# Effective Channel Hardening in an Indoor Multiband Scenario

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## Abstract

We evaluate channel hardening for a large scale antenna system by means of indoor channel measurements over four frequency bands, \(1.472 \,\hbox {GHz}\), \(2.6 \,\hbox {GHz}\), \(3.82 \,\hbox {GHz}\) and \(4.16 \,\hbox {GHz}\). NTNU’s Reconfigurable Radio Network Platform has been used to record the channel estimates for 40 radio links to a 64 element array with wideband antennas in a rich scattering environment. We examine metrics for channel hardening, namely, the coherence bandwidth, the rms delay spread and the normalized effective subcarrier power, for the effective channel perceived by a single user after precoding and superposition in the downlink. We describe these metrics analytically and demonstrate them with measured data in order to characterize the rate of hardening of the effective channel as the number of antenna elements at the base station increases. The metrics allow for direct insight into the benefits of channel hardening with respect to radio system requirements.

## Keywords

Channel hardening Massive MIMO Delay spread## 1 Introduction

Massive multiple input multiple output (MIMO) systems are envisioned as a key feature of the next generation of communication systems which provide large sum capacity as well as spectral and energy efficiency, while simultaneously serving multiple users. Some of the theoretical properties [2, 12, 18] have been empirically shown through recent measurement campaigns [9, 10, 14]. In these analyses, keeping the number of users 10 folds less than number of base station (BS) antennas is considered good practice.

The large scale antenna systems are being investigated as contenders for wireless sensor networks (WSNs) to offer mass connectivity with high reliability in 5G paradigm. In these systems, a BS equipped with a large number of antennas serves a very large number of sensor nodes such as ships, automobiles, trains, engines and robots, which are categorically referred to as user equipments (UEs). As many of these applications are safety critical, the robustness of the wireless communication links is vital. Channel hardening could be exploited, in order to establish reliable links between BS and UEs. By definition, the channel hardens when by increasing the number of BS antennas, the deviation of the gain of the perceived channel at each UE decreases and the channel gain value becomes deterministic [11].

From a radio system perspective, the desirability of channel hardening at sensor nodes is twofold: The signals traveling from the BS to each single UE are precoded at the antenna elements and filtered by the physical channel and then will superpose and form what is referred to as effective channel. When the effective channel hardens, it is as if the propagation happened through a quasi-deterministic flat-fading equivalent channel by averaging out the small fading effects, as a result forming a reliable link from the BS to the UE. Considering channel hardening in the delay domain, complex equalization and estimation at the UE side can be avoided because the effective channel collapses to a single tap due to the delay dispersion being smaller then the delay tap resolution.

In [8, 15], the authors have looked at channel hardening for the subcarriers using maximum ratio combining (MRC) in the uplink (UL). The standard deviation is used to examine the dispersion of the channel. Alternatively, the work in [3] formulates the concept of an effective (equivalent) received channel at a single UE by using time-reversal precoding. Here, the measured channels between the UE and the BS are used to examine temporal focusing by using the delay spread and the strongest tap power distribution. In [16] the measured channels between a 128 antenna BS and 36 UEs have been used to evaluate the root mean square (rms) delay spread of the effective combined channel for three common linear precoding schemes.

In this paper, we refer to the perceived channel at a single user in the downlink (DL) as an *effective channel*. This is the channel formed by precoding, propagation and superposition of signals from each BS antenna element. A calibration of the transmitter and receiver chains at the BS is necessary, such that the reciprocity assumption holds [7].

Channel hardening is considered from two points of view: firstly, as a property that causes the effective channel transfer function (CTF) between the UE and the BS to become more deterministic, secondly as a property to focus the received signal in the delay domain as the number of BS antennas increases. We illustrate these properties in the effective channel in order to determine how many antennas are sufficient to achieve a certain level of channel hardening. This will allow for the remaining BS antennas to be considered as contributors to achieve a multi-user system by using the remaining degrees of freedom to orthogonalize the effective channels.

We base our analysis on a channel dependent precoding which weights the signal at the antenna elements and relies on exploiting channel reciprocity in the DL to form a matched filter combination when observed by the UE. The aim of the precoding is to guarantee the best average signal to noise ratio (SNR) at the UE under the assumption that the channel state information (CSI) of the channel is perfectly known and no co-channel interference exists.

We formulate the normalized effective subcarrier power in order to examine the flatness of the effective channel at the UE. Additionally, the coherence bandwidth as well as the effective delay spread are analyzed. All three metrics have been characterized with measurement data at 4 frequency bands: \(1.472 \,\hbox {GHz}\), \(2.6 \,\hbox {GHz}\), \(3.82 \,\hbox {GHz}\) and \(4.16 \,\hbox {GHz}\). These constitute the commonly considered frequency range for 5G and WSN systems and highlight potential frequency dependencies.

