**6. Test-bed implementation of the iterative transceiver filter design and power control**

We, in this section, first present an iterative algorithm for joint transceiver filter design and power control proposed in [27] and then explain how this algorithm is implemented on KTH four-multi test-bed and finally present measurement results.

#### **6.1. Iterative transceiver filter design and power control algorithm**

In an interference network, each user can affect the received SINR at its corresponding destination through the choice of beamforming and receiving filters as well as the transmitting power. Considering single stream transmissions, the received SINR at the *k*th destination is computed as

$$\text{SINR}\_k\{\mathbf{u}\_k, \mathbf{v}\_k, P\_k\} = \frac{P\_k \mathbf{u}\_k^\* \mathbf{H}\_{kk} \mathbf{v}\_k \mathbf{v}\_k^\* \mathbf{H}\_{kk}^\* \mathbf{u}\_k}{I F\_k + N\_0} \tag{18}$$

where *IFk* is given by (11). Given the beamforming and receiving filters, from Equation (18) the minimum transmitting power required to maintain a fixed rate *Rtar,k* (assuming that the interference can be regarded as Gaussian noise) at *k*th destination can be found as

$$P\_k = \beta\_k \{ \mathbf{u}\_{k'} \mathbf{v}\_{k'} \boldsymbol{\gamma}\_k \} := \frac{\boldsymbol{\gamma}\_k \left( \boldsymbol{I} \mathbf{F}\_k + \mathbf{N}\_0 \right)}{\mathbf{u}\_k^\* \mathbf{H}\_{kk} \mathbf{v}\_k \mathbf{v}\_k^\* \mathbf{H}\_{kk}^\* \mathbf{u}\_k} \, \tag{19}$$

where γk=2Rtar ,k −1 is the minimum required SNR in an AWGN channel. The total power of the interference, *IFk* which appears in the nominator is a function of transmitting powers at all the interfering transmitters. It means that increasing the power of one source causes higher level of interference at non-corresponding destinations and therefore the other sources need to transmit with higher power as well. In [28], it is shown that if the relation between the powers *Pk*, *k*=1,..., *K* satisfies a set of conditions then either there is an unique solution for the powers that can be found iteratively starting from any initial value or no solution exists. It is easy to prove that Equation (19) meets the conditions in [28] and therefore optimal transmitting powers can be found distributively after a sufficient number of iterations.

Inspired by Max-SINR algorithm and the aforementioned power control algorithm, an iterative transceiver design and power control algorithm was proposed in [27]. A brief version of the algorithm is presented on the next page for the sake of completeness. The algorithm is composed of three update phases in each iteration such that the receiving filters, transmission powers and beamforming filters are sequentially updated. The receiving and beamforming filters are optimized to deliver the maximum SINR at the destinations in the forward com‐ munication direction and at the sources in the reverse direction, respectively according to the concept of Max-SINR algorithm. On the other hand, in the power update phase, the powers are set to the minimum values needed for maintaining a fixed rate communication. The transmission power is upper bounded by *P*max. This algorithm assumes that accurate CSI is obtained at terminals. An extensions of the algorithm when terminals have access to only noisy CSI is proposed in [29].


**Initialize:v**k(0), Pk(0), k ∈{1, ..., K}, n=1.

**repeat**

**6. Test-bed implementation of the iterative transceiver filter design and**

We, in this section, first present an iterative algorithm for joint transceiver filter design and power control proposed in [27] and then explain how this algorithm is implemented on KTH

In an interference network, each user can affect the received SINR at its corresponding destination through the choice of beamforming and receiving filters as well as the transmitting power. Considering single stream transmissions, the received SINR at the *k*th destination is

SINR , , ( ) *k k kk k k kk k*

interference can be regarded as Gaussian noise) at *k*th destination can be found as

<sup>+</sup> = = **u v**

 g

powers can be found distributively after a sufficient number of iterations.

