NR - Downlink Codebook

The \(\mathbf{W}_1\) in \eqref{eq:w} can be further expressed as

\begin{align*} \mathbf{W}_1 &= \begin{bmatrix} \mathbf{B} & \\ & \mathbf{B} \end{bmatrix} \\ &= \mathbf{I}_2 \otimes \mathbf{B}, \end{align*}

where \(\mathbf{B} \triangleq \begin{bmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_L \end{bmatrix} \in \mathbb{C}^{N_1N_2 \times L}\) are the candidate beams/vectors over the whole bandwidth.

• For rank 1 or 2, \(L\) can be configured to 1 or 4 by CodebookMode. Otherwise, \(L\) is fixed to 1.
• Component columns of \(\mathbf{B}\) are spatially correlated.
• There are two structures for \(\mathbf{B}\).
  • \(\mathbf{b}_i \in \mathcal{F}_{N_1 \times N_2}^{O_1 \times O_2}\), \(i = 1, \ldots, L\).
  • If \(P_{\text{CSI-RS}} \ge 16\) and rank equals 3 or 4, each column can be expressed as

\begin{align} \mathbf{b}_i(\theta) = \begin{bmatrix}1 \\ \theta \end{bmatrix} \otimes \widetilde{\mathbf{b}}_i, \quad |\theta|=1, \widetilde{\mathbf{b}}_i \in \mathcal{F}_{N_1/2 \times N_2}^{O_1 \times O_2}.\label{eq:occ} \end{align}

For the sake of inter-layer interference avoidance, the precoding vectors for multi-rank transmission should be orthogonal. To this end, following 3 methods have been utilized, and the methods for each rank value can be summarized in Table 1.

1. Orthogonal vectors, e.g., DFT beams.
2. Opposite co-phasing between the both polarizations.
3. Use shorter DFT vectors instead and apply a length-2 orthogonal cover code (OCC) on them. For a vector as in \eqref{eq:occ}, \(\forall \theta, |\theta|=1, \mathbf{b}_i(\theta) \perp \mathbf{b}_i(-\theta)\), \(i = 1, \ldots, L\).

Table 1: Methods of keeping inter-layer orthogonal

Rank	Methods
2 (\(L = 1\))	1 or 2 beam(s) in \(\mathcal{F}_{N_1 \times N_2}^{O_1 \times O_2}\) + co-phasing
2 (\(L = 4\))	2 beams in \(\mathcal{F}_{N_1 \times N_2}^{O_1 \times O_2}\) + co-phasing
3, 4 (\(P_{\text{CSI-RS}} < 16\))	2 beams in \(\mathcal{F}_{N_1 \times N_2}^{O_1 \times O_2}\) + co-phasing
3, 4 (\(P_{\text{CSI-RS}} \ge 16\))	1 beam in \(\mathcal{F}_{N_1/2 \times N_2}^{O_1 \times O_2}\) + 2x-OCC + co-phasing
5, 6	3 beams in \(\mathcal{F}_{N_1 \times N_2}^{O_1 \times O_2}\) + co-phasing
7, 8	4 beams in \(\mathcal{F}_{N_1 \times N_2}^{O_1 \times O_2}\) + co-phasing

In the CSI feedback, PMI selection can be limited within a subset of the entire codebook, a.k.a. codebook subset restriction.

In 2-port case, i.e., nrOfAntennaPorts = two, the restriction is indicated by twoTX-CodebookSubsetRestriction, which is essentially a length-6 bitmap. The 4 least significant bits indicate the availability of the 4 codewords for rank-1 transmission, and the 2 most significant bits correspond to the 2 codewords for rank-2 transmission.

For more than two ports scenarioes, the codebook subset restriction is performed at the stage of \(\mathbf{W}_1\) selection. It is indicated by n1-n2, a length-\(N_1O_1N_2O_2\) bitmap.

Besides codebook subset restriction, rank adaptation is also restricted, indicated by typeI-SinglePanel-ri-Restriction. The parameter is a length-8 bitmap, with each bit indicating the availability of a rank candidate.

