## 5 Multiplexing Readout
### Time-Division and Code-Division Multiplexing  
SQUID-based time-division multiplexing (TDM) is, at the moment, the most mature technology to read-out arrays of TES microcalorimeters. In TDM, the TESs are dc-biased. Each TES is coupled to its own first-stage SQUID. \(N\) rows of first-stage SQUIDs are switched  
Figure 12: X-ray spectra of the Fe\({}^{50}-Ka\) complex at 5.9 keV demonstrating excellent energy resolution with SRON high aspect ratio TESs: (**a**) \(80\times 20\,\upmu\)m\({}^{2}\), \(R_{N}=100\,\upmu\)m\(\Omega\) and bias frequency 2.5 MHz, and (**b**) \(80\times 10\,\upmu\)m\({}^{2}\), \(R_{N}=200\,\upmu\Omega\) and bias frequency 3.5 MHz. The dashed green line is the natural lineshape of the complex.  
on sequentially and \(M\) columns of SQUIDs are read out in parallel. One single bias line is used for all the \(N\) TESs in one column and each TES cannot be individually biased at its optimal bias point. For this reason, a highly uniform TES array is required.  
The latest generation of TDM architecture developed at NIST has an overall system bandwidth of 6 MHz bandwidth, which allows to switch the rows faster with transient of 128 ns and total row time of 160 ns. A three-stage SQUID configuration is used with a input flux noise of 0.19 \(\mu\Phi_{0}/\sqrt{Hz}\). More details of the state-of-the-art of the TDM architecture, including the description of a practical X-ray spectrometer for beamline science, can be found in [9; 133]. Recent 40-row TDM experiments with X-IFU-like TESs make use of new shunt resistors and the optimized first-stage SQUIDs recently developed at NIST [125]. The achieved 1x40-row (32 TES rows + 8 repeats of the last row) energy resolution with the 32 TESs was 1.91 \(\pm\) 0.01 eV at 1.5 keV, 2.23 \(\pm\) 0.02 eV at 5.9 keV, 2.40 \(\pm\) 0.02 eV at 6.9 keV, and 3.44 \(\pm\) 0.04 eV at 11.9 keV [128]. A NIST 8-columns\(\times\)32-row TDM system with a prototype of a X-IFU kilo-pixel array from NASA-GSFC has shown an exquisite average energy resolution of 2.16 \(\pm\) 0.006 eV at 6.9 keV [134]. The instrument will be soon delivered to LNLN EBIT for astrophysics experiments. These results are compliant with the stringent Athena X-IFU requirements.  
The main disadvantage of TDM is the increase of SQUID noise with the \(\sqrt{N}\) due to aliasing, into the signal bandwidth, of the Nyquist noise above the rate at which all the \(N\) pixels in a column are measured and below the measurement bandwidth. This sets a fundamental limit on the maximum number of pixels measurable with high resolution in multiplexing mode. Code Division Multiplexing (CDM) [135] provides a way to reduce at a negligible level the SQUID noise aliasing effect and has the big advantage of being compatible with the existing TDM read-out infrastructure. In CDM, rather than measuring one TES at a time, all the dc-biased TESs in a column are read out simultaneously at each time step. The signals from the TESs are encoded with a Walsh basis and summed with equal weight, but different polarity patterns. The design and fabrication of the SQUID multiplexing chip is rather complex, since the encoding is achieved by a flux-summing architecture of \(N\) different microcalorimeters in \(N\) different SQUIDs with different combinations of coupling polarity. CDM is also susceptible to single-point failure mechanisms which can result in an unconstrained demodulation matrix. However, potential solutions do exist [136]. Excellent performance has been reported so far with a 32-channel flux-summed CDM [10] with a mean energy resolution of 2.77 \(\pm\) 0.02 eV over 30 working sensors. The multiplexed noise level and signal slew rate achieved were sufficient to allow the readout of more than 40 pixels per column. Even a larger multiplexing number could be achieved with the recent technological improvement in the fabrication of the shunt resistors chip and the first-stage SQUIDs [125].