Imperfect and Unmatched CSIT is Still Useful for the Frequency Correlated MISO Broadcast Channel
Abstract
Since MaddahAli and Tse showed that the completely stale transmitterside channel state information (CSIT) still benefits the Degrees of Freedom (DoF) of the MultipleInputMultipleOutput (MISO) Broadcast Channel (BC), there has been much interest in the academic literature to investigate the impact of imperfect CSIT on DoF region of time correlated broadcast channel. Even though the research focus has been on time correlated channels so far, a similar but different problem concerns the frequency correlated channels. Indeed, the imperfect CSIT also impacts the DoF region of frequency correlated channels, as exemplified by current multicarrier wireless systems.
This contribution, for the first time in the literature, investigates a general frequency correlated setting where a twoantenna transmitter has imperfect knowledge of CSI of two singleantenna users on two adjacent subbands. A new scheme is derived as an integration of ZeroForcing Beamforming (ZFBF) and the scheme proposed by MaddahAli and Tse. The achievable DoF region resulted by this scheme is expressed as a function of the qualities of CSIT. ^{1}^{1}1The project HARP acknowledges the financial support of the Seventh Framework Program for Research of the European Commission under grant number: 318489
I Introduction
In downlink multiuser multipleinputmultipleoutput (MUMIMO) communications, the latency and inaccurate CSIT degrade the DoF when conventional precoding techniques such as ZFBF are employed. Strategies to exploit imperfect feedback to enhance DoF region has therefore attracted a lot of attention. The completely stale CSIT was first studied by MaddahAli and Tse. In their contribution [1], an optimal peruser DoF of in a twouser setup was achieved by a simple transmission scheme (denoted as MAT scheme in the sequel). That result was extended later in [2] and [3], by accounting for imperfect current CSIT. The optimal DoF was derived and expressed as a function of the quality of the current CSIT. The achievablility was shown using a scheme that bridges the DoF found in [1] and [2][3].
In [4] and [5], unequal quality of current CSIT per user was investigated. The optimal bound found is a superset of the results in [2] and [3], revealing that an asymmetric DoF is achieved by each user. Moreover, [6] has studied the imperfect delayed CSIT, suggesting that it can be used as good as perfect delayed CSIT.
To date, all the works only focus on the time correlated channel. However, the DoF region of frequency correlated channels is also impacted by the imperfect CSIT. In current multicarrier communication systems, the CSIT is measured and reported by users on a persubband basis. In practice, each user only reports its CSI on a group of predefined subbands, which might provide few information about the channel of the subbands outside the group because of the weak correlation between different subbands. In this paper, we investigate, for the first time in the literature, the DoF of a general twosubband based frequency correlated broadcast channel with arbitrary imperfect CSIT (see Section II). Our contributions are summarized as follows:

Derive an achievable DoF of a twouser and twosubband based scenario as a function of the quality of the CSIT,

Design a novel transmission strategy, motivated by MAT and ZFBF, that achieves the DoF region.
The rest of this paper is organized as follows. The system model is introduced in Section II and the achievable DoF region is given in Section III. The DoF achieved via reusing MAT and ZFBF is identified in Section IV and a novel transmission scheme is introduced. Section V concludes the paper.
The following notations are used throughout the paper. Bold lower letters stand for vectors whereas a symbol not in bold font represents a scalar. and represent the transpose and conjugate transpose of a matrix or vector respectively. denotes the orthogonal space of the channel vector . refers to the expectation of a random variable, vector or matrix. is the norm of a vector. represents the magnitude of a scalar. corresponds to , where is supposed to be the SNR throughout the paper and logarithms are in base . denotes the power of while and represent the rate of achieved at receiver 1 and 2 respectively.
Ii System Model
We consider a twouser broadcast channel with two transmit antennas and one antenna per user. The related parameters are defined as follows. and are the channel states in subband of user 1 and user 2 respectively. Denoting the transmit signal vector in subband as , subject to a persubband based power constraint , the observations at receiver 1 and 2, and respectively, can be written as
(1)  
(2) 
where and are unit power AWGN noise. Signal vector is a function of the symbol vectors for user 1 and user 2, denoted as and respectively. is a twoelement symbol vector containing and . Similarly, is defined.
