{h}}^{(2)T}}[1]{\bf
s}^{(3)}$ $\displaystyle=L^{(2,1)}[1]+L^{(2,2)}[3]+L^{(2,3)}[1].$ (43)

Finally, for user 3, if we denote ${{\bf\tilde{h}}^{(3)T}}[n]={{\bf
h}^{(3)T}}[n]{\bf V}^{(3)}[n]$ and
$L^{(3,3)}[n]={{\bf\tilde{h}}^{(3)T}}[n]{\bf s}^{(3)}$ for $n=2,3$, the
received signals at time slot 2 and 3 are given by

$\displaystyle{y}^{(3)}[2]$ $\displaystyle=\sum_{k=1}^{3}{{\bf
h}^{(3)T}}[2]{\bf V}^{(k)}[2]{\bf s}^{(k)}$ $\displaystyle={{\bf
h}^{(3)T}}[2]{\bf V}^{(1)}[2]{\bf s}^{(1)}+{{\bf h}^{(2)T}}[2]{\bf
V}^{(2)}[2]{\bf s}^{(2)}$ $\displaystyle+{{\bf h}^{(3)T}}[2]{\bf
V}^{(3)}[2]{\bf s}^{(3)}$ $\displaystyle={{\bf{h}}^{(3)T}}[1]{\bf
s}^{(1)}+{{\bf{h}}^{(3)T}}[1]{\bf s}^{(2)}+{{\bf{\tilde{h}}}^{(3)T}}[2]{\bf
s}^{(3)}$ $\displaystyle=L^{(3,1)}[1]+L^{(3,2)}[1]+L^{(3,3)}[2],$ (44)
$\displaystyle{y}^{(3)}[3]$ $\displaystyle=\sum_{k=1}^{3}{{\bf
h}^{(3)T}}[3]{\bf V}^{(k)}[3]{\bf s}^{(k)}$ $\displaystyle={{\bf
h}^{(3)T}}[3]{\bf V}^{(1)}[3]{\bf s}^{(1)}+{{\bf h}^{(3)T}}[3]{\bf
V}^{(2)}[3]{\bf s}^{(2)}$ $\displaystyle+{{\bf h}^{(3)T}}[3]{\bf
V}^{(3)}[3]{\bf s}^{(3)}$ $\displaystyle={{\bf{h}}^{(3)T}}[1]{\bf
s}^{(1)}+{{\bf{h}}^{(3)T}}[1]{\bf s}^{(2)}+{{\bf{\tilde{h}}}^{(3)T}}[3]{\bf
s}^{(3)}$ $\displaystyle=L^{(3,1)}[1]+L^{(3,2)}[1]+L^{(3,3)}[3].$ (45)

####  III-A3 Decoding

To explain the decoding process, let us consider the received signals at user
1. In time slot 1, user 1 acquired knowledge of the interference signals in
the form of $L^{(1,2)}[1]$ and $L^{(1,3)}[1]$. From phase two, user 1 received
the same interference signals $L^{(1,2)}[1]$ and $L^{(1,3)}[1]$ at time slot 2
and 3 as shown in (40) and (41). Therefore, to decode the desired signals, the
observation made during phase two are subtracted from the observation in phase
one, resulting in the equations:

$\displaystyle y^{(1)}[1]\\!-\\!y^{(1)}[2]$
$\displaystyle=L^{(1,1)}[1]+L^{(1,2)}[1]+L^{(1,3)}[1]$
$\displaystyle-L^{(1,1)}[2]-L^{(1,2)}[1]-L^{(1,3)}[1]$
$\displaystyle=L^{(1,1)}[1]-L^{(1,1)}[2]$ $\displaystyle=\left({\bf
h}^{(1)T}[1]-{\bf h}^{(1)T}[2]{\bf V}^{(1)}[2]\right){\bf s}^{(1)},$ (46)
$\displaystyle y^{(1)}[1]\\!-\\!y^{(1)}[3]$
$\displaystyle=L^{(1,1)}[1]+L^{(1,2)}[1]+L^{(1,3)}[1]$
$\displaystyle-L^{(1,1)}[3]-L^{(1,2)}[1]-L^{(1,3)}[1]$
$\displaystyle=L^{(1,1)}[1]-L^{(1,1)}[3]$ $\displaystyle=\left({\bf
h}^{(1)T}[1]-{\bf h}^{(1)T}[3]{\bf V}^{(1)}[3]\right){\bf s}^{(1)}.$ (47)

After removing the interference signals, the effective channel input-output
relationship for user 1 during the three time slots is given by

$\displaystyle\left[\\!\\!\begin{array}[]{c}{y}^{(1)}[1]-{y}^{(1)}[2]\\\
{y}^{(1)}[1]-y^{(1)}[3]\\\
\end{array}\\!\\!\right]\\!\\!=\\!\\!\underbrace{\left[\begin{array}[]{c}{\bf
h}^{(1)T}[1]-{\bf h}^{(1)T}[2]{\bf V}^{(1)}[2]\\\ {\bf h}^{(1)T}[1]-{\bf
h}^{(1)T}[3]{\bf V}^{(1)}[3]\\\ \end{array}\right]}_{{\bf
H}^{(1)}_{\textrm{eff}}}\left[\\!\\!\begin{array}[]{c}s^{(1)}_{1}\\\
s^{(1)}_{2}\\\ \end{array}\\!\\!\right].$ (54)

Since precoding matrix ${\bf V}^{(1)}[n]$ for $n=2,3$ was designed
independently of channel ${\bf h}^{(1)T}[1]$, the elements of the effective
channel vector observed at time slot 2 and 3, i.e.,
$\left[\tilde{h}^{(1)}_{1}[2],\tilde{h}^{(1)}_{2}[2]\right]={{\bf
h}^{(1)T}}[2]{\bf V}^{(1)}[2]$ and
$\left[\tilde{h}^{(1)}_{1}[3],\tilde{h}^{(1)}_{2}[3]\right]={{\bf
h}^{(1)T}}[3]{\bf V}^{(1)}[3]$ are also statistically independent random
variables. This implies that the three channel vectors, ${\bf h}^{(1)T}[1]$,
${{\bf h}^{(1)T}}[2]{\bf V}^{(1)}[2]$, and ${{\bf h}^{(1)T}}[3]{\bf
V}^{(1)}[3]$ are linearly independent. Therefore, $\textrm{rank}\left({\bf
H}^{(1)}_{\textrm{eff}}\right)=2$ with a probability of one because all
channel elements are drawn from a continuous distribution. As a result, user 1
decodes two desired symbols within three time slots. In the same way, user 2
and user 3 are able to retrieve a linear combination of their desired symbols
by removing the interference signals and can use the same decoding method.
Since the transmitter has delivered two independent symbols for its intended
user in three channel uses, a total $d=\frac{6}{3}=2$ DoF are achieved.

###  III-B The Relationship between STIA and Wireless Index Coding

In the above example, we showed that the proposed STIA achieves the optimal
DoF for the 3-user $2\times 1$ vector broadcast channel without using current
CSI over all channel time slots. We can interpret STIA from a wireless index
coding perspective. The index coding problem was introduced in [25] and has
been studied in subsequent work [26]-[28]. Suppose that a transmitter has a
set of information messages $\mathcal{W}=\\{W_{1},W_{2},\ldots,W_{K}\\}$ for
multiple receivers and each receiver wishes to receive a subset of
$\mathcal{W}$ while knowing some another subset of $\mathcal{W}$ as side-
information. The index coding problem is to design the best encoding strategy
at the transmitter, which minimizes the number of transmissions while ensuring
that all receivers can obtain the desired messages. To illustrate, consider
the case where a transmitter with a single antenna serves two single antenna
users. Suppose each user has the message for the other user as side-
information, i.e., user 1 has $W_{2}$ and user 2 has $W_{1}$. In this case, an
index coding method is to send the linear sum of two messages $W_{1}+W_{2}$ to
both receivers. Since user 1 and 2 receive
$y^{(1)}[1]=h^{(1)}_{1}[1](W_{1}+W_{2})$ and
$y^{(2)}[1]=h^{(2)}_{1}[1](W_{1}+W_{2})$ ignoring the noise and assuming the
flat fading channel, each receiver can extract the desired message by using
side-information as $\hat{W}_{1}=y^{(1)}[1]/h^{(1)}_{1}[1]-W_{2}$ and
$\hat{W}_{2}=y^{(2)}[1]/h^{(2)}_{1}[1]-W_{1}$. STIA mimics this index coding
algorithm in a more general sense by using current and outdated CSI jointly.
This is because, during phase one, each user acquires side-information in a
form of a linear combination of all transmitted data symbols where the linear
coefficients are created by a wireless channel. During the second phase, the
transmitter constructs the precoding matrices by using outdated and current
CSI so that each user can exploit the received interference equations in the
first phase as side-information, which results in minimizing the number of
transmissions; it leads to an increase in the DoF.