*P*

*P*

\* \*\*

**uH vvH u**

( ) <sup>0</sup>

*k kk k k kk k IF N*

−1 is the minimum required SNR in an AWGN channel. The total power of

*k*

*IF N* <sup>=</sup> <sup>+</sup>

where *IFk* is given by (11). Given the beamforming and receiving filters, from Equation (18) the minimum transmitting power required to maintain a fixed rate *Rtar,k* (assuming that the

\* \*\* ( ) ,, : , *k k*

the interference, *IFk* which appears in the nominator is a function of transmitting powers at all the interfering transmitters. It means that increasing the power of one source causes higher level of interference at non-corresponding destinations and therefore the other sources need to transmit with higher power as well. In [28], it is shown that if the relation between the powers *Pk*, *k*=1,..., *K* satisfies a set of conditions then either there is an unique solution for the powers that can be found iteratively starting from any initial value or no solution exists. It is easy to prove that Equation (19) meets the conditions in [28] and therefore optimal transmitting

Inspired by Max-SINR algorithm and the aforementioned power control algorithm, an iterative transceiver design and power control algorithm was proposed in [27]. A brief version of the algorithm is presented on the next page for the sake of completeness. The algorithm is composed of three update phases in each iteration such that the receiving filters, transmission powers and beamforming filters are sequentially updated. The receiving and beamforming filters are optimized to deliver the maximum SINR at the destinations in the forward com‐ munication direction and at the sources in the reverse direction, respectively according to the concept of Max-SINR algorithm. On the other hand, in the power update phase, the powers

g 0

**u v** (18)

**uH vvH u** (19)

four-multi test-bed and finally present measurement results.

**6.1. Iterative transceiver filter design and power control algorithm**

*k k kk*

*k k k kk*

b

*P*

**power control**

64 Contemporary Issues in Wireless Communications

computed as

where γk=2Rtar ,k **Update receiver filtering vector:**

$$\mathbf{u}\_{k}(n) = \arg\max\_{\mathbf{u}\_{k}} \left\{ \text{SINR}\_{k} \left( \mathbf{u}\_{k'}, \mathbf{v}\_{k}(n-1), P\_{k}(n-1) \right) \right\} \tag{20}$$

**Update transmission power:**

$$P\_k(n) = \min \left\{ \beta\_k \left( \mathbf{u}\_k(n), \mathbf{v}\_k(n-1), \boldsymbol{\gamma}\_k \right), P\_{\text{max}} \right\} \tag{21}$$

$$\text{Reverse the communication direction:} \begin{cases} \mathbf{u}\_k^r (n-1) = \mathbf{v}\_k (n) \\ \mathbf{H}\_{ij}^r = \mathbf{H}\_{ji}^\*, \; j \in \{1, \dots, K\} \\ p\_k^r (n-1) = P\_F \end{cases}$$

**Update transmitter beamforming vector:**

$$\mathbf{u}\_k^r(n) = \arg\max\_{\mathbf{u}\_k^r} \left\{ \text{SINR}\_k^r \left( \mathbf{u}\_{k'}^r, \mathbf{v}\_k^r(n-1), P\_k^r(n-1) \right) \right\} \tag{22}$$

Set beamforming vector **v**k(n)=**u**kr (n) and n =n+ 1

**until**n = N

#### **6.2. Transmitted frame structure**

The air interface of the network is designed based on OFDM modulation using KTH fourmulti's modulation and coding toolbox. Coding rate of 1/2 and 16QAM modulation was chosen for transmitting fixed-rate data streams though the air interface. The transmitted frame structure is depicted in Fig. 4. In our experiment, each frame consists of 20 payload symbols and either two or three reference signals (RS) (i.e. pilot symbols). The payload symbols are *concurrently* transmitted from all the sources while each stream is beamformed with its corresponding beamforming filter instructed by Algorithm 1. Although the overhead of pilot symbols is significant in this case, we note that the number of payload symbols could be larger

depending on the coherence time of the indoor channel. In the implementation in Section 7 there are in fact thousands of payload symbols with no additional pilots. Thus the overhead could be reduced to one percent for the environment in Fig. 5. "Book˙Chapter˙MFZNS" — 2014/10/14 — 15:09 — page 17 — #17

Three types of RS are employed in the network, which are referred to as *channel state information RS (CSI-RS)*, *demodulation RS (DM-RS)* and *power RS (P-RS)*. During the pilot transmission all the sub-carriers of the OFDM symbol is filled with known QAM symbols. We next explain each type of these reference signals:

17

Fig. 4: Transmitted frame structure in IA-PC scheme. **Figure 4.** Transmitted frame structure in IA-PC scheme.

power is not fixed and is set by the power control algorithm.

scaling factor α. In each frame, the scaling factor is computed as

α =

these reference signals: • CSI-RS: The received noisy CSI-RS at the destinations are exploited to estimate the corre-CSI-RS: The received noisy CSI-RS at the destinations are exploited to estimate the corre‐ sponding channel matrices to enable execution of Algorithm 1. The CSI-RS are transmitted *orthogonally*; i.e., one CSI-RS is transmitted from each transmit antenna in the network while the other antennas are silent. To enhance the quality of the channel estimations the CSI-RS are scaled such that the associated QAM symbol has the maximum transmit power *P*max.

