**3.1. Channel training for interference alignment**

In practice, destinations can acquire CSI through a pilot-based channel training scheme. For example, consider block fading channel model in which channel gains are constant within one fading block and change to independent values in the subsequent blocks. The length of each block coincides with the coherence time of channel denoted as *T*.

Terminals first need to learn the channels within each fading block, and next use the estimated channels to perform their transmission. A pilot-assisted interference alignment scheme is proposed in [12] which perform these tasks. According to this scheme, transmission within each fading block is conducted in two phases: *pilot transmission phase* and *data transmission phase*. These two phases have the duration of *T*τ=α*T* and *Td*=(1-α)*T*, respectively, where α ∈ [*K*/ *T*, 1] called *channel sharing factor* is a design parameter. A larger α leads to more accurate channel estimation at the expense of less time left for data transmission. The transmission power of the pilot symbols in general can be different from that of the data symbols. Let *P*, *Pd*, and *P*τ denote the average transmission power, the average power of data symbols, and the average power of pilot symbols, respectively. Source S*<sup>l</sup>* (*l* ∈ {1,..., *K*}) sends *T*τ known pilot symbols with power *P*τ. Then, the destination D*<sup>k</sup>* (*k* ∈ {1,..., *K*}) applies a minimum mean square error (MMSE) estimator to obtain an estimate of the channel gain. As shown in Fig. 2 this channel estimate is used at the receiver filter and the decoder to recover the desired message [13].

A more accurate estimation of the channel can be obtain by allocating more transmission power for training symbols which implies that a lower power is left for data transmission. The achievable rate region by taking into account this noisy CSI is computed in [12]. According to Proposition 2 in [12], the optimum power allocation which maximizes the achievable rate region is

$$P\_{d, \text{opt}} = \beta\_{\text{opt}} P\_{\prime} P\_{\text{r}, \text{opt}} = K \left( \left( 1 - \left( 1 - \alpha \right) \beta\_{\text{opt}} \right) / \alpha \right) P\_{\prime} \tag{2}$$

where

$$\mathcal{B}\_{\text{opt}} = \frac{1}{1 - \alpha} \left( 1 + \sqrt{\frac{1 + KP \,/\, (1 - \alpha)}{1 + PT \,/\, N\_0}} \right)^{-1} \,\, \text{}.\tag{3}$$

If *P* ≫ 1, then the

$$
\beta\_{\rm opt} \approx \frac{1/(1-\alpha)}{1 + \sqrt{\text{KN}\_0/T(1-\alpha)}}.\tag{4}
$$

This result recommends that in large networks more power should be allocated to the channel training instead of the data transmission. The intuition behind this result is that in large networks the performance loss due to imperfect interference alignment as a consequence of imperfect CSI becomes more important. Thus, it is recommended to allocate more power to pilot symbols instead of data symbols to acquire CSI more accurately.

#### **3.2. Channel state information feedback**

As we have discussed in the previous section, destinations can acquire CSI through a pilotbased channel training scheme. The destinations then can send the estimated CSI to the sources via channel state feedbacks. They can transmit either un-quantized CSI (analog feedback) or quantized CSI (digital feedback) via feedback channels. In the following, we briefly review some key results for different type of feedback schemes.

*1) Analog Feedback:* The destinations can obtain an estimate of the incoming channels according to the scheme mentioned in Section 3.1. Then, they may transmit the analog value of the estimated channels over the feedback channel. Let the function *f* in Fig. 2 to be defined as *f*(*x*)=*x*, and assume error-free feedback channels. According to Theorem 3 in [12], in a singleinput single-output (SISO) time-varying *K*-user network with coherence time *T*, the achievable sum DoF is

$$d\_{\Sigma} = \min\left\{ K\{1 - K/T\} / 2, T / 8 \right\}. \tag{5}$$

This result is achieved when the number of the users selected to be active is

$$K\_{\rm opt} = \min\left\{ K, T \mid \mathcal{D} \right\}. \tag{6}$$

This recommends that, in large networks (*K* > *T* /2) when no CSI is a priori available at terminals, first select a subset of the users and next perform channel training and interference alignment within the set of active users.