###  III-C Extensions of STIA Algorithm

We proposed the STIA algorithm as a way to take advantage of current and past
CSI in the vector broadcast channel. Now, we make some remarks about how the
STIA algorithm can be modified and extended to deal with more complex networks
and practical aspects.

Remark 1 (Extension into interference networks [20] ): The proposed algorithm
is directly applicable to the $K$-user MISO interference channel where each
transmitter has $K-1$ antennas. The reason is that the beamforming matrices
used in the second phase are independently generated without knowledge of the
other user’s data symbols. In other words, data sharing among the different
transmitters is not required to apply the STIA in the $K$-user MISO
interference channel. For example, the $\frac{3}{2}$ of DoF gain is also
achievable for the $3$-user MISO interference channel with the periodic
feedback model when $N_{\textrm{t}}=2$.

Remark 2 (Transmission power constraint): Since it is assumed that the
transmitter sends a signal with large enough power in the DoF analysis, the
beamforming solutions containing matrix inversion do not violate the transmit
power constraint. In practice, however, the proposed STIA algorithm needs to
be modified so that the power constraint is satisfied. This modification may
incur the sum-rate performance loss but does not affect the DoF gain. The sum-
rate maximization or mean square error minimization problems of the STIA
algorithm given transmit power constraint can be investigated in future work.

Remark 3 (Effective channel estimation in multi-carrier systems): In the STIA
algorithm, each receiver requires the effective channel during phase two. For
example, user $1$ has to know $\\{{\bf h}^{(1)T}[2]{\bf V}^{(1)}[2]\\}$ and
$\\{{\bf h}^{(1)T}[3]{\bf V}^{(1)}[3]\\}$ when it decodes the desired signal
as shown in (54). This effective channel knowledge may be acquired at the
receiver if the transmitter sends this information as data to the user by
spending other downlink channel resources, i.e., a feed-forward information.
This feed-forward, however, is a source of overhead in the system. Assuming a
multi-carrier system, effective channel knowledge at the receiver can be
obtained without feed-forward processing. The main idea for the effective
channel estimation at the receiver is to use a demodulation reference signal
and an interference cancellation technique.

Let us explain this by giving an example that uses a multi-carrier system,
which allows channel coherence in the frequency domain to be exploited.
Consider a multi-carrier system where the channel values are the same across
channel coherence band $B_{\textrm{c}}$, i.e., ${\bf h}^{(1)}[t,f_{j}]={\bf
h}^{(1)}[t]$ for $t\in\\{2,3\\}$ and $j\in\\{1,2,\ldots,B_{\textrm{c}}\\}$.
When the transmitter sends the training signal (demodulation reference signal)
${\bf t}[2,f_{j}]$ by the same precoding method used for data signal as
${\bf\tilde{t}}[2,f_{j}]=\sum_{k=1}^{K}{\bf V}^{(k)}[2,f_{j}]{\bf t}[2,f_{j}]$
through subcarrier $f_{j}$ where $j\in\\{1,2,\ldots,B_{\textrm{t}}\\}$ and
$B_{\textrm{t}}<B_{\textrm{c}}$. The received signal at time slot $2$ and
subcarrier $f_{j}$ is given by

$\displaystyle y^{(1)}[2,f_{j}]$ $\displaystyle={\bf
h}^{(1)T}[2,f_{j}]{\bf\tilde{t}}[2,f_{j}]$ $\displaystyle=\sum_{k=1}^{3}{\bf
h}^{(1)T}[2,f_{j}]{\bf V}^{(k)}[2,f_{j}]{\bf t}[2,f_{j}]$ $\displaystyle={\bf
h}^{(1)T}[2,f_{j}]{\bf V}^{(1)}[2]{\bf t}[2,f_{j}]+2{\bf
h}^{(1)T}[1,f_{j}]{\bf t}[2,f_{j}],$ (55)

where the last equality comes from the fact that ${\bf h}^{(1)T}[1,f_{j}]={\bf
h}^{(1)T}[2,f_{j}]{\bf V}^{(k)}[2,f_{j}]$ for $k\in\\{2,3\\}$ by the STIA
algorithm. Note that each receiver has knowledge of training signal ${\bf
t}[2,f_{j}]$ and the channel at time slot 1 ${\bf h}^{(1)T}[1,f_{j}]$. Hence,
user 1 can extract out the desired equation for the channel estimation at
$f_{j}$ subcarrier as $y^{(1)}[2,f_{j}]-2{\bf h}^{(1)T}[1,f_{j}]{\bf
t}[2,f_{j}]={\bf h}^{(1)T}[2,f_{j}]{\bf V}^{(1)}[2,f_{j}]{\bf t}[2,f_{j}]$
where $j\in\\{1,2,\ldots,B_{\textrm{t}}\\}$. As an example, the receiver could
estimate the effective channel ${\bf\tilde{h}}^{(1)}=({\bf
h}^{(1)T}[2,f_{j}]{\bf V}^{(1)}[2,f_{j}])^{*}$ by solving the equation,

$\displaystyle\left[\\!\\!\\!\begin{array}[]{c}y^{(1)*}[2,f_{1}]\\!-\\!(2{\bf
h}^{(1)T}[1,f_{1}]{\bf t}[2,f_{1}])^{*}\\\ \vdots\\\
y^{(1)*}[2,f_{B_{\textrm{t}}}]\\!-\\!(2{\bf
h}^{(1)T}[1,f_{B_{\textrm{t}}}]{\bf t}[2,f_{B_{\textrm{t}}}])^{*}\\\
\end{array}\\!\\!\\!\\!\right]$
$\displaystyle\\!\\!=\\!\left[\\!\\!\\!\\!\begin{array}[]{c}{\bf
t}^{*}[2,f_{1}]\\\ \vdots\\\ {\bf t}^{*}[2,f_{B_{\textrm{t}}}]\\\
\end{array}\\!\\!\\!\right]{\bf\tilde{h}}^{(1)}.$ (62)

Since the size of effective channel ${\bf\tilde{h}}^{(1)}$ is $2\times 1$, if
the transmitter uses an independent training signal ${\bf t}^{*}[2,f_{j}]$
over $B_{\textrm{t}}\geq 2$ subcarriers among total $B_{\textrm{c}}$
subcarriers, it is estimated reliably, if the noise effect is ignored.
Therefore, the STIA algorithm can be implemented without forwarding the
effective channel to the users. Further, the overhead required for channel
estimation is the same as that required for a multiuser MISO system. This
implies that the STIA algorithm does not require additional overhead for
learning the effective channel in multi-carrier systems.

##  IV An Inner Bound of Sum-DoF Region

In this section, we characterize an achievable sum-DoF region as a function of
feedback frequency for the $K$-user $(K-1)\times 1$ MISO broadcast channel.
The main result is established in the following theorem.

