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The Core Idea of JSDM

The central insight of JSDM is a separation of timescales. The spatial covariance $\mathbf{R}_k$ changes slowly (seconds to minutes), while the instantaneous channel $\mathbf{h}_k$ fluctuates on the coherence time (milliseconds). JSDM decomposes precoding into two stages: a pre-beamforming matrix $\mathbf{B}_g$ designed from the slow-changing covariance (one per group), and an inner MU-MIMO precoder $\mathbf{P}_g$ designed from the fast-changing effective channel (which lives in a much lower dimension). The result is a system that delivers near-optimal MU-MIMO performance with drastically reduced CSI overhead.

Definition:
Two-Stage JSDM Precoding

Consider a base station with $N_t$ antennas serving $K$ users partitioned into $G$ groups. The JSDM precoder for user $k \in \mathcal{S}_g$ is

$\mathbf{v}_{k} = \mathbf{B}_g \mathbf{p}_k$

where:

$\mathbf{B}_g \in \mathbb{C}^{N_t \times r_g}$ is the pre-beamforming matrix for group $g$ , designed from the long-term covariance eigenspace of the group. The columns of $\mathbf{B}_g$ span (approximately) the dominant eigenspace of $\bar{\mathbf{R}}_g$ .
$\mathbf{p}_k \in \mathbb{C}^{r_g}$ is the inner precoding vector for user $k$ , computed from the reduced-dimension effective channel $\tilde{\mathbf{h}}_k = \mathbf{B}_g^H \mathbf{h}_k \in \mathbb{C}^{r_g}$ .

The composite precoding matrix for all users in group $g$ is $\mathbf{W}_{g} = \mathbf{B}_g \mathbf{P}_g$ where $\mathbf{P}_g = [\mathbf{p}_k]_{k \in \mathcal{S}_g} \in \mathbb{C}^{r_g \times |\mathcal{S}_g|}$ .

Pre-beamforming matrix

The outer-stage matrix $\mathbf{B}_g \in \mathbb{C}^{N_t \times r_g}$ in JSDM that projects onto the covariance eigenspace of group $g$ . It is computed from long-term statistics and does not require instantaneous CSI. It reduces the effective channel dimension from $N_t$ to $r_g$ .

Inner MU-MIMO precoder

The inner-stage precoding matrix $\mathbf{P}_g \in \mathbb{C}^{r_g \times |\mathcal{S}_g|}$ that handles multi-user interference within group $g$ . It operates on the effective channel $\tilde{\mathbf{H}}_k = \mathbf{B}_g^H \mathbf{H}_{k}$ of dimension $r_g \ll N_t$ , enabling standard MU-MIMO techniques (ZF, MMSE) at reduced complexity.

Related: {{Ref:Gloss Pre Beamformer}}

Theorem: JSDM Received Signal Model

Under JSDM two-stage precoding, the received signal at user $k \in \mathcal{S}_g$ is

$y_k = \underbrace{\mathbf{h}_k^H \mathbf{B}_g \mathbf{p}_k}_{\text{desired signal}} \; + \underbrace{\sum_{j \in \mathcal{S}_g \setminus \{k\}} \mathbf{h}_k^H \mathbf{B}_g \mathbf{p}_j}_{\text{intra-group interference}} \; + \underbrace{\sum_{g' \neq g} \sum_{j \in \mathcal{S}_{g'}} \mathbf{h}_k^H \mathbf{B}_{g'} \mathbf{p}_j}_{\text{inter-group interference}} \; + \; w_k$

where $w_k \sim \mathcal{CN}(0, \sigma^2)$ . Using the effective channel $\tilde{\mathbf{h}}_k = \mathbf{B}_g^H \mathbf{h}_k$ , the intra-group terms simplify to $\tilde{\mathbf{h}}_k^H \mathbf{p}_j$ , which is a standard $r_g$ -dimensional MU-MIMO problem.