sponding channel matrices to enable execution of Algorithm 1. The CSI-RS are transmitted *orthogonally*; i.e., one CSI-RS is transmitted from each transmit antenna in the network while DM-RS: The DM-RS are used to compute the effective channel by taking into account the transmit and receive filters. Therefore they need to be stream-dedicated and be processed by the same pre-coder as the payload symbols of the corresponding stream. In this way, their power is not fixed and is set by the power control algorithm.

the other antennas are silent. To enhance the quality of the channel estimations the CSI-RS are scaled such that the associated QAM symbol has the maximum transmit power Pmax. • DM-RS: The DM-RS are used to compute the effective channel by taking into account the P-RS: Algorithm 1 is constructed to select the minimum possible transmission power to minimize the interference at the destinations. This hence reduces the power of DM-RS and may lead to a poor estimation of cross-channels, which is not favorable. To tackle this problem, P-RS is introduced where the amplitudes of the CSI-RS are scaled after the pre-coder by a scaling factor α. In each frame, the scaling factor is computed as

transmit and receive filters. Therefore they need to be stream-dedicated and be processed by

the same pre-coder as the payload symbols of the corresponding stream. In this way, their

• P-RS: Algorithm 1 is constructed to select the minimum possible transmission power to min-

imize the interference at the destinations. This hence reduces the power of DM-RS and may

lead to a poor estimation of cross-channels, which is not favorable. To tackle this problem,

PR-S is introduced where the amplitudes of the DM-RS are scaled after the pre-coder by a

where Pk,j is the transmit power of the jth sub-carrier of kth source. The destinations therefore

need to get informed about the scaling factor. This is achieved by having node 1 (the master

node) repeat its first DM-RS but now scaled with α. This enables all destinations to make

robust estimate of α (it is assumed that all destinations can hear node 1). The factor α is also

quantized into a discrete set of values to avoid that α introduces estimation errors.

October 14, 2014 DRAFT

Pmax

, (23)

maxk,j Pk,j

Interference Alignment — Practical Challenges and Test-bed Implementation http://dx.doi.org/10.5772/59200 67

$$\alpha = \sqrt{\frac{P\_{\text{max}}}{\max\_{k,j} P\_{k,j}}},\tag{23}$$

18

where *Pk,j* is the transmit power of the *j*th sub-carrier of *k*th source. The destinations therefore need to get informed about the scaling factor. This is achieved by having node 1 (the master node) repeat its first DM-RS but now scaled with α. This enables all destinations to make robust estimate of α (it is assumed that all destinations can hear node 1). The factor α is also quantized into a discrete set of values to avoid that α introduces estimation errors.

Fig. 5: Measurement environment map. **Figure 5.** Measurement environment map.

#### *C. Measurement results* **6.3. Measurement results**

17

depending on the coherence time of the indoor channel. In the implementation in Section 7 there are in fact thousands of payload symbols with no additional pilots. Thus the overhead

"Book˙Chapter˙MFZNS" — 2014/10/14 — 15:09 — page 17 — #17

Three types of RS are employed in the network, which are referred to as *channel state information RS (CSI-RS)*, *demodulation RS (DM-RS)* and *power RS (P-RS)*. During the pilot transmission all the sub-carriers of the OFDM symbol is filled with known QAM symbols. We next explain

10 OFDM

symbols

could be reduced to one percent for the environment in Fig. 5.

10 OFDM

symbols

payload

power is not fixed and is set by the power control algorithm.

scaling factor α. In each frame, the scaling factor is computed as

**Figure 4.** Transmitted frame structure in IA-PC scheme.

power is not fixed and is set by the power control algorithm.

scaling factor α. In each frame, the scaling factor is computed as

α =

CSI-RS DM-RS

no transmission

Fig. 4: Transmitted frame structure in IA-PC scheme.

• CSI-RS: The received noisy CSI-RS at the destinations are exploited to estimate the corre-

DM-RS: The DM-RS are used to compute the effective channel by taking into account the transmit and receive filters. Therefore they need to be stream-dedicated and be processed by the same pre-coder as the payload symbols of the corresponding stream. In this way, their

P-RS: Algorithm 1 is constructed to select the minimum possible transmission power to minimize the interference at the destinations. This hence reduces the power of DM-RS and may lead to a poor estimation of cross-channels, which is not favorable. To tackle this problem, P-RS is introduced where the amplitudes of the CSI-RS are scaled after the pre-coder by a

scaled such that the associated QAM symbol has the maximum transmit power *P*max.