###### Theorem 1

For the $K$-user $(K-1)\times 1$ MISO broadcast channel with the periodic CSI
feedback (Model 1), the achievable sum-DoF region is characterized as a
function of feedback frequency $\omega$,

$\displaystyle d(\omega)\\!=\\!\left\\{\\!\\!\\!\begin{array}[]{l
l}(K-1)\omega+1,\\!\\!\\!\\!&\\!\\!\\!\\!\quad\textrm{for}\quad\\!\\!0\leq\omega\leq\frac{K-2}{2(K-1)},\\\
a(K)\omega+b(K),\\!\\!\\!\\!&\\!\\!\\!\\!\quad\textrm{for}\quad\\!\\!\frac{K-2}{2(K-1)}\leq\omega\leq\frac{K-1}{K},\\\
K-1\\!\\!\\!\\!&\\!\\!\\!\\!\quad\textrm{for}\\!\\!\quad\frac{K-1}{K}\leq\omega\leq
1.\\\ \end{array}\right.$ (66)

where $a(K)=\frac{K(K-1)(K-2)}{(K-1)(K-2)+K}$ and
$b(K)=\frac{K(K-1)}{(K-1)(K-2)+K}$.

###### Proof:

We prove Theorem 1 by four different transmission methods, each of which
achieves four different corner points of the region as shown in Fig. 3. Since
time sharing can be used to achieve the lines connecting each point, we focus
on characterizing the conner points.

###  IV-A Achievablity of Point A and D

Achievability for the corner point A when $\omega=0$ is shown by TDMA. Also,
achievability for the point D when $\omega=1$ is proven by ZF. When receivers
do not send back CSI to the transmitter, i.e., $\omega=0$, then the TDMA
method is used to achieve $d(0)=1$ [10]. At the other extreme, when the
transmitter has current CSI over all time slots at $\omega=1$, it is possible
to achieve the cut-set bound $d(1)=\min\\{N_{\textrm{t}},K\\}=K-1$ DoF by the
ZF method.

###  IV-B Achievability of Point B

Let us consider $\omega=\frac{K-2}{2K-2}$, where the non-CSI feedback duration
$T_{\textrm{n}}$ has $K$ time slots and the CSI feedback duration
$T_{\textrm{f}}$ has $K-2$ time slots. For this case, we show that
$d\left(\frac{K-2}{2K-2}\right)=\frac{K(K-1)}{2K-2}=\frac{K}{2}$ DoF are
achievable by STIA using $K$ time slots in the first phase and $K-2$ time
slots in the second phase.

####  IV-B1 Phase One

The first phase has $K$ time slots. Since the transmitter does not have
channel knowledge in this phase, it sends $N_{\textrm{t}}=K-1$ independent
data symbols using spatial multiplexing on $N_{\textrm{t}}$ antennas without
using any precoding so that each user receives one linear equation for
$N_{\textrm{t}}$ desired symbols while overhearing $K-1$ other transmissions,
each of which contains $K-1$ undesired information symbols. During time slot
$n$, the transmitter sends the desired symbols for user $n$, ${\bf x}[n]={\bf
s}^{(n)}$ through $N_{\textrm{t}}$ antennas. Hence, the received signal at
user $k$ in this phase is given by

$\displaystyle{y}^{(k)}[n]$ $\displaystyle={{\bf h}^{(k)T}}[n]{\bf
s}^{(n)},\quad n\in\\{1,2\ldots,K\\},$ (67)

where $k\in\\{1,2\ldots,K\\}$ and the noise term was dropped for simplicity
because it does not affect the DoF calculation.

####  IV-B2 Phase Two

The second phase spans $K-2$ time slots. In this phase, the transmitter has
CSI for the current and past time periods because each user sends back both
outdated and current CSI from the $K+1$ time slots in phase one.

The main objective of phase two is for the transmitter to provide each user
with additional observations that can be used to build additional $K-2$
linearly independent equations in the desired symbols. Since each user has one
observation from phase one containing its $K-1$ symbols, if it obtains
additional $K-2$ linearly independent equations from this phase, then
$N_{\textrm{t}}=K-1$ desired symbols can be decoded at each user. To deliver
$N_{\textrm{t}}$ desired symbols to each user through phase two, the
transmitted signal at time slot $n$ is given by

$\displaystyle{\bf x}[n]=\sum_{k=1}^{K}{\bf V}^{(k)}[n]{\bf s}^{(k)},$ (68)

where $n\in\\{K+1,K+2,\ldots,2K-2\\}$. Notice that the transmitter repeatedly
sends the same symbol vector ${\bf s}^{(k)}$ regardless of time index $n$, but
using a different beamforming matrix ${\bf V}^{(k)}[n]$, which varies
according to time index. When the transmitter sends the signal in (68) at time
slot $n$, the received signal at user $\ell$ is given by

$\displaystyle{y}^{(\ell)}[n]$ $\displaystyle={{\bf
h}^{(\ell)T}}[n]\sum_{k=1}^{K}{\bf V}^{(k)}[n]{\bf s}^{(k)}$
$\displaystyle=\underbrace{{{\bf h}^{(\ell)T}}[n]{\bf V}^{(\ell)}[n]{\bf
s}^{(\ell)}}_{{L}^{(\ell,\ell)}[n]}\\!+\\!\\!\\!\sum_{k:k\neq\ell}^{K}\underbrace{{{\bf
h}^{(\ell)T}}[n]{\bf V}^{(k)}[n]{\bf s}^{(k)}}_{{L}^{(\ell,k)}[n]},$ (69)

where $\ell\in\\{1,2\ldots,K\\}$. Since the transmitter sends a linear
combination of $K(K-1)$ data symbols for all the users in each time slot, each
user sees not only one desired term i.e., ${L}^{(\ell,\ell)}[n]$ but also
receives $K-1$ interference terms, i.e., ${L}^{(\ell,k)}[n]$ for
$k\in\\{1,2,\ldots,K\\}/\\{\ell\\}$. To decode the $K-1$ desired symbols
reliably in high SNR, user $\ell$ should be able to remove the interference
equations ${L}^{(\ell,k)}[n]$ for $k\in\\{1,2,\ldots,K\\}/\\{\ell\\}$.

Recall that during the first phase, user $\ell$ overheard the interfering
symbols for user $k$, ${\bf s}^{(k)}$, at the $k$-th time slot in the form of
${{\bf h}^{(\ell)T}}[k]{\bf s}^{(k)}=L^{(\ell,k)}[k]$. Therefore, if user
$\ell$ sees the same linear combinations of the interfering symbols ${\bf
s}^{(k)}$ at the $n$-th time repeatedly, i.e., ${{\bf h}^{(\ell)T}}[n]{\bf
V}^{(k)}[n]{\bf s}^{(k)}={{\bf h}^{(\ell)T}}[k]{\bf s}^{(k)}$, then it can
remove this interference by subtracting the previously observed signal as
side-information. Therefore, it is important to design the beamforming
matrices so that the users see the same interference in the first and second
phases. To accomplish this goal, the beamforming matrices should satisfy the
space-time interference alignment condition:

$\displaystyle\underbrace{\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[k]\\\
\vdots\\\ {{\bf h}^{(k-1)T}}[k]\\\ {{\bf h}^{(k+1)T}}[k]\\\ \vdots\\\ {{\bf
h}^{(K)T}}[k]\\\ \end{array}\right]}_{(K-1)\times N_{\textrm{t}}}$
$\displaystyle=\underbrace{\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[n]\\\
\vdots\\\ {{\bf h}^{(k-1)T}}[n]\\\ {{\bf h}^{(k+1)T}}[n]\\\ \vdots\\\ {{\bf
h}^{(K)T}}[n]\\\ \end{array}\right]}_{(K-1)\times
N_{\textrm{t}}}\underbrace{{\bf V}^{(k)}[n]}_{N_{\textrm{t}}\times
N_{\textrm{t}}},$ (82) $\displaystyle\Rightarrow{\bf{H}}_{k}^{c}[k]$
$\displaystyle={\bf{H}}_{k}^{c}[n]{\bf V}^{(k)}[n],$ (83)

where $k\in\\{1,2,\ldots,K\\}$, $n\in\\{K\\!+\\!1,K\\!+\\!2,\ldots,2K-2\\}$
and, ${\bf{H}}_{k}^{c}[n]$ and ${\bf{H}}_{k}^{c}[k]$ represent an augmented
channel matrix of all users excepting user $k$ at time slot $n$ and $k$,
respectively. This inter-user interference alignment condition implies that
the signal for user $k$, which interferes with other users excluding user $k$
during the second phase, should be the same as the interference signal
overheard by the other users in time slot $k$ in the first phase. In other
words, the transmitter delivers ${\bf s}^{(k)}$ using the beamforming matrix
${\bf V}^{(k)}[n]$ in such a way that the other interfering receivers
(excluding user $k$) see the same linear combination of the interfering data
${\bf s}^{(k)}$, which they previously observed at time slot $k$ in the first
phase. Notice that the rank of ${\bf{H}}_{k}^{c}[n]$ becomes $K-1$ with
probability one because $N_{\textrm{t}}=K-1$ and all channel values are
selected from a continuous distribution. Using this fact, the beamforming
matrix for user $k$, ${\bf V}^{(k)}\in\mathbb{C}^{N_{\textrm{t}}\times
N_{\textrm{t}}}$, satisfying the interference alignment condition in (83) is
constructed as

