Channel Estimation in MIMO Systems with
One-bit Spatial Sigma-delta ADCs

Millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems have become very popular for sensing and wireless communications beyond 5G [2, 3, 4]. While the abundant spectrum available at the mmWave frequency bands enables higher cellular data rates and precise positioning, links at mmWave frequencies are very sensitive to blockages and have significantly higher path loss. These issues are alleviated by beamforming with very large antenna arrays, typically packed in small areas. MIMO systems operating at mmWave frequencies, commonly referred to as massive MIMO systems, are either single-user MIMO (SU-MIMO) systems with multi-antenna user equipment (UE) and a base station (BS) having a large antenna array or multi-user MIMO (MU-MIMO) systems with many single antenna UEs communicating with a BS having a large array.

High-resolution analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) for every antenna in the array significantly increase the radio frequency (RF) complexity and power consumption of massive MIMO systems. Low-resolution quantizers (e.g., 1-bit) are thus preferred albeit their deteriorated performance [5, 6, 7]. Sigma-delta ($\Sigma\Delta$) quantization is a popular technique frequently used to increase the effective resolution of low-resolution quantizers [8]. In a 1-bit $\Sigma\Delta$ quantizer, the time-domain signal is first oversampled at a rate significantly higher than the Nyquist rate. Then the difference between the input and the 1-bit quantized output, i.e., the quantization noise, is fed back in time by adding it to the input at the next time instance. This operation leads to noise shaping with the quantization noise pushed to higher temporal frequencies. This means that the effective quantization noise is negligible for a low-pass signal, and it would be as if the signal were quantized by a high-resolution quantizer. This classical architecture to increase the effective resolution of time-domain signals by using a simple 1-bit quantizer with feedback has been recently adapted to the spatial domain [9, 10] and is receiving steady attention for multi-antenna communications [11, 12, 13, 14, 15].

In a 1-bit spatial $\Sigma\Delta$ quantizer, oversampling and feedback are performed in the spatial domain, i.e., across antennas. To perform spatial oversampling, the antenna elements of an array are placed less than half wavelength apart. The quantization noise of each antenna is fed back along with the input of the next antenna. Analogous to its temporal counterpart, a spatial $\Sigma\Delta$ quantizer pushes and shapes the quantization noise to higher spatial frequencies away from the array broadside. In other words, the quantization noise at lower spatial frequencies in a spatial $\Sigma\Delta$ quantizer is reduced as if it was arising from a higher-resolution quantizer [9, 10]. By introducing phase shifts to the quantization noise before feedback allows angle steering so that the quantization noise (respectively, the effective resolution) will be lower (respectively, higher) for signals arriving around the steering angle [12, 14]. Therefore, with angle steering, it is possible to obtain a higher effective resolution for signals of interest in a spatial sector of certain width centered around any desired angle. In addition, a careful selection of the quantization voltage level assigned to 1-bit quantized signals significantly improves the inference performance when working with spatial $\Sigma\Delta$ quantizers.

In essence, 1-bit spatial $\Sigma\Delta$ quantization is an attractive architecture for massive MIMO systems, requiring only 1-bit quantizers per antenna element with feedback across the elements. However, feedback and 1-bit quantization make the channel estimation required for beamforming and symbol detection very challenging. This work focuses on channel estimation in MIMO systems with 1-bit spatial $\Sigma\Delta$ quantizers.

In this section, we describe the architecture of a multi-channel first-order 1-bit spatial $\Sigma\Delta$ ADC with angle steering. Let us denote the input and output of an $N_{\rm r}$ channel 1-bit spatial $\Sigma\Delta$ ADC, at time $t$, as ${{\mathbf{x}}}(t)=[x_{1}(t),x_{2}(t),\dots,x_{N_{\rm r}}(t)]^{\raisebox{1.2pt}{$\rm\scriptstyle T$}}\in{\mathbb{C}}^{N_{\rm r}}$ and ${{\mathbf{y}}}(t)=[y_{1}(t),y_{2}(t),\dots,y_{N_{\rm r}}(t)]^{\raisebox{1.2pt}{$\rm\scriptstyle T$}}\in{\mathbb{C}}^{N_{\rm r}}$, respectively.

