### 3.1 Proposed precoding design

In this paper, we are interested in the problem of the sum rate maximization under total RAU power constraints. On the *n*th subcarrier, the SINR of downlink UE *k* is

$$\begin{aligned} {\gamma _{k,n}} = \frac{{{{\left| {{\mathbf {h}}_{k,n}^H{{\mathbf {w}}_k}} \right| }^2}}}{{\sum \nolimits _{u \ne k}^K {{{\left| {{\mathbf {h}}_{k,n}^H{{\mathbf {w}}_u}} \right| }^2}} + \sigma _k^2}} \end{aligned}$$

(6)

Then, the design problem is given by

$$\begin{aligned} &\mathop {\max }\limits _{{{\mathbf {w}}_k}} \,\sum \limits _{n = 1}^N {\sum \limits _{k = 1}^K {{{\log }_2}\left( {1 + {\gamma _{k.n}}} \right) } } \\&\hbox {s.t.}\,\sum \limits _{k = 1}^K {{{\left\| {{{\mathbf {w}}_k}} \right\| }^2}} \leqslant {P_{\max }},\forall k \end{aligned}$$

(7)

Since problem (7) is NP-hard, the globally optimal design mainly acts as a theoretical benchmark rather than a practical solution. Herein, motivated by [13, 14], we develop a low-complexity algorithm that satisfies the necessary optimal conditions of (7).

According to the monotonicity of the logarithmic function, (7) is equivalent to

$$\begin{aligned}&\mathop {\max }\limits _{{{\mathbf {w}}_k}} \,\prod \limits _{n = 1}^N {\prod \limits _{k = 1}^K {\left( {1 + {\gamma _{k,n}}} \right) } }\\&\hbox {s.t.}\,\sum \limits _{k = 1}^K {{{\left\| {{{\mathbf {w}}_k}} \right\| }^2}} \leqslant {P_{\max }},\forall k \end{aligned}$$

(8)

which can be equivalently rewritten as

$$\begin{aligned}&\mathop {\max }\limits _{{{\mathbf {w}}_k},{{\mathbf {t}}_{k,n}}} \,\prod \limits _{n = 1}^N {\prod \limits _{k = 1}^K {{f_{k,n}}} } \end{aligned}$$

(9a)

$$\begin{aligned}&\hbox {s.t.}\,\sum \limits _{k = 1}^K {{{\left\| {{{\mathbf {w}}_k}} \right\| }^2}} \leqslant {P_{\max }},\forall k \end{aligned}$$

(9b)

$$\begin{aligned}&{\gamma _{k,n}} \geqslant {f_{k,n}} - 1,\forall k \end{aligned}$$

(9c)

$$\begin{aligned}&{f_{k,n}} \geqslant 1,\forall k \end{aligned}$$

(9d)

Note that the objective function in (9) admits an SOC representation. Since (9b) and (9d) are already convex forms, we mainly focus on dealing with the constraints in (9c).

First, (9c) can be rewritten as

$$\begin{aligned} \sum \limits _{u \ne k}^K {{{\left| {{\mathbf {h}}_{k,n}^H{{\mathbf {w}}_u}} \right| }^2}} + \sigma _k^2 \leqslant \frac{{{\mathbf {w}}_k^H{{\varvec{\Lambda }} _{k,n}}{{\mathbf {w}}_k}}}{{{f_{k,n}} - 1}} \end{aligned}$$

(10)

where \({{\varvec{\Lambda }} _{k,n}} ={\mathbf {h}}_{k,n}{{\mathbf {h}}_{k,n}^H}\). We can see that (10) is also non-convex, since the right side of (10) has the form of quadratic-over-linear, it can be replaced by its first-order expansions [15]. Thus, we define

$$\begin{aligned} J({\mathbf {w}},f,\Lambda ) = \frac{{{{\mathbf {w}}^H}{\varvec{\Lambda }} {\mathbf {w}}}}{{f - \alpha }} \end{aligned}$$

