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Why Codebooks, Not Steering Vectors?

In Sections 20.1-20.2 we assumed that $\mathbf{F}_{\text{RF}}$ is a continuous variable to be optimized. In deployed mmWave systems, however, the analog phase-shifter states are selected from a discrete codebook $\mathcal{F} = \{\mathbf{v}^{(1)}, \ldots, \mathbf{v}^{(|\mathcal{F}|)}\}$ of pre-designed beams. There are three reasons.

First, the channel is unknown. Before a data link exists, the base station and user must agree on a beam pair through a training (beam-alignment) phase. A finite codebook defines the set of candidates to try. Second, the RF hardware itself supports only a discrete set of phase states. The codebook must respect this hardware constraint. Third, a well-designed codebook offers a hierarchical structure: coarse beams first, refined beams after, which reduces training overhead from $O(N_t)$ (exhaustive) to $O(\log_2 N_t)$ .

Definition:
DFT Codebook

The DFT codebook for a uniform linear array (ULA) of $N_t$ antennas with $\lambda/2$ spacing consists of $M$ unit-norm beams

$\mathbf{v}^{(m)} = \frac{1}{\sqrt{N_t}} \begin{pmatrix} 1 \\ e^{-j 2\pi m / M} \\ e^{-j 2\pi (2m) / M} \\ \vdots \\ e^{-j 2\pi (N_t-1)m/M} \end{pmatrix}, \quad m = 0, 1, \ldots, M - 1.$

The $m$ -th beam points toward the angle $\theta_m$ satisfying $\pi\sin\theta_m = 2\pi m / M$ , equivalently $\sin\theta_m = 2m/M \bmod 2$ . When $M = N_t$ the beams are orthogonal and tile the angular range $\theta \in [-\pi/2, \pi/2]$ uniformly in $\sin\theta$ . The codebook size $M$ is the angular resolution.

DFT beams are the columns of the $M \times N_t$ DFT matrix. Their angular beamwidth is $\sim 2/N_t$ radians - inversely proportional to the aperture. The number of orthogonal beams is capped at $N_t$ , so oversampling ( $M > N_t$ ) produces overlapping beams but finer pointing resolution.

,

Definition:
5G NR Type I and Type II Codebooks

5G NR Type I codebooks are DFT-like with oversampling: each codeword is a Kronecker product of a horizontal DFT vector and a vertical DFT vector, selected from two independent codebooks of oversampling factors $O_1, O_2$ . A rank- $r$ precoder is formed by $r$ such DFT beams combined with a co-phasing coefficient. The codebook size is $O_1 O_2 N_1 N_2 \cdot 4$ for rank-1 transmission, where $N_1 \times N_2$ is the panel shape.

5G NR Type II codebooks are linear combinations of $L \leq 4$ orthogonal DFT beams per polarization, with amplitudes and phases reported per beam. The feedback overhead scales as $L \cdot (\log_2 O_1 O_2 + b_A + b_P)$ bits, where $b_A$ is the amplitude quantization and $b_P$ the phase quantization. Type II is used for high-rank multi-user MIMO; Type I is used for single-user rank reporting. Both are specified in 3GPP TS 38.214.

The choice of codebook is one of the most impactful design decisions in 5G NR. Type I is simpler and sufficient for high-mobility single-user cases. Type II enables the fine-grained MU-MIMO precoding of Chapters 6-8 at the cost of higher feedback overhead, and is the target of most 5G-Advanced (Release 18) enhancements.

,

Exhaustive Beam Sweep (Baseline)

Complexity:

O(M_{\text{tx}} M_{\text{rx}})

training symbols

Input: Tx codebook

\mathcal{F}_{\text{tx}}

of size

M_{\text{tx}}

,

Rx codebook

\mathcal{F}_{\text{rx}}

of size

M_{\text{rx}}

.

Output: Best beam pair

(\hat{m}, \hat{n})

.

1. for

m = 1, \ldots, M_{\text{tx}}

do

2.

\quad

Tx transmits training sequence with beam

\mathbf{v}_{\text{tx}}^{(m)}

3.

\quad

for

n = 1, \ldots, M_{\text{rx}}

do

4.

\qquad

Rx receives with beam

\mathbf{v}_{\text{rx}}^{(n)}

; measures

r_{m,n} = |\mathbf{v}_{\text{rx}}^{(n)\,H} \mathbf{H} \mathbf{v}_{\text{tx}}^{(m)}|^2

5.

\quad

end for

6. end for

7.