Different from type I codebook, in type II codebook based PMI selection, each precoding vector, corresponding to a layer, is explicitly expressed as a weighted sum of an incomplete orthogonal basis vectors. E.g., the precoding vector for \(l\) th layer on \(s\) th subband can be written as

\begin{align*} \mathbf{w}_{l,s} = \begin{bmatrix} \mathbf{B} & \\ & \mathbf{B} \end{bmatrix} \begin{bmatrix} \mathbf{c}_{l,s,1} \\ \mathbf{c}_{l,s,2} \end{bmatrix}, \quad l = 1, 2; s = 1, 2, \ldots, S; \end{align*}

where

• \(S\) is the number of subbands.
• \(\mathbf{B} \triangleq \begin{bmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_L \end{bmatrix}\) is the orthogonal basis, \(\mathbf{b}_m \in \mathbb{C}^{N_1N_2 \times 1}\), \(m = 1, 2, \ldots, L\). In 4-port, i.e., \(P_{\text{CSI-RS}} = 4\) case, \(L\) is fixed to 2; Otherwise, \(L\) can be configured to 2, 3, or 4 by numberOfBeams. The basis vectors are essentially a subset of a complete DFT basis vectors.
  • The complete basis is indicated by a pair of oversampling factors, i.e., \(i_{1,1} = \begin{bmatrix} q_1 & q_2 \end{bmatrix}\), \(q_1 \in \{0, 1, \ldots, O_1 - 1\}\), \(q_2 \in \{0, 1, \ldots, O_2 - 1\}\).
  • The selected subset out of the basis is indicated by \(i_{1, 2}\) out of \(C_{N_1N_2}^L\) candidates.
• \(\mathbf{c}_{l,s,r} = \begin{bmatrix} c_{l,s,r,1} \\ c_{l,s,r,2} \\ \vdots \\ c_{l,s,r,L} \end{bmatrix} \in \mathbb{C}^{L\times 1}, r = 1, 2\). \(c_{l,s,r,m}\) is the corresponding complex-valued weight of \(\mathbf{b}_m\), \(m = 1, 2, \ldots, L\). Each weight can be further defactorized into following 3 parts.

  • Wideband amplitude \(\mathbf{p}_{l,r}^{(1)} \triangleq \begin{bmatrix} p_{l,1,r}^{(1)} & p_{l,2,r}^{(1)} & \cdots & p_{l,L,r}^{(1)} \end{bmatrix} \in \mathbb{C}^{1 \times L}\), \(r=1,2\), indicated by \(i_{1,4,l}\). All the wideband amplitudes are normalized to the strongest weights/coefficients, whose index is indicated by \(i_{1,3,l} = L r_{l,\max} + m_{l,\max}\), where

    \begin{align*} (m_{l,\max}, r_{l,\max}) = \arg\max_{\substack{m=1,2,\ldots,L;\\r=1,2.}} p_{l,m,r}^{(1)}. \end{align*}
  • Subband amplitude \(\mathbf{p}_{l,r}^{(2)} \triangleq \begin{bmatrix} p_{l,1,r}^{(2)} & p_{l,2,r}^{(2)} & \cdots & p_{l,L,r}^{(2)} \end{bmatrix} \in \mathbb{C}^{1 \times L}\), \(r=1,2\), indicated by \(i_{2,2,l}\).
  • Subband phase \(\varphi_l\) indicated by \(i_{2,1,l}\).

Particularly, for the strongest weight/coefficient, its amplitudes, irrespective of wideband or subband, are considered equal to the highest quantization level by default; Further its phase is identified as the reference for other coefficients. Therefore, the amplitudes and phase corresponding to the strongest coefficients are not reported.

The granularities of the information discussed above can be listed in Table 2.