(11)

where \({\varvec{\Lambda }} \ge 0\) and \(f \ge \alpha\). We obtain the first-order Taylor expansion of (11) about a certain point \(\left( {{{\mathbf {w}}^{(a)}},{f^{(a)}}} \right)\) as

$$\begin{aligned}&J({\mathbf {w}},f,{\mathbf {w}}_{}^{(a)},f_{}^{(a)},{\varvec{\Lambda }} ,\alpha ) \\&\quad = \frac{{2\Re \left\{ {{{\left( {{\mathbf {w}}_{}^{\left( a \right) }} \right) }^H}{\varvec{\Lambda }} {\mathbf {w}}} \right\} (f_{}^{\left( a \right) } - \alpha ) - {{\left( {{\mathbf {w}}_{}^{\left( a \right) }} \right) }^H}{\varvec{\Lambda }} {\mathbf {w}}_{}^{\left( a \right) }(f - \alpha )}}{{{{\left( {f_{}^{\left( a \right) } - \alpha } \right) }^2}}} \end{aligned}$$

(12)

From the above analysis, we can transform the constraint of (9c) into a convex form

$$\begin{aligned} \sum \limits _{u \ne k}^K {{{\left| {{\mathbf {h}}_{k,n}^H{{\mathbf {w}}_u}} \right| }^2}} + \sigma _k^2 \leqslant J({{\mathbf {w}}_k},{f_k},{\mathbf {w}}_k^{(a)},f_k^{(a)},{{\varvec{\Lambda }} _k},1) \end{aligned}$$

(13)

Finally, the original problem (8) can be reformulated as a convex approximate problem (14) which can be solved in Algorithm 1.

$$\begin{aligned}&\mathop {\max }\limits _{\left\{ {{{\mathbf {w}}_k},{{\mathbf {t}}_{k,n}}} \right\} } \,\prod \limits _{n = 1}^N {\prod \limits _{k = 1}^K {{f_{k,n}}} } \\&\hbox {s.t.}\,\left( {9b} \right) ,\,\left( {9d} \right) ,\,\left( {13} \right) \end{aligned}$$

(14)

### 3.2 Downlink channel estimation

In this section, considering the impact of DSDs on the accuracy of cell-free massive MIMO-OFDM system channel estimation, we propose a high-precision multi-RB downlink channel estimation scheme, which provides the necessary channel information for multi-RB precoding optimization. In the pilot design of the 5G NR [16], CSI-RS is mainly used for channel sounding to obtain the path loss and DSD between UEs and RAUs. DMRS is mainly used for the demodulation of uplink and downlink data. The least squares (LS) algorithm is usually used to estimate the initial channel response of the pilot position, and then, interpolation filtering is performed in the frequency domain to obtain the channel response of the data position [17].

Regarding the CSI-RS of cell-free system, in uplink, sounding reference signal (SRS) is used to collect DSDs and large-scale fading; in downlink, tracking reference signal (TRS), a special CSI-RS, is used to track the phase deviation on the received signal.

For SRS, per UE tracking can be achieved in the 5G NR protocol. Therefore, the uplink channel estimation can be implemented in CPU. For downlink channel estimation, the signal overhead for per AP tracking is too large, and the implementation complexity is too high for UEs. Therefore, to estimate the statistical properties of downlink channels, we choose to use a composite channel tracking method, which is discussed in the following specific steps.

The specific steps of the multi-RB channel estimation we proposed are as follows:

First, all RAUs transmit orthogonal CSI-RS, UE *k* estimates large-scale fading, \(\beta _{\mathrm{{1}}}^{1/2},\beta _2^{1/2},...,\beta _M^{1/2}\) and DSDs, \({\tau _{\mathrm{{1}}}},{\tau _2},...,{\tau _M}\) of the overall channel with all RAUs based on the reference signal. It is worth mentioning that all RAUs can also send the same CSI-RS pilot, that is, the CSI-RS pilot adopts a single-port design. At this time, the receiver only estimates the composite signal from all RAUs to the UE. Specifically, multiple RAUs are regarded as multipath signals, and the power and delay of each path are estimated on UE *k* based on the distinguishable time-domain multipath signals. The single-port design can effectively reduce the pilot overhead and design complexity, but it will produce additional channel time/frequency selectivity. In the case of sufficient estimation accuracy, these two methods are theoretically equivalent.