This manuscript is organized as follows: in Sect. 2 the measurement campaign which was carried out in an indoor area with industrial profile forming a quasi-static scenario is reviewed. The acquired data corresponds to 40 spatial sample points characterizing UL channels to a 64 element array in the above-mentioned frequency bands. The details of this campaign are reported in [6]. In Sect. 3, the concept of the effective channel is introduced and the measured channel data is used to analytically form the effective channels. The metrics of hardening are described and evaluated in Sect. 4. We show that coherence bandwidth ceases to be a practical measure for effective channel evaluation. Lastly, the conclusions are presented in Sect. 5.

## 2 Measurement Description

The investigation reported in this paper is based on the measured data acquired during a campaign carried out at the Norwegian University of Technology and Science (NTNU) in December 2017 using the Reconfigurable Radio Network Platform (ReRaNP). The details of the UEs and the BS including channel estimate acquisition are described in [6].

### 2.1 Measurement Setup

For our BS, we used two modules of the NTNU massive MIMO testbed with 32 National Instruments (NI) USRP-2943R devices. Two RF chains exist in each Universal Software Radio Peripheral (USRP) unit, as a result we have 64 radio chains at the BS. These units are controlled by one CPU which is running the NI LabVIEW Communications MIMO Application Framework. They form a time division duplex (TDD) system with a Long Term Evolution (LTE)-like physical layer with \(20 \,\hbox {MHz}\) operational bandwidth [1].

Massive MIMO testbed parameters

Parameter | Value |
---|---|

# of BS antennas | 64 used |

# of UEs | 8 |

Center frequency | 1.2–6 GHz |

Bandwidth of operation | \(20 \,\hbox {MHz}\) |

Baseband sampling rate | \(30.72 \, \hbox {MS/s}\) |

Subcarrier spacing | \(15 \,\hbox {kHz}\) |

# of subcarriers | 1200 |

FFT size | 2048 |

Frame duration | \(10 \,\hbox {ms}\) |

Subframe duration | \(1 \,\hbox {ms}\) |

Slot duration | \(0.5 \,\hbox {ms}\) |

TDD periodicity | \(1 \,\hbox {ms}\) |

### 2.2 Measurement Scenario

The measurement campaign was carried out in an indoor space with industrial profile in presence of glass, stone and metal reflecting surfaces. As depicted in Fig. 2 the BS was set up at the balcony at the end of the long hall while the cart containing the 8 UEs was placed on a wheeled cart at around 15 meter distance from the BS. The antennas of the UEs are positioned in a semi-circle configuration as shown in Fig. 3. They are directed away from the BS to suppress the line of sight (LOS) links and to ensure reflected links are more accentuated. The approximate distance between the UE cart and the main reflectors at the end of the hall is around \(30 \,\hbox {m}\). As shown in the campaign photos in Figs. 3 and 4, there exist many reflecting surfaces, such as concrete walls, window glasses, metal lamp posts and metal bars (at the end of the hallway) which form a rich scattering environment. For each frequency band, the UE cart was positioned along 5 pre-marked locations within \(1 \,\hbox {m}\) diameter to obtain more sampling points of the environment.

### 2.3 Channel Estimate Acquisition

The system has *M* antennas on the BS side, *K* UEs and uses orthogonal frequency division multiplexing (OFDM) with 1200 usable subcarriers distributed over a \(20 \,\hbox {MHz}\) band. Each user *k* transmits pilot symbols on a unique set of *F* subcarriers (in total 100 subcarriers for each UE) during the channel estimate acquisition to ensure orthogonality between the pilot symbols. These symbols are used to estimate the channel coefficient \(G_{mk}[f]\), with *m*, *k* and *f* as BS antenna index, user index and subcarrier index, respectively, by a least-squares method.

*n*denotes the delay bin. This representation corresponds directly to a tapped delay line.

### 2.4 Measurement Results

## 3 Effective Channel Concept

In TDD massive MIMO systems, the UL channels between BS and UEs are estimated at the BS by using a set of orthogonal pilot symbols sent by each UE and received at each antenna element in the arrays, as depicted in Fig. 7a. Given reciprocity holds for the transmitter and receiver chains in the coherence bandwidth of the system, these channel estimates denoted by \(H_{mk}[f]\) are used to calculate the precoding weights at each antenna element. If no co-channel interference is assumed, the system can be considered as a multiple input single output (MISO) system and the *effective channel* perceived at the UE is formed by superposition of these individual channels. In other words, the UE, unaware of any beam forming or precoding, observes a DL SISO channel from the BS which is the *effective channel*.

As shown in Fig. 7b, from the UE’s perspective, all the signals from the BS form an effective channel according to Eq. (4).

*k*in the DL

*k*and the DL symbol to user

*k*is \(X_{l}^{\mathrm {BS}}[f]\). The second term constitutes the multiuser interference and \(N_{k}[f]\) is the additive noise at user

*k*. To illustrate the properties of the effective channel we choose the weights to be the complex conjugate of the CTF coefficients with a normalization of \(\sqrt{M}\) to impose an average power constraint

*M*the difference disappears as \(\sum _M \left| {H_{mk}[f]}\right| ^2 \approx M E\{\left| {H_{mk}[f]}\right| ^2\}=M\) due to the self averaging property of the large array. The chosen normalization for the matched filtering does maximize the average SNR in the DL and simplifies the time domain analysis in Sects. 4.1 and 4.2 as it directly transfers to time reversal precoding.

