Ferkans — Interactive Telecom Tutor

How Much Do We Lose by Not Having Full CSI?

The practical appeal of JSDM is clear: reduced CSI overhead. But the fundamental question remains: what is the rate penalty? If JSDM sacrifices too much sum rate compared to full-CSI precoding, the overhead savings are not worthwhile. The analysis in this section shows that the penalty is remarkably small — and in the large-array regime, it vanishes entirely.

Definition:
JSDM Sum Rate

The achievable sum rate of JSDM with ZF inner precoding and equal power allocation is

$R_{\text{sum}}^{\text{JSDM}} = \sum_{g=1}^{G} \sum_{k \in \mathcal{S}_g} \log_2\!\left(1 + \text{SINR}_k^{\text{JSDM}}\right)$

where

$\text{SINR}_k^{\text{JSDM}} = \frac{\frac{\ntn{tx\_power}}{K} |\tilde{\mathbf{h}}_k^H \mathbf{p}_k|^2}{\frac{\ntn{tx\_power}}{K} \sum_{g' \neq g} \sum_{j \in \mathcal{S}_{g'}} |\mathbf{h}_k^H \mathbf{B}_{g'} \mathbf{p}_j|^2 + \sigma^2}$

and $\mathbf{P}_g$ is the ZF precoder applied to the effective channel $\tilde{\mathbf{H}}_g$ .

Theorem: Asymptotic Optimality of JSDM

Under the one-ring channel model with a ULA of $N_t$ antennas, $G$ groups with non-overlapping angular supports, and fixed $K$ and group sizes $|\mathcal{S}_g|$ , the per-user rate gap between full-CSI ZF and JSDM-ZF satisfies

$\frac{1}{K}\left(R_{\text{sum}}^{\text{full-ZF}} - R_{\text{sum}}^{\text{JSDM-ZF}}\right) \to 0 \quad \text{as } N_t \to \infty.$

In words: JSDM is asymptotically optimal in the large-array regime. The rate loss vanishes because (i) inter-group interference decays as $O(1/N_t)$ , and (ii) the pre-beamformer captures all the channel energy as the covariance eigenspace becomes sharp.

As $N_t$ grows, two things happen simultaneously. First, the angular resolution of the array improves, making the group eigenspaces more cleanly separated — the DFT beams become narrower, and the energy leaking from one group's region to another shrinks. Second, the covariance eigenvalues become more concentrated — the effective rank $r_g$ stabilizes while the signal energy captured by $\mathbf{B}_g$ approaches $100\%$ of the total. Together, these effects mean that the JSDM precoder converges to the full-CSI precoder in terms of rate.

Proof

Inter-group interference bound

For non-overlapping angular supports, the inter-group leakage satisfies $\mathbb{E}\left[\|\mathbf{h}_k^H \mathbf{B}_{g'}\|^2\right] = \text{tr}(\mathbf{B}_{g'}^H \mathbf{R}_k \mathbf{B}_{g'}) \leq \lambda_{\max}(\mathbf{R}_k) \|\mathbf{U}_{g'}^H \mathbf{U}_k\|_F^2.$ For a ULA with non-overlapping angular supports, $\|\mathbf{U}_{g'}^H \mathbf{U}_k\|_F^2 = O(1/N_t)$ by the concentration of the DFT sidelobes.

Signal energy preservation

The signal energy through the pre-beamformer is $\mathbb{E}[|\tilde{\mathbf{h}}_k^H \mathbf{p}_k|^2] = \text{tr}(\mathbf{B}_g^H \mathbf{R}_k \mathbf{B}_g) \cdot \text{(ZF gain factor)}.$ As $N_t \to \infty$ with fixed angular spread, the dominant eigenvalues of $\mathbf{R}_k$ grow as $O(N_t)$ (the array gain), and $\text{tr}(\mathbf{B}_g^H \mathbf{R}_k \mathbf{B}_g) / \text{tr}(\mathbf{R}_k) \to 1$ .

Rate convergence

The SINR gap satisfies $\text{SINR}_k^{\text{full-ZF}} - \text{SINR}_k^{\text{JSDM-ZF}} = O(1)$ (bounded) while both SINRs grow as $O(N_t)$ . Therefore $R_k^{\text{full-ZF}} - R_k^{\text{JSDM-ZF}} = \log_2\!\left(\frac{\text{SINR}_k^{\text{full-ZF}}}{\text{SINR}_k^{\text{JSDM-ZF}}}\right) \to 0. \quad \blacksquare$

Sum Rate: Full CSI vs. JSDM vs. Per-Group ZF

Compare the ergodic sum rate of three precoding strategies as a function of $N_t$ : (1) full-CSI ZF with perfect channel knowledge, (2) JSDM with ZF inner precoder, and (3) per-group ZF without inter-group coordination. Observe how the JSDM curve converges to full CSI as $N_t$ grows.

Parameters

K

8

G

2

SNR (dB)10

\Delta\theta

(deg)15

Theorem: Inter-Group Interference Leakage Bound

For a ULA with $N_t$ antennas and half-wavelength spacing, if groups $g$ and $g'$ have angular supports $[\theta_g^-, \theta_g^+]$ and $[\theta_{g'}^-, \theta_{g'}^+]$ with angular separation $\delta = \min(|\theta_g^- - \theta_{g'}^+|, |\theta_{g'}^- - \theta_g^+|) > 0$ , the normalized inter-group leakage satisfies

$\frac{1}{N_t} \text{tr}(\mathbf{B}_{g'}^H \mathbf{R}_k \mathbf{B}_{g'}) \leq \frac{C(\delta)}{N_t}$

for all $k \in \mathcal{S}_g$ , where $C(\delta) > 0$ is a constant depending on the angular separation $\delta$ and the angular power spectrum but not on $N_t$ .