Second, considering that there are *P* equally spaced DMRS in the coherent bandwidth, after LS, the frequency-domain signal collected by UE *k* is

$$\begin{aligned} {{\mathbf {y}}_{N_P}} = {\left[ {{y_{N_1}},{y_{N_2}},...,{y_{N_P}}} \right] ^T} \in {{\mathbb {C}}^{P \times 1}} \end{aligned}$$

(15)

For convenience of representation, the subscript *k* is omitted in the following notation. On subcarrier *p*,

$$\begin{aligned} {y_{N_p}} = \sum \limits _{m = 1}^M {\beta _{m}^{1/2}{\Theta _{m,{N_p}}}{\mathbf {g}}_{m,{N_p}}^T{{\mathbf {w}}_m}} + {z_{N_p}} \end{aligned}$$

(16)

The precoding vector of the channel between UE *k* and RAU *m* is \({\mathbf {w}}_m\). Assuming that the frequency-domain channel remains unchanged on these *P* resource elements, the frequency-domain signal collected by UE *k* can be modeled as

$$\begin{aligned} \left[ {\begin{array}{*{20}{c}} {{y_{N_1}}}\\ \vdots \\ {{y_{N_P}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\Theta _{1,{N_1}}}}&{} \cdots &{}{{\Theta _{M,{N_1}}}}\\ \vdots &{} \ddots &{} \vdots \\ {{\Theta _{1,{N_P}}}}&{} \cdots &{}{{\Theta _{M,{N_P}}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {\beta _{1}^{1/2}{\mathbf {g}}_{1,{N_1}}^T{{\mathbf {w}}_{1}}}\\ \vdots \\ {\beta _{M}^{1/2}{\mathbf {g}}_{M,{N_p}}^T{{\mathbf {w}}_{M}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{z_{N_1}}}\\ \vdots \\ {{z_{N_P}}} \end{array}} \right] \end{aligned}$$

(17)

wherein

$$\begin{aligned} {\varvec{\Omega }} = \left[ {\begin{array}{*{20}{c}} {{\Theta _{1,{N_1}}}}&{} \cdots &{}{{\Theta _{M,{N_1}}}}\\ \vdots &{} \ddots &{} \vdots \\ {{\Theta _{1,{N_P}}}}&{} \cdots &{}{{\Theta _{M,{N_P}}}} \end{array}} \right] \end{aligned}$$

(18)

$$\begin{aligned} {\mathbf {f}} = {\left[ {\beta _{1}^{1/2}{\mathbf {g}}_{1,{N_1}}^T{{\mathbf {w}}_{1}},...,\beta _{M}^{1/2}{\mathbf {g}}_{M,{N_P}}^T{{\mathbf {w}}_{M}}} \right] ^T} = {\left[ {{f_1},...,{f_M}} \right] ^T} \end{aligned}$$

(19)

Subsequently, using minimum mean square error (MMSE) channel estimation, the cross-correlation matrix of \({{\mathbf {y}}_{n_P}}\) and \({\mathbf {f}}\) is

$$\begin{aligned} {{\mathbf {R}}_{{\mathbf {f}}{{\mathbf {y}}_{N_P}}}} = \mathrm{E}\left[ {{\mathbf {f}}{\mathbf {y}}_{N_P}^H} \right] = {{\mathbf {C}}_{\mathbf {f}}}{\varvec{\Omega }}^H \end{aligned}$$

(20)

and the autocorrelation matrix of \({{\mathbf {y}}_{N_P}}\) is

$$\begin{aligned} {{\mathbf {R}}_{{{\mathbf {y}}_{N_P}}{\mathbf {y}}_{N_P}}} = \mathrm{E}\left[ {{{\mathbf {y}}_{N_P}}{\mathbf {y}}_{N_P}^H} \right] = {{\varvec{\Omega }} }{{\mathbf {C}}_{\mathbf {f}}}{\varvec{\Omega }}^H + \frac{{{\sigma ^2}}}{A}{{\mathbf {I}}_P} \end{aligned}$$

(21)

where \({{\mathbf {C}}_{\mathbf {f}}} = diag\left( {\sigma _{{f_1}}^2,...,\sigma _{{f_M}}^2} \right) = diag\left( {{\beta _1},...,{\beta _M}} \right)\) is the covariance matrix of \({\mathbf {f}}\), *A* is the pilot power.