The channels are characterized as follows. and are mutually independent and identically distributed with zero mean and unit covariance matrix ( and ). The imperfect CSIT of user 1 is denoted as while the imperfect CSIT of user 2 is , each with the error vector of and . The variances of the error vectors are and .
The CSIT setting in this twosubband based scenario is illustrated in Figure 1.
User 1 estimates its channel information in the first subband using pilots and feeds it back as while user 2 reports its CSI in the second subband as . As shown, we assume the qualities of and are identical and expressed using a parameter, , which is defined as
(3) 
As the CSI of two adjacent subbands are correlated, the transmitter can predict the channel information of the unreported subband. To be specific, with the knowledge of , the channel condition of the second subband of user 1, , is predicted. Similarly, is predicted based on the knowledge of . The qualities of these two predicted channel states are characterized as , which is defined as
(4) 
Remarks: 1) and vary within the range of , where represents no CSIT whereas stands for perfect CSIT; 2) The quality of the predicted CSIT, and , is bounded by the quality of and , namely ; 3) We assume that this twosubband scenario can be repeated an infinite number of times; 4) The transmitter and both users have the knowledge of and . Besides, each receiver has perfect knowledge of local CSI; 5) It is important to note the quantities and .
Throughout the paper, we define a perchanneluse based DoF, which is expressed as
(5) 
where is the rate achieved by user over channel uses^{2}^{2}2 channel uses may physically refer to subbands with full transmission power or transmitting symbols using power of in a single subband. In this paper, it takes the latter understanding..
Iii DoF Region with General CSIT Pattern
Theorem 1.
In a frequency correlated MISO BC with imperfect CSIT shown in Figure 1, an achievable DoF region is characterized as a polygon composed of the following corner points:
(6) 
When is fixed, points C and D (shown by circle points, see Figure 2) move closer to each other as increases. For , points C and D join at point E (or F, see Figure 2). If continues increasing, the DoF region will not expand any further. This reveals that the CSIT with quality satisfying can be as good as . Specifically, the DoF region achieved by MAT [1] with and (composed of point F, and , see Figure 2) can be actually achieved by and .
Moreover, if we fix and increase , all the points will move either upwards or to the right. When reaches , point A will join point C while points B and D overlap. The DoF region can be simply achieved by doing ZFBF plus superposition coding.
In addition, the maximum sum DoF is achieved by the diamond point E, and star point F, for the case . Otherwise, it is obtained by the circle points C and D.
Iv Achievability
Iva Motivations
In this part, we briefly revisit two existing schemes, MAT and ZFBF. Their achievable rates in frequency correlated BC will be identified and analyzed. Their suboptimalities will motivate the derivation of the novel transmission strategy.
IvA1 Reusing MAT scheme and Extensions
In [1], the transmission of MAT finishes in three time slots, during which the transmit signal and received signals are
where and . User 1 receives its desired symbol vector in but overhears in . The decoding is enabled once the transmission at slot 3 is completed, where the sum of the overheard interference is retransmitted. After decoding received in and subtracting , user 1 obtains an additional independent observation of its desired symbol vector, . Hence, user 1 can decode . Similarly, user 2 can decode . In this way, four symbols are successfully transmitted in three slots, resulting in the symmetric DoF of .
However, among all the six CSI in these three time slots, only two of them are in fact employed, namely and . Equivalently, we can reuse MAT in the scenario shown in Figure 1 provided that and . The sum of the overheard interference, , is reconstructed and retransmitted using an extra channel use (subband 3). The CSI of this extra channel use does not have to be known at the transmitter.
When , the transmit power should be adjusted because the overheard interferences generated at each user are reconstructed with nonnegligible error at the transmitter. Specifically, after subtracting from , is obtained plus a residue interference, , where . To make the residue interference drowned by the noise, the transmission power of in subband 1 should be reduced to . In this way, channel use is employed per subband, during which, both and achieve the rate resulting in the sum DoF over channel uses.
