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Beyond CRB: Optimising for Reconstruction Quality

The ISAC literature (Chapter 29) optimises the tradeoff between communication rate and sensing accuracy, measured by the CRB. But the CRB measures estimation accuracy for a single parameter — it does not capture the quality of a reconstructed image. A waveform that minimises the CRB for delay estimation may produce a poor image if it concentrates k-space coverage in one direction.

The final step in the convergence of communication, sensing, and imaging is to optimise the entire pipeline end-to-end: from waveform design, through sensing and reconstruction, to the communication feedback. This section develops the imaging-aware ISAC framework, where the objective is reconstruction quality (NMSE, SSIM) rather than CRB.

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Definition:
Joint Communication-Sensing-Imaging (JCSI) Framework

The JCSI framework optimises the entire pipeline:

$\max_{\mathbf{W}, \boldsymbol{\alpha}} \quad R(\mathbf{W}) \quad \text{s.t.} \quad \mathcal{J}_{\mathrm{img}}(\boldsymbol{\alpha}, \mathbf{W}) \leq \epsilon_{\mathrm{img}}, \quad \|\mathbf{W}\|_F^2 \leq P_t$

where:

$\mathbf{W}$ is the joint transmit waveform (precoding + pilots),
$\boldsymbol{\alpha}$ are the reconstruction algorithm parameters (regularisation weights, denoiser parameters),
$R(\mathbf{W})$ is the communication sum-rate,
$\mathcal{J}_{\mathrm{img}}(\boldsymbol{\alpha}, \mathbf{W})$ is the imaging quality metric (e.g., expected NMSE of the reconstructed scene),
$P_t$ is the transmit power budget.

The key difference from standard ISAC is that $\mathcal{J}_{\mathrm{img}}$ depends on the reconstruction algorithm, not just on the waveform — creating a bilevel optimisation: the outer level designs the waveform, the inner level reconstructs the image.

Standard ISAC uses CRB as the sensing metric, which is algorithm-independent but does not account for the quality of the reconstructed image. Imaging-aware ISAC replaces CRB with a reconstruction-dependent metric, making the design problem harder but more relevant.

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Definition:
Rate-Imaging Pareto Frontier

The rate-imaging Pareto frontier is the set of achievable (rate, imaging quality) pairs:

$\mathcal{P} = \{(R, \mathcal{J}) : \exists\, \mathbf{W} \text{ s.t. } R(\mathbf{W}) \geq R,\; \mathcal{J}_{\mathrm{img}}(\mathbf{W}) \leq \mathcal{J},\; \|\mathbf{W}\|_F^2 \leq P_t\}.$

Points on the frontier are Pareto-optimal: no waveform can improve one metric without degrading the other.

At the extremes:

Communication-only: $\mathbf{W}$ maximises rate (e.g., SVD precoding), imaging quality is poor.
Imaging-only: $\mathbf{W}$ minimises $\mathcal{J}_{\mathrm{img}}$ (e.g., orthogonal pilot grid), communication rate is zero.

The frontier between these extremes characterises the fundamental tradeoff.

Theorem: Gap Between CRB-Optimal and Imaging-Optimal Waveforms

For a scene with $K$ scatterers and an OFDM waveform with $N_f$ subcarriers, the waveform that minimises the CRB for delay estimation concentrates power on the edge subcarriers (maximum bandwidth):

$\mathbf{W}_{\mathrm{CRB}} = \arg\min_{\mathbf{W}} \mathrm{CRB}(\tau) \implies |W_{k}|^2 \propto \delta(k - 1) + \delta(k - N_f).$

The waveform that minimises imaging NMSE distributes power across subcarriers to ensure uniform k-space coverage:

$\mathbf{W}_{\mathrm{img}} = \arg\min_{\mathbf{W}} \mathrm{NMSE}(\hat{\boldsymbol{\sigma}}) \implies |W_{k}|^2 = P_t / N_f \quad \forall k.$

The NMSE gap between the two waveforms is:

$\mathrm{NMSE}(\mathbf{W}_{\mathrm{CRB}}) / \mathrm{NMSE}(\mathbf{W}_{\mathrm{img}}) = O(N_f / K)$

which can be $> 10$ dB for typical parameters ( $N_f = 1024$ , $K = 10$ ).

CRB cares about the sharpest possible estimate of a single parameter (delay). Imaging cares about reconstructing a full scene, which requires uniform frequency coverage. Edge-only power gives excellent delay resolution but leaves most of k-space unilluminated, degrading the image.

Proof

CRB-optimal waveform

The CRB for delay estimation under an OFDM waveform is $\mathrm{CRB}(\tau) = \frac{1}{\text{SNR} \cdot \sum_k (2\pi f_k)^2 |W_k|^2}$ . This is minimised by placing all power at the frequencies with largest $|f_k|$ : the edge subcarriers.

Imaging-optimal waveform

The imaging NMSE for a sparse scene under LASSO depends on the coherence of the sensing matrix columns. Uniform power $|W_k|^2 = P_t/N_f$ minimises the coherence (analogous to RIP), yielding the best NMSE.

