Technical Field

The invention relates in general to analog-to-digital (A/D) converter devices, in particular, A/D con- verter devices with densely connected analog networks and multiple control loops.

Background Art

The proposed A/D converter device is related to A/D converter devices as in [1, 2, 3, 4, 5, 6] and, more particularly, to control-bounded converter devices as in [7, 8, 9], A general system description of a control-bounded converter device is shown in Fig. 1. For control-bounded converter devices, the continuous-time input signal u(i) (the signal to be converted) is fed into an analog continuous-time dynamical system, which provides amplification (in certain frequency bands) and interacts with a digital control and estimation unit. The digital control and estimation unit ensures, with suitable digital control signals, that all state variables (voltages and/or currents) of the analog system remain within their proper limits. Based on these control signals, the digital control and estimation unit produces a digital estimate ύ(ί) as described in [7] and [9].

A primary advantage of control-bounded converter devices is that large amplification in desired fre- quency bands can be achieved without concerns for stability.

An example of a control-bounded converter device with local control is shown in Fig. 3 [8, 9], In this example, the linear dynamical system is a chain of integrators. Each integrator is controlled individually with the help of a one-bit flash analog-to-digital (A/D) converter and a zero-order hold digital-to-analog (D/A) converter. A proposed circuit implementation of each integrator is shown in Fig. 2. The transfer function (i.e., the Laplace transform of the impulse response) from the input u(i) to the xw(t)-th state is this means, for a sinusoidal input signal with a frequency of / we expect the signal-to-noise ratio (SNR) performance of this converter device to be proportional to where and Ω _BW is the frequency band of interest.

The chain of integrators converter device, of Fig. 3, achieves performance similar to that of an over- sampling converter device such as Δ∑ modulators [1]. In fact the multi stage noise shaping (MASH) DS modulator and the chain of integrator converter device both share performance scaling attributes and their chain like structure,

Disclosure of the Invention

The problem solved by the recent invention is to provide a more versatile architecture of this type. This problem is solved by the A/D-converter device of claim 1. Accordingly, the A/D-converter device has

- an analog converter device input for a continuous-time input signal u(t), with t being the time

- a digital converter device output representing an estimate u(t) of the input signal u(t)

- a plurality of N integrators, with each integrato } having o a current or voltage input carrying a current or voltage o a current or voltage output carrying a current or voltage wherein said current or voltage outputs define a state vector

- a digital-to-analog converter converting a digital control signal s[k] to an analog control signal s (t)

- an analog-to-digital converter converting said state vector x(t) via a real valued linear transforma- tion G to a digital control observation s[k]

- an interconnect network interconnecting said analog converter device input u(t), said analog con- trol signal s (t), and said state vector x(t) to said current or voltage inputs with wherei and A, B, and G are all real valued matrices

- a digital control and estimation unit adapted to generate said digital converter device output and said digital control signal s[k] using said digital control observation wherein at least one of the following conditions apply:

(ai) the row rank of the matrices M = (A, B) and Τ are equal, and for every pair of permuta- tion matrices the matrix has at least one m _d , _e = 0 for d < e — 1 or d > e + 1, and/or

(a ₂ ) the columns of G form a set of vectors ε such that the span of £ and the span of ε \ {γ} are identical for any Υ ∈ ε.

A converter device as described above overcomes the limitations of the chain of integrator converter device, e.g. of the type of Figure 3, and may offer favorable properties including robustness against disturbances and imperfections such as thermal noise, clock jitter, and component mismatch, power efficiency, configurability and multi-input conversion.

Specific advantages may e.g. include the following.

- Interconnections between states of the state vector may suppress the sensitivity to component mismatch, limit cycles and increase robustness against thermal noise.

- Controlling in many overlapping subspaces, as follows from an overcomplete set of column vectors in G, suppress sensitivity in component mismatch as well as increases the performance of the whole converter device. - Circuit resources can be shared between multiple conversion processes.