Table 2: The granularity of the payload feedback

Entry	Granularity
Orthogonal basis vectors (\(i_{1,1}\) and \(i_{1,2}\))	Common to all layers/polarizations/subbands
The index of the strongest coefficient (\(i_{1,3,l}\))	Per-layer
Wideband amplitude (\(i_{1,4,l}\))	Per-layer/polarization
Subband amplitude (\(i_{2,2,l}\))	Per-layer/polarization/subband
Subband phase (\(i_{2,1,l}\))	Per-layer/polarization/subband

The final PMI payload for feedback is comprised of two parts,

\begin{align*} i_1 & = \begin{bmatrix} i_{1,1} & i_{1,2} & i_{1,3,l} & i_{1,4,l} \end{bmatrix}; \\ i_2 &= \begin{cases} \begin{bmatrix} i_{2,1,l} \end{bmatrix}, & \color{blue}{subbandAmplitude = false}; \\ \begin{bmatrix} i_{2,1,l} & i_{2,2,l} \end{bmatrix}, & \color{blue}{subbandAmplitude = true}; \end{cases} \end{align*}

where \(l = 1, \ldots, v\).

For type II codebook, the codebook subset restriction and rank restriction are configured by n1-n2-codebookSubsetRestriction and typeII-RI-Restriction, respectively.

For type II codebook based PMI selection, the detailed procedure can be summarized as following steps.

1. Orthogonal basis selection.
   • According to the configured CSI-RS, perform channel estimation and obtain the channel matrix on each CSI-RS sample, i.e., \(\mathbf{H}_k \in \mathbb{C}^{R_x \times P_{\text{CSI-RS}}}, k \in \mathcal{K}\), where \(\mathcal{K}\) is the set of indices of CSI-RS samples throughout the bandwidth and \(R_x\) is the number of TXRU equipped at the target user. For polarization separation, the channel matrix can be further written as \(\mathbf{H}_k = \begin{bmatrix} \mathbf{H}_{k,1} & \mathbf{H}_{k,2} \end{bmatrix}\), \(\mathbf{H}_{k,r} \in \mathbb{C}^{R_x \times P_{\text{CSI-RS}}/2}, r = 1, 2\).

   • Compute the average covariance matrix over the wideband.

     \begin{align*} \mathbf{C} \triangleq \dfrac{1}{| \mathcal{K}|} \sum_{k \in \mathcal{K}} \sum_{r=1}^{2} \mathbf{H}_{k,r}^H \mathbf{H}_{k,r}. \end{align*}

   • Select the optimal orthogonal basis from the \(O_1O_2C_{N_1N_2}^L\) candidates, i.e.,

     \begin{align*} \mathbf{B}_{\text{opt}} = \arg\max_{\mathbf{B}} \text{Tr}(\mathbf{B}^H \mathbf{C} \mathbf{B}). \end{align*}
2. Calculate per-subband per-layer per-polarization target vectors.

   • For \(s\) th subband, calculate the target vector \(\mathbf{w}_{l, s}\) for \(l\) th layer, which is \(l\) th column of \(\mathbf{W}_s\), the eigen matrix of the average covariance matrix, i.e.,

     \begin{align*} \sum_{k \in \mathcal{K}_s}\mathbf{H}_k^H \mathbf{H}_k = \mathbf{W}_s \mathbf{\Lambda}_s \mathbf{W}_s^H, \end{align*}

     where \(\mathcal{K}_s\) is the set of indices of the CSI-RS samples in \(s\) th subband, and \(\mathbf{\Lambda}_s\) is the corresponding diagonal matrix with the eigen values in the diagonal axis in a descending order, \(s = 1, 2, \ldots, S\), where \(S\) is the number of subbands.

   • For \(l\) th layer on \(s\) th subband, the per-polarization target vectors, \(\mathbf{w}_{l,s,1}\) and \(\mathbf{w}_{l,s,2}\), are the two subvectors of \(\mathbf{w}_{l,s}\), i.e., \(\mathbf{w}_{l,s} = \begin{bmatrix} \mathbf{w}_{l,s,1} \\ \mathbf{w}_{l,s,2} \end{bmatrix}\), \(\mathbf{w}_{l,s,1}, \mathbf{w}_{l,s,2} \in \mathbb{C}^{P_{\text{CSI-RS}}/2 \times 1}\), \(s = 1, 2, \ldots, S\).
3. Weights determination, quantization, and feedback.