$\displaystyle{{\bf{V}}^{(k)}}[n]={{\bf{H}}_{k}^{c}[n]}^{-1}{\bf{H}}_{k}^{c}[k].$
(84)

Now, we consider decodability achieved by using the designed
${{\bf{V}}^{(k)}}[n]$. To make the exposition concrete, we focus on the
decoding process at user 1. The concatenated received signals at user $1$
during the both phases are given by

$\displaystyle\left[\\!\\!\\!\begin{array}[]{c}{y}^{(1)}[1]\\\ {y}^{(1)}[2]\\\
\vdots\\\ {y}^{(1)}[K]\\\ \hline\cr{y}^{(1)}[K\\!+\\!1]\\\
{y}^{(1)}[K\\!+\\!2]\\\ \vdots\\\ {y}^{(1)}[2K\\!-2]\\\
\end{array}\\!\\!\right]\\!\\!=\\!\\!\left[\\!\\!\begin{array}[]{c}{{\bf
h}^{(1)T}}[1]{\bf s}^{(1)}\\\ {{\bf h}^{(1)T}}[2]{\bf s}^{(2)}\\\ \vdots\\\
{{\bf h}^{(1)T}}[K]{\bf s}^{(k)}\\\ \hline\cr{{\bf
h}^{(1)T}}[K\\!+\\!1]\sum_{k=1}^{K}{{\bf{V}}^{(k)}}[K+1]{\bf s}^{(k)}\\\ {{\bf
h}^{(1)T}}[K\\!+\\!2]\sum_{k=1}^{K}{{\bf{V}}^{(k)}}[K+2]{\bf s}^{(k)}\\\
\vdots\\\ {{\bf h}^{(1)T}}[2K\\!-\\!2]\sum_{k=1}^{K}{{\bf{V}}^{(k)}}[2K-2]{\bf
s}^{(k)}\\\ \end{array}\\!\\!\right].$ (101)

Since we designed the transmit beamforming matrix ${{\bf{V}}^{(k)}}[n]$ so
that it satisfies the space-time interference alignment conditions in (84), it
is possible for user 1 to eliminate interference signals received during the
second phase using the overheard equations during the first phase as side-
information. For example, at the $\ell$-th time slot for
$\ell\in\\{K+1,K+2,\ldots,2K-2\\}$, user 1 receives ${\bf
h}^{(1)T}[\ell]\sum_{k=2}^{K}{\bf{V}}^{(k)}[\ell]{\bf s}^{(k)}$ interference
signal. Since user 1 already received the same interference signals from time
slot 2 to time slot $K$, i.e., $\left\\{{\bf h}^{(1)T}[2]{\bf s}^{(2)},{\bf
h}^{(1)T}[3]{\bf s}^{(3)},\ldots,{\bf h}^{(1)T}[K]{\bf s}^{(K)}\right\\}$, it
can remove all the interference signals observed in time slot $\ell$. After
interference cancellation, the effective received signal at user 1 is given by

$\displaystyle\left[\\!\\!\\!\\!\begin{array}[]{c}{y}^{(1)}[1]\\\
{y}^{(1)}[K\\!+\\!1]\\!-\\!\sum_{k=2}^{K}{y}^{(1)}[k]\\\
{y}^{(1)}[K\\!+\\!2]\\!-\\!\sum_{k=2}^{K}{y}^{(1)}[k]\\\ \vdots\\\
{y}^{(1)}[2K-2]\\!-\\!\sum_{k=2}^{K}{y}^{(1)}[k]\\\
\end{array}\\!\\!\\!\\!\right]$
$\displaystyle\\!\\!\\!=\\!\\!\\!\underbrace{\left[\\!\\!\\!\\!\begin{array}[]{c}{{\bf
h}^{(1)T}}[1]\\\ {{\bf h}^{(1)T}}[K+1]{{\bf{V}}^{(1)}}[K\\!+\\!1]\\\ {{\bf
h}^{(1)T}}[K+2]{{\bf{V}}^{(1)}}[K\\!+\\!2]\\\ \vdots\\\ {{\bf
h}^{(1)T}}[2K-2]{{\bf{V}}^{(1)}}[2K\\!-\\!2]\\\
\end{array}\\!\\!\\!\right]}_{N_{\textrm{t}}\times N_{\textrm{t}}}\\!\\!{\bf
s}^{(1)},$ (112) $\displaystyle\Rightarrow{\bf\hat{y}}^{(1)}$
$\displaystyle={\bf{\hat{H}}}^{(1)}{\bf s}^{(1)},$ (113)

where ${\bf\hat{y}}^{(1)}$ and ${\bf\hat{H}}^{(1)}$ denote the effective
received signal vector and channel matrix at user 1 after interference
cancellation. The effective channel matrix for ${\bf{\hat{H}}}^{(1)}$ has a
full rank of $N_{\textrm{t}}$ almost surely because ${\bf{V}}^{(1)}[K+1]$,
${\bf{V}}^{(1)}[K+2]$, $\ldots$, and ${\bf{V}}^{(k)}[2K-2]$ are independently
constructed with respect to ${{\bf h}^{(1)T}}[1]$, ${{\bf h}^{(1)T}}[K+1]$,
${{\bf h}^{(1)T}}[K+2]$, $\ldots$, and ${{\bf h}^{(1)T}}[2K-2]$. Therefore, in
the high SNR regime, the receiver can decode the desired $N_{\textrm{t}}$
symbols $s^{(1)}_{k}$ for $k\in\\{1,2\ldots,N_{\textrm{t}}\\}$ using a ZF
receiver. By the same argument, it is possible to obtain $N_{\textrm{t}}=K-1$
data symbols over $2K-2$ time slots when $\omega=\frac{K-2}{2K-2}$ for the
other users as well. As a result,
$d(\frac{K-2}{2K-2})=\frac{K(K-1)}{2K-2}=\frac{K}{2}$ DoF are achieved using
periodic feedback.

###  IV-C Achievability of Point C

Consider point C where the non-CSI feedback duration equals to one time slot,
i.e., $T_{\textrm{n}}=1$, and the CSI feedback duration comprises of $K-1$
time slots, i.e., $T_{\textrm{f}}=K-1$. The achievability proof for point C is
also shown by the STIA method.

####  IV-C1 Phase One

Only one time slot is used in phase one. In this time slot, the transmitter
sends $N_{\textrm{t}}=K-1$ independent symbols to their corresponding users at
the same time. The main objective in this phase is for each user to see an
equation in the form of a superposition of all transmitted data symbols, which
will be used as side-information for decoding. The transmitted signal in time
slot $1$ is given by

$\displaystyle{\bf x}[1]=\sum_{k=1}^{K}{\bf s}^{(k)},$ (114)

1 where ${\bf
s}^{(k)}=\left[s^{(k)}_{1},s^{(k)}_{2},\ldots,s^{(k)}_{K-1}\right]^{T}$. The
received signal at user $j$ is given by

$\displaystyle y^{(j)}[1]$ $\displaystyle={\bf h}^{(k)T}[1]\sum_{k=1}^{K}{\bf
s}^{(k)}$ $\displaystyle={L}^{(j,j)}[1]+\sum_{k=1,k\neq j}^{K}{L}^{(j,k)}[1],$
(115)

where noise is neglected for the DoF analysis as before.