IvA2 Conventional approachZFBF
ZFBF is one of the conventional interference mitigation techniques that achieve MUMIMO transmission. The transmitter precodes two symbols and (intended to user 1 and 2 respectively) using the knowledge of CSIT of both users. The transmission signal in subband 1 is expressed as
(7) 
where and , resulting in the received signals
(8)  
(9) 
As the qualities of and are and respectively, and are drowned by the noise. In this way, the rate achieved by and are and respectively. The amount of channel use in subband 1 is . Similarly, the same transmission is applicable in subband 2 by switching the power of each user’s symbol. Hence, the sum DoF is during these channel uses.
IvA3 Analysis and motivation
Sum Rate  Channel  Pre  

()  Uses  coding  
MAT  No  
ZFBF  Yes  
Objective  Yes 
The comparison between MAT and ZFBF are presented in Table I. ZFBF saves channel uses while it incurs a rate loss of compared to MAT.
When is small, MAT outperforms ZFBF in sum DoF. In this case, ZFBF precoding works inefficiently in rejecting the interference potentially seen by user 2 in subband 1, the transmit power of in (7) is therefore significantly limited, resulting in low DoF. Similarly, user 2 achieves low rate in subband 2. However, MAT transmits two symbols to each user in turn. The CSIT with quality is exploited to provide confident side information over an extra channel use. The DoF is therefore boosted up.
When approaches , ZFBF works well in rejecting the interference potentially seen by both users in each subband. The sum rate achieved by ZFBF therefore approximates as , resulting in a higher sum DoF than MAT by saving the extra channel uses. However, MAT incurs a loss because the CSIT with quality is wasted during the channel uses.
Intuitively, given a certain value of , a better sum DoF can be obtained by a strategy that optimally balances the employment of CSIT with quality and the usage of extra channel use. This objective strategy can be designed as the integration of ZFBF and MAT. It would outperform ZFBF by employing a small fraction of extra channel use to perform overheard interference cancellation. At the same time, as precoding is introduced, the amount of extra channel use could be reduced compared to MAT while the sum rate remains . The amount of extra channel use would be a function of and , bridging ZFBF and MAT. When , the transmission scheme would be upgraded to ZFBF; for , it would collapse to pure MAT. Bearing this in mind, we derive the novel transmission block in the following section.
IvB Building New Transmission Blocks
Following the aforementioned motivation, the main features of this strategy are presented in the last row of Table I. It is a combination of ZFBF and MAT in terms of precoding and the number of symbols transmitted to each user per subband.
The transmission signals in subband 1 and 2 are respectively expressed as
(10)  
(11) 
where two private symbols () are transmitted to user 2 and one private symbol () is sent to user 1 in subband 1. Precoding is also considered in , where is precoded with , and are projected to the orthogonal space of and respectively. The new symbols in are similarly encoded and transmitted.
Besides, and are two pieces of the quantized overheard interference, encoded with the rates, and respectively. and are common messages that should be decoded by both users. The power and rate allocation are presented in Table II.
Symbols  Power  Encoding rate ()  

Subband 1  
Subband 2  
The received signals at each receiver in each subband are expressed as
(12)  
(13)  
(14)  
(15) 
where and are the overheard interferences generated at user 1 in subband 1 and at user 2 in subband 2 respectively,
(16)  
(17) 
Note that the power stated below each term is obtained asymptotically, which is merely valid at high SNR. Since , the term in is drowned by the noise in (12). Similarly, in vanishes.
Hence, the overheard interference and are composed of and respectively, which are then possible to be detected at receiver 1 and 2 respectively (will be discussed in Section IVC). In this way, when reconstructing the sum of the overheard interferences, the channel component, for instance, in (16), can be dropped. As a consequence, in contrast to MAT where is rebuilt and sent, we reconstruct the sum of the symbols and as
(18) 
can be generated from a codebook . Since and are encoded with same rate, we assume they are generated from the same codebook, denoted as . Moreover, we design as a set close to the arithmetic plus^{3}^{3}3 close to arithmetic plus is defined as, if and , then .. In this way, can be generated from as well, with the encoding rate, , identical to that for and
Furthermore, as motivated by [4] and [6], we split into two parts, and as
(19) 
each is encoded from a codebook and respectively, which are independent to each other. The encoding rates of these two codebooks are subject to
(20)  
(21) 
Hence, can be considered as a product set of and . and are decoded separately in two parallel channels, can therefore be perfectly reconstructed by combining them.