   • For \(l\) th layer in \(r\) th polarization, the wideband amplitude can be computed according to

     \begin{align*} p_{l,m,r}^{(1)} = |\mathbf{b}_m^H \mathbf{w}_{l,r}^{(1)}|, \quad l = 1, \ldots, v; r = 1, 2; \end{align*}

     where \(\mathbf{w}_{l,r}^{(1)}\) is \(l\) th column of \(\mathbf{W}_r^{(1)}\),

     \begin{align*} \sum_{k\in \mathcal{K}} \mathbf{H}_{k,r}^H \mathbf{H}_{k,r} = \mathbf{W}_r^{(1)} \mathbf{\Lambda}_r \left[\mathbf{W}_r^{(1)} \right]^H, \quad r = 1, 2. \end{align*}

   • The index of the strongest coefficient can be obtained as \(i_{1,3,l} = Lr_{l,\max}+m_{l,\max}\), where

     \begin{align*} (m_{l,\max}, r_{l,\max}) = \arg\max_{\substack{m=1, \ldots, L;\\r=1,2.}} p_{l,m,r}^{(1)}, \quad l = 1, \ldots, v. \end{align*}

   • For \(l\) th layer on \(s\) th subband, in \(r\) th polarization, the complex-valued coefficient of \(\mathbf{b}_m\) can be obtained by

     \begin{align*} c_{l,s,r,m} = \mathbf{b}_m^H \mathbf{w}_{l,s,r}, \quad s = 1, \ldots, S; r = 1, 2; m = 1, \ldots, L. \end{align*}

   • Then, the per-subband per-polarization amplitude coefficient is

     \begin{align*} p_{l,s,r,m}^{(2)} = \frac{|c_{l,s,r,m}|}{p_{l,m_{l,\max}, r_{l,\max}}^{(1)} p_{l,m,r}^{(1)}}, \quad s = 1, 2, \ldots, S; r = 1, 2; m = 1, 2, \ldots, L; \end{align*}

     and the per-subband per-polarization phase is

     \begin{align*} \varphi_{l,s,r,m}^{(2)} = \angle \frac{c_{l,s,r,m}}{\mathbf{b}_{m_{l,\max}}^H \mathbf{w}_{l,r}^{(1)}}, \quad s = 1, 2, \ldots, S; r = 1, 2; m = 1, 2, \ldots, L. \end{align*}
   • Quantization and feedback.

In release 16, type II codebook and type II port selection codebook are further enhanced. In order to reduce the overhead for subband-wise PMI feedback, for each layer, the target vectors throughout all the subbands are transformed into the time domain by an IDFT operation. Then, a fraction of stronger taps/vectors are kept and each of them is factorized as a weighted sum of the orthogonal basis vectors. Eventually, the weights are quantized and reported to the network. With the introduction of the frequency-domain compression, the overhead is remarkably reduced. Even for rank-3 and rank-4 feedback, the overheads are not intolerable any more. Therefore, rank-3 and rank-4 reporting are also supported.

In mathematical sense, the precoding vectors for \(l\) th layer throughout all the subbands can be expressed as

\begin{align*} \begin{bmatrix} \mathbf{w}_{l,1} & \mathbf{w}_{l,2} & \cdots & \mathbf{w}_{l,N_3} \end{bmatrix} = \begin{bmatrix} \mathbf{B} & \\ & \mathbf{B} \end{bmatrix} \begin{bmatrix} \mathbf{c}_{l,1,1} & \mathbf{c}_{l,2,1} & \cdots & \mathbf{c}_{l,M_v,1} \\ \mathbf{c}_{l,1,2} & \mathbf{c}_{l,2,2} & \cdots & \mathbf{c}_{l,M_v,2}\end{bmatrix} \mathbf{W}_{\text{DFT}}^H, \quad l = 1, \ldots, v; \end{align*}