####  IV-C2 Phase Two

The remaining $K-1$ time slots are used in phase two. The objective in phase
two is for the transmitter to send the information signal in such a way that
each user sees the same interference shape observed at time slot $1$. To
accomplish this goal, the transmit precoder for carrying the data symbols for
user $k$ at time slot $n\in\\{2,\ldots,K\\}$, is computed as

$\displaystyle{\bf V}^{(k)}[n]$ $\displaystyle=\left[\begin{array}[]{c}{{\bf
h}^{(1)T}}[n]\\\ \vdots\\\ {{\bf h}^{(k-1)T}}[n]\\\ {{\bf h}^{(k+1)T}}[n]\\\
\vdots\\\ {{\bf h}^{(K)T}}[n]\\\
\end{array}\right]^{-1}\left[\begin{array}[]{c}{{\bf h}^{(1)T}}[1]\\\
\vdots\\\ {{\bf h}^{(k-1)T}}[1]\\\ {{\bf h}^{(k+1)T}}[1]\\\ \vdots\\\ {{\bf
h}^{(K)T}}[1]\\\ \end{array}\right],$ (128)

where $k\in\\{1,2\ldots,K\\}$. Effectively, the current channel is equalized
and forced to mimic the channel corresponding to the old CSI available in time
slot 1. The received signal at user $k$ at time slot $n$ is given by

$\displaystyle{y}^{(k)}[n]$ $\displaystyle={\bf h}^{(k)T}[n]\sum_{j=1}^{K}{\bf
V}^{(j)}[n]{\bf s}^{(j)}$ $\displaystyle={L}^{(k,k)}[n]+\sum_{j\neq
k}^{K}{L}^{(k,j)}[n]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}{L}^{(k,k)}[n]+\sum_{j\neq
k}^{K}{L}^{(k,j)}[1],$ (129)

where (a) comes from the fact that during the second phase, i.e.,
$n\in\\{2,3,\ldots,K\\}$, user $k$ observes the same interference pattern
$\sum_{j\neq k}^{K}{L}^{(k,j)}[n]=\sum_{j\neq k}^{K}{L}^{(k,j)}[1]$ for
$n\in\\{2,\ldots,K\\}$.

####  IV-C3 Interference Cancellation

Because of the special choice of precoding matrix in (128), the receiver is
able to subtract interference received from the received signal during the
second phase using the saved equation from the first phase. Using the received
signal in each time slot of phase 2, the receiver constructs

$\displaystyle{y}^{(k)}[n]-{y}^{(k)}[1]$ $\displaystyle={\bf
h}^{(k)T}[n]\sum_{j=1}^{K}{\bf V}^{(j)}[n]{\bf s}^{(j)}$ $\displaystyle-{\bf
h}^{(k)T}[1]\sum_{j=1}^{K}{\bf V}^{(j)}[1]{\bf s}^{(j)}$
$\displaystyle={L}^{(k,k)}[n]+\sum_{j=1,j\neq k}^{K}{L}^{(k,j)}[n]$
$\displaystyle-{L}^{(k,k)}[1]-\sum_{j=1,j\neq k}^{K}{L}^{(k,j)}[1]$
$\displaystyle={L}^{(k,k)}[n]-{L}^{(k,k)}[1],\quad n\in\\{2,3,\ldots,K\\}.$
(130)

Stacking all $K-1$ equations together into a linear system gives

$\displaystyle\\!\\!\\!\\!\\!\left[\\!\\!\\!\begin{array}[]{c}{y}^{(k)}[2]-{y}^{(k)}[1]\\\
{y}^{(k)}[3]-{y}^{(k)}[1]\\\ \vdots\\\ {y}^{(k)}[K]-{y}^{(k)}[1]\\\
\end{array}\\!\\!\\!\right]\\!\\!\\!$
$\displaystyle=\\!\\!\\!\left[\\!\\!\begin{array}[]{c}{L}^{(k,k)}[2]-{L}^{(k,k)}[1]\\\
{L}^{(k,k)}[3]-{L}^{(k,k)}[1]\\\ \vdots\\\ {L}^{(k,k)}[K]-{L}^{(k,k)}[1]\\\
\end{array}\right]$ (139)
$\displaystyle\\!\\!\\!=\\!\\!\left[\\!\\!\\!\begin{array}[]{c}{\bf
h}^{(k)T}[2]{\bf V}^{(k)}[2]-{\bf h}^{(k)T}[1]\\\ {\bf h}^{(k)T}[3]{\bf
V}^{(k)}[3]-{\bf h}^{(k)T}[1]\\\ \vdots\\\ {\bf h}^{(k)T}[K]{\bf
V}^{(k)}[K]-{\bf h}^{(k)T}[1]\\\ \end{array}\\!\\!\\!\right]{\bf s}^{(k)}$
(144) $\displaystyle=\\!{\bf\bar{H}}^{(k)}{\bf s}^{(k)},$ (145)

where $k\in\\{1,2,\ldots,K\\}$. Now recall the following facts. First, the
precoding matrices ${\bf V}^{(k)}[n]$ in (128) are independently generated
regardless of the direct channel ${\bf h}^{(k)T}[n]$ for
$n\in\\{2,3\ldots,K\\}$. Second, the channel elements come from a continuous
random distribution. Using these two facts, it follows that ${\bf
h}^{(k)T}[n]{\bf V}^{(k)}[n]$ are statistically independent. Further, since
${\bf h}^{(k)T}[n]{\bf V}^{(k)}[n]$ and ${\bf h}^{(k)T}[1]$ are linear
independent for $\forall n$ and $\forall k$, it follows that
$\textrm{rank}\left({\bf\bar{H}}^{(k)}\right)=K-1$ almost surely. Therefore,
by applying a ZF decoder, user $k$ obtains ${\bf s}^{(k)}$. Consequently, for
the $K$-user $(K-1)\times 1$ MISO broadcast channel, each transmitter delivers
$(K-1)$ data symbols to the corresponding user over $K$ time slots when
$\omega=\frac{K-1}{K}$, which leads to achieve
$d\left(\frac{K-1}{K}\right)=K-1$ DoF. ∎

Remark 4 (Least Squares STIA): It is possible to extend the STIA algorithm to
$N_{\textrm{t}}<(K-1)$, at the expense of DoF. As shown in (84) and (128),
perfect STIA is accomplished when $N_{\textrm{t}}\geq(K-1)$. If
$N_{\textrm{t}}<(K-1)$, the matrix in (128) is not invertible. Consequently,
we propose to find the precoder that minimizes but not necessarily eliminates
the residual interference in the alignment condition. By minimizing the error
residual in the interference alignment condition, $\min_{{\bf
V}^{(k)}[n]}\parallel{\bf{H}}_{k}^{c}[k]-{\bf{H}}_{k}^{c}[n]{\bf
V}^{(k)}[n]\parallel_{F}$, the least squares STIA (LS-STIA) is obtained as

$\displaystyle{{\bf\hat{V}}^{(k)}_{LS}}[n]=\left({\bf{H}}_{k}^{c}[n]^{*}{{\bf{H}}_{k}^{c}[n]}\right)^{-1}{{\bf{H}}_{k}^{c}[n]}^{*}{\bf{H}}_{k}^{c}[k].$
(146)

Although LS-STIA solution in (146) does not achieve the same DoF performance
compared to the perfect STIA in (84) and (128), it could be a robust solution
to the alignment error when $N_{\textrm{t}}<(K-1)$. Precisely quantifying the
DoF degradation and practical achievable rates is a topic for future work.