The pre-beamformer $\mathbf{B}_g$ acts as a spatial filter that focuses on group $g$ 's angular region. Users in other groups are (ideally) suppressed because their eigenspaces are approximately orthogonal. The remaining intra-group interference is handled by the inner precoder $\mathbf{P}_g$ using standard MU-MIMO techniques on the effective channel.

Proof

Transmit signal structure

The transmitted signal is $\mathbf{x} = \sum_{g=1}^{G} \mathbf{B}_g \mathbf{P}_g \mathbf{s}_g$ where $\mathbf{s}_g = [s_k]_{k \in \mathcal{S}_g}$ are the data symbols for group $g$ .

Received signal at user $k \in \mathcal{S}_g$

User $k$ receives $y_k = \mathbf{h}_k^H \mathbf{x} + w_k = \mathbf{h}_k^H \sum_{g'=1}^{G} \mathbf{B}_{g'} \mathbf{P}_{g'} \mathbf{s}_{g'} + w_k.$

Separate intra- and inter-group terms

Isolating the group $g$ contribution: $y_k = \underbrace{\mathbf{h}_k^H \mathbf{B}_g \mathbf{P}_g \mathbf{s}_g}_{\text{own group}} + \underbrace{\sum_{g' \neq g} \mathbf{h}_k^H \mathbf{B}_{g'} \mathbf{P}_{g'} \mathbf{s}_{g'}}_{\text{other groups}} + w_k.$ The own-group term further decomposes into desired signal plus intra-group interference. The inter-group term is small when the eigenspaces $\mathcal{R}(\mathbf{B}_g)$ and $\mathcal{R}(\mathbf{B}_{g'})$ are approximately orthogonal. $\blacksquare$

Definition:
Effective Reduced-Dimension Channel

For user $k \in \mathcal{S}_g$ , the effective channel after pre-beamforming is

$\tilde{\mathbf{h}}_k \triangleq \mathbf{B}_g^H \mathbf{h}_k \in \mathbb{C}^{r_g}.$

If $\mathbf{B}_g = \mathbf{U}_g^{(r_g)}$ (the dominant eigenvectors of $\bar{\mathbf{R}}_g$ ), then

$\tilde{\mathbf{h}}_k \sim \mathcal{CN}\!\left(\mathbf{0}, \, (\mathbf{U}_g^{(r_g)})^H \mathbf{R}_k \, \mathbf{U}_g^{(r_g)}\right).$

The inner precoder $\mathbf{P}_g$ operates on the stacked effective channel matrix $\tilde{\mathbf{H}}_g = [\tilde{\mathbf{h}}_k]_{k \in \mathcal{S}_g}^H \in \mathbb{C}^{|\mathcal{S}_g| \times r_g}$ , which is a standard $|\mathcal{S}_g|$ -user, $r_g$ -antenna MU-MIMO channel.

Key Takeaway

JSDM separates precoding into two stages operating on different timescales: the pre-beamformer $\mathbf{B}_g$ uses long-term statistics (slow), and the inner precoder $\mathbf{P}_g$ uses the effective $r_g$ -dimensional channel (fast). This separation is the key to reducing CSI overhead from $O(N_t)$ to $O(r_g)$ per group.

Example: JSDM with Two Angular Groups

A base station with $N_t = 64$ ULA antennas serves $K = 8$ users in two groups: Group 1 has 4 users arriving from $[10°, 30°]$ with effective rank $r_1 = 5$ , and Group 2 has 4 users arriving from $[50°, 70°]$ with effective rank $r_2 = 5$ . Design the JSDM precoder and compare with full-dimensional ZF.

Solution

Pre-beamformer design

Compute $\bar{\mathbf{R}}_1$ by averaging the covariances of Group 1 users. Eigendecompose: $\bar{\mathbf{R}}_1 = \mathbf{U}_1 \boldsymbol{\Lambda}_1 \mathbf{U}_1^H$ . Set $\mathbf{B}_1 = \mathbf{U}_1^{(5)}$ (the 5 dominant eigenvectors). Similarly for Group 2.

Effective channels

For each user $k$ in group $g$ , compute $\tilde{\mathbf{h}}_k = \mathbf{B}_g^H \mathbf{h}_k \in \mathbb{C}^5$ . Stack into $\tilde{\mathbf{H}}_g \in \mathbb{C}^{4 \times 5}$ .

Inner precoder

Apply ZF within each group: $\mathbf{P}_g = \tilde{\mathbf{H}}_g^H (\tilde{\mathbf{H}}_g \tilde{\mathbf{H}}_g^H)^{-1}$ (with appropriate power normalization). This is a $5 \times 4$ matrix — a standard 4-user, 5-antenna ZF problem.

CSI comparison

Full ZF requires each user to feed back a $64$ -dimensional channel vector. JSDM requires each user to feed back a $5$ -dimensional effective channel vector. Total CSI per coherence interval: full ZF = $8 \times 64 = 512$ complex scalars; JSDM = $8 \times 5 = 40$ complex scalars — a 12.8 $\times$ reduction.

JSDM Pre-Beamformer Beampattern

Visualize the beampattern formed by the pre-beamforming matrix $\mathbf{B}_g$ for each user group. The pre-beamformer focuses energy on the angular region of its assigned group while suppressing other directions. Adjust the angular separation to see when inter-group interference becomes significant.

Parameters

N_t

64

\theta_1

(deg)-20

Center angle of Group 1

\theta_2

(deg)30

Center angle of Group 2

\Delta\theta

(deg)15

r_g

6

Effective rank per group

When Does JSDM Work Well?

JSDM achieves its largest gains when the group eigenspaces are approximately orthogonal: $\mathbf{U}_g^H \mathbf{U}_{g'} \approx \mathbf{0}$ for $g \neq g'$ . This occurs when (i) the angular separation between groups exceeds the angular spread, and (ii) $N_t$ is large enough that the array can resolve the angular regions. With a ULA, the DFT structure of the steering vectors ensures near-orthogonality when the angular support regions do not overlap — a condition that holds in many practical massive MIMO scenarios.

🎓CommIT Contribution(2013)

JSDM: Joint Spatial Division and Multiplexing

A. Adhikary, J. Nam, J.-Y. Ahn, G. Caire — IEEE Trans. Inf. Theory, vol. 59, no. 10, pp. 6441–6463

JSDM was introduced by Adhikary, Nam, Ahn, and Caire as a practical framework for FDD massive MIMO. The key contribution is the two-stage precoding architecture that separates the spatial processing into a long-term pre-beamformer (based on covariance eigenspaces) and a short-term inner precoder (based on reduced-dimension effective channels). This decomposition achieves two goals simultaneously: (1) it reduces the CSI feedback overhead from $O(N_t K)$ to $O(\sum_g r_g |\mathcal{S}_g|)$ , making FDD massive MIMO feasible, and (2) it provides inter-group interference suppression "for free" through the spatial separation of group eigenspaces. The paper shows that JSDM is asymptotically optimal in the large-array regime — the rate loss compared to full-CSI precoding vanishes as $N_t \to \infty$ — and provides practical group formation and beam selection algorithms.

jsdmfdd-massive-mimotwo-stage-precodingcsi-reductionspatial-divisionView Paper →

JSDM Two-Stage Precoding

Complexity: Stage 1:

O(G \cdot N_t^{2} \cdot r_g)

(eigendecomposition, amortized over

T_{\text{cov}}

). Stage 2:

O(G \cdot r_g^2 \cdot |\mathcal{S}_g|)

(ZF inversion per group, done every

T_c

).