As presented in Table II, and are respectively superposed on the private symbols transmitted in subband 1 and 2. However, their power and , should not exceed the power constraint . This constraint can be expressed as
(22) 
As a consequence, the transmission is subject to the relationship between (21) and (22).
First, in the case of , namely , the power of and does not exceed the persubband power constraint by simply setting . Moreover, we can superimpose a common message on in subband 1 and on in subband 2 using power stated in Table II, which is scaled with . Second, when , the power constraint is still satisfied but no common messages is transmitted since . Third, for the case of , namely , the value of and are bounded by as in (22). Therefore, has to be divided into three pieces as
(23) 
with the rates given by
(24)  
(25) 
The transmission of requires an extra channel use in another subband. Next, we will discuss the achievabilities of each point in Figure 2 depending on the requirement of extra channel use.
IvC Case I: Achieving Points C and D
In this case, is split into two parts and no extra channel use is required. Messages and are transmitted provided that . The decoding procedure is described as follows.
IvC1 Stage IDecode and
Revisiting (12) and , the received power of is at each receiver. Successive interference cancelation (SIC) is selected as the decoding strategy. is decoded at the first stage treating all the other symbols as noise. Consequently, the rates of achieved at user 1 and user 2 are and , respectively. These two rates are equal to , which is asymptotically for infinite . Similarly, achieves the rate in and .
IvC2 Stage IIDecode , and obtain
As and are independently encoded and sent in subband 1 and 2 respectively, they can be decoded separately at both receivers. After that, is obtained by combining them.
In and , is decoded at the second stage of SIC, where has been completely subtracted. Treating all the component to the r.h.s. of in (12) and (13) as noise, is decoded with the rate of and by user 1 and user 2 respectively. Both quantities are equal to , which is at high SNR. Similarly, is decoded with rate in subband 2. After that, and have been successfully decoded so that is completely recovered.
IvC3 Stage IIIDecode and
Employing SIC as the decoding strategy, and are decoded from and respectively.
Let us introduce a notation, , representing the signal after subtracting and as
(26)  
(27)  
(28) 
where is given in (16) and results from merging and . Treating as noise, can be decoded with the rate , which is asymptotically equal to at high SNR. After that, is seen by subtracting from as
(29) 
The rate of is . Similarly, employing SIC to results in and .
IvC4 Stage IVDecode and
As has been recovered perfectly in Stage II, each user can get access to . From (18), the rate of is obtained as , where having the knowledge of is the prerequisite. As was successfully decoded at user 1 in Stage III, it can be completely removed from (18), resulting in . Similarly the rate of at receiver 2 is .
After decoding , is decodable from . Denoting , merging and the noise into , we have
(30) 
is obtained by removing from , resulting in the rate . Similarly, is decoded with the rate .
To sum up, the DoF achieved in these two subbands are
(31)  
(32) 
where we assume and are intended to user 1 so that point D is achieved. Similarly, point C is achieved if and are intended to user 2.
IvD Case II: Achieving Point E
In this case, we remind the reader of the discussion in Section IVB that is split into three parts and an extra channel use is required to transmit . Besides, no common message is transmitted.
To achieve point E, we repeat the transmission blocks in (10) and (11) for times and employ one additional subband, namely subband , to finalize the transmissions of , where refers to the third piece of overheard interference generated in subband and . The rate of is denoted as and we assume . The quality of CSIT in subband is identical to subband 1.
The transmission in subband is expressed as
(33) 
with the power and rate allocation presented in Table III.
Symbols  Power  Encoding rate 

Considering and the transmit power, the received signal at user 1 is given as
(34) 
where all the symbols are decodable using SIC. Specifically, after are decoded, are decoded treating all the components to the r.h.s. of it in (34) as noise. The rate achieved for is derived as
(35)  
(36)  
(37) 
After decoding