2.1 System model
We consider a single-cell FDD massive MIMO system where an
M-antenna base-station serves
K single-antenna users through downlink beamforming. With downlink channel as
\({\textbf{H}} \in {\mathbb {C}}^{K \times M}\) and beamforming weights matrix as
\({\textbf{W}} \in {\mathbb {C}}^{M \times K}\), the received signals at the mobile user can be written as
$$\begin{aligned} {\textbf{y}} = {\textbf{H}} {\textbf{W}} {\textbf{x}} + {\textbf{n}}, \end{aligned}$$
(1)
where
\({\textbf{x}} \in {\mathbb {C}}^{K}\) is the transmitted signals and
\({\textbf{n}} \in {\mathbb {C}}^K\) is the additive noises that follow standard complex Gaussian distribution.
To design beamforming weights \({\textbf{W}}\) for effective downlink multi-user beamforming, e.g., conjugate beamforming or zero-forcing beamforming, the key step is to estimate downlink channel state information at the base-station side. Therefore, next, we first present details of FDD massive MIMO channels, including propagation domain channel model and measured channels, and then show potential errors of existing scalable channel estimation schemes in FDD massive MIMO.
2.3 Scalable channel estimation in FDD massive MIMO
To investigate the potential error of downlink channel estimation, here we focus on the single-user case only. Without loss of generality, the user downlink channel is denoted as
\({\textbf{h}}_{\textrm{D}} \in {\mathbb {C}}^{M}\). Based on the propagation domain channel model, as shown in Eq.
2, downlink channel model can be rewritten as
$$\begin{aligned} {\textbf{h}}_{\textrm{D}} = \sum _{i=1}^{I} \sum _{j=1}^{J_i} \sqrt{g_{ij}} e^{j\phi _{\textrm{D}ij}} {\textbf{a}}\left( \theta _{ij},\varphi _{ij} \right) = {\textbf{S}} {\textbf{b}}, \end{aligned}$$
(5)
where
$$\begin{aligned} {\textbf{S}} = \begin{bmatrix} {\textbf{a}}\left( \theta _{11},\varphi _{11}\right)&\cdots&{\textbf{a}}\left( \theta _{1J_1},\varphi _{1J_1}\right)&\cdots&{\textbf{a}}\left( \theta _{I1},\varphi _{I1}\right)&\cdots&{\textbf{a}}\left( \theta _{IJ_I},\varphi _{IJ_I}\right) \end{bmatrix} \end{aligned}$$
(6)
denotes full propagation domain and
$$\begin{aligned} {\textbf{b}} = \begin{bmatrix} \sqrt{g_{11}} e^{j\phi _{\textrm{D}11}}&\cdots&\sqrt{g_{1J_1}} e^{j\phi _{\textrm{D}1J_1}}&\cdots&\sqrt{g_{I1}} e^{j\phi _{\textrm{D}I1}}&\cdots&\sqrt{g_{IJ_I}} e^{j\phi _{\textrm{D}IJ_I}} \end{bmatrix}^T \end{aligned}$$
(7)
coefficients in the propagation domain.
In scalable channel estimation schemes that exploit channel low-dimensional domain, there are two main steps. First, estimate the low-dimensional propagation domain
\(\hat{{\textbf{S}}} \in {\mathbb {C}}^{M \times L}, L < M\) for downlink training, where
L is the number of training vectors in the estimated domain. Second, estimate domain coefficients
\(\hat{{\textbf{b}}} \in {\mathbb {C}}^L\) via downlink training and uplink feedback. After that, the downlink channel can be reconstructed as
$$\begin{aligned} \hat{{\textbf{h}}}_{\textrm{D}} = \hat{{\textbf{S}}} \hat{{\textbf{b}}}. \end{aligned}$$
(8)
Previous works focus on domain coefficients estimation as in the second step only and assume perfect knowledge of the channel propagation domain. However, the error in propagation domain estimation
\(\hat{{\textbf{S}}}\) also contributes to channel estimation error. Therefore, it is important to investigate the impact of propagation domain estimation error.