Quantization noise from each antenna is fed back along with the input of the next antenna to realize the $\Sigma\Delta$ architecture in space. Due to the presence of the feedback, the quantization noise in a spatial $\Sigma\Delta$ ADC may become unbounded, overloading the output [22, 12]. To prevent overloading, the input signal, $x\in{\mathbb{C}}$, is clipped using ${\mathcal{L}}_{c}(\cdot)$ with clipping level $c>0$ as

where the operator $\left\{x\right\}_{c}$ is defined as $\left\{x\right\}_{c}=\max\{x,c\}$ and $\jmath=\sqrt{-1}$. The clipped signal is then quantized using a 1-bit quantizer ${\mathcal{Q}}_{b}[\cdot]$ with quantization voltage level $b$ as

Let $r_{n}(t)$ denote the input to channel $n$ of the ADC at time $t$. Then the corresponding output $y_{n}(t)$ is given by

where $e_{n}(t)=y_{n}(t)-r_{n}(t)$ is the quantization error. The quantization error at channel $n$ is phase shifted by $\varphi$ and added to the input of channel $n+1$ as

where $\varphi=2\pi d\sin{\psi}$ with $\psi$ being the steering angle and $d$ being the inter-element spacing in the array in wavelengths. Since there is no feedback to the first channel, $e_{0}(t)=\leavevmode\nobreak\ 0$. From (1) and (2), we have the recursion

We are interested in estimating parameters underlying the input signal ${{\mathbf{x}}}(t)$ from the output of the 1-bit spatial $\Sigma\Delta$ ADC ${{\mathbf{y}}}(t)$. This is a challenging problem because of the cascade of two non-linearities in (II). Moreover, the information loss introduced by the 1-bit spatial $\Sigma\Delta$ ADC is mainly determined by the clipping level $c$ and the quantization level $b$. For a given quantization level $b$, we can prevent the system from overloading by choosing the clipping voltage $c$ as [12]

A small value of $c$ leads to severe loss in the input information due to clipping, which cannot be compensated later by any choice of the quantization level $b$. Hence, it is necessary to choose a sufficiently large $c$ to avoid information loss due to clipping. In this work, we propose to choose $c$ such that ${\rm Pr}\left(\|{\mathcal{L}}_{c}[{{\mathbf{x}}}(t)]-{{\mathbf{x}}}(t)\|_{2}>\delta\right)\leq\epsilon$ for appropriately selected constants $\delta,\epsilon>0$. See Section V-C for more details on the selection of $c$. With such a choice of $c$, the output of the clipper can be approximated as ${\mathcal{L}}_{c}[x_{n}(t)]\approx x_{n}(t),\>n=1,2,\ldots,N_{\rm r}$. Then the 1-bit spatial $\Sigma\Delta$ recursion in (II) simplifies to

where the $N_{\rm r}\times N_{\rm r}$ lower triangular matrices ${\mathbf{U}}$ and ${\mathbf{V}}$ are defined as

with ${\mathbf{I}}$ being the $N_{\rm r}\times N_{\rm r}$ identity matrix. The architecture of the first-order 1-bit spatial $\Sigma\Delta$ ADC is shown in Fig. 1.

Quantization is a non-linear and irreversible operation that makes its statistical analysis complicated. To simplify the analysis of a quantizer, the usual approach is to linearize the quantizer and account for the error due to linearization through additive noise, which is often assumed to be uniformly distributed and uncorrelated with the input of the quantizer [23, 22, 10, 11, 12]. However, this assumption is reasonable only for a multi-level quantizer with many levels and when the input has a sufficiently large dynamic range. This classical approach of modeling the approximation error due to linearization as additive uniform noise is not suitable for 1-bit quantizers. Therefore, in what follows, we develop a noise model for 1-bit spatial $\Sigma\Delta$ ADCs.

To begin with, we first express the input-output relation of a spatial $\Sigma\Delta$ ADC in terms of the ${\rm floor}[\cdot]$ function by drawing inspiration from [22], where a similar expression for the deterministic error in a temporal 1-bit $\Sigma\Delta$ quantizer was developed. Next, we propose to linearize the ${\rm floor}[\cdot]$ function by interpreting it as a multi-level quantizer to compute the second-order statistics of the quantization noise that is useful for decorrelating observations when solving parametric estimation and detection problems involving 1-bit spatial $\Sigma\Delta$ ADCs.

The error $e_{n}(t)$ in (1) due to 1-bit spatial $\Sigma\Delta$ quantization admits a closed-form expression as given in the next Lemma.