###  IV-D Analysis of the DoF Gain from Outdated CSI Feedback

This section quantifies the DoF gain due to outdated CSI feedback relative to
a baseline strategy when each user does not feedback CSI to the transmitter
during the non-feedback period $T_{\textrm{n}}$, and sends back current CSI
only during the feedback period $T_{\textrm{f}}$. The baseline transmission
strategy is to select alternatively a transmission method between the ZF
method when CSIT is available and the TDMA method when CSIT is not available.
The achievable sum-DoF region $d_{\textrm{ZF-TDMA}}(\omega)$ by this baseline
transmission strategy is given by

$\displaystyle d_{\textrm{ZF-TDMA}}(\omega)$
$\displaystyle=(K-1)\omega+(1-\omega)$
$\displaystyle=(K-2)\omega+1\quad\textrm{for}\quad\omega\in[0,1].$ (147)

By computing the different between $d(\omega)$ in Theorem 1 and
$d_{\textrm{ZF-TDMA}}(\omega)$, we observe how much DoF gain we can obtain
from additional outdated CSI feedback in the periodic CSI feedback as shown in
Fig. 3. There are two interesting observations. One is that outdated CSI
feedback always improves the DoF gain over all feedback frequency in the range
$0<\omega<1$. Further, when feedback the frequency is larger or equal to
$\frac{K-1}{K}$, outdated CSI feedback does not degrade the DoF gain
performance. This DoF gain comes from joint exploitation of current and
outdated CSI, which allows each receiver to exploit its overheard side-
information to achieve DoF gains compared to the baseline case.

###  IV-E Comprison with an Outer Bound [30]

Recently, a DoF outer bound result was established in [30] for the alternating
CSIT model [29] in the vector broadcast channel. Interestingly, it was shown
in [30] that the transmitter requires to use perfect CSI during the
$\min\\{N_{\textrm{t}},K\\}/K$ fraction of time in minimunm to achieve the
$\min\\{N_{\textrm{t}},K\\}$ cut-set DoF by a new converse proof technique.
From this converse argument, the proposed region in Theorem 1 has the DoF
optimality for $\omega\in[\frac{K-1}{K},1]$ and $\omega=0$. For
$0<\omega<\frac{K-1}{K}$, however, the DoF gap between the outer bound result
in [30] and the proposed region exists as illustrated in Fig. 3. Therefore,
finding the optimal sum-DoF region over the range of $0<\omega<\frac{K-1}{K}$
is still an open problem. Further, one interesting open problem is to
investigate the DoF region in the case when $N_{\textrm{t}}<K-1$. We
conjecture that increasing the number of users would help to decrease the
feedback frequency requirement from the observation in the outer bound result
[30]. As shown in Remark 4, however, the proposed STIA algorithm does not
satisfy the perfect alignment condition for the $N_{\textrm{t}}<K-1$ case due
to a feasibility condition issue leading to DoF loss. Therefore, to show
tightness of this outer bound in the periodic CSI feedback framework, a new
achievable scheme would be needed for the $N_{\textrm{t}}<K-1$ case.

##  V CSI Feedback Delay-DoF Gain Trade-Off

In this section, we characterize the achievable sum-DoF region for the
periodic CSI feedback Model 2 in Section II. We quantify the relationship
between the CSI feedback delay and the sum-DoF gain for the vector broadcast
channel leveraging the concept of STIA and using the results in [11]. Through
this trade-off analysis, we provide insights into the interplay between CSI
feedback delay and system performance from a sum-DoF gain perspective.

We devote to prove the following theorem in this section.

###### Theorem 2

_For the $K$-user $(K-1)\times 1$ MISO broadcast channel with periodic CSI
feedback (Model 2), the achievable trade-off region between CSI feedback delay
and the sum-DoF gain is given by_

$\displaystyle d(\gamma)\\!=\\!\\!\left\\{\\!\\!\begin{array}[]{l
l}K-1,&\\!\\!\\!\\!\\!\\!\quad\textrm{for}\quad 0\leq\gamma\leq\frac{1}{K},\\\
\left(\\!-K\\!+\\!\frac{Kc(K)}{K-1}\\!\right)\\!\gamma\\!+\\!K\\!-\\!\frac{c(K)}{K\\!-\\!1},&\\!\\!\\!\\!\\!\\!\quad\textrm{for}\quad\frac{1}{K}<\gamma\leq
1,\\\ c(K),&\\!\\!\\!\\!\\!\\!\quad\textrm{for}\quad\gamma\geq 1.\\\
\end{array}\right.$ (151)

_where $c(K)=\frac{K-1}{1+\frac{1}{2}+\cdots+\frac{1}{K-1}}$._

###  V-A Achievability Proof for Theorem 2

The main technical challenge in this proof is to show that
$d(\frac{1}{K})=\min\\{N_{\textrm{t}},K\\}=K-1$. The other region
$d(\gamma)=K-1$ for $0<\gamma\leq\frac{1}{K}$ follows from this result using
time sharing between the proposed STIA and ZF. Further, for the region of
$\frac{1}{K}<\gamma\leq 1$, achievability of
$d(\gamma)=\left\\{-K+\frac{Kc(K)}{K-1}\right\\}\gamma+K-\frac{c(K)}{K-1}$ can
be shown by using time sharing between the proposed STIA and the Maddah-Ali-
Tse (MAT) method in [11].

Without loss of generality, we assume that the channel coherence
$T_{\textrm{c}}$ has $K$ time slots and the feedback delay time is just one
time slot $T_{\textrm{fb}}=1$ so that $\gamma=\frac{1}{K}$. Under this
assumption, if the user sends back CSI at the first time slot of the $t$-th
channel block ${t}_{1}$, then the transmitter can have current CSI at
$\\{{t}_{2},{t}_{3},\ldots,{t}_{K}\\}$ time slots due to one time slot
feedback delay. Under this channel knowledge assumption, we show that
$d(\frac{1}{K})=N_{\textrm{t}}$ of DoF are achievable asymptotically if
$N_{\textrm{t}}=K-1$. Consider $n+K-1$ channel blocks, which are comprised of
a total of $Kn+K(K-1)$ time slots so that
$S_{t}=\\{1,2,3,4,\ldots,Kn+K(K-1)\\}$ for transmission. Since we assume that
the normalized feedback delay is $\frac{1}{K}$, the total time slot set can be
decomposed two subsets, $S_{c}$ with $|S_{c}|=(K-1)(n+K-1)$ and $S_{d}$ with
$|S_{d}|=n+K-1$. Here, $S_{c}$ and $S_{d}$ represent time slot sets
corresponding to the case where the transmitter has current and delayed CSI.
Now, define $n$ time slot sets, $\left\\{I_{1},I_{2},\ldots,I_{n}\right\\}$,
each of which has $K$ elements for STIA algorithm, i.e.,
$I_{\ell}=\\{k_{1},k_{2},\ldots,k_{K}\\}$ where $k_{1}\in S_{d}$, $k_{j}\in
S_{c}$ for $j\in\\{2,3,\ldots,K\\}$, and $k_{\ell}$ and $k_{j}$ belong to a
different channel coherence time block when $\ell\neq j$. Further, let us
define two different time slot sets $I_{\textrm{ZF}}$ for ZF and
$I_{\textrm{TDMA}}$ for TDMA transmission. Since the used time slots for STIA
transmission are $|I_{1}\cup I_{2}\cup\cdots\cup I_{n}|=Kn$, the time slot set
for ZF and TDMA is defined as

$\displaystyle I_{\textrm{ZF}}=S_{c}-(I_{1}\cup I_{2}\cup\cdots\cup I_{n})$
$\displaystyle I_{\textrm{TDMA}}=S_{d}-(I_{1}\cup I_{2}\cup\cdots\cup I_{n}),$

where $|I_{\textrm{ZF}}|=(K-1)^{2}$ and $|I_{\textrm{TDMA}}|=K-1$. For
example, as shown in Fig. 5, when $K=3$ and $n=3$, there are a total of
$3n+6=15$ time resources. The temporal resources can be partitioned into five
indexed sets defined as $I_{1}=\\{1,5,9\\}$, $I_{2}=\\{4,8,12\\}$,
$I_{3}=\\{7,11,15\\}$, $I_{\textrm{ZF}}=\\{2,3,6,14\\}$, and
$I_{\textrm{TDMA}}=\\{10,13\\}$. The key idea in this proof is to apply
different transmission strategies: the proposed STIA, ZF and TDMA according to
different subsets of time slots.

