In this section, we propose a new approach to design the precoding codebook \(\mathcal {P}\) based on the DFRST matrix.

### Problem formulation

Consider the total effective power \(\|\mathbf {HP}_{i}\|_{F}^{2}\) in (2), where \(\mathbf {P}_{i} \in \mathcal {P}\). The loss in received channel power is expressed as \(\min _{\mathbf {P}\in \mathcal {P}}\left (\|\mathbf {HP}_{opt}\|_{F}^{2}-\|\mathbf {HP}_{i}\|_{F}^{2}\right)\), where **P**
_{
opt
} is the optimal precoding matrix. As a measure of distortion, the average of the loss in received channel power is given as \(E \left \{\min _{\mathbf {P}\in \mathcal {P}}\left (\|\mathbf {HP}_{opt}\|_{F}^{2}-\|\mathbf {HP}_{i}\|_{F}^{2}\right) \right \}\).

Since \(\|\mathbf {HP}_{opt}\|_{F}^{2} \geq \|\mathbf {HP}_{i}\|_{F}^{2}\) for all \(\mathbf {P}\in \mathcal {U}(M_{t},M)\), the loss in received channel power is always non-negative and upper-bounded by [7]

$$\begin{array}{@{}rcl@{}} &&\min_{\mathbf{P}\in \mathcal{P}}\left(\left\|\mathbf{HP}_{opt}\|_{F}^{2}-\|\mathbf{HP}_{i}\right\|_{F}^{2}\right) \\ & \leq & \lambda_{1}^{2}\{\mathbf{H}\} E\left\{\min_{\mathbf{P}\in \mathcal{P}} \frac{1}{2} \left\|\mathbf{P}_{opt} \mathbf{P}_{opt}^{*}- \mathbf{P}_{i}\mathbf{P}_{i}^{*} \right\|_{F}^{2} \right\}, \end{array} $$

(5)

where \(\lambda _{1}^{2}\{\mathbf {H}\}\) denotes the the largest singular value of **H**.

Since \(\lambda _{1}^{2}\{\mathbf {H}\}\) is determined by the channel gain, the loss in received channel power can be minimized by doing the minimization \(E\left \{\min _{\mathbf {P}\in \mathcal {P}} \frac {1}{2} \left \|\mathbf {P}_{opt} \mathbf {P}_{opt}^{*}- \mathbf {P}_{i} \mathbf {P}_{i}^{*} \right \|_{F}^{2}\right \}\). This minimization problem equivalents to maximize the minimum subspace distance between any pair {**P**
_{
k
},**P**
_{
t
}} of codebook column spaces [8, 20, 21]. In another word, the codebook \(\mathcal {P}=\{\mathbf {P}_{1},\mathbf {P}_{2},\ldots,\mathbf {P}_{Q}\}\) is designed to maximize

$$\begin{array}{@{}rcl@{}} \gamma = \min_{1\leq k \neq t \leq Q} \frac{1}{\sqrt{2}} \left\|\mathbf{P}_{k} \mathbf{P}_{k}^{*}-\mathbf{P}_{t} \mathbf{P}_{t}^{*} \right\|_{F}. \end{array} $$

(6)

### Proposed codebook design approach

In this subsection, we propose a new codebook design approach as follows.

**Firstly, construct the precoding matrix **
**P**
_{
R
}. The *M*
_{
t
}×*M* matrix **P**
_{
R
} is formed by any *M* columns of the *M*
_{
t
}×*M*
_{
t
} DFRST matrix as follows,

$$\begin{array}{@{}rcl@{}} {\begin{aligned} P_{R}\,=\,\sqrt{\frac{2}{M_{t}+1}} \left(\begin{array}{ccccc} \sin\frac{\pi}{M_{t}+1}& \cdots & \sin\frac{t\pi}{M_{t}+1} & \cdots &\sin\frac{M\pi}{M_{t}+1}\\ \sin\frac{2\pi}{M_{t}+1}& \cdots & \sin\frac{2t\pi}{M_{t}+1} & \cdots &\sin\frac{2M\pi}{M_{t}+1}\\ \vdots & & & \cdots &\vdots\\ \sin\frac{(M_{t}-1)\pi}{M_{t}+1}& \cdots & \sin\frac{t(M_{t}-1)\pi}{M_{t}+1} & \cdots &\sin\frac{(M_{t}-1)M\pi}{M_{t}+1}\\ \sin\frac{M_{t}\pi}{M_{t}+1}& \cdots & \sin\frac{tM_{t}\pi}{M_{t}+1} & \cdots & \sin\frac{M_{t} M \pi}{M_{t}+1} \end{array} \right), \end{aligned}} \end{array} $$

(7)

where 1≤*t*≤*M*.

**Secondly, determine the integers **
\(\protect \phantom {\dot {i}\!}\mathbf {u}=[u_{1},u_{2}, \ldots, u_{M_{t}-1}]\) Following the literature [7, 17], let

$$\begin{array}{@{}rcl@{}} \mathbf{\Theta}= \left(\begin{array}{cccccc} 1 & & \\ &e^{-j2u_{1}} & &\\ & & \ddots & & \\ & & &e^{-j2(i-1)u_{i-1}}& &\\ & & & &\ddots\\ &&& & & e^{-j2(M_{t}-1)u_{M_{t}-1}} \end{array} \right). \end{array} $$

(8)

and

$$\begin{array}{@{}rcl@{}} \mathbf{P}_{i}=\mathbf{\Theta}^{i-1}\mathbf{P}_{R}, \end{array} $$

(9)

where *i*=1,2,…,*Q*. Then, we can choose \(u_{1},u_{2},\ldots, u_{M_{t}-1}\phantom {\dot {i}\!}\) from the set \(\phantom {\dot {i}\!}\mathcal {Z}=\{ \mathbf {u}\in \mathbb {Z}^{M_{t}-1} | \forall k,0 \leq u_{k} \leq Q-1 \}\) by

$$ \mathbf{u}=\arg\max_{\mathcal{Z}}\min_{2\leq i \leq Q} \frac{1}{\sqrt{2}} \|\mathbf{P}_{R} \mathbf{P}_{R}^{*}-\mathbf{P}_{i} \mathbf{P}_{i}^{*} \|_{F}. $$

(10)

where \(\phantom {\dot {i}\!}\mathcal {Z}=\{ \mathbf {u}\in \mathbb {Z}^{M_{t}-1} | \forall k,0 \leq u_{k} \leq Q-1 \}\) denotes the set that contains all the possible combinations of \(\phantom {\dot {i}\!}\mathbf {u}=\left [u_{1},u_{2},\ldots, u_{M_{t}-1}\right ] \), for any *k*∈ [ 1,*M*
_{
t
}−1], *u*
_{
k
}∈ [ 0,*Q*−1].

The precoding codebook \(\mathcal {P}\) can be stored at the transmitter/receiver using ⌈(*M*
_{
t
}−1) log2*Q*⌉ bits as only \(u_{1},u_{2},\ldots, u_{M_{t}-1}\phantom {\dot {i}\!}\) must be stored. Then, the chosen precoding matrix **P**
_{
i
} at the transmitter, corresponding to the feedback index *i*, can be easily calculated by computing **Θ**
^{i−1}
**P**
_{
R
},*i*=2,3,…,*Q*. The proposed codebook design approach can be described by Algorithm 1. Note that the proposed approach is to design the codebook for the precoding system, and this procedure can be completed off-line. Therefore, the proposed approach can be applied into the different scenarios, including different transmit antennas, receive antennas, and modulation scheme. Hence, the proposed approach can be applied for vast number of data transmission.