## 5 Multiplexing Readout
### MHz-Frequency-Division Multiplexing  
The MHz Frequency-division multiplexing (MHz-FDM) scheme is fundamentally different from the other multiplexing architectures from the fact that the TES itself, and not the SQUID, is the modulating element in the multiplexing process. The detectors are voltage biased with sinusoidal carriers at different frequencies in the 1 to 5 MHz range and they operate as amplitude modulators [137]. MHz-FDM is under development for the read-out of large array of X-ray calorimeters and infra-red bolometers for future space mission and used on ground-based CMB experiments [138; 139]. In MHz-FDM, each TES is coupled to a superconducting high-\(Q\)\(LC\) bandpass filter [55; 56; 140; 141], which has the double function of limiting the information bandwidth and providing a single carrier, stiff ac voltage bias to the TES. The signals from many TESs are then summed at the input of one low-temperature first-stage single SQUID current amplifier, amplified further by a second-stage SQUID array, operated at higher temperature, and finally demodulated at room temperature by digital electronics. The SQUID signal is linearized by using a very robust baseband feedback (BBFB) scheme [57; 142; 143] or other digital active nulling configuration [144]. The major advantage of the MHz-FDM architecture is the relative simplicity of the cold electronics, which only requires a limited amount of SQUID amplifiers, and its robustness to electromagnetic interferences. The fabrication of high-\(Q\) lithographic \(LC\) filters has been, for many years, a major obstacle to the practical implementation of the MHz-FDM. With the state-of-the-art of the \(LC\)-filters fabrication, a very high yield can be achieved, as well as a high accuracy of the frequency definition and a compact design [55]. FDM makes as well a better use of the available bandwidth with respect to the other traditional scheme. Moreover, in an instrument like X-IFU for example, tens of pixels can be readout with a single SQUID amplifier, without the need of more dynamic range than for the readout of a single pixel [145]. The details of the MHz-FDM system for a X-IFU-like instrument, using for each channel a two-stage SQUID amplifier developed at VTT [146], can be found in [12]. The first-stage SQUID has an input inductance smaller than \(3\,\mathrm{nH}\), a bandwidth \(>8\,\mathrm{MHz}\) and a input current noise \(<3\,\mathrm{pA}/\sqrt{\mathrm{Hz}}\), for a total power dissipation smaller than \(2\,\mathrm{nW}\). One of the major issues related to the MHz-FDM readout is the behaviour of a TES under MHz bias, which is strongly influenced by the non-linear Josephson effects describe in Section 2. Only recently, the physical processes involved in an ac bias TES have been better understood. The current development undergoing at SRON (see Section 4.2) is leading to an optimal TES design that will reduce these non-ideal effects to a negligible level. Moreover, non-linearities in the readout chain, due to, for example, the SQUID amplifiers or DACs, can generate intermodulation lines within the detector bandwidth degrading its performance. To minimize the effect of these non-linearities and to make efficient use of the available readout bandwidth, the LC resonators are usually designed to be on a regular grid. Despite the high accuracy of the lithographic processes, however, the observed resonant frequencies have a spread of about few kHz around the designed values. The important recent breakthrough that solves this issue is the experimental demonstration at SRON of the frequency shift algorithm (FSA), a very elegant approach to electronically tune the resonant frequency of the cold resonators by means of a feedback on the modulated bias signal [147]. The performance of a new implementation of the FSA, scalable for multi-pixel operation, has been recently shown by [PERSON] et al. [126].