- Configurable analog networks may enable conversion where the power consumption, the frequency tuning, mismatch suppression, and effective resolution can be instantaneously adapted. This leads to a conversion principle that utilize resources after demand.

Brief Description of the Drawings

The invention will be better understood and objects other than those set forth above will become apparent when consideration is given to the following detailed description thereof. This description makes reference to the annexed drawings, wherein:

Fig. 1 shows the general architecture of a control-bounded converter device as in [8],

Fig. 2 shows an operational amplifier (op-amp) realization of a first order integrator conversion mod- ule.

Fig. 3 shows the chain of integrator converter device from [8],

Fig. 4 shows the general system schematic.

Fig. 5 shows a particular implementation where the control observation Τ and feedback matrix A share a pre-transformation Φ.

Fig. 6 describes a particular differential circuit implementation of a braided chain converter device.

Fig. 7 shows the particular realization of the resistor network from Fig. 6. Namely, the l-th differen- tial input is connected to the k- th differential output via the l, k-th resistor pair in the matrix.

Fig. 8: shows an oscillator conversion module where the state is two dimensional and the linear dynamic system has a circular feedback pattern.

Fig. 9 shows the amplitude response of a leap frog converter device (solid line) in comparison to an integrator-chain converter device (dashed line).

Fig. 10 shows the amplitude response of an oscillator chain converter device.

Fig. 11 shows a particular realization of a configurable control-bounded converter device using dif- ferential operational amplifiers.

Modes for Carrying Out the Invention

We begin by reviewing the operation principle of control-bounded converter devices according to prior art as in Fig. 1, 2, and 3. We then proceed to a number of examples that illustrate additional specific advantages of the disclosed invention.

1 The Operating Principle of a Control-Bounded Converter Device

Fig. 1 shows the general system description of a control-bounded converter device. As all the devices presented here in are control-bounded converter devices as in Fig. 1, we will start by further describing the general operating principle of such a converter device and in particular the role of the digital control and estimation unit. As shown in Fig. 1, the system has two parts: the analog linear dynamical sys- tem and the digital control and estimation unit. The linear dynamical system takes a continuous-time input signal u(i) (the signal to be converted), and the control and estimation unit provides a digital esti- mate/representation ύ(ί) of u(i). Furthermore, the linear dynamical system is interactive as it provides N analog observable states and M digital control inputs. As described in [7], the control and estima- tion unit ensures that all internal states of the linear dynamical system remains within proper limits at all times by observing sampled and quantized representations of the observable states and feeding back suitable control signals. Furthermore, the linear dynamical system is such that it provides amplification for certain frequency bands. It follows that the control and estimation unit implicitly collects information about the input signal u(t). Subsequently, the control and estimation unit can produce a digital estimate u(t) as further described in the reconstruction-problem section below. More precisely, the control and estimation unit produces a digital representation of a continuous-time signal ύ(ί) that approximates u(i). As an example this could be the samples of ύ(ί ) evaluated with finite precision at uniformly spaced time instances.

Fig. 3 shows a prior art example of a control-bounded converter device [8] called a chain of inte- grators. This example is described in detail here as it demonstrates a concrete implementation of the control-bounded converter device principle outlined above. In this example the analog linear dynami- cal system is made up of N analog integrators connected in a chain with the help of the multiplicative weights β ₁ , . . . analog summers. Furthermore, the digital control and estimation unit observe each analog state xe(t) through the multiplicative weight k and a one-bit flash analog-to-digital converter that samples and quantizes the analog-continuous time signal into a digital representation [ ] From this representation the digital control and estimation unit produce a control signal [ ) (in this case a simple copy of ) that is fed back into the analog linear dynamical system through a digital-to-analog conversion followed by an amplitude scaling K _l . In a circuit implementation the analog continuous-time states x ₁ (t), . . . , x _N (t) represent physical quantities such as currents or voltages. To illustrate this, a particular circuit implementation of a node in the chain of integrators is illustrated in Fig. 2

Here the control and input signal summation is performed by two resistors connecting to a virtual ground of an op-amp with capacitive feedback. The amplifications are Furthermore, the digital output is formed by sampling the output voltage potential and quantizing it using a one-bit quantizer. The D/A conversion is a zero-order hold circuit which essentially maps the 1-bit digital representation to two predefined voltages for the duration of a clock period.