where

• \(N_3\) is the DFT size in the configured bandwidth part (BWP).
• \(v\) is the rank number.
• \(\mathbf{w}_{l,f} \in \mathbb{C}^{P_{\text{CSI-RS}} \times 1}\) is the precoding vector for \(l\) th layer on \(f\) th PMI sample, \(f = 1, 2, \ldots, N_3\).
• \(\mathbf{B} \in \mathbb{C}^{N_1N_2 \times L}\) is the orthogonal basis, whose linear weighted sum is used to approximate target vectors. Just as in release 15, for enhanced type II codebook, the basis is indicated by \(i_{1,1}\) and \(i_{1,2}\); While for enhanced type II port selection codebook, the basis is signalled by \(i_{1,1}\). Additionally, parameters \(L\), together with the subsequent \(p_v\) and \(\beta\) are jointly configured by ParamCombination.
• \(\mathbf{W}_{\text{DFT}} \in \mathbb{C}^{N_3 \times M_v}\), where
  • \(M_v = \lceil p_v N_3/R \rceil\) is the number of strongest target taps/vectors selected for each layer.
    • \(R \in \{1, 2\}\) is the PMI granularity within each subband, which is configured by numberOfPMISubbandsPerCQISubband. In other words, there are \(R\) precoders for a regular subband.
    • Particularly, when \(R=2\), for the first or last subbands in a configured BWP, if the number of PRB's inside is larger than \(N_{\text{PRB}}^{\text{SB}}/2\), there are two PMI values within the subband; Otherwise, there is only one PMI value within the subband; where \(N_{\text{PRB}}^{\text{SB}}\) is subband size in terms of PRB, configured by subbandSize.
  • Due to the sparsity in the time domain, only the strongest \(M_v\) taps are chosen out of the \(N_3\) ones. The columns in \(\mathbf{W}_{\text{DFT}}\) stands for the target taps picked out of the \(N_3\) candidates, indicated by \(i_{1,5}\) and \(i_{1,6,l}\).
    • If \(N_3 > 19\), the candidate taps are first constrained to a window, delimited by a starting position \(M_{\text{initial}}\) (\(i_{1,5}\)) and a length \(2M_v\). Then, \(M_v\) taps are further selected out of the window for final feedback (\(i_{1,6,l}\)).
    • If \(N_3 \le 19\), \(M_v\) taps are directly selected out of the \(N_3\) candidates (\(i_{1,6,l}\)¹).
• \(\mathbf{c}_{l,t,r} \in \mathbb{C}^{L \times 1}\) is the coefficients of \(t\) th time-domain tap for \(l\) th layer in \(r\) th polarization, \(t = 1, 2, \ldots, M_v; r = 1, 2\).
  • After tap selection, each tap vector is expanded as a linear weighted sum of the vectors in \(\mathbf{B}\). Totally \(2LM_v\) coefficients are yielded and quantized.
  • For the sake of overhead reduction, only the non-zero coefficients are reported, indicated by a length-\(2LM_v\) bitmap, \(i_{1,7,l}\). What is worthy of attention, following two limitations are imposed on the selection:
    • For each layer, the number of non-zero coefficients should be no more than \(K_0\), where \(K_0 = \lceil \beta 2LM_v \rceil\);
    • The total number of non-zero coefficients for all the layers should be no more than \(2K_0\).
  • The non-zero coefficients indicated by the bitmap are quantized into amplitudes (\(i_{2,4,l}\)) and phases (\(i_{2,5,l}\)) with the strongest coefficient (\(i_{1,8,l}\)) as the reference.

The granularities of the information discussed above can be listed in Table 3.

Table 3: The granularity of the payload feedback

Entry	Granularity
Orthogonal basis vector (\(i_{1,1}\) and \(i_{1,2}\))	Common to all layers/polarizations/taps
Tap window delimiter (\(i_{1,5}\))	Common to all layers/polarizations
Tap selection (\(i_{1,6,l}\))	Per-layer
Coefficient selection (\(i_{1,7,l}\))	Per-layer/polarization/tap
The index of the strongest coefficient (\(i_{1,8,l}\))	Per-layer
The amplitude of the strongest tap (\(i_{2,3,l}\))	Per-layer/polarization
Tap amplitude (\(i_{2,4,l}\))	Per-layer/polarization/tap
Tap phase (\(i_{2,5,l}\))	Per-layer/polarization/tap

The final PMI payload for feedback is comprised of two parts,

\begin{align*} i_1 &= \begin{bmatrix} i_{1,1} & i_{1,2} & i_{1,5} & i_{1,6,l} & i_{1,7,l} & i_{1,8,l} \end{bmatrix}, \\ i_2 &= \begin{bmatrix} i_{2,3,l} & i_{2,4,l} & i_{2,5,l} \end{bmatrix}; \end{align*}

where \(l = 1, \ldots, v\).