CSI-RS: The received noisy CSI-RS at the destinations are exploited to estimate the corre‐ sponding channel matrices to enable execution of Algorithm 1. The CSI-RS are transmitted *orthogonally*; i.e., one CSI-RS is transmitted from each transmit antenna in the network while the other antennas are silent. To enhance the quality of the channel estimations the CSI-RS are

sponding channel matrices to enable execution of Algorithm 1. The CSI-RS are transmitted

*orthogonally*; i.e., one CSI-RS is transmitted from each transmit antenna in the network while

the other antennas are silent. To enhance the quality of the channel estimations the CSI-RS are

transmit and receive filters. Therefore they need to be stream-dedicated and be processed by

the same pre-coder as the payload symbols of the corresponding stream. In this way, their

• P-RS: Algorithm 1 is constructed to select the minimum possible transmission power to min-

imize the interference at the destinations. This hence reduces the power of DM-RS and may

lead to a poor estimation of cross-channels, which is not favorable. To tackle this problem,

PR-S is introduced where the amplitudes of the DM-RS are scaled after the pre-coder by a

where Pk,j is the transmit power of the jth sub-carrier of kth source. The destinations therefore

need to get informed about the scaling factor. This is achieved by having node 1 (the master

node) repeat its first DM-RS but now scaled with α. This enables all destinations to make

robust estimate of α (it is assumed that all destinations can hear node 1). The factor α is also

quantized into a discrete set of values to avoid that α introduces estimation errors.

October 14, 2014 DRAFT

Pmax

, (23)

these frames.

maxk,j Pk,j

scaled such that the associated QAM symbol has the maximum transmit power Pmax.

• DM-RS: The DM-RS are used to compute the effective channel by taking into account the

P-RS

each type of these reference signals:

66 Contemporary Issues in Wireless Communications

S3,Ant2 S3,Ant1 S2,Ant2 S2,Ant1 S1,Ant2 S1,Ant1

these reference signals:

The test-bed measurement was performed in KTH signal processing department which floor map is illustrated in Fig. 5. The measurement environment is categorized as an indoor office. In this experiment only non-line-of-sight (non-LOS) scenarios were investigated by placing source and destination nodes in the corridor and inside the nearby offices, receptively. The receive antenna gains also decreased by connecting 10 dB attenuators to them in order to avoid saturation of receive PAs. The measurement was done in 100 batches. In each batch a random placement of the destination nodes in the area marked by colored circles in the figure were measured. The signals transmitted The test-bed measurement was performed in KTH signal processing department which floor map is illustrated in Fig. 5. The measurement environment is categorized as an indoor office. In this experiment only non-line-of-sight (non-LOS) scenarios were investigated by placing source and destination nodes in the corridor and inside the nearby offices, receptively. The receive antenna gains also decreased by connecting 10 dB attenuators to them in order to avoid saturation of receive power amplifiers. The measurement was done in 100 batches. In each batch a random placement of the destination nodes in the area marked by colored circles in the figure were measured. The signals transmitted according to two different schemes were measured sequentially in each batch. In the first scheme, referred to as *noPC*, the iterative interference alignment was implemented according to [17] for benchmarking. In the second scheme, referred to as *PC*, transmission powers and beamforming filters were computed according to Algorithm 1 and MMSE receiving filters were applied at the destinations. Each scheme was run with 28 frames inter-spaced 0.15 seconds. The statistics of the first frames of both schemes were not taken into account since there is no feedback information at these frames.

according to two different schemes were measured sequentially in each batch. In the first scheme,

referred to as *noPC*, the iterative interference alignment was implemented according to [17] for

benchmarking. In the second scheme, referred to as *PC*, transmission powers and beamforming

filters were computed according to Algorithm 1 and MMSE receiving filters were applied at the

destinations. Each scheme was run with 28 frames inter-spaced 0.15 seconds. The statistics of the

first frames of both schemes were not taken into account since there is no feedback information at

High power may push the terminals' PA to work in their non-linear region. Non-linearities in the

transmit-receive chain degrades the performance of the system by introducing *distortion noise* into the

system. Distortion noise is usually modeled as a Gaussian noise whose power increases by increasing

the transmission power of the source nodes [30]. In order to make sure that the reduction of transmit

power is not the only cause for the performance improvement, four different levels of transmission

Table II shows the average performance of the two schemes for the 100 measurement batches. In

this measurement the PC scheme's target SINR γ<sup>k</sup> was set to 18 dB for all the users. The performances

were compared in the sense of bit-error rate (BER) and transmitted power. The table shows that the

PC scheme has the lowest BER, although its average transmit power is much lower than the noPC

October 14, 2014 DRAFT

power were tested in noPC scheme that is each 7 frames were transmitted with a different power.

