Ferkans — Interactive Telecom Tutor

The Factorization Problem

Sections 20.1-20.3 argued that the hybrid architecture is the right answer at mmWave. We now face the central algorithmic problem: given a target fully-digital precoder $\mathbf{W}_{\text{opt}}$ derived from the channel (for example, the top- $K$ right singular vectors of $\mathbf{H}$ ), find $\mathbf{F}_{\text{RF}}$ (constant modulus) and $\mathbf{F}_{\text{BB}}$ (unconstrained) such that

$(\mathbf{F}_{\text{RF}}, \mathbf{F}_{\text{BB}}) = \arg\min_{\mathbf{F}_{\text{RF}}, \mathbf{F}_{\text{BB}}} \|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\|_F^2$

subject to $|[\mathbf{F}_{\text{RF}}]_{m,n}| = 1/\sqrt{N_t}$ and $\|\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\|_F^2 \leq P_t$ .

This problem is NP-hard in general because of the constant-modulus constraint. The algorithmic insight of the mmWave literature is to exploit the sparsity of the mmWave channel: since $\mathbf{H}$ has only $L$ dominant propagation paths, $\mathbf{W}_{\text{opt}}$ lives in a low-dimensional subspace spanned by transmit steering vectors. Restricting $\mathbf{F}_{\text{RF}}$ to have columns from a dictionary of steering vectors turns the problem into a sparse recovery problem solvable by OMP - and yields optimal solutions whenever $N_{\text{RF}} \geq L$ .

Definition:
Sparse mmWave Channel Model

The narrowband mmWave channel from a $N_t$ -element transmitter to a $N_r$ -element receiver with $L$ dominant paths is

$\mathbf{H} = \sqrt{\frac{N_tN_r}{L}} \sum_{\ell=1}^{L} \alpha_\ell \, \hat{\mathbf{a}}(\phi_\ell^r) \, \mathbf{a}(\phi_\ell^t)^H,$

where $\alpha_\ell \sim \mathcal{CN}(0, 1)$ is the complex path gain, $\phi_\ell^t, \phi_\ell^r$ are the angles of departure and arrival, and $L$ is typically 1-5 at 28 GHz in an outdoor LOS/NLOS mix. Stacking the steering vectors as columns of $\mathbf{A}_t = [\mathbf{a}(\phi_1^t), \ldots, \mathbf{a}(\phi_L^t)]$ and similarly for $\mathbf{A}_r$ , the channel factors as

$\mathbf{H} = \mathbf{A}_r \, \mathbf{\Sigma} \, \mathbf{A}_t^H,$

with $\mathbf{\Sigma}$ diagonal containing the scaled gains. The effective rank is $\min(L, N_t, N_r)$ and is usually equal to $L$ .

The key observation is that $\mathbf{W}_{\text{opt}}$ (the top right singular vectors of $\mathbf{H}$ ) lies in the column span of $\mathbf{A}_t$ . So a good hybrid precoder should use columns of $\mathbf{F}_{\text{RF}}$ drawn from $\mathbf{A}_t$ . Discretizing the angles to a grid yields a finite dictionary, and OMP selects the best $N_{\text{RF}}$ columns.

,

Spatially Sparse Precoding via OMP

Complexity:

O(N_{\text{RF}} \cdot G \cdot N_t \cdot K)

, dominated by dictionary correlation

Input: Target precoder

\mathbf{W}_{\text{opt}} \in \mathbb{C}^{N_t \times K}

,

Tx steering-vector dictionary

\mathbf{A}_t \in \mathbb{C}^{N_t \times G}

,

number of RF chains

N_{\text{RF}}

.

Output:

\mathbf{F}_{\text{RF}}, \mathbf{F}_{\text{BB}}

1.

\mathbf{F}_{\text{RF}} \leftarrow [\,]

(empty)

2.

\mathbf{W}_{\text{res}} \leftarrow \mathbf{W}_{\text{opt}}

3. for

i = 1, 2, \ldots, N_{\text{RF}}

do

4.

\quad

\boldsymbol{\Psi} \leftarrow \mathbf{A}_t^H \mathbf{W}_{\text{res}}

5.

\quad

k^{\star} \leftarrow \arg\max_{k=1,\ldots,G} \|\boldsymbol{\Psi}[k, :]\|_2^2

6.

\quad

\mathbf{F}_{\text{RF}} \leftarrow [\mathbf{F}_{\text{RF}}, \mathbf{A}_t[:, k^{\star}]]

7.

\quad

\mathbf{F}_{\text{BB}} \leftarrow (\mathbf{F}_{\text{RF}}^H \mathbf{F}_{\text{RF}})^{-1} \mathbf{F}_{\text{RF}}^H \mathbf{W}_{\text{opt}}

(LS fit)

8.