The above expressions are derived in the appendix. Let us collect the output of all the channels of the 1-bit spatial $\Sigma\Delta$ ADC in a vector ${{\mathbf{y}}}(t)$ and rewrite (5) as

From Lemma 1, we have

where ${\boldsymbol{\nu}}=\mu{\mathbf{V}}\boldsymbol{1}\in{\mathbb{C}}^{N_{\rm r}}$ with $\mu=0.5+\jmath 0.5$, $\boldsymbol{1}$ is the length-$N_{\rm r}$ column vector with all ones, and the fractional part function is applied elementwise. Using the fact that $\langle x\rangle=x-{\rm floor}[x],\>\forall\>x\in\mathbb{R}$, and from (7), the deterministic relation between the input and output of a 1-bit spatial $\Sigma\Delta$ ADC is given by

where we have transformed the non-linearity due to the 1-bit quantization ${\mathcal{Q}}_{b}[\cdot]$ to the non-linear ${\rm floor[\cdot]}$ function.

Now we leverage the fact that the input-output relation of a ${\rm floor}[\cdot]$ function is similar to that of a multi-level quantizer with a step size of one. For the spatial $\Sigma\Delta$ ADC channels (corresponding to antenna elements) with larger indices that are away from the $1$st channel, the dynamic range of the real and imaginary parts of $[0.5b^{-1}{\mathbf{U}}{{\mathbf{x}}}(t)+{\boldsymbol{\nu}}]_{n}$ is large when compared to $[0,1)$. This means that for antenna elements with indices away from the $1$st element, the ${\rm floor}[\cdot]$ function acts like a multi-level quantizer with an input having a large dynamic range. Therefore, it is now reasonable to model the approximation error due to the linearization of the ${\rm floor}[\cdot]$ function as additive noise, ${{\mathbf{w}}}(t)\in\leavevmode\nobreak\ \mathbb{C}^{N_{\rm r}}$, which is uniformly distributed and uncorrelated with the input for channels having larger indices, i.e., we have

with the real and imaginary parts of $[{{\mathbf{w}}}(t)]_{n}\in[0,1)$. Substituting in (9) yields

where ${{\mathbf{q}}}(t)=2b{\mathbf{U}}^{-1}({{{\mathbf{w}}}}(t)+\mu\boldsymbol{1})$, which is an affine transformation of ${{\mathbf{w}}}(t)$, is also uniformly distributed and $[{{\mathbf{q}}}(t)]_{n}$ is uncorrelated with the input $[{{\mathbf{x}}}(t)]_{n}$ for larger channel (or antenna) indices $n$. In massive MIMO systems with large number of antennas, there are many antennas for which $x_{n}(t)$ and $[{{\mathbf{q}}}(t)]_{n}$ are uncorrelated. The correlation of ${[{{\mathbf{q}}}(t)]}_{n}=y_{n}(t)-x_{n}(t)$ with $x_{n}(t)$, i.e., $\mathbb{E}[x_{n}(y_{n}-x_{n})^{*}]$ for an antenna array having $N_{\rm r}=128$ elements with an inter-element spacing of one eighth the signal wavelength is illustrated in Fig. † ‣ 2(a).

The covariance matrix of the noise vector ${{\mathbf{q}}}(t)$ in (10) is given by ${\mathbf{R}}_{q}={\mathbb{E}}\left[{{\mathbf{q}}}(t){{\mathbf{q}}}^{\raisebox{1.0pt}{$\rm\scriptscriptstyle H$}}(t)\right]=\frac{2b^{2}}{3}{\mathbf{U}}^{-1}{\mathbf{U}}^{\raisebox{1.0pt}{-$\rm\scriptscriptstyle H$}}$. When $\psi=0$, the matrix ${\mathbf{U}}^{-1}$ with ones on the main diagonal, $-1$ on the first sub-diagonal, and zeros elsewhere, is a spatial high-pass filter, which shapes the quantization noise to higher spatial frequencies. We illustrate the angular power spectrum of the quantization noise, i.e., $\mathbb{E}[|{{\mathbf{a}}}_{\rm BS}^{\raisebox{1.0pt}{$\rm\scriptscriptstyle H$}}(\theta)({{\mathbf{y}}}(t)-{{\mathbf{x}}}(t))|^{2}]$, for different values of the steering angle in Fig. † ‣ 2(b), where ${{\mathbf{a}}}_{\rm BS}(\theta)=[1,e^{-\jmath 2\pi d{\rm sin}(\theta)},\ldots,e^{-\jmath(N_{\rm r}-1)2\pi d{\rm sin}(\theta)}]^{\raisebox{1.2pt}{$\rm\scriptstyle T$}}$ is the steering vector of the uniform linear array (ULA) having $N_{\rm r}$ elements at the BS with an inter-element spacing $d$ wavelengths. We can see that the quantization noise is very small for the directions around the steering angle.