Figure 5: Periodic CSI feedback (Model 2) when $n=3$, $K=3$, and
$\gamma=\frac{1}{3}.$

####  V-A1 STIA

We first show that $nK(K-1)$ DoF are achievable using the proposed STIA
algorithm when $nK$ time slots are used i.e, $|I_{1}\cup I_{2}\cup\cdots\cup
I_{n}|=Kn$. This proof is identical to the achievability proof for Point C in
Theorem 1, which was shown in Section IV. Since for given $I_{\ell}$, the
transmitter is able to use $K-1$ current and outdated CSI over a total $K$
slots, the normalized delay parameter $\gamma=\frac{1}{K}$ is equivalently
interpreted in terms of the feedback frequency $\omega$ with the relationship
of $\omega=\frac{K-1}{K}=1-\gamma$. From this equivalence, we conclude that
$K(K-1)$ DoF are achievable by spending each time slot set $I_{\ell}$ with $K$
elements for $\ell\in\\{1,2,\ldots,n\\}$.

####  V-A2 TDMA and ZF

Since $Kn$ time slots have been used for STIA among the total temporal
resources$Kn+K(K-1)$, $I_{\textrm{ZF}}$ and $I_{\textrm{TDMA}}$ time slot sets
remain where $|I_{\textrm{ZF}}|=(K-1)^{2}$ and $|I_{\textrm{TDMA}}|=K-1$.
Here, we use an alternative transmission method depending on CSIT knowledge.
The transmitter sends data by using TDMA because CSIT is not available due to
feedback delay. Alternatively, the transmitter delivers multiple data streams
by using ZF because the transmitter is able to use CSI. From this method, the
achievable DoF is $d(\frac{1}{K})=(K-1)(K-1)^{2}+(K-1)$ in the remaining
$K(K-1)$ time slots.

####  V-A3 Asymptotic the Sum-DoF Gain

We have divided the total time resource $S_{t}$ with $|S_{t}|=Kn+K(K-1)$ into
three different groups and applied a different transmission method in each
group. Hence, using time sharing, the total sum-DoF gain is computed as

$\displaystyle d\left(\frac{1}{K}\right)$
$\displaystyle=\\!\frac{\underbrace{(K-1)Kn}_{\textrm{STIA}}+\underbrace{(K-1)(K-1)^{2}}_{\textrm{ZF}}+\underbrace{(K-1)}_{\textrm{TDMA}}}{Kn+K(K-1)}$
$\displaystyle=\frac{(K-1)Kn+(K-1)^{3}+(K-1)}{Kn+K(K-1)}.$ (152)

Therefore, for a given $K$, as $n$ goes to infinity, the sum-DoF gain
asymptotically achieves

$\displaystyle\lim_{n\rightarrow\infty}d\left(\frac{1}{K}\right)=K-1,$ (153)

which completes the proof.

###  V-B Special Case: Three-user Two-Antenna Vector Broadcast Channel

In this section, we provide an exact characterization of the CSI feedback
delay-DoF gain trade-off region for the 3-user $2\times 1$ MISO broadcast
channel. Achievability is shown by the result in Theorem 2 and the enhanced
DoF result in Theorem 5 [11]. Further, a converse is proven using the result
in [30] by considering a minor modification for a block fading channel.

###### Corollary 1

The optimal CSI feedback delay-DoF gain trade-off region for the 3-user
$2\times 1$ MISO broadcast channel is given by

$\displaystyle d^{*}(\gamma)=\left\\{\begin{array}[]{l
l}2,&\quad\textrm{for}\quad 0\leq\gamma\leq\frac{1}{3},\\\
-\frac{3}{4}\gamma+\frac{9}{4},&\quad\textrm{for}\quad\frac{1}{3}<\gamma\leq
1,\\\ \frac{3}{2},&\quad\textrm{for}\quad\gamma\geq 1.\\\ \end{array}\right.$
(157)

####  V-B1 Achievability

Achievability of this corollary can be proved by applying time sharing
technique between STIA and the enhanced transmission method proposed in
Theorem 5 [11].

####  V-B2 Converse

For a proper converse argument, let us consider the following lemma given by
[30].

###### Lemma 1

For the 3-user $2\times 1$ MISO broadcast channel with identical channel
coherence patterns and the same normalized feedback delay $\gamma$ across the
users, the following DoF bound holds:

$\displaystyle d^{(1)}+d^{(2)}+d^{(3)}\leq\frac{9}{4}-\frac{3}{4}\gamma.$
(158)

###### Proof:

A detailed proof is provided in [30] for the fast fading scenario where the
channel gains change independently across time for all users. Here, we
consider a minor modification for a block fading channel where all users have
identical channel coherence patterns. The idea of the converse proof in [30]
is to covert the original 3-user $2\times 1$ MISO broadcast channel into a
2-user $2\times 1$ MISO physically degraded broadcast channel by allowing
cooperation between user 2 and user 3 and providing the channel output signal
of user 1 with the cooperating user group. Using the fact that feedback does
not increase the capacity of the physically degraded broadcast channel and its
capacity depends on the marginal distribution only, it is possible to create
$\min\\{N_{\textrm{t}},K\\}-1=1$ artificial receivers that can decode the same
message of user 1. By applying the differential entropy bound techniques used
in [30], the bounds follows

$\displaystyle 2d^{(1)}+d^{(2)}+d^{(3)}\leq 2+\alpha,$ (159) $\displaystyle
d^{(1)}+2d^{(2)}+d^{(3)}\leq 2+\alpha,$ (160) $\displaystyle
d^{(1)}+d^{(2)}+2d^{(3)}\leq 2+\alpha,$ (161)

where $\alpha$ denotes the fraction of time instances where the perfect CSIT
is available. By replacing $\alpha=1-\gamma$ and adding three inequalities in
(159), (160), and (161), the inequality of (158) results, completing the
proof. ∎

Using Lemma 1 and the cut-set bound
$d^{(1)}+d^{(2)}+d^{(3)}\leq\min\\{K,N_{\textrm{t}}\\}=2$, it concludes that
the region in (48) is the optimal.

Figure 6: The optimal CSI feedback delay-DoF gain trade-off for the 3-user $2\times 1$ MISO broadcast channel. TABLE I: Comparison of three different interference alignment algorithms Scheme | Fading type | Required CSIT | Feed-forward overhead
---|---|---|---
Blind IA [23] | Staggered block fading | Fading autocorrelation function | No
MAT | General fading | Outdated CSI | CSIR for other receivers
STIA | General fading | Outdated and current CSI | No

Remark 5 (Sum-DoF Region Comparison): We will compute the achievable CSI
feedback delay-DoF gain tradeoff curves achieved by time sharing among
conventional schemes, and compare them against the proposed optimal tradeoff
curve. By time sharing between ZF and TDMA, it is possible to show that the
$d_{\textrm{ZF-TDMA}}(\gamma)=-\gamma+2$ curve is achievable for
$0\leq\gamma\leq 1$. Similarly, if we consider the time sharing method between
ZF and MAT, then the $d_{\textrm{ZF-MAT}}(\gamma)=-\frac{1}{2}\gamma+2$ of
curve is achieved for $0\leq\gamma\leq 1$. As illustrated in Fig. 6, time
sharing between the proposed transmission method and the MAT method in Theorem
5 [11] achieves a higher CSI feedback delay-DoF trade-off region. For
instance, the proposed algorithm achieves the 2 of DoF when
$\gamma=\frac{1}{3}$, which is the $\frac{1}{3}$ of DoF gain over ZF-TDMA and
$\frac{1}{6}$ of DoF gain over ZF-MAT.