(\hat{m}, \hat{n}) \leftarrow \arg\max_{m,n} r_{m,n}

Each round is one pilot symbol and one RSRP measurement. For $M_{\text{tx}} = M_{\text{rx}} = 64$ , this is 4096 pilot symbols per alignment event - minutes at low SNR, seconds at high SNR. IEEE 802.11ad uses this scheme with small codebooks; 5G NR and 802.11ay use hierarchical refinement instead.

Hierarchical Beam Search

Complexity:

O(\log_2 M)

training symbols

Input: Hierarchical codebook with

K

levels, level

k

of size

2^k

Output: Best fine beam index

\hat{m}

1.

\mathcal{A} \leftarrow \{1, 2, \ldots, 2^K\}

# candidate set, starts with all fine beams

2. for

k = 1, 2, \ldots, K

do

3.

\quad

Partition

\mathcal{A}

into two halves

\mathcal{A}_0, \mathcal{A}_1

4.

\quad

Tx sends with wide beam covering

\mathcal{A}_0

; Rx measures

r_0

5.

\quad

Tx sends with wide beam covering

\mathcal{A}_1

; Rx measures

r_1

6.

\quad

\mathcal{A} \leftarrow \mathcal{A}_{\arg\max(r_0, r_1)}

7. end for

8.

\hat{m} \leftarrow

the single remaining index in

\mathcal{A}

The hierarchical search trades depth for breadth: $2K$ measurements instead of $M = 2^K$ . The catch is that the wide-beam levels must maintain enough beamforming gain to be detectable - not trivial, because a beam twice as wide has half the gain. Hierarchical codebook design is a separate research problem; see Xiao et al. (2016) for one practical solution.

Theorem: Quantization Loss of a DFT Codebook

Let $\mathbf{v}^{\star}$ be the optimal continuous-phase transmit beamformer for a rank-1 LOS channel of direction $\theta_t$ . Let $\mathbf{v}^{(m^{\star})}$ be the closest DFT codeword of size $M$ , selected to minimize $|\theta_t - \theta_m|$ . The beamforming-gain loss satisfies

$\frac{|\mathbf{v}^{(m^{\star})\,H} \mathbf{a}(\theta_t)|^2}{|\mathbf{v}^{\star\,H} \mathbf{a}(\theta_t)|^2} \geq \text{sinc}^2\!\left(\frac{N_t}{M}\right),$

where $\text{sinc}(x) = \sin(\pi x)/(\pi x)$ . In particular, with $M = N_t$ (orthogonal DFT codebook) the worst-case loss is $\text{sinc}^2(1) \approx 0$ , realized at the midpoint between two codewords; doubling $M$ reduces the worst-case loss to about 1 dB.

The DFT beampattern in the angular direction is a sinc-shaped lobe of width $2/N_t$ . If the actual target direction falls exactly between two codebook beams, the loss is governed by the sinc sidelobe at the midpoint - a deep null at half-codebook spacing. Oversampling (larger $M$ ) moves the midpoint closer to the main lobe of the nearest codeword.

Show Hint

Write $\mathbf{v}^{(m^{\star})}$ and $\mathbf{a}(\theta_t)$ as DFT-like vectors and compute their inner product.

Use the closed form $\sum_{k=0}^{N_t-1} e^{jk\Delta} = e^{j(N_t-1)\Delta/2}\sin(N_t\Delta/2)/\sin(\Delta/2)$ .

The worst case is $\Delta = 2\pi/M$ where $\Delta$ is the phase increment mismatch.

Proof

Inner product as a Dirichlet kernel

Write the ULA steering vector with phase increment $\psi = \pi \sin\theta_t$ and the DFT codeword with increment $\psi_m = 2\pi m / M$ . Then $\mathbf{v}^{(m)\,H} \mathbf{a}(\theta_t) = \frac{1}{N_t}\sum_{k=0}^{N_t-1} e^{jk(\psi - \psi_m)}$ , which is a Dirichlet kernel.

Worst-case mismatch

The worst case occurs at the midpoint between two DFT codewords, $\psi - \psi_{m^{\star}} = \pi/M$ . Evaluating the Dirichlet kernel: $|\text{inner product}|^2 = \frac{1}{N_t^{2}}\frac{\sin^2(N_t\pi/(2M))}{\sin^2(\pi/(2M))}$ .

Sinc approximation for large $\ntn{ntx}$

For $N_t/M$ not too large, $\sin(\pi/(2M)) \approx \pi/(2M)$ , so the ratio to the optimal $N_t^{2}/N_t^{2} = 1$ simplifies to $\text{sinc}^2(N_t/M)$ , where the extra factor of $1/2$ is absorbed in the argument convention. At $M = N_t$ the bound is $\text{sinc}^2(1) \approx 0$ ; at $M = 2N_t$ it is $\text{sinc}^2(1/2) = (2/\pi)^2 \approx 0.405$ , a loss of about 3.9 dB. Hence orthogonal DFT codebooks are marginal; oversampling by $2\times$ to $4\times$ is standard in deployed systems. $\blacksquare$

Hierarchical Beam Codebook Patterns

Visualize the beam patterns of a hierarchical codebook across levels. Each level doubles the angular resolution (halves the beamwidth). The plot shows the angular gain versus direction $\theta$ for a selectable codebook size and level.

Parameters

N_t

32

Hierarchy level

k

3

Level $k$ has $2^k$ beams covering the aperture

Beam index3

Example: Training Overhead: Exhaustive vs. Hierarchical

A mmWave link uses $N_t = 128$ transmit antennas and $N_r = 16$ receive antennas, each with its own codebook. Compare the training overhead (in pilot symbols) of exhaustive beam sweep and hierarchical refinement. Assume no beam ambiguity.

Solution

Exhaustive overhead

$M_{\text{tx}} = 128$ , $M_{\text{rx}} = 16$ , giving $M_{\text{tx}} \cdot M_{\text{rx}} = 2048$ pilot symbols per alignment event.

Hierarchical overhead at the Tx

Hierarchy depth $K_{\text{tx}} = \log_2 128 = 7$ , requiring $2 K_{\text{tx}} = 14$ symbols per Tx search.

Hierarchical overhead at the Rx

Hierarchy depth $K_{\text{rx}} = \log_2 16 = 4$ , requiring $2 K_{\text{rx}} = 8$ symbols. Jointly: $14 + 8 = 22$ symbols if the searches are sequential.

Saving

Hierarchical refinement uses $22$ symbols vs $2048$ for exhaustive - a 93x reduction. At 1 ms per symbol this is the difference between 22 ms and 2 s, critical for mobility-induced re-alignment. $\blacksquare$

Common Mistake: Hierarchical Search Needs Enough SNR Per Level

Mistake:

Hierarchical beam search sounds strictly better than exhaustive sweep, so one might conclude to always use it.

Correction:

A wide beam covering $2^{K-k}$ fine directions has array gain $\N_t/2^{K-k}$ , not $\N_t$ . At the coarsest level the gain may be $8$ - $16$ dB below the best beam, and if the per-symbol SNR is insufficient, the wide-beam measurement collapses into noise and the wrong half is chosen. At very low SNR (deep shadow, initial access from afar) an exhaustive sweep with the highest-gain beams can be more reliable despite its overhead.

Beam Alignment

The process by which a transmitter and receiver agree on a pair of analog beams prior to data transmission. In mmWave and sub-THz systems, beam alignment dominates the link-acquisition time because the high path loss demands beamforming gain before any reliable communication can occur.

Why This Matters: Beam Management in 5G NR

5G NR implements beam alignment through the SSB/CSI-RS beam-sweeping protocol of Section 6.1 in TS 38.213. The base station transmits Synchronization Signal Blocks (SSBs) on a sequence of beams during the P1 procedure, allowing the UE to select the best one. Fine refinement follows in procedures P2 (Tx refinement) and P3 (Rx refinement). The number of SSBs per burst is up to 64, directly reflecting the DFT codebook size used at initial access. Failure of beam tracking during mobility triggers a beam-failure recovery procedure - the mmWave counterpart of a handover.

Key Takeaway

Beam codebooks are not a compromise forced by hardware - they are the enabling data structure that makes mmWave link setup tractable. An oversampled DFT codebook provides the angular resolution; a hierarchical structure reduces training overhead from $O(N_t)$ to $O(\logN_t)$ ; and the 3GPP Type I/II codebooks embed both into a deployable standard. Understanding codebook design is understanding where 5G mmWave actually happens.

Codebook Design for Beam Alignment

Why Codebooks, Not Steering Vectors?

Definition: DFT Codebook

Definition: 5G NR Type I and Type II Codebooks

Exhaustive Beam Sweep (Baseline)

Hierarchical Beam Search

Theorem: Quantization Loss of a DFT Codebook

Inner product as a Dirichlet kernel

Worst-case mismatch

Sinc approximation for large $\ntn{ntx}$

Hierarchical Beam Codebook Patterns

Parameters

Example: Training Overhead: Exhaustive vs. Hierarchical

Exhaustive overhead

Hierarchical overhead at the Tx

Hierarchical overhead at the Rx

Saving

Common Mistake: Hierarchical Search Needs Enough SNR Per Level

Beam Alignment

Why This Matters: Beam Management in 5G NR

Key Takeaway

Definition:
DFT Codebook

Definition:
5G NR Type I and Type II Codebooks