Before we end this section, we make the following two remarks. Although in the development of the noise model, the spatial $\Sigma\Delta$ ADC is steered to the array broadside with $\psi=0$ for ease of exposition, introducing a phase shift in the feedback does not alter the uncorrelatedness between the input and the quantization noise as can be seen in Fig. † ‣ 2(a). Next, for the ${\rm floor}[\cdot]$ function in (9) to behave as a multi-level quantizer, the quantization voltage level $b$ plays an important role. When $b$ is very large, the dynamic range of the real and imaginary parts of the argument of the ${\rm floor}[\cdot]$ function, i.e., $0.5b^{-1}\mathbf{U}\mathbf{x}(t)+{\boldsymbol{\nu}}$, can be very small for which the uniform noise or uncorrelated noise assumption might fail. Hence, carefully selecting $b$ (as discussed in Section V-C) becomes crucial.

Next, we use the developed spatial $\Sigma\Delta$ signal and noise models to estimate MIMO channels with angular models.

In this paper, we consider the two commonly encountered SU-MIMO and MU-MIMO settings in MIMO communications. In the SU-MIMO setting, a single UE with $N_{\rm t}$ antennas communicates with a multi-antenna BS, whereas in the MU-MIMO setting, $K$ single antenna UEs communicate with a multi-antenna BS. In both these settings, the BS has a ULA with $N_{\rm r}$ antennas and it receives and processes uplink training pilots.

Let us collect the uplink training pilots in ${\mathbf{S}}=[{{\mathbf{s}}}(1),{{\mathbf{s}}}(2),\ldots,{{\mathbf{s}}}(T)]$, where $T$ is the pilot length. Without loss of generality, let us assume that the columns of the pilot matrix have unit norm. Let $P$ denote the total transmit power at the UE and ${\mathbf{H}}$ denote the MIMO channel matrix. The signal received at the BS prior to quantization, denoted as ${\mathbf{X}}=[{{\mathbf{x}}}(1),{{\mathbf{x}}}(2),\ldots,{{\mathbf{x}}}(T)]\in{\mathbb{C}}^{N_{\rm r}\times T}$, can be expressed as

where $P$ is also the uplink SNR of the system and ${\mathbf{Z}}\in{\mathbb{C}}^{N_{\rm r}\times T}$ is the additive white Gaussian receiver noise matrix with entries $[{\mathbf{Z}}]_{ij}\sim{\mathcal{C}}{\mathcal{N}}(0,1)$. The received signal is then quantized using an $N_{\rm r}$-channel 1-bit spatial $\Sigma\Delta$ ADC to obtain ${\mathbf{Y}}=[{{\mathbf{y}}}(1),{{\mathbf{y}}}(2),\ldots,{{\mathbf{y}}}(T)]\in{\mathbb{C}}^{N_{\rm r}\times T}$, which is given by

Next, we parameterize ${\mathbf{H}}$ by assuming a narrowband spatially sparse model for MU-MIMO and SU-MIMO systems.

In this section, we present the proposed algorithm for channel estimation with angular models in SU-MIMO systems having 1-bit spatial $\Sigma\Delta$ ADCs at the BS. Specifically, we propose a two-step algorithm to estimate the channel parameters $(\boldsymbol{\theta},\boldsymbol{\phi},\boldsymbol{\alpha})$ from uplink pilots. In the first step, we estimate the AoAs (i.e., $\boldsymbol{\theta}$) and the path gains (i.e., $\boldsymbol{\alpha}$) using precoded uplink pilots, which excite all the angles uniformly. Next, in the second step, to estimate the AoDs, i.e., $\boldsymbol{\phi}$, we select precoders from a codebook using a recursive bisection procedure that leverages 1-bit feedback between the BS and UE to reduce the number of channel uses and thus the channel estimation overhead.

As we discuss later, it is crucial to carefully select the clipping voltage levels to benefit from the advantages of the 1-bit spatial $\Sigma\Delta$ ADCs. The proposed two-step channel estimation procedure is developed keeping in mind the dependence of the unknown parameters on voltage level selection.