Remark 6: The proposed CSI feedback delay-DoF gain trade-off shows that if
users send CSI to the transmitter within 33$\%$ of the channel coherence time,
then the system performance is not degraded from a DoF perspective. Based on
previous work [16]-[18], it was understood that even a small amount of CSI
feedback delay resulted in degraded DoF. Our result confirms the intuition
that a delay corresponding to a reasonable fraction of the coherence time does
not change the sum-DoF scaling. We are able to precisely quantify how much
delay is allowed - finding that it corresponds to one third of the channel
coherence time.

Example: Consider a LTE system using $f=2.1$ GHz carrier frequency, which
serves users with mobility of $v=3$ km/h (walking speed). In this case, the
channel coherence time is roughly calculated as
$T_{\textrm{c}}\simeq\frac{c}{8fv}=21.4$ msec (two radio frames) where $c$
denotes the speed of light. Therefore, if the users can feedback CSI within
7.133 msec (7 subframes), then there is no performance loss from a sum-DoF
point-of-view. Based on this observation, the proposed STIA algorithm can be
interpreted as a CSI delay robust transmission algorithm.

##  VI Comparisons with Similar Interference Alignment Techniques

In this section, we compare the proposed transmission method with the blind
interference alignment [23] and the MAT method in [11]. Our STIA algorithm has
similarities with blind interference alignment [23], [24] and the MAT
algorithm [11], [12]. During phase one, none of the transmission algorithms
require CSI and simply send multiple data streams without preprocessing. The
only difference occurs in the second phase. Each algorithm performs alignment
in the second phase in a slightly different way. Blind interference alignment
exploits a special channel coherence structure called staggered block fading
at the transmitter. To realize blind interference alignment, each user may
feedback the fading autocorrelation function of its channel. Using blind
interference alignment with receive antenna switching, it was shown that
$\frac{N_{\textrm{t}}K}{N_{\textrm{t}}+K-1}$ DoF gain is achieved in [24].
Therefore, for $N_{\textrm{t}}=K-1$ and $N_{\textrm{r}}=1$ case, blind
interference alignment and the proposed STIA both achieve the same DoF gain of
$\frac{K(K-1)}{2K-2}=\frac{K}{2}$ even if the channel knowledge assumption and
fading scenarios are different.

Unlike blind interference alignment, the MAT algorithm realizes the alignment
in the second phase by using delayed CSI under a general i.i.d. fading channel
assumption. The basic idea of the MAT algorithm is to exploit the outdated CSI
to swap the received equations that they have already seen during the first
phase between users. By swapping the received equations, each user decodes the
desired data symbols while eliminating the aligned interference signal. To
apply the MAT algorithm, each user should feedback CSI whenever their channel
changes so that the transmitter can track all the channel variations. Further,
each receiver needs to know the CSI of other receivers since the newly
obtained linear combination of the desired symbols is obtained by swapping
during the second phase. Therefore, to learn the CSI of other receivers, the
transmitter must forward the CSI of other receivers to each user, which may
cause additional overhead.

The proposed STIA exploits both outdated and current CSI offered by the
periodic feedback mechanism under a general i.i.d. fading channel assumption.
Due to the requirement for current CSI at some instant, it places a more
restrictive feedback delay requirement compared to the MAT algorithm [11].
Also, it incurs the same CSI feedback overhead as the MAT method because CSI
must be sent back to the transmitter when it changes. The proposed algorithm,
however, avoids the need for additional feed-forward overhead. This reason is
that the effective channel can be estimated using the demodulated reference
signal as explained in Remark 3. We summarize the similarity and differences
of the three different transmission algorithms as in Table I.

##  VII Conclusion

We proposed a new STIA algorithm that exploits both current and outdated CSI
jointly for the MISO broadcast channel with two different periodic CSI
feedback models. For the CSI feedback frequency limited model, we
characterized the achievable sum-DoF region as a function of the feedback
frequency, and showed that the joint exploitation of current and outdated CSI
provides a significant sum-DoF gain compared to the case when the transmitter
uses current CSI only. For the CSI feedback link delay limited model, we
derived a trade-off region between CSI feedback delay and the sum-DoF gain for
the MISO broadcast channel using the proposed STIA algorithm and leveraging
results in [11]. Based on this trade-off, we provided insights into the
interplay between CSI feedback delay and system performance from a DoF gain
perspective.

## References

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Namyoon Lee is a Ph. D student in the Department of Electrical and Computer
Engineering at The University of Texas at Austin. He received his M.Sc. degree
in Electrical Engineering from KAIST, Daejeon in 2008 and B.Sc degree in Radio
and Communication Engineering from Korea University, Seoul, Korea in 2006.
From 2008 to 2011, he was with Samsung Advanced Institute of Technology (SAIT)
and Samsung Electronics Co. Ltd. in Korea, where he studied next generation
wireless communication networks and involved standardization activities of the
3GPP LTE-A (Long Term Evolution Advanced). He also was graduate intern in
advanced network topologies team within the Wireless Communications Lab (WCL)
at Intel Labs, Santa Clara, CA, USA in 2012. His primal research interest is
to developing and analyzing future wireless communication systems using tools
including network information theory and stochastic geometry. Mr. Lee was a
recipient of the 2009 Samsung Best Paper Award. He also was awarded several
fellowships, including the Graduate Student Research Fellowship from the Korea
Science and Engineering Foundation (KOSEF) in 2006; the Korea Government
Fellowship from 2006 to 2007; the Kwanjeong Educational Foundation Fellowship
in 2011-2014; and MCD Fellowship in 2012.
---
Robert W. Heath, Jr. is a Professor in the Department of Electrical and
Computer Engineering at The University of Texas at Austin and is a member of
the Wireless Networking and Communications Group where he directs the Wireless
Systems Innovations Laboratory. He received his B.S.E.E. (1996) and his
M.S.E.E. (1997) degrees from the University of Virginia, and the Ph.D.E.E.
(2002) degree from Stanford University. From 1998 to 1999, he was a Senior
Member of the Technical Staff at Iospan Wireless Inc, San Jose, CA where he
played a key role in the design and implementation of the physical and link
layers of the first commercial MIMO-OFDM communication system. The proposed
broadband wireless access system employed OFDM modulation, MIMO technology,
and adaptive space-time modulation. From 1999 to 2001 he served as a Senior
Consultant for Iospan Wireless Inc. In 2003 he founded MIMO Wireless Inc, a
consulting company dedicated to the advancement of MIMO technology. Since
January 2002, he has been with the Department of Electrical and Computer
Engineering at The University of Texas at Austin where he serves as an
Associate Professor and is affiliated with the Wireless Networking and
Communications Group. He has consulted for a variety of international
companies. Prof. Heath’s current research include all aspects of MIMO
communication including antenna design, practical receiver architectures,
limited feedback techniques, mobility management, and scheduling algorithms as
well as cognitive radio, 60GHz wireless communication, and body area networks.
He teaches courses on a variety of topics including Wireless Communication,
Digital Signal Processing, Space-Time Communication, and Statistical Signal
Processing, to support his research program in wireless communication systems.
He has published over 150 refereed conference and journal papers, been awarded
20 patents, and is the author of the forthcoming book Advanced MIMO
Communication. Dr. Heath has been an Editor for the IEEE Transactions on
Communication and an Associate Editor for the IEEE Transactions on Vehicular
Technology. He is a member of the Signal Processing for Communications
Technical Committee in the IEEE Signal Processing Society. He was a technical
co-chair for the 2007 Fall Vehicular Technology Conference, is the general
chair of the 2008 Communication Theory Workshop, is a co-organizer and co-
technical-chair of the 2009 Signal Processing for Wireless Communications
Workshop, and is co-technical-chair of the 2010 International Symposium on
Information Theory. He is the recipient of the David and Doris Lybarger
Endowed Faculty Fellowship in Engineering and is a registered Professional
Engineer in Texas.
---