High power may push the terminals' power amplifier to work in their non-linear region. Nonlinearities in the transmit-receive chain degrades the performance of the system by introducing *distortion noise*into the system. Distortion noise is usually modeled as a Gaussian noise whose power increases by increasing the transmission power of the source nodes [30]. In order to make sure that the reduction of transmit power is not the only cause for the performance improvement, four different levels of transmission power were tested in noPC scheme that is each 7 frames were transmitted with a different power.

Table 2 shows the average performance of the two schemes for the 100 measurement batches. In this measurement the PC scheme's target SINR *γk* was set to 18 dB for all the users. The performances were compared in the sense of bit-error rate (BER) and transmitted power. The table shows that the PC scheme has the lowest BER, although its average transmit power is much lower than the noPC scheme with the best BER performance.


**Table 3.** Measurement results.

**Figure 6.** Empirical CDF of SINR. The solid line represents the PC scheme and the dashed lines denote the noPC scheme with different transmit powers (Ptr).

Empirical cumulative distribution function (CDF) of the received SINR for two schemes is plotted in Fig 6. This plot reveals the reason for the low BER of PC scheme, despite its low transmit power. The received SINRs in this scheme are concentrated around the target value while in the noPC scheme they are distributed over a wider range. Having SINR higher than the target value while the transmit rate is fixed leads to the waste of energy and on the other hand SINRs lower than the target increase the probability of error and therefore the BER.

20

*Power saving gain* at each frame is computed as the ratio of the transmit power in noPC scheme with 7.1 dBm average power and the total power transmitted in the PC scheme with 18 dB target SINR. Fig. 7 shows the empirical CDF of the power saving gain. As the empirical CDF implies, implementation of Algorithm 1 in PC scheme leads to at least 4dB gain in 90% of the measurements. Fig. 7 also shows that in 10% of the measurements gains higher than 13 dB was observed. "Book˙Chapter˙MFZNS" — 2014/10/14 — 15:09 — page 20 — #20

The benefit of power control in the PC scheme is in fact two-fold. By decreasing the transmit power, while retaining the target SINR, not only less interference is received at the destinations but also the distortion noise due to transceiver impairments decreases.

Fig. 7: Emperical CDF of transmit power saving gain. **Figure 7.** Emperical CDF of transmit power saving gain.

High power may push the terminals' power amplifier to work in their non-linear region. Nonlinearities in the transmit-receive chain degrades the performance of the system by introducing *distortion noise*into the system. Distortion noise is usually modeled as a Gaussian noise whose power increases by increasing the transmission power of the source nodes [30]. In order to make sure that the reduction of transmit power is not the only cause for the performance improvement, four different levels of transmission power were tested in noPC scheme that is

Table 2 shows the average performance of the two schemes for the 100 measurement batches. In this measurement the PC scheme's target SINR *γk* was set to 18 dB for all the users. The performances were compared in the sense of bit-error rate (BER) and transmitted power. The table shows that the PC scheme has the lowest BER, although its average transmit power is

**scheme noPC PC**

FER 0.6856 0.1700 0.0528 0.0561 0.0071 BER 0.0815 0.0124 0.0020 0.0030 2.2 × 10−4

0 6 12 18 24 30 36

SINR (dB)

**Figure 6.** Empirical CDF of SINR. The solid line represents the PC scheme and the dashed lines denote the noPC

Empirical cumulative distribution function (CDF) of the received SINR for two schemes is plotted in Fig 6. This plot reveals the reason for the low BER of PC scheme, despite its low transmit power. The received SINRs in this scheme are concentrated around the target value while in the noPC scheme they are distributed over a wider range. Having SINR higher than the target value while the transmit rate is fixed leads to the waste of energy and on the other hand SINRs lower than the target increase the probability of error and therefore the BER.

Average power (dBm) -12.9 -3.4 1.1 7.1 -1

Average SINR (dB) 10.9 20 24.3 26.7 18.5

each 7 frames were transmitted with a different power.