Since we aim to quantify the
propagation domain modeling error—normalized error of approximating downlink channel with uplink dominant angles, next we ask and answer two questions on propagation domain estimation corresponding to the imperfectness of the two aforementioned assumptions that will affect the normalized error:
1
What will be the normalized error if utilizing dominant angles in the propagation domain instead of all angles to approximate the downlink channel?
2
What will be the extra normalized error if utilizing uplink dominant angles instead of downlink ones to approximate the downlink channel?
The answers to the questions depend on channel properties only. Therefore, to answer the questions, we seek to start from the fundamentals, i.e., FDD massive MIMO channels, to investigate the normalized error of approximating downlink channel with uplink dominant angles. We use a combination of a numerical and experimental approach. For the numerical approach, we employ the spatial channel model to formulate modeling error and examine the scalability with the base-station array size and dependency on the channel angle spread; for the experimental approach, we further validate the observations of modeling error based on measured channels.
2.4 Modeling error definition
In this section, we characterize the normalized error of approximating downlink channel with uplink dominant propagation domain, defined as
modeling error, by answering the two questions brought up in Sect.
2.3. Each question corresponds to one source of error, with the first one denoted as
approximation error and the second one denoted as
mismatch error.
For single-user case, the resulting rate with conjugate beamforming based on estimated channel
\(\hat{{\textbf{h}}}_{\textrm{D}} \in {\mathbb {C}}^{M}\) is
$$\begin{aligned} \begin{aligned} R \left( \hat{{\textbf{h}}}_{\textrm{D}}, {\textbf{h}}_{\textrm{D}} \right)&= {\mathbb {E}} \left[ \log \left( 1 + P \frac{ \Vert \hat{{\textbf{h}}}_{\textrm{D}}^H {\textbf{h}}_{\textrm{D}} \Vert ^2 }{\Vert \hat{{\textbf{h}}}_{\textrm{D}}\Vert ^2} \right) \right] \\&= {\mathbb {E}} \left[ \log \left( 1 + P \underbrace{\Vert {\textbf{h}}_{\textrm{D}}\Vert ^2}_{P_{\textrm{channel}}} \frac{ \Vert \hat{{\textbf{h}}}_{\textrm{D}}^H {\textbf{h}}_{\textrm{D}} \Vert ^2 }{\Vert \hat{{\textbf{h}}}_{\textrm{D}}\Vert ^2 \Vert {\textbf{h}}_{\textrm{D}}\Vert ^2} \right) \right] \end{aligned} \end{aligned}$$
(9)
where
P denotes the downlink transmission power for the user. From Eq. (
9), maximizing rate is same as minimizing the following
normalized error
$$\begin{aligned} E = 1 - \frac{ \Vert \hat{{\textbf{h}}}_{\textrm{D}}^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert \hat{{\textbf{h}}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert }. \end{aligned}$$
(10)
Note
\(E \in [0,1]\), where
\(E=0\) occurs when there is no channel estimation error and
\(E=1\) occurs when the estimated channel is orthogonal to the actual channel, a worst-case scenario. Thus, smaller
E is better.
We focus on propagation domain estimation and assume genie-aided domain coefficients training. Consider estimated propagation domain
\(\hat{{\textbf{S}}} \in {\mathbb {C}}^{M \times L}, L < M\), the genie-aided estimated domain coefficients will be
$$\begin{aligned} \hat{{\textbf{b}}} = \hat{{\textbf{S}}}^{\dag } {\textbf{h}}_{\textrm{D}} \end{aligned}$$
(11)
based on least-square estimator, where
\(\hat{{\textbf{S}}}^{\dag }\) stands for the pseudo-inverse matrix of
\(\hat{{\textbf{S}}}\). To evaluate estimated propagation domain
\(\hat{{\textbf{S}}}\), based on Eq.