\quad

\mathbf{W}_{\text{res}} \leftarrow \dfrac{\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}}{\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\|_F}

9. end for

10.

\mathbf{F}_{\text{BB}} \leftarrow \dfrac{\sqrt{P_t}}{\|\mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}}\|_F} \mathbf{F}_{\text{BB}}

(power normalization)

OMP is a greedy residual-matching algorithm: at each iteration, it selects the steering vector most correlated with the current residual and absorbs it into $\mathbf{F}_{\text{RF}}$ , then re-fits $\mathbf{F}_{\text{BB}}$ in least-squares sense. The outer power normalization in line 10 ensures the total transmit power does not exceed $P_t$ .

Theorem: Optimality of OMP Under Ideal Sparsity

Suppose the target precoder can be written as $\mathbf{W}_{\text{opt}} = \mathbf{A}_t \mathbf{Z}$ for some matrix $\mathbf{Z} \in \mathbb{C}^{L \times K}$ , where $\mathbf{A}_t$ is a dictionary of $L$ orthonormal (or weakly correlated) transmit steering vectors. If the dictionary is contained in the OMP search set and $N_{\text{RF}} \geq L$ , then OMP recovers $\mathbf{F}_{\text{RF}} = \mathbf{A}_t$ and $\mathbf{F}_{\text{BB}} = \mathbf{Z}$ , achieving $\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F = 0$ .

Each OMP iteration reduces the residual by projecting out the best dictionary direction. If the target is exactly a linear combination of $L$ dictionary vectors and those are orthogonal, $L$ iterations peel them off one by one; the residual vanishes at iteration $L$ .

Show Hint

In each iteration, compute $\boldsymbol{\Psi} = \mathbf{A}_t^H \mathbf{W}_{\text{res}}$ and show that its rows are zero except at the remaining true support.

Show that the LS fit step exactly recovers the coefficients of the selected dictionary atoms.

Induction on the iteration count: at iteration $i$ , the residual lies in the span of the $L - i + 1$ unused dictionary vectors.

Proof

Orthogonality of $\mathbf{A}_t$

By hypothesis $\mathbf{A}_t^H \mathbf{A}_t = \mathbf{I}_L$ , so the coefficient matrix satisfies $\mathbf{A}_t^H \mathbf{W}_{\text{opt}} = \mathbf{Z}$ . Non-zero rows of $\mathbf{Z}$ correspond to active dictionary indices, zero rows to inactive ones.

First iteration selects an active index

At iteration 1, $\boldsymbol{\Psi} = \mathbf{A}_t^H \mathbf{W}_{\text{opt}} = \mathbf{Z}$ . The row-norm argmax selects an index $k^{\star}_1$ where $\|\mathbf{Z}[k^{\star}_1, :]\|_2^2 > 0$ , which is by assumption an active index $\ell_1 \in \{1, \ldots, L\}$ .

LS fit and residual orthogonality

After line 7, $\mathbf{F}_{\text{BB}}[1, :] = \mathbf{Z}[\ell_1, :]$ exactly, and the residual $\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}$ has the $\ell_1$ -row of its $\mathbf{A}_t^H$ projection equal to zero. The residual lies in the span of the remaining $L - 1$ active dictionary vectors.

Induction and termination

By induction, after $i$ iterations the residual lies in the span of $L - i$ dictionary vectors. At iteration $L$ the residual reaches zero; for $i > L$ the argmax picks an inactive index whose coefficient vanishes. Hence $N_{\text{RF}} = L$ suffices and $\mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}} = \mathbf{W}_{\text{opt}}$ exactly. $\blacksquare$

Alternating Minimization (MO-AltMin)

Complexity:

O(T_{\text{iter}} \cdot N_t^{2} \cdot N_{\text{RF}})

per outer loop, with

T_{\text{iter}} \sim 20

-

50

Input: Target

\mathbf{W}_{\text{opt}}

, initial

\mathbf{F}_{\text{RF}}^{(0)}

with constant-modulus entries.

Output: Hybrid precoder

(\mathbf{F}_{\text{RF}}, \mathbf{F}_{\text{BB}})

.

1.

t \leftarrow 0

2. repeat

3.

\quad

\mathbf{F}_{\text{BB}}^{(t+1)} \leftarrow (\mathbf{F}_{\text{RF}}^{(t)\,H} \mathbf{F}_{\text{RF}}^{(t)})^{-1} \mathbf{F}_{\text{RF}}^{(t)\,H} \mathbf{W}_{\text{opt}}

# LS step (unconstrained)

4.

