Fig. 10: Robustness of the digital twin to poor signal-to-noise ratio.

a End-to-end test flow. EM denotes electromagnetic, B₀ is the static magnetic field, B₁ is the radio frequency magnetic field, FID denotes the free induction decay, BSS denotes blind source separation, p₀ is the optimal control pulse, p is the cooperative pulse, and F_diag is the diagonal of the coupling matrix F. b Example results for running the digital twin with 2 channels, with a signal-to-noise ratio of 10 dB, and a minimal Δf = 0.02 ppm (10 Hz). The left, middle, and right panels show the original ¹H spectrum, the decomposed ¹H spectrum, and the fidelity (η) of the cooperative pulse, respectively. ν₀ is the resonance offset, ν₁ is the radio frequency amplitude, and \(\overline{\eta }\) is the averaged fidelity. c Mixing matrix criterion (C_A) as a function of the signal-to-noise ratio. d Mean fidelity of the cooperative pulse as a function of the signal-to-noise ratio. The mean fidelities of optimal control pulse are \(\overline{{\eta }_{1}}=0.9952\) for channel 1 and \(\overline{{\eta }_{2}}=0.9951\) for channel 2. The two channels are indicated as chn. 1 and chn. 2 in (b) and (d).