10, we consider the normalized error of estimated downlink channel utilizing the estimated propagation domain, which is formulated as
$$\begin{aligned} \begin{aligned} E (\hat{{\textbf{S}}})&= 1 - \frac{ \Vert \hat{{\textbf{h}}}_{\textrm{D}}^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert \hat{{\textbf{h}}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert } \\&= 1 - \frac{ \Vert (\hat{{\textbf{S}}} \hat{{\textbf{b}}}) ^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert \hat{{\textbf{S}}} \hat{{\textbf{b}}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert } \\&= 1 - \frac{ \Vert (\hat{{\textbf{S}}} \hat{{\textbf{S}}}^{\dag } {\textbf{h}}_{\textrm{D}}) ^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert \hat{{\textbf{S}}} \hat{{\textbf{S}}}^{\dag } {\textbf{h}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert }. \end{aligned} \end{aligned}$$
(12)
We aim to quantify modeling error—the normalized error of approximating downlink channel with uplink dominant propagation domain. To analyze modeling error, we need to obtain both downlink domain of low dimensionality and uplink one. In propagation domain, the low dimensionality part is constructed from array response vectors corresponding to dominant channel angles. Therefore, first, we try to construct downlink dominant propagation domain
\({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}, L_d \ll M\), where
M is the number of antennas at the base-station and
\(L_d\) is the number of dominant angles. The downlink dominant angle set is defined as
$$\begin{aligned} \left\{ \left( \theta _{\textrm{d}1},\varphi _{\textrm{d}1} \right) , ..., \left( \theta _{\textrm{d}L_d},\varphi _{\textrm{d}L_d} \right) \right\} = \text {arg} \,\,\,\, \underset{\left\{ \left( \theta _{1},\varphi _{1} \right) , ..., \left( \theta _{L_d},\varphi _{L_d} \right) \right\} }{\text {minimize}} \,\,\,\, \left\| {\textbf{h}}_{\textrm{D}} - {\textbf{A}}_{\textrm{d}} {\textbf{A}}_{\textrm{d}}^{\dag } {\textbf{h}}_{\textrm{D}}\right\| _2, \end{aligned}$$
(13)
where
\({\textbf{A}}_{\textrm{d}} = \begin{bmatrix} {\textbf{a}}\left( \theta _{1},\varphi _{1}\right)&\cdots&{\textbf{a}}\left( \theta _{L_d},\varphi _{L_d}\right) \end{bmatrix}\) and
\({\textbf{A}}_{\textrm{d}}^{\dag }\) stands for the pseudo-inverse matrix of
\({\textbf{A}}_{\textrm{d}}\). We extract downlink dominant angles from full downlink CSI
\({\textbf{h}}_{\textrm{D}} \in {\mathbb {C}}^{M}\) utilizing maximum likelihood estimator [
26]. Then, the downlink dominant propagation domain based on dominant angles is constructed as
$$\begin{aligned} {\textbf{S}}_{\textrm{d}} = \begin{bmatrix} {\textbf{a}}\left( \theta _{\textrm{d}1},\varphi _{\textrm{d}1}\right)&{\textbf{a}}\left( \theta _{\textrm{d}2},\varphi _{\textrm{d}2}\right)&\cdots&{\textbf{a}}\left( \theta _{\textrm{d}L_d},\varphi _{\textrm{d}L_d}\right) \end{bmatrix}, \end{aligned}$$
(14)
where
\({\textbf{a}}\) is the array response vector defined in Eq. (
3).
Second, to construct uplink dominant propagation domain
\({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\), similarly, we extract uplink dominant angles
\(\left\{ \left( \theta _{\textrm{u}1},\varphi _{\textrm{u}1} \right) , ..., \left( \theta _{\textrm{u}L_d},\varphi _{\textrm{u}L_d} \right) \right\}\) from full uplink CSI
\({\textbf{h}}_{\textrm{U}} \in {\mathbb {C}}^{M}\). Then, the uplink dominant propagation domain is constructed as
$$\begin{aligned} {\textbf{S}}_{\textrm{u}} = \begin{bmatrix} {\textbf{a}}\left( \theta _{\textrm{u}1},\varphi _{\textrm{u}1}\right)&{\textbf{a}}\left( \theta _{\textrm{u}2},\varphi _{\textrm{u}2}\right)&\cdots&{\textbf{a}}\left( \theta _{\textrm{u}L_d},\varphi _{\textrm{u}L_d}\right) \end{bmatrix}. \end{aligned}$$
(15)
Corresponding to the two questions brought up in Sect.