Input: Channel covariances

\{\mathbf{R}_k\}_{k=1}^{K}

, grouping

\{\mathcal{S}_g\}_{g=1}^{G}

, instantaneous channels

\{\mathbf{h}_k\}

Output: Precoding vectors

\{\mathbf{v}_{k}\}_{k=1}^{K}

Stage 1: Pre-beamforming (long-term, updated every $T_{\text{cov}}$ )

1. for

g = 1, \ldots, G

do

2.

\quad

Compute group covariance:

\bar{\mathbf{R}}_g = \frac{1}{|\mathcal{S}_g|} \sum_{k \in \mathcal{S}_g} \mathbf{R}_k

3.

\quad

Eigendecompose:

\bar{\mathbf{R}}_g = \mathbf{U}_g \boldsymbol{\Lambda}_g \mathbf{U}_g^H

4.

\quad

Set

\mathbf{B}_g = \mathbf{U}_g^{(r_g)}

(first

r_g

eigenvectors)

5. end for

Stage 2: Inner precoding (short-term, updated every $T_c$ )

6. for

g = 1, \ldots, G

do

7.

\quad

for

k \in \mathcal{S}_g

do

8.

\quad\quad

Compute effective channel:

\tilde{\mathbf{h}}_k = \mathbf{B}_g^H \mathbf{h}_k \in \mathbb{C}^{r_g}

9.

\quad

end for

10.

\quad

Stack:

\tilde{\mathbf{H}}_g = [\tilde{\mathbf{h}}_k]_{k \in \mathcal{S}_g}^H \in \mathbb{C}^{|\mathcal{S}_g| \times r_g}

11.

\quad

Compute inner precoder:

\mathbf{P}_g = f_{\text{prec}}(\tilde{\mathbf{H}}_g)

(e.g., ZF, MMSE)

12. end for

Composite precoding:

13. for

k \in \mathcal{S}_g

do

14.

\quad

\mathbf{v}_{k} = \mathbf{B}_g \mathbf{p}_k

where

\mathbf{p}_k

is column of

\mathbf{P}_g

15. end for

The critical advantage is that Stage 2 operates on $r_g$ -dimensional channels rather than $N_t$ -dimensional ones. For typical massive MIMO parameters ( $N_t = 64$ , $r_g = 6$ ), the inner precoding complexity is reduced by a factor of $(64/6)^2 \approx 114$ .

Common Mistake: Ignoring Inter-Group Interference

Mistake:

Assuming that the pre-beamformer $\mathbf{B}_g$ completely eliminates inter-group interference, i.e., treating $\mathbf{h}_k^H \mathbf{B}_{g'} = \mathbf{0}$ for $g' \neq g$ .

Correction:

Inter-group interference is small but not zero for finite $N_t$ . The leakage $\|\mathbf{h}_k^H \mathbf{B}_{g'}\|^2$ depends on the overlap between eigenspaces and decreases as $O(1/N_t)$ under favorable conditions. In a finite system, this residual interference should be accounted for in rate analysis, and power allocation across groups should be optimized to manage it.

Why This Matters: JSDM and 5G NR FDD Operation

JSDM directly addresses the CSI feedback bottleneck that has limited FDD massive MIMO in practice. In 5G NR, the Type II codebook (3GPP TS 38.214) follows a remarkably similar two-stage structure: a wideband component (analogous to the pre-beamformer $\mathbf{B}_g$ ) captures the dominant spatial directions, while a subband component (analogous to the inner precoder) refines the beamforming per frequency chunk. The conceptual lineage from JSDM to the 5G codebook design is direct and acknowledged in the standards literature.

JSDM Two-Stage Precoding Geometry

User groups with distinct angular spreads are separated by pre-beamforming matrices

\mathbf{B}_g

that project onto each group's eigenspace. The inner MU-MIMO precoder

\mathbf{P}_g

then handles residual intra-group interference in the reduced-dimension effective channel. The animation shows angular groups being isolated and then resolved, illustrating the two-stage JSDM architecture.

The JSDM Framework