\quad

for

m = 1, \ldots, N_t

,

n = 1, \ldots, N_{\text{RF}}

do

5.

\qquad

[\mathbf{F}_{\text{RF}}^{(t+1)}]_{m,n} \leftarrow \dfrac{1}{\sqrt{N_t}} \cdot \text{sign}\!\left([\mathbf{W}_{\text{opt}} \mathbf{F}_{\text{BB}}^{(t+1)\,H}]_{m,n}\right)

# phase-only projection

6.

\quad

end for

7.

\quad

t \leftarrow t + 1

8. until

\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}^{(t)}\mathbf{F}_{\text{BB}}^{(t)}\|_F^2

converges

Alternating minimization (AltMin) alternates between the unconstrained optimum over $\mathbf{F}_{\text{BB}}$ (least squares) and the constrained optimum over $\mathbf{F}_{\text{RF}}$ (entry-wise phase-only projection). Each step is monotonically non-increasing in the objective, so convergence to a stationary point is guaranteed. The fixed point may be a local minimum, so multiple random initializations are used in practice. The "MO-AltMin" variant of Yu et al. (2016) replaces the phase-only projection with a manifold-optimization step that is provably globally convergent.

Alternating Minimization Convergence

Trace the objective function $\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F^2$ versus iteration for alternating minimization on a synthetic target precoder. The number of RF chains controls the fundamental approximation error; under $N_{\text{RF}} \geq L$ the residual drops to machine precision.

Parameters

N_t

64

N_{\text{RF}}

4

L

(true channel paths)3

Iterations40

Hybrid vs Digital Energy Efficiency

Compare the energy efficiency (bits/sec/Hz/W) of a fully-digital and a hybrid architecture as the number of antennas grows. The fully-digital architecture scales its power linearly in $N_t$ ; the hybrid architecture scales only in $N_{\text{RF}}$ . The crossover point is where hybrid becomes the better choice.

Parameters

SNR (dB)10

N_{\text{RF}}

8

P_{\text{RF}}

(W/chain)1

Example: OMP-Based Hybrid Precoding: A Small Case

Consider a 28 GHz channel with $N_t = 32$ , $N_r = 4$ , $L = 2$ paths at azimuths $\phi^t_1 = -30^\circ$ , $\phi^t_2 = 20^\circ$ . The fully-digital precoder aims to transmit $K = 2$ streams along these two directions. Design an OMP-based hybrid precoder with $N_{\text{RF}} = 2$ RF chains.

Solution

Dictionary construction

Build a $32 \times 128$ angular dictionary by sampling $\phi = -90^\circ$ to $+90^\circ$ in steps of $\approx 1.4^\circ$ . Each dictionary column is $\mathbf{a}(\phi_k)$ .

First OMP iteration

Compute correlations $\mathbf{A}_t^H \mathbf{W}_{\text{opt}}$ ; the row with largest $\ell_2$ norm corresponds to the dictionary index closest to $-30^\circ$ (or $+20^\circ$ , depending on path power). Add this column to $\mathbf{F}_{\text{RF}}$ and re-fit $\mathbf{F}_{\text{BB}}$ .

Second OMP iteration

The residual is now orthogonal to the first selected vector. The argmax selects the dictionary entry near the other angle. Adding it produces $\mathbf{F}_{\text{RF}} = [\mathbf{a}(-30^\circ), \mathbf{a}(20^\circ)]$ up to the dictionary grid resolution.

Final residual

With $N_{\text{RF}} = 2 = L$ and the dictionary containing the true steering vectors (exactly or to within grid resolution), the residual $\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F$ is zero modulo the dictionary discretization - typically below $10^{-2}$ in normalized units. $\blacksquare$

🔧Engineering Note

Dictionary Granularity in Practice

The theoretical optimality of OMP (Theorem TOptimality of OMP Under Ideal Sparsity) assumes the true steering directions lie on the dictionary grid. In practice they do not, and the residual floor is governed by the grid resolution. A dictionary of $G = N_t$ directions (one per orthogonal DFT beam) is typically too coarse; $G = 4 N_t$ or more is standard, trading correlation complexity ( $O(G N_t)$ per iteration) for approximation accuracy. For $N_t = 256$ , a dictionary of $G = 1024$ is a reasonable balance. Atomic norm minimization and grid-free methods (e.g., root-MUSIC-style approaches) avoid the discretization entirely but add optimization complexity; they are rarely used in real-time hybrid precoding.