*2) Digital Feedback:* Digital channel state feedback strategies in which each destination quantize the incoming channels and sends the index of the quantized channel over feedback channels has been studied in several works (e.g. [14-16]). It is shown that the same DoF as when perfect CSI is available can be obtained, provided that the destinations a priori know channels and the feedback signals' rate is proportional to log *P*, where *P* is the transmission power of each source. That is,

$$N\_f \sim \log P\_f \tag{7}$$

where *Nf* is the number of feedback bits. It should be noticed that in practice where destinations do not know incoming channels a priori, part of the radio resources need to be allocated for channel training and a smaller DoF will be achievable (cf. [12]). This part is discussed in more detail in section 7.1.

#### **3.3. Iterative interference alignment**

channel estimate is used at the receiver filter and the decoder to recover the desired message

A more accurate estimation of the channel can be obtain by allocating more transmission power for training symbols which implies that a lower power is left for data transmission. The achievable rate region by taking into account this noisy CSI is computed in [12]. According to Proposition 2 in [12], the optimum power allocation which maximizes the achievable rate

*P PP K <sup>d</sup>*,opt opt ,opt , 11 / , (( ( ) opt) ) *P*

1 1/

1 1 / (1 ) 1 .

0 1 / (1 ) . 1 / (1 ) *KN T* a

This result recommends that in large networks more power should be allocated to the channel training instead of the data transmission. The intuition behind this result is that in large networks the performance loss due to imperfect interference alignment as a consequence of imperfect CSI becomes more important. Thus, it is recommended to allocate more power to

As we have discussed in the previous section, destinations can acquire CSI through a pilotbased channel training scheme. The destinations then can send the estimated CSI to the sources via channel state feedbacks. They can transmit either un-quantized CSI (analog feedback) or quantized CSI (digital feedback) via feedback channels. In the following, we briefly review

*1) Analog Feedback:* The destinations can obtain an estimate of the incoming channels according to the scheme mentioned in Section 3.1. Then, they may transmit the analog value of the estimated channels over the feedback channel. Let the function *f* in Fig. 2 to be defined as *f*(*x*)=*x*, and assume error-free feedback channels. According to Theorem 3 in [12], in a singleinput single-output (SISO) time-varying *K*-user network with coherence time *T*, the achievable

*KP PT N*


ab

0

a

a  a

1


(2)

(3)

t= = --

a

opt

pilot symbols instead of data symbols to acquire CSI more accurately.

**3.2. Channel state information feedback**

some key results for different type of feedback schemes.

b

b

opt

b

[13].

56 Contemporary Issues in Wireless Communications

region is

where

If *P* ≫ 1, then the

sum DoF is

In this part we present an iterative algorithm referred to as *leakage minimization algorithm* and an extension for it called *Max-SINR* which both are proposed in [17]. Consider a three-user multiple-input multiple-output (MIMO) interference network where each terminal is equip‐ ped with M antennas. For presentation simplicity here we assume M to be even. The result when M is odd is provided in [4]. It has been shown in [4] that the achievable DoF of each source–destination pair is M / 2. To achieve this DoF, the transmitter-side beamformers and the receiver-side filters should be designed. In the following, we will briefly present the algorithm proposed in [17] to compute the beamformers and filters. We assume each terminal can acquire only local channel side information, i.e. knowledge about the channels which are directly connected to it throughout training of the channel. Destination D*k* can obtain **H***kl* and source S*k* can obtain **H***lk*, *l* ∈ {1, 2, 3}, where **H***lk* is the forward channel from S*k* to D*<sup>l</sup>* and **H***Rlk* is the reverse channel from *Dl* to S*k*. The iterative computation of the beamformers and the filters occur in the training phase. After the convergence of the computed filters and beamformers to the interference alignment solutions, the data transmission starts. Let **V***k* denote an M×M / 2 transmitter-side beamforming matrix where its columns are the orthogonal basis of the transmitted signal space of S*k*. The transmitted signal of S*k* is

$$\mathbf{x}\_k = \mathbf{V}\_k \mathbf{\bar{x}}\_{k\prime} \tag{8}$$

where each element of the M / 2×1 vector x¯ k represents an independently encoded Gaussian codeword with power 2Pk / M. Each codeword is beamformed with the corresponding column of **V***k*.