Gap

The CRB-optimal waveform effectively uses 2 frequencies (edges), giving a sensing matrix with $2N_a$ measurements for $Q$ unknowns. The imaging-optimal uses all $N_f N_a$ measurements. The NMSE ratio scales as $N_f/(2K) \cdot \log(Q/K)$ . $\blacksquare$

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Communication Rate vs. Imaging Quality Pareto Frontier

Explore the Pareto frontier between communication sum-rate and imaging reconstruction NMSE. The curve shows the achievable tradeoff for different waveform designs.

Compare the CRB-optimal point (good delay estimation, poor imaging) with the imaging-optimal point (good reconstruction, reduced rate) and the Pareto-optimal points in between.

Parameters

N_t

(antennas)64

N_f

(subcarriers)256

K

(users)4

SNR (dB)20

Example: Joint Waveform Design for ISAC with Imaging

A 64-antenna base station with 256 OFDM subcarriers at 28 GHz serves 4 users while imaging a scene with $K_s = 8$ scatterers. The power budget is $P_t = 30$ dBm and $\text{SNR} = 20$ dB. Design the waveform allocation between communication and imaging, and compute the achievable rate and NMSE.

Solution

Communication-only baseline

All power to communication: SVD precoding to 4 users. Sum-rate: $R = 4 \times \log_2(1 + 20 \times 64/4) \approx 4 \times 8.3 = 33.2$ bps/Hz. Imaging NMSE: very high ( $> 0$ dB) — no dedicated sensing.

Imaging-only baseline

All power to orthogonal pilots: $256$ subcarriers with uniform power. NMSE $\approx K_s/(N_f \cdot \text{SNR}) = 8/(256 \times 100) = 3.1 \times 10^{-4}$ ( $-35$ dB). Communication rate: $R = 0$ .

Joint design (80/20 split)

Allocate 80% of power to communication, 20% to sensing pilots spread across 64 subcarriers.

Communication: $R \approx 0.8 \times 33.2 = 26.6$ bps/Hz (80% of max).

Imaging: 64 pilot subcarriers with $0.2 P_t$ total. Effective sensing SNR: $0.2 \times 100 = 20$ (13 dB). NMSE $\approx 8/(64 \times 20) = 6.25 \times 10^{-3}$ ( $-22$ dB).

This is a good operating point: 80% of communication rate retained, imaging NMSE 13 dB worse than optimal but still sufficient for beam prediction and environment mapping.

Definition:
End-to-End Learning for JCSI

The JCSI problem can be solved via end-to-end learning: jointly optimise the waveform $\mathbf{W}$ , the reconstruction algorithm parameters $\boldsymbol{\alpha}$ , and the communication precoder $\mathbf{V}$ by minimising a combined loss:

$\mathcal{L} = -\mu R(\mathbf{W}, \mathbf{V}) + (1-\mu)\,\mathcal{J}_{\mathrm{img}}(\boldsymbol{\alpha}, \mathbf{W})$

where $\mu \in [0, 1]$ controls the rate-imaging tradeoff.

The key technical challenge is that the reconstruction algorithm (LASSO, OAMP) contains non-differentiable operations ( $\ell_1$ prox, hard thresholding). Deep unfolding (Chapter 18) addresses this by unrolling the algorithm into a differentiable computation graph, enabling end-to-end gradient-based optimisation.

This is the natural culmination of the deep unfolding ideas from Chapter 18: the unrolled reconstruction network is not just used for imaging, but is embedded in a larger end-to-end system that jointly optimises communication and imaging.

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End-to-End JCSI Optimisation

Complexity:

O(E \cdot B \cdot L \cdot Q^2)

where

L

= unrolled iterations,

Q

= image size

Input: Channel realisations

\{\mathbf{H}_t\}

, scene ground truth

\{\boldsymbol{\sigma}_t\}

, tradeoff

\mu

Output: Waveform

\mathbf{W}^*

, reconstruction parameters

\boldsymbol{\alpha}^*

, precoder

\mathbf{V}^*

1. Initialise:

\mathbf{W} \leftarrow

uniform power,

\boldsymbol{\alpha} \leftarrow

LASSO defaults,

\mathbf{V} \leftarrow

ZF precoder

2. for epoch

= 1, \ldots, E

do

3.

\quad

Sample batch

\{(\mathbf{H}_b, \boldsymbol{\sigma}_b)\}_{b=1}^{B}

4.

\quad

Forward pass:

5.

\quad\quad

Compute sensing returns:

\mathbf{y}_{b} = \mathbf{A}(\mathbf{W})\boldsymbol{\sigma}_b + \mathbf{w}_{b}

6.

\quad\quad

Reconstruct:

\hat{\boldsymbol{\sigma}}_b = f_{\boldsymbol{\alpha}}(\mathbf{y}_{b}, \mathbf{A}(\mathbf{W}))

(unrolled OAMP)

7.

\quad\quad

Compute rate:

R_b = \log_2\det(\mathbf{I} + \text{SNR}\,\mathbf{H}_b\mathbf{V}\mathbf{V}^H\mathbf{H}_b^H)

8.