Practical Constraints

•
Dictionary too coarse: loss proportional to $\text{sinc}^2(N_t/G)$
•
Dictionary too fine: OMP correlation step becomes the runtime bottleneck
•
Recommended: $G = 2N_t$ to $4N_t$ for typical mmWave arrays

Common Mistake: AltMin Gets Stuck in Local Minima

Mistake:

Because AltMin is a descent method with guaranteed convergence, one might assume it finds the global optimum of the hybrid factorization problem.

Correction:

Alternating minimization converges to a stationary point, not a global optimum. The constant-modulus constraint set is non-convex, and the objective has many local minima. In practice one runs AltMin from 10-50 random initializations and keeps the best result, or uses the manifold-optimization variant (MO-AltMin) which has stronger convergence guarantees. OMP, in contrast, is deterministic given the dictionary and is optimal when the channel is truly sparse - but suffers more from grid mismatch in the off-grid regime.

🎓CommIT Contribution(2022)

CommIT Multiuser Multibeam Array-Fed Architecture

G. Bartoli, R. Abdolee, G. Caire — IEEE Trans. Wireless Communications

A distinctive CommIT contribution to hybrid beamforming is the array-fed reflector architecture introduced by Bartoli, Abdolee, and Caire (2022). Instead of placing the phase-shifter network between the RF chains and a planar active array, a small active array ( $N_{\text{RF}}$ elements) illuminates a passive parabolic or lens reflector. The reflector implements a fixed (frequency-independent) angular-to-spatial mapping: each point on the focal plane launches a pencil beam in a distinct direction. Multi-user, multi-beam transmission is then achieved by placing one active element per user beam, with the digital precoder handling inter-beam interference.

The architecture has two striking properties. First, it eliminates the phase-shifter network and its insertion loss, replacing it with a lossless quasi-optical structure. Second, the hardware cost scales with the number of users, not the aperture: a reflector with 1 m diameter at 300 GHz has a beamforming aperture of $\sim 10^5$ antennas-equivalent while using only $N_{\text{RF}} = 8$ active chains. This is the same principle behind Starlink's V2 user terminals and is increasingly the design point for sub-THz 6G research. Chapter 21 develops the related array-fed RIS concept.

hybrid-beamformingmmwavesub-thzcommitarray-fedView Paper →

Quick Check

Under what condition is OMP-based hybrid precoding provably exact (zero residual) for an $L$ -path mmWave channel?

$N_{\text{RF}} = L$ and the dictionary contains the true steering vectors

$N_{\text{RF}} = N_t$ , regardless of the dictionary

$N_{\text{RF}} \geq 2K$ , by Theorem 20.1

$L = 1$ (rank-1 LOS only)

Correction:

N_{\text{RF}} = L

and the dictionary contains the true steering vectors

Under sparsity, $N_{\text{RF}} = L$ iterations suffice: each OMP step peels off one path from the residual. Dictionary containment is essential to avoid grid mismatch.

AltMin Residual Trajectory Animation

Animated version of the AltMin convergence plot. The objective $\|\mathbf{W}_{\text{opt}} - \mathbf{F}_{\text{RF}}\mathbf{F}_{\text{BB}}\|_F^2$ is traced iteration by iteration for two configurations side by side: $N_{\text{RF}} \geq L$ (green, drops to machine precision) and $N_{\text{RF}} < L$ (red, plateaus at a positive floor).

Parameters

N_t

64

N_{\text{RF}}

3

L

3

Iterations40

Key Takeaway

Two algorithmic paradigms dominate hybrid precoder design. OMP-based sparse precoding exploits the mmWave channel's angular sparsity and is provably optimal when $N_{\text{RF}} \geq L$ and the dictionary contains the true steering vectors. Alternating minimization (AltMin / MO-AltMin) works on arbitrary channels but converges to local minima and needs careful initialization. Both are descent methods; the choice between them is governed by whether channel sparsity can be assumed.

OMP and Alternating Minimization

The Factorization Problem

Definition: Sparse mmWave Channel Model

Spatially Sparse Precoding via OMP

Theorem: Optimality of OMP Under Ideal Sparsity

Orthogonality of $\mathbf{A}_t$

First iteration selects an active index

LS fit and residual orthogonality

Induction and termination

Alternating Minimization (MO-AltMin)

Alternating Minimization Convergence

Parameters

Hybrid vs Digital Energy Efficiency

Parameters

Example: OMP-Based Hybrid Precoding: A Small Case

Dictionary construction

First OMP iteration

Second OMP iteration

Final residual

Dictionary Granularity in Practice

Common Mistake: AltMin Gets Stuck in Local Minima

CommIT Multiuser Multibeam Array-Fed Architecture

Quick Check

AltMin Residual Trajectory Animation

Parameters

Key Takeaway

Definition:
Sparse mmWave Channel Model