68 Contemporary Issues in Wireless Communications

**Table 3.** Measurement results.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

scheme with different transmit powers (Ptr).

much lower than the noPC scheme with the best BER performance.

Ptr = −12.9 dBm Ptr = −3.4 dBm Ptr = 1.1 dBm Ptr = 7.1 dBm SINRtar = 18 dB

#### FEEDBACK In this section we describe the implementation of interference using limited digital feedback (see **7. Test-bed implementation of the interference alignment with compressed feedback**

VII. TEST-BED IMPLEMENTATION OF THE INTERFERENCE ALIGNMENT WITH COMPRESSED

Section III-B2). Rather than using a uniform quantizer, we use a modified form of the MIMO matrix compression of the IEEE802.11ac standard. The propagation scenario and testbed is the same as used in Section VI i.e. a system with three (K = 3), 2×2 MIMO links. However, the transmission power is higher than in previous sections and the system is therefore limited by interference and hardware impairments rather than thermal noise [25], [26]. A second difference from the previously presented results in Section VI, is the performance criterion. Rather than using bit-error rate (BER) we here use transmission rate as the criterion. The transmission rate is obtained by probing each link with ten different coding and modulation schemes (MCS). The rate is determined by finding the MCS with In this section we describe the implementation of interference using limited digital feedback (see Section 3.2). Rather than using a uniform quantizer, we use a modified form of the MIMO matrix compression of the IEEE802.11ac standard. The propagation scenario and test-bed is the same as the one used in Section 6 i.e. a system with three (*K*=3), 2 × 2 MIMO links. However, the transmission power is higher than in previous sections and the system is therefore limited by interference and hardware impairments rather than thermal noise [25], [26]. A second difference from the previously presented results in Section 6, is the performance criterion. Rather than using BER we here use transmission rate as the criterion. The transmission rate is obtained by probing each link with ten different coding and modulation schemes (MCS). The rate is determined by finding the MCS with the highest rate which does not incur any frame errors.

the highest rate which does not incur any frame errors. The section is organised as follows. Section VII-A describes the feedback compression of the channel state information assuming a system with K links and M antennas in each source and The section is organised as follows. Section 7.1 describes the feedback compression of the CSI assuming a system with *K* links and M antennas in each source and destination node, while the measurement results are presented in Section 7.2.

The feedback scheme described in the standard IEEE802.11ac resembles the feedback method for slowly time-varying single-user MIMO channels presented in [31]. In this scheme a singular value

October 14, 2014 DRAFT

H = WΛF<sup>∗</sup>. (24)

destination node, while the measurement results are presented in Section VII-B.

decomposition of the MIMO channel H (for a certain sub-carrier) is first performed as

*A. Compression of IEEE802.11ac and adaptation to interference alignment*

#### **7.1. Compression of IEEE802.11ac and adaptation to interference alignment**

The feedback scheme described in the standard IEEE802.11ac resembles the feedback method for slowly time-varying single-user MIMO channels presented in [31]. In this scheme a singular value decomposition (SVD) of the MIMO channel **H** (for a certain sub-carrier) is first performed as

$$\mathbf{H} = \mathbf{W} \mathbf{A} \mathbf{F}^\* \,. \tag{24}$$

The destination then feeds back, in compressed form, the complex unitary matrix **F** and the real diagonal matrix **Λ**, while the complex unitary matrix **W** does not need to be fed back. To realize this we may imagine that each destination pre-multiplies its received signal with **W**<sup>∗</sup>. The effective channel then becomes H ͜ =ΛF\* which is a channel which can be reconstructed by the sources. In practice, the destination will not pre-multiply its received signal with **W**<sup>∗</sup>, since almost every receive technique is invariant to such a linear unitary transform.

In the case of centralised interference alignment, knowledge of the channels between all sources and destinations is required to obtain all the beamformers **V***k*. We assume here that sources are connected to a common back-bone for exchange of CSI and synchronization. If we directly apply the compression scheme presented above for the single-user case so that each destination Dk, compresses the matrices **H***k*,*<sup>l</sup>* for *l* ∈ {1,…., *K*} it is clear that there is no single unitary matrix to be applied at the destination to transform all involved K channels into H ͜ k ,l =Λk ,l Fk ,l \* such that all channels can be obtained from the compressed feedback from user *k*.