\quad\quad

Compute imaging loss:

\mathcal{J}_b = \|\hat{\boldsymbol{\sigma}}_b - \boldsymbol{\sigma}_b\|^2/\|\boldsymbol{\sigma}_b\|^2

9.

\quad

Loss:

\mathcal{L} = \frac{1}{B}\sum_b [-\mu R_b + (1-\mu)\mathcal{J}_b]

10.

\quad

Backward pass:

\nabla_{\mathbf{W},\boldsymbol{\alpha},\mathbf{V}} \mathcal{L}

via backpropagation through the unrolled graph

11.

\quad

Update: Adam step on

(\mathbf{W}, \boldsymbol{\alpha}, \mathbf{V})

12. end for

13. return

(\mathbf{W}^*, \boldsymbol{\alpha}^*, \mathbf{V}^*)

The unrolled OAMP in step 6 must be differentiable. The soft thresholding operator $\mathcal{S}_\lambda(\cdot)$ is already differentiable; the OAMP Onsager correction is a linear operation. The key non-trivial gradient is through the sensing matrix $\mathbf{A}(\mathbf{W})$ , which depends on the waveform.

Sensing Metrics: CRB vs. Imaging Quality

Property	CRB	Imaging NMSE	Imaging SSIM
What it measures	Single-parameter estimation bound	Global pixel-wise error	Perceptual structural similarity
Algorithm-dependent?	No (Fisher information only)	Yes (depends on reconstruction)	Yes (depends on reconstruction)
Captures scene structure?	No	Partially	Yes
Optimisation difficulty	Closed-form (matrix trace)	Bilevel (waveform + reconstruction)	Bilevel (waveform + reconstruction)
Suitable for imaging?	No — misses k-space coverage	Yes — standard metric	Yes — correlates with perception

Quick Check

Why can the CRB-optimal waveform produce a poor image even though it achieves the best delay estimation accuracy?

It concentrates power at edge frequencies, leaving most of k-space unilluminated

It uses too much transmit power

It has lower SNR than the imaging-optimal waveform

Correction:

It concentrates power at edge frequencies, leaving most of k-space unilluminated

CRB for delay is minimised by maximising the effective bandwidth (edge-frequency power). But imaging requires uniform k-space coverage across all frequencies. Edge-only power leaves gaps.

Common Mistake: Using CRB as an Imaging Objective

Mistake:

Optimising an ISAC waveform for CRB minimisation and expecting good image reconstruction.

Correction:

CRB is a point-estimation metric: it measures how well you can estimate a single parameter (delay, angle). It does not account for the quality of a full scene reconstruction, which depends on k-space coverage, algorithm choice, and scene structure.

For imaging applications, use reconstruction-aware metrics (NMSE, SSIM, LPIPS) as the sensing objective. The CRB-optimal and imaging-optimal waveforms can differ by $> 10$ dB in NMSE (Theorem 30.3).

Historical Note: From Radar-Communication Coexistence to Joint Design

The relationship between radar and communication has evolved through three phases. In the 2000s, the focus was coexistence: ensuring that radar and communication systems sharing the same spectrum do not interfere with each other (spectrum sharing, interference mitigation). In the 2010s, the dual-function paradigm emerged: a single waveform serves both purposes (ISAC). The CRB-rate tradeoff became the standard design criterion.

The 2020s are bringing the third phase: joint design that accounts for the full imaging pipeline, not just point estimation. This reflects the recognition that sensing in 6G is not just "detecting a target at range $r$ " but "reconstructing a 3D environment map" — an imaging problem that requires different design criteria than the CRB.

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Why This Matters: From ISAC to JCSI in 6G Standards

The 3GPP Release 19 study item on "AI/ML for NR air interface" includes sensing-assisted communication as a use case. Current proposals focus on CRB-based waveform design, but the imaging community's push toward reconstruction-aware metrics is beginning to influence standardisation. The JCSI framework of this section represents the research frontier: optimising waveforms for image quality rather than point estimation. We expect this to mature toward Release 20/21 as digital twin concepts gain traction in 6G discussions.

Joint Communication-Sensing-Imaging (JCSI)

An extension of ISAC that optimises the joint system for image reconstruction quality (NMSE, SSIM) rather than point-estimation accuracy (CRB). The optimisation is bilevel: the outer level designs the waveform, the inner level runs the reconstruction algorithm.

Related: {{Ref:Def Joint Csic}}

Imaging-Aware ISAC

An ISAC design philosophy where the sensing metric accounts for the quality of the reconstructed image, not just the accuracy of individual parameter estimates. This leads to different optimal waveforms than CRB-based ISAC, with more uniform k-space coverage.

Related: {{Ref:Thm Imaging Crb Gap}}

Key Takeaway

The JCSI framework goes beyond ISAC by optimising for reconstruction quality rather than CRB. The CRB-optimal waveform can produce images $> 10$ dB worse than the imaging-optimal waveform. End-to-end learning via deep unfolding jointly optimises the waveform, the reconstruction algorithm, and the communication precoder. The rate-imaging Pareto frontier characterises the fundamental tradeoff, with practical operating points achieving $\sim 80\%$ of maximum rate while maintaining good imaging quality.

Joint Communication-Sensing-Imaging Optimization