We take different subsets over the *M* base station antennas, in order to form several effective channels and use them to determine how the different metrics behave for increasing number of antennas in the following section.

## 4 Channel Hardening Metrics

In the frequency domain, channel hardening is regarded as flat fading of the effective CTF over a large bandwidth. In the delay domain the channel hardening implies that the strong contributions of the effective channel impulse response (CIR) are confined to a single delay tap, with reliable tap power for most realizations. Since the rms delay spread of the effective CIR determines the necessity for an equalizer in the UE design, with sufficient channel hardening, the UE receiver could be simplified. The next three subsections describe the figures of merit for characterizing channel hardening in both delay and frequency domains.

### 4.1 Power Variation of the Effective Channel

Figure 8 shows the empirical cumulative distribution functions (CDFs) of \(\mathfrak {Q}_{k}[f]\). The combinations are formed from 1, 4, 16 and 64 antenna elements over all measured frequency bands, with channels drawn from similar subsets of consecutive close antenna elements in the subarrays. The observed statistics of the channel is practically the same in the range 1.5–4.2 GHz. Furthermore, the variation of \(\mathfrak {Q}_{k}[f]\) reduces with increasing *M*. Considering the link budget, channel hardening would reduce the fading to \(2 \,\hbox {dB}\) for \(90 \,\%\) of the observed effective channels with 64 antennas at the BS.

### 4.2 Effective PDP and Coherence Bandwidth

In this section, first we derive an analytical form for the effective power delay profile (PDP) as a function of the PDP of the UL channels. Weights given by Eq. (6) are used to implement time reversal precoding in the DL. The frequency correlation is calculated as the Fourier transform of the effective PDP, then coherence bandwidth can be obtained as a metric for evaluation of the effective channel.

*k*at all antenna elements, the channel can be described as complex normal distributed \(h_{mk}[n] \sim {\mathbb {C}\mathcal {N}}\left( 0, p_{k}[n] \right)\) where \(p_{k}[n] = {\mathbb {E}}\left\{ {\left| {h_{mk}[n]}\right| ^2}\right\}\) is the PDP for all the channels from user

*k*to the array. With the normalization of path loss and large scale fading the average channel gain is \(\small \sum p_{k}[n]=1\). The use of the weights from Eq. (6) in Eq. (4) corresponds to applying the matched filter in the time domain,

*M*when these weights are used in a precoding filter. An instantaneous power constraint would give the normalization with \(\sqrt{\sum _{M}\sum _{n} \left| {h_{mk}[n]}\right| ^2}\), which for large

*M*approximately is \(\sqrt{M}\) as

*k*is denoted \(R_{k}[\Delta f]\) and is the Fourier transform of the PDP \(p_{k}[n]\) [5]. The correlation function \(\mathfrak {R}_{k}[\Delta f]\) for the effective channel \(\mathfrak {h}_{k}[n]\) is hence the Fourier transform of \(\mathfrak {p}_{k}[n]\). The PDP \(\mathfrak {p}_{k}[n]\) consist of two terms. The convolution \(p_{k}[-n] *p_{k}[n]\) corresponds in the frequency domain to the absolute square of the frequency correlation, that is \(\left| {R_{k}[\Delta f]}\right| ^2\). The peak \(\delta _{n} M\) in Eq. (12) at zero lag corresponds to an offset for all frequencies. Hence, with a normalization so that \(\mathfrak {R}_{k}[0]=1\) we obtain

As the variability of the frequency response \(\mathfrak {H}_{k}[f]\) reduces with *M* the coherence bandwidth will increase simply because all the subcarriers will have just minor differences in amplitude around a dominating average.

### 4.3 The Effective Delay Spread

*M*. For large

*M*this will make it possible to have low complexity receivers in the DL as the channel effectively becomes a single tap channel.

*M*as

## 5 Conclusion

Measured data from a quasi-static indoor radio environment over four frequency bands in the range from \(1.4 \, \, \hbox {to} \,\, 4.2 \, \hbox {GHz}\) are used to study channel hardening properties in massive MIMO when using time-reversal precoding. The measurements contain 40 single user realizations against a 64 element antenna array BS to elucidate channel hardening in both the delay and the frequency domain. The observed statistical properties for the channels are practically the same over the studied frequency bands. The coherence bandwidth is demonstrated to have limited merit as a measure for the effective DL channel. The rms delay spread and the normalized subcarrier power of the effective radio channel are described as physically motivated figures of merit for low complexity single tap receivers

## Notes

### Acknowledgements

ReRaNP is supported by the Norwegian Research Council within the National Financing Initiative for Research Infrastructure. Further support was provided by Telenor ASA within the project Testing RAN Options for 5G.

### Funding

Funding was provided by Norges Forskningsråd (Grant No. 245699).

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