To overcome this problem, in the system implementation we have based the feedback from destination *k* on the big matrix, **H**[*k*] obtained by concatenating the channel matrices **H***k*,*<sup>l</sup>* for *l*=1,…, *K* column-wise. Thus in this case **H**[*k*] , is computed as,

$$\mathbf{H}^{[k]} = \left[\mathbf{H}\_{k,1'}, \dots, \mathbf{H}\_{k,K}\right]. \tag{25}$$

Thus destination Dk now feedback a compressed version of the right-hand side eigenvectors of **H**[k] i.e. the real diagonal matrix **Λ**[k] and the complex unitary matrix **F**[*k*] of the SVD

$$\mathbf{H}^{\{k\}} = \mathbf{W}^{\{k\}} \mathbf{L}^{\{k\}} \left(\mathbf{F}^{\{k\}}\right)^{\*}.\tag{26}$$

The size of matrix **Λ**[k] is *M × M* while **F**[*k*] is *KM × M*. Just as in the single-link case we can now imagine that the destination pre-multiplies its received signal with (**W**[*k*] ) <sup>∗</sup>. The transmitter side is then able to obtain knowledge of the effective channel **H** ͜ k =Λ k (F k )\* . From this combined channel the constituent sub-channels can be pulled out from the corresponding columns e.g. the second channel H ͜ k ,2 corresponds to columns *M+*1,..., 2*M* of H ͜ k. The reconstructed channels can then be used in place of the actual channels in any transmit beamformer and receive filter algorithm.

**7.1. Compression of IEEE802.11ac and adaptation to interference alignment**

͜ =ΛF\*

[ ]

of **H**[k] i.e. the real diagonal matrix **Λ**[k] and the complex unitary matrix **F**[*k*]

imagine that the destination pre-multiplies its received signal with (**W**[*k*]

is then able to obtain knowledge of the effective channel **H**

almost every receive technique is invariant to such a linear unitary transform.

as

H ͜ k ,l =Λk ,l

Fk ,l

The effective channel then becomes H

70 Contemporary Issues in Wireless Communications

destination Dk, compresses the matrices **H***k*,*<sup>l</sup>*

*l*=1,…, *K* column-wise. Thus in this case **H**[*k*]

The size of matrix **Λ**[k] is *M × M* while **F**[*k*]

͜ k

the second channel H

destination *k* on the big matrix, **H**[*k*]

The feedback scheme described in the standard IEEE802.11ac resembles the feedback method for slowly time-varying single-user MIMO channels presented in [31]. In this scheme a singular value decomposition (SVD) of the MIMO channel **H** (for a certain sub-carrier) is first performed

The destination then feeds back, in compressed form, the complex unitary matrix **F** and the real diagonal matrix **Λ**, while the complex unitary matrix **W** does not need to be fed back. To realize this we may imagine that each destination pre-multiplies its received signal with **W**<sup>∗</sup>.

the sources. In practice, the destination will not pre-multiply its received signal with **W**<sup>∗</sup>, since

In the case of centralised interference alignment, knowledge of the channels between all sources and destinations is required to obtain all the beamformers **V***k*. We assume here that sources are connected to a common back-bone for exchange of CSI and synchronization. If we directly apply the compression scheme presented above for the single-user case so that each

unitary matrix to be applied at the destination to transform all involved K channels into

To overcome this problem, in the system implementation we have based the feedback from

,1 , ,, . *<sup>k</sup> k kK* = ¼ é ù

Thus destination Dk now feedback a compressed version of the right-hand side eigenvectors

channel the constituent sub-channels can be pulled out from the corresponding columns e.g.

,2 corresponds to columns *M+*1,..., 2*M* of H

\* such that all channels can be obtained from the compressed feedback from user *k*.

, is computed as,

**H WF** = L \* . (24)

which is a channel which can be reconstructed by

for *l* ∈ {1,…., *K*} it is clear that there is no single

of the SVD

<sup>∗</sup>. The transmitter side

. From this combined

. The reconstructed channels

obtained by concatenating the channel matrices **H***k*,*<sup>l</sup>* for

ë û **HH H** (25)

( ) [] [] \* [] [] . *k kk k* **H WL F** <sup>=</sup> (26)

͜ k

͜ k =Λ k (F k)\*

is *KM × M*. Just as in the single-link case we can now

)