Let **U**<sup>k</sup> be an M×M / 2 receiver-side filtering matrix whose columns are the orthogonal basis of the desired signal subspace at D*k*. The filter output of D*k* is

$$\overline{\mathbf{y}}\_k = \mathbf{U}\_k^\* \mathbf{y}\_{k'} \tag{9}$$

where **y**k is the received vector. The transmitter-side beamforming matrices and the receiverside filtering matrices should satisfy the following interference alignment conditions:

$$\mathbf{U}\_{k}^{\prime}\mathbf{H}\_{kj}\mathbf{V}\_{j} = 0, \forall j \neq k:\ j, k \in \{1, 2, 3\},$$

$$\text{rank}\left(\mathbf{U}\_{k}^{\prime}\mathbf{H}\_{kk}\mathbf{V}\_{k}\right) = \frac{\mathbf{M}}{2}, \forall k:k \in \{1, 2, 3\}.\tag{10}$$

If global CSI is available, the beamforming and filtering matrices can be designed such that these conditions are satisfied. However, with the lack of global CSI if we choose the beam‐ formers and the filters randomly, with high probability only the second condition in (10) will be satisfied. Consequently, some interference remains at the destinations. The total power of interference at D*k* is

$$IF\_k = \operatorname{Tr} \Big[ \mathbf{U}\_k^\* \mathbf{Q}\_k \mathbf{U}\_k \Big]\_{\prime} \tag{11}$$

where

$$\mathbf{Q}\_k = \sum\_{j=1, j \neq k}^{K} \frac{2P\_j}{M} \mathbf{H}\_{kj} \mathbf{V}\_j \mathbf{V}\_j^\* \mathbf{H}\_{kj}^\* \tag{12}$$

is the covariance matrix of interference at D*k*. Clearly, *IFk*=0 only if the beamformers and the filters satisfy conditions in (10). However, D*k* can utilize the local channel side information to minimize this received interference power by optimizing its receiver-side filter. Therefore, assuming that the beamformers are fixed, the receiver-side filter **U**k is the solution of the following problem:

$$\min\_{\mathbf{U}\_k, \mathbf{U}\_k \mathbf{U}\_k^\* = I\_{\frac{M}{2}}} IF\_k. \tag{13}$$

The solution is given in [17]:

*<sup>k</sup>* , *k k* **x Vx** = (8)

\* , *<sup>k</sup> k k* **y Uy** <sup>=</sup> (9)

/ M. Each codeword is beamformed with the corresponding column

{ }

**UH V** (10)

**UQU** (11)

= ¹ *<sup>M</sup>* **<sup>Q</sup>** <sup>=</sup> <sup>å</sup> **H VV H** (12)

, min . *k kk M <sup>k</sup> IF* **U UU** <sup>=</sup>*<sup>I</sup>* (13)

0, : , 1,2,3 ,

*j k jk*

= "¹ Î

*k k*

If global CSI is available, the beamforming and filtering matrices can be designed such that these conditions are satisfied. However, with the lack of global CSI if we choose the beam‐ formers and the filters randomly, with high probability only the second condition in (10) will be satisfied. Consequently, some interference remains at the destinations. The total power of

represents an independently encoded Gaussian

where each element of the M / 2×1 vector

58 Contemporary Issues in Wireless Communications

Pk

the desired signal subspace at D*k*. The filter output of D*k* is

\*

*k kj j*

**UH V**

*k kk k*

\*

codeword with power 2

interference at D*k* is

following problem:

where

of **V***k*.

x¯ k

Let **U**<sup>k</sup> be an M×M / 2 receiver-side filtering matrix whose columns are the orthogonal basis of

where **y**k is the received vector. The transmitter-side beamforming matrices and the receiver-

side filtering matrices should satisfy the following interference alignment conditions:

( ) { }

\* Tr , *k kkk IF* <sup>=</sup> é ù ë û

2 *<sup>j</sup> k kj j j kj*

*P*

1,

*j jk*

<sup>K</sup> \* \*

is the covariance matrix of interference at D*k*. Clearly, *IFk*=0 only if the beamformers and the filters satisfy conditions in (10). However, D*k* can utilize the local channel side information to minimize this received interference power by optimizing its receiver-side filter. Therefore, assuming that the beamformers are fixed, the receiver-side filter **U**k is the solution of the