Finally, the MMSE estimate of \({\mathbf {f}}\) is [18]

$$\begin{aligned} \hat{{\mathbf {f}}} = {{\mathbf {R}}_{{\mathbf {f}}{{\mathbf {y}}_{N_P}}}}{\mathbf {R}}_{{{\mathbf {y}}_{N_P}}{{\mathbf {y}}_{N_P}}}^{ - 1}{{\mathbf {y}}_{N_P}} = {\left( {\frac{\psi }{\zeta }{{\mathbf {C}}_{\mathbf {f}}}^{ - 1} + {\varvec{\Omega }}^H{\mathbf \Omega }} \right) ^{ - 1}}{\varvec{\Omega }}^H{{\mathbf {y}}_{N_P}} \end{aligned}$$

(22)

where \(\psi = E\left[ {{{\left| {{\mathbf {h}}_{k,n}^H{{\mathbf {w}}_k}} \right| }^2}} \right] /A\) is the average signal-to-pilot power ratio, and \(\zeta = E\left[ {{{\left| {{\mathbf {h}}_{k,n}^H{{\mathbf {w}}_k}} \right| }^2}} \right] /{\sigma ^2}\) is the average signal-to-noise ratio. Through DMRS interpolation, the MMSE estimate of UE *k* can be obtained as

$$\begin{aligned} {\hat{{\mathbf {y}}}_{N_\varphi }} = {\varvec{\Phi }}{\hat{{\mathbf {f}}}} = {\varvec{\Phi }} {\left( {\frac{\psi }{\zeta }{{\mathbf {C}}_{\mathbf {f}}}^{ - 1} + {\varvec{\Omega }}^H{{\varvec{\Omega }}}} \right) ^{ - 1}}{\varvec{\Omega }}^H{{\mathbf {y}}_{N_P}} \end{aligned}$$

(23)

wherein

$$\begin{aligned} {\varvec{\Phi }} = \left[ {\begin{array}{*{20}{c}} {{\Theta _{1,1}}}&{} \cdots &{}{{\Theta _{M,1}}}\\ \vdots &{} \ddots &{} \vdots \\ {{\Theta _{1,{N_\varphi }}}}&{} \cdots &{}{{\Theta _{M,{N_\varphi }}}} \end{array}} \right] \end{aligned}$$

(24)

Through calculation, the mean square error of channel estimation is

$$\begin{aligned} {{\hbox {MSE}_{{{\hat{{\mathbf {y}}}}}_{{N_\varphi }}}}} = \frac{1}{N_\varphi }\mathrm{E}\left[ {{{\left( {{{\mathbf {y}}}_{{N_\varphi }}} - {{\hat{{\mathbf {y}}}}}_{{N_\varphi }} \right) }^H}\left( {{{\mathbf {y}}}_{{N_\varphi }}} - {{\hat{{\mathbf {y}}}}}_{{N_\varphi }} \right) } \right] = \frac{1}{N_\varphi }tr\left( {{C_{\hat{{\mathbf {f}}}}}{{\varvec{\Phi }} ^H}{\varvec{\Phi }} } \right) \end{aligned}$$

(25)

where \({{\mathbf {C}}_{\hat{{\mathbf {f}}}}} = {({\mathbf {C}}_{\hat{{\mathbf {f}}}}^{ - 1} + \frac{\zeta }{\psi }{\varvec{\Omega }}^H{{\varvec{\Omega }}})^{ - 1}}\) is the covariance matrix of the estimated channel \({\hat{{\mathbf {f}}}}\).