Similarly, for enhanced type II codebook, codebook subset restriction and rank restriction are configured by n1-n2-codebookSubsetRestriction-r16 and TypeII-ri-Restriction-r16, respectively; For enhanced type II port selection codebook, the rank restriction is indicated by TypeII-PortSelection-ri-Restriction-r16.

For enhanced type II codebook, the procedure for PMI feedback is similar to that of type II codebook, which can be listed as follows.

1. Orthogonal basis selection, exactly the same as for type II codebook.
2. Calculate the \(N_3\) target vectors in each polarization for each layer.

   • Calculate the target vector \(\mathbf{w}_{l,f}\) for \(l\) th layer \(f\) th PMI sample in the frequency domain, which is \(l\) th column of \(\mathbf{W}_f\), the eigen matrix of the average covariance matrix, i.e.,

     \begin{align*} \sum_{k \in \mathcal{K}_f} \mathbf{H}_k^H \mathbf{H}_k = \mathbf{W}_f \mathbf{\Lambda}_f \mathbf{W}_f^H, \end{align*}

     where \(\mathcal{K}_f\) is the set of indices of the CSI-RS samples corresponding to \(f\) th PMI, and \(\mathbf{\Lambda}_f\) is the corresponding diagonal matrix with the eigen values in the diagonal axis in a descending order, \(f=1,2,\ldots,N_3\).

   • For \(f\) th PMI sample of \(l\) th layer, the per-polarization target vector, \(\mathbf{w}_{l,f,1}\) and \(\mathbf{w}_{l,f,2}\), are the two subvectors of \(\mathbf{w}_{l,f}\), i.e., \(\mathbf{w}_{l,f} = \begin{bmatrix} \mathbf{w}_{l,f,1} \\ \mathbf{w}_{l,f,2} \end{bmatrix}\), \(\mathbf{w}_{l,f,1}, \mathbf{w}_{l,f,2} \in \mathbb{C}^{P_{\text{CSI-RS}}/2 \times 1}\), \(f = 1, 2, \ldots, N_3\).

3. IDFT-based frequency-domain compression. Taking \(l\) th layer for instance, frequency-domain compression can be illustrated by

   \begin{align*} \begin{bmatrix} \widetilde{\mathbf{w}}_{l,1,1} & \widetilde{\mathbf{w}}_{l,2,1} & \cdots & \widetilde{\mathbf{w}}_{l,M_v,1} \\ \widetilde{\mathbf{w}}_{l,1,2} & \widetilde{\mathbf{w}}_{l,2,2} & \cdots & \widetilde{\mathbf{w}}_{l,M_v,2} \end{bmatrix} = \begin{bmatrix} \mathbf{w}_{l,1,1} & \mathbf{w}_{l,2,1} & \cdots & \mathbf{w}_{l,N_3,1} \\ \mathbf{w}_{l,1,2} & \mathbf{w}_{l,2,2} & \cdots & \mathbf{w}_{l,N_3,2} \end{bmatrix} \mathbf{W}_{\text{DFT}}^{*}, \quad l = 1, \ldots, v, \end{align*}

   where \(\widetilde{\mathbf{w}}_{l,t,r}\) is \(t\) th tap vector in the time domain in \(r\) th polarization for \(l\) th layer, \(t = 1, 2, \ldots, M_v; r = 1, 2\). In this way, \(M_v\) strongest taps are selected.

4. Weights determination and quantization for the resultant time-domain tap vectors.
5. Feed back the non-zero coefficients, as the same as in type II codebook based PMI feedback.