2.3, there are two factors that affect modeling error. First, to keep the low dimensionality of propagation domain under channel training overhead constraints, only
\(L_d\) dominant angles-based response vectors constructed propagation domain
\({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\) is utilized for downlink channel approximation. As a result, there will still be certain channel estimation error due to the approximation; we denote this normalized channel estimation error as
approximation error. Following the definition of normalized channel estimation error in Eq.
10, approximation error is formulated as
$$\begin{aligned} \begin{aligned} E_{\textrm{approximation}}&= E ({\textbf{S}}_{\textrm{d}}) \\&= 1 - \frac{ \Vert ({\textbf{S}}_{\textrm{d}} {\textbf{S}}_{\textrm{d}}^{\dag } {\textbf{h}}_{\textrm{D}}) ^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert {\textbf{S}}_{\textrm{d}} {\textbf{S}}_{\textrm{d}}^{\dag } {\textbf{h}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert }, \end{aligned} \end{aligned}$$
(16)
where
\({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\) is the downlink dominant propagation domain as illustrated in Eq.
14. As evident from the above equation, approximation error will be in the range from 0 to 1. When the downlink dominant propagation domain
\({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\) gets closer to downlink channel in antenna domain, approximation error will decrease and get closer to 0.
Second, since downlink propagation domain information is not available before any downlink channel training, uplink channel-inferred dominant propagation domain instead of actual downlink channel dominant propagation domain is utilized for downlink channel training. Consequently, there will be extra normalized channel estimation error due to the uplink and downlink dominant propagation domain mismatch; we denote this normalized channel estimation error as
mismatch error, which is formulated as
$$\begin{aligned} \begin{aligned} E_{\textrm{mismatch}}&= E ({\textbf{S}}_{\textrm{u}}) - E ({\textbf{S}}_{\textrm{d}}) \\&= \frac{ \Vert ({\textbf{S}}_{\textrm{d}} {\textbf{S}}_{\textrm{d}}^{\dag } {\textbf{h}}_{\textrm{D}}) ^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert {\textbf{S}}_{\textrm{d}} {\textbf{S}}_{\textrm{d}}^{\dag } {\textbf{h}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert } - \frac{ \Vert ({\textbf{S}}_{\textrm{u}} {\textbf{S}}_{\textrm{u}}^{\dag } {\textbf{h}}_{\textrm{D}}) ^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert {\textbf{S}}_{\textrm{u}} {\textbf{S}}_{\textrm{u}}^{\dag } {\textbf{h}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert }, \end{aligned} \end{aligned}$$
(17)
where
\({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\) is the uplink dominant propagation domain as illustrated in Eq.
15. As evident from the above equation, mismatch error will also be in the range from 0 to 1. When the uplink channel-inferred dominant propagation domain
\({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\) gets closer to actual downlink channel one
\({\textbf{S}}_{\textrm{d}} \in {\mathbb {C}}^{M \times L_d}\), mismatch error will decrease and get closer to 0.
Combining approximation error and mismatch error, the total normalized error of approximating downlink channel with uplink dominant propagation domain
\({\textbf{S}}_{\textrm{u}} \in {\mathbb {C}}^{M \times L_d}\), i.e., modeling error, is formulated as
$$\begin{aligned} \begin{aligned} E_{\textrm{modeling}}&= E_{\textrm{approximation}} + E_{\textrm{mismatch}} \\&= E({\textbf{S}}_{\textrm{u}}) \\&= 1 - \frac{ \Vert ({\textbf{S}}_{\textrm{u}} {\textbf{S}}_{\textrm{u}}^{\dag } {\textbf{h}}_{\textrm{D}}) ^H {\textbf{h}}_{\textrm{D}} \Vert }{\Vert {\textbf{S}}_{\textrm{u}} {\textbf{S}}_{\textrm{u}}^{\dag } {\textbf{h}}_{\textrm{D}}\Vert \Vert {\textbf{h}}_{\textrm{D}}\Vert }. \end{aligned} \end{aligned}$$
(18)
As evident from the above equation, modeling error will be in the range from 0 to 1. When the uplink channel-inferred dominant propagation domain
\({\textbf{S}}_{\textrm{u}}\) gets closer to downlink channel in antenna domain, the modeling error will get closer to 1.