The leakage decreases as $1/N_t$ because the array can more precisely resolve angular directions as it grows. The constant $C(\delta)$ captures the sidelobe level of the pre-beamformer pattern evaluated at the angular region of the other group — wider separation means lower sidelobes and smaller $C(\delta)$ .

Proof

Express leakage in the angular domain

For a ULA with DFT pre-beamformer, $\text{tr}(\mathbf{B}_{g'}^H \mathbf{R}_k \mathbf{B}_{g'}) = \sum_{n \in \mathcal{I}_{g'}} \mathbf{f}_n^H \mathbf{R}_k \mathbf{f}_n$ where $\mathbf{f}_n$ is the $n$ -th DFT vector.

Bound each term

Since $k \in \mathcal{S}_g$ , the angular power spectrum of user $k$ is concentrated in the angular region of group $g$ . By the Dirichlet kernel bound, $|\mathbf{f}_n^H \mathbf{a}(\theta)|^2 \leq \frac{1}{N_t} \cdot \frac{1}{\sin^2(\pi(\nu_n - \nu(\theta)))}$ for DFT beam $n$ and direction $\theta$ outside group $g'$ 's region.

Sum and normalize

Summing over $n \in \mathcal{I}_{g'}$ and integrating over the angular power spectrum of user $k$ yields the $O(1/N_t)$ bound with $C(\delta)$ depending on the minimum distance $|\nu_n - \nu(\theta)|$ between the DFT beams of group $g'$ and the angular support of user $k$ . $\blacksquare$

Example: Sum Rate Gap for Finite $N_t$

Consider $K = 8$ users in $G = 2$ groups with angular separation $\delta = 20°$ and $\text{SNR} = 10$ dB. Estimate the sum rate gap between full-CSI ZF and JSDM-ZF for $N_t \in \{32, 64, 128\}$ .

Solution

Monte Carlo setup

Generate channels from the one-ring model with angular spread $\Delta\theta = 15°$ . Group 1: center $-20°$ , Group 2: center $+20°$ . Compute the group covariances and eigenspaces.

Full-CSI ZF rate

With full channel knowledge, ZF precoding: $\mathbf{W} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ . Average the sum rate over 1000 channel realizations.

JSDM rate

Pre-beamform with $\mathbf{B}_g = \mathbf{U}_g^{(r_g)}$ , compute effective channels $\tilde{\mathbf{H}}_g$ , apply per-group ZF. Include inter-group interference in the SINR.

Results

Typical values: at $N_t = 32$ , the gap is $\sim 3$ bits/s/Hz ( $\sim 8\%$ relative). At $N_t = 64$ , the gap shrinks to $\sim 1.2$ bits/s/Hz ( $\sim 3\%$ ). At $N_t = 128$ , the gap is $\sim 0.4$ bits/s/Hz ( $< 1\%$ ). The convergence is fast — by 128 antennas, JSDM performs nearly identically to full CSI.

Key Takeaway

JSDM achieves the same sum rate as full-CSI precoding in the large-array regime, but with CSI overhead that scales with the effective channel rank $r_g$ rather than the array size $N_t$ . This is the fundamental result that makes FDD massive MIMO feasible.

Historical Note: Degrees of Freedom and the BC Capacity Region

2003–2013

The information-theoretic foundation for JSDM's sum-rate analysis rests on the broadcast channel (BC) capacity region, established by Weingarten, Steinberg, and Shamai (2006) for the MIMO Gaussian BC. Caire and Shamai (2003) had earlier shown that dirty-paper coding (DPC) achieves the capacity region. JSDM achieves a rate close to this capacity with linear precoding by exploiting the spatial structure — the pre-beamformer provides a form of "structural DPC" where inter-group interference is pre-cancelled through spatial separation rather than through non-linear encoding.

🔧Engineering Note

Power Allocation Across Groups

Equal power allocation across all users is suboptimal when groups have different effective ranks or different average channel strengths. Optimal power allocation solves a convex problem:

$\max_{\{p_k \geq 0\}} \sum_k \log_2(1 + \text{SINR}_k(p_k)) \quad \text{s.t.} \quad \sum_k p_k \leq \ntn{tx\_power}.$

In practice, a simple heuristic allocates power proportional to the effective rank: $p_g \propto r_g / \sum_{g'} r_{g'}$ . This ensures that groups with higher spatial dimensionality (more users or wider angular spread) receive proportionally more power. The gain over equal allocation is typically 0.5–1.5 dB in sum rate.

Practical Constraints

•
Per-antenna power constraints may further restrict the allocation
•
Unequal group sizes require careful fairness-aware power control

Precoding Strategy Comparison

Strategy	CSI Required	Feedback per User	Inter-Group Interference	Complexity
Full-CSI ZF	Instantaneous $\mathbf{H}_{k} \in \mathbb{C}^{N_t}$	$N_t$ complex scalars	None (joint processing)	$O(K^{2} N_t)$
JSDM-ZF	Covariance $\mathbf{R}_k$ + effective $\tilde{\mathbf{h}}_k \in \mathbb{C}^{r_g}$	$r_g$ complex scalars	$O(1/N_t)$ , vanishes asymptotically	$O(G \cdot r_g^2 \|\mathcal{S}_g\|)$
Per-Group ZF	Same as JSDM-ZF	$r_g$ complex scalars	Not suppressed	$O(G \cdot r_g^2 \|\mathcal{S}_g\|)$
MRT (no CSI at Tx)	Only covariance $\mathbf{R}_k$	0	Full interference	$O(K N_t)$

Sum-Rate Analysis