The IEEE 802.11ac feedback compression scheme starts by rotating the phase of the columns of **F**[*k*] in order the last row of this matrix to have real and positive elements. These phase rotations do not need to be sent to the sources, since these rotations only amount to a phase rotation of the signals which can be undone at the destination. In the next step, **F**[*k*] , is multiplied by a diagonal matrix to have the first column become real and positive as

$$\mathbf{F}^{[k]} \leftarrow \mathbf{F}^{[k]} \; \mathbf{diag}(\exp(j\phi\_{1,1}), \dots, \exp(j\phi\_{\text{KM}-1,1}), 1). \tag{27}$$

These angles *ϕ*1,1,…, *ϕKM −*1,1 are then quantised uniformly, each with b*<sup>ϕ</sup>* bits. Then, real-valued Givens rotations are utilised to successively zero out the second to the *KM-*th element of the first column of **F**[*k*] . For the latter rotations the angles ψ2,1,..., ψ*KM*,1 are used. The angles lie between 0 and π/2 and are quantised uniformly with *b*<sup>ψ</sup> bits. This procedure is repeated in a similar fashion for the remaining columns of **F**[*k*] . More details are presented in [31] and [32] as well as in the Matlab/Octave functions available at http://people.kth.se/∼perz/packV/.

Since we use OFDM there is one **H**[*k*] matrix for each user and sub-carrier. In the IEEE 802.11ac standard, the parameter Ng is defined. This parameter determines the frequency domain granularity of the feedback. If Ng=1, the feedback of **F**[*k*] is done on every sub-carrier. If Ng=2, the feedback is only done on every other sub-carrier. In the standard, the values 1, 2 and 4 have been defined for Ng. In our measurements the values 8, 16 and 38 have also been considered, since this would significantly reduce the number of feedback bits. The angle resolution parameters *bϕ* and *b*<sup>ψ</sup> are defined in the IEEE 802.11ac standard as either *bϕ*=5 and *b*ψ=7 or *bϕ*=7 and *b*ψ=9. In the presented results only the latter value pair is used. The total number of bits required to feed back the **F**[*k*] matrix is given by

$$m\_{\mathbf{b}} = \left( \left( 2KM - 1 \right) M - M^2 \right) \left( b\_{\phi} + b\_{\varphi} \right) / 2 \tag{28}$$

The number of bits can be reduced by a further (*K* − 1)*bϕ* bits. We can see this by dividing the **F**[*k*] into sub-matrices of size *M × M* as

$$\mathbb{E}\left(\mathbf{F}^{\{k\}}\right)^{\*} = \left[ \left(\mathbf{F}\_{1}^{\{k\}}\right)^{\*}, \dots, \left(\mathbf{F}\_{K}^{\{k\}}\right)^{\*} \right]. \tag{29}$$

Since the signals fromsource,*l*,onlypropagates throughsub-matrix**F**[*k*] *l* ,wemayfreelymultiply sub-matrices 1,..., *K* − 1 by one phasor each. By doing so, we may set *ϕ*1,1,..., *ϕ1+(K−2)M,1* to zero, and thus we do not need to transmit them over the feedback channels. The last sub-matrix can not be rotated since then the last row of **F**[*k*] would no longer be real and positive.

The elements of **Λ**[k] are divided by the noise standard deviation. The so obtained value can be regarded as the SNR of a corresponding stream. The reporting of these SNRs is done separately per stream, and is done in two steps. In the first step, the average SNR in dB over the whole frequency band, i.e. for all singular values of the narrowband channels for which SNR is reported, is fed back to S*k*. This value is uniformly quantised with 8 bits in the range from 10 dB to 53.75 dB. In the second step, the difference in dB between the SNR of the reported sub-carrier and the average SNR is computed and is uniformly quantised with 4 bits in the range from 8 dB to 7 dB. In S*k*, the SNRs of all the reported sub-carriers are first reconstructed. Then, linear interpolation (in dB) is deployed to obtain the SNR for all the sub-carriers for which **F**[*k*] is fed back. With these two entities at hand, the channel matrices **H**[k] are recon‐ structed and used to compute the desired pre-coders. For the sub-carriers where there is no feedback available, the pre-coding of the nearest reported sub-carrier is utilized, i.e. no matrix interpolation method is used.

A practical problem which occurred during the early experimentation was that the SNR sometimes exceeded 53.75 dB. This happened due to the high transmission power and short range. First this was handled by reporting the maximum value 53.75 dB whenever this happened. However, this resulted in making the reconstructed channel at the transmitter highrank although the estimated channel at the receiver were in fact low-rank-which was very detrimental to the performance. To circumvent this problem, an offset is subtracted from SNR values when this condition happens.