Here, we focus on modeling error in the propagation domain, while similar error analysis can be applied to other domain-based channel characterization. Also, we want to emphasize that modeling error is determined by FDD massive MIMO channel properties only and thus scheme-independent. But, modeling error is an important and necessary part to analyze and evaluate the performance of scalable FDD massive MIMO channel estimation schemes.
Modeling error quantifies the normalized estimation error of approximating downlink channel with uplink dominant angle response vectors. As expected, the modeling error will result in beamforming performance impact due to channel estimation error. To understand the performance impact of modeling error, here we first derive the beamforming rate loss corresponding to modeling error.
To quantify the beamforming performance impact of modeling error, as illustrated in Section
2.4, we focus on the single-user case and evaluate single-user beamforming achievable rate with conjugate beamforming. When the base-station has the perfect downlink CSI
\({\textbf{h}}_{\textrm{D}}\) available, the achievable rate with conjugate beamforming will be
$$\begin{aligned} R \left( {\textbf{h}}_{\textrm{D}}, {\textbf{h}}_{\textrm{D}} \right) = {\mathbb {E}} \left[ \log \left( 1 + P \Vert {\textbf{h}}_{\textrm{D}}\Vert ^2 \right) \right] . \end{aligned}$$
(19)
And utilizing approximated downlink channel with uplink dominant angles
\({\textbf{h}}_{\textrm{uD}}\) defined as
$$\begin{aligned} {\textbf{h}}_{\textrm{uD}} = {\textbf{S}}_{\textrm{u}} {\textbf{S}}_{\textrm{u}}^{\dag } {\textbf{h}}_{\textrm{D}}, \end{aligned}$$
(20)
where
\({\textbf{S}}_{\textrm{u}}\) includes uplink dominant angle response vectors as shown in Eq.
15. With conjugate beamforming, the achievable rate will be
$$\begin{aligned} R \left( {\textbf{h}}_{\textrm{uD}}, {\textbf{h}}_{\textrm{D}} \right) = {\mathbb {E}} \left[ \log \left( 1 + P \Vert {\textbf{h}}_{\textrm{D}}\Vert ^2 \frac{ \Vert {\textbf{h}}_{\textrm{uD}}^H {\textbf{h}}_{\textrm{D}} \Vert ^2 }{\Vert {\textbf{h}}_{\textrm{uD}}\Vert ^2 \Vert {\textbf{h}}_{\textrm{D}}\Vert ^2} \right) \right] . \end{aligned}$$
(21)
Then, the rate gap between beamforming based on perfect downlink CSI
\({\textbf{h}}_{\textrm{D}}\) and beamforming based on approximated downlink channel
\({\textbf{h}}_{\textrm{uD}}\), denoted as
rate loss, is formulated as
$$\begin{aligned} \begin{aligned} \Delta R&= R \left( {\textbf{h}}_{\textrm{D}}, {\textbf{h}}_{\textrm{D}} \right) - R \left( {\textbf{h}}_{\textrm{uD}}, {\textbf{h}}_{\textrm{D}} \right) \\&= {\mathbb {E}} \left[ \log \left( \frac{1 + P \Vert {\textbf{h}}_{\textrm{D}}\Vert ^2}{1 + P \Vert {\textbf{h}}_{\textrm{D}}\Vert ^2 \left( 1 - E_{\textrm{modeling}}\right) ^2} \right) \right] \end{aligned}. \end{aligned}$$
(22)
As evident from the above equation, the modeling error will affect the rate loss and larger modeling error will lead to larger rate loss.