> \* 2

="Î

<sup>M</sup> rank , : 1,2,3 . <sup>2</sup>

$$\mathbf{U}\_k^d = \mathbf{v}^d \mathbf{[Q}\_k \mathbf{J}\_l \qquad d = 1, \dots, \frac{\mathbf{M}}{2},\tag{14}$$

where **U**k d denotes the *d*th column of **U**k and *ν<sup>d</sup>* [**A**] is the eigenvector corresponding to the *d*th lowest eigenvalue of **A**.

To optimize the transmitter-side beamformers the reciprocity of the channels can be exploited to obtain CSI at sources. For instance, destinations can transmit training sequences over the *reverse channels* (channels from destinations to sources) which are separated from the *forward channels* (channels from sources to destinations) in time via time-division duplex (TDD). The *reciprocity* of the forward and reverse channels is assumed, i.e. **H**kl r =**H**lk \* (∀l, k ∈{1, 2, ..., K}). For this purpose, the destination D*k* performs beamforming and S*<sup>k</sup>* perform filtering in the reverse direction with matrices **V**k d and **U**k, respectively. These matrices can be selected to perform interference alignment in the reverse direction. Since, the reciprocity holds, if we choose **U**k r =**V**k and **V**k r =**U**k, the solutions of the interference alignment problem in the reverse direction are equivalent to those in the forward direction. In a similar way as in the forward direction, the reverse direction filters are computed as,

$$\mathbf{v}\left(\mathbf{U}\_k^c\right)^d = \mathbf{v}^d \mathbf{[}\mathbf{Q}\_k^c\text{]} \quad d = 1, \ldots, \frac{\mathbb{1}}{2}.\tag{15}$$

where **Q**<sup>r</sup> <sup>k</sup> is the covariance matrix of interference at Sk and is computed in a similar way as in (12) with the reverse direction channels and beamformers. We can set the beamforming matrices in the forward direction as **V**k=**U**k r and repeat this optimization procedure until the beamforming matrices and the receiving filters converge. The convergence of this algorithm is shown in [17]. Next, the sources beamform their data using the computed beamforming matrices and the destinations decode their desired signals by applying associated filters. An extension of this iterative algorithm is proposed in [17] which instead of minimizing leakage at each destination tries to maximize signal-to-noise-plus-interference ratio (SINR) corre‐ sponding to each srteam. This algorithm is referred to as *Max-SINR algorithm* in the literature. According to *Max-SINR algorithm*, the receiver filtering vector **U**k d instead of the one in (14) is selected to be a MMSE filter as follows

$$\mathbf{U}\_k^d = \frac{\left(\mathbf{Q}\_k^d\right)^{-1} \mathbf{H}\_{kk} \mathbf{V}\_k^d}{\left\| \left(\mathbf{Q}\_k^d\right)^{-1} \mathbf{H}\_{kk} \mathbf{V}\_k^d \right\|\_2},\tag{16}$$

$$\mathbf{Q}\_{k}^{d} = \sum\_{j=1}^{K} \sum\_{l=1}^{M\_{l}^{d}} \frac{2P\_{j}}{M} \mathbf{H}\_{kj} \mathbf{V}\_{j}^{l} \left(\mathbf{V}\_{j}^{l}\right)^{\*} \mathbf{H}\_{kj}^{\*} - \frac{2P\_{j}}{M} \mathbf{H}\_{kk} \mathbf{V}\_{k}^{d} \left(\mathbf{V}\_{k}^{d}\right)^{\*} \mathbf{H}\_{kk}^{\*} + N\_{0} \mathbf{I}\_{M} \mathbf{I}\_{M}.\tag{17}$$

where **I**M/2 is M/2 × M/2 identity matrix and N0 is the noise power. Similarly, the transmitter beamforming vectors are updated in the reverse transmissions. In the following sections, we will present test-bed implementation of algorithms which use Max-SINR algorithm for computing the transmitter and receiver filters.