2 System Description

To fully describe and distinguish the proposed converter devices from converter devices such as in Fig. 3, we now introduce a more general system description for control-bounded converter devices, depicted in Fig. 4. Namely, the analog linear dynamical system of any control-bounded converter device can be described by its governing differential equations where A e R ^{N x N} is the feedback matrix, B e R ^N is the input vector, and G 6 E ^NxM is the control in- put matrix. Furthermore, the state vector x(f) = . . . , x^{t )) is made up of the analog states of the converter device, the input signal u(t) is the signal to be converted, and the control contribution corresponds to a vector of all control signals after an element-wise digital-to-analog conversion step. To avoid ambiguity we insist on the components of the state vector to represent physical quantities such as individual currents and voltages (with reference to a common ground) at different points of the associated circuit. Note that the system matrices A, B, G, and G might contain time varying elements. The digital control and estimation unit is divided into two parts: The digital control (DC) and the digital estimation (DE). The DC observes a sampled and quantized version of the state vector via the linear transformation ^N . We refer to this observation as the control observation where T> represents some finite set. Based on this (and possibly previous) observations the digital control outputs a multidimensional control signal ^M where V is some finite set. The control signal can interact with the analog linear dynamical system in various ways as f all can be changed within a predefined finite set of configurations. Additionally, the DC also outputs a control contribution s(t) as previously described in the context of the chain of integrator converter device example Fig. 3. Simultaneously, based on the control signal, the DE produces an estimate u(t). In contrast to the analog part of the system, all digital interactions are done in synchronization with a global clock.

As an example the chain of integrators example from Fig. 3 results in the static system description where I _N is a N-by-N identity matrix. Additionally, the DC simply copies the control observation as control signal s[kc].

2.1 The Technology

Using the notation above, we recognize the disclosed converter devices as those that uphold at least one of the following conditions:

- The row rank of the matrices ( ) are equal, larger than one, and for every pair of permutation matrices (P, P) the matrix has at least one for k < l — 1 or k > l + 1.

- The columns of G form an overcomplete set of vectors. An overcomplete set of vectors U is such that the span of U and the span of are identical for any u £ hi where the span of U is the set consisting of all linear combinations of the elements of U, These conditions amount to the same idea as they both provide redundancy. Specifically, the first condition enhances robustness by ensuring a distributed dynamical system dividing the signal paths over multiple physical nodes. The second condition enhances robustness by ensuring a distributed control that divides the control effort over multiple overlapping control paths.

Specific examples highlighting the features of such technologies will be presented in Section 5. 2.2 Transfer Function Matrix

The system description additionally provides a direct way of computing the transfer function matrix G(s) which consists of the Laplace transforms of the impulse responses between the input signal(s) and state variables. The transfer function matrix is determined by the linear equation system

2.3 Multiple Inputs

It is straightforward to extend the system model for multiple inputs by turning the input vector into a matrix where is the number of inputs. Additionally, control-bounded converte r devices with multiple inputs enables analog-to-digital conversion where circuit components are shared among multiple conversion tasks,

3 Effective Control

We call a DC effective if it can ensure that the converter device maintains a bounded state vector at all times. There are multiple ways to ensure this, as the DC can interact with the linear dynamical system in various ways. However, for a bounded input signal an effective control is such that where is a set of all state vectors elementwise bounded by the real number ( ) is provided by the DC given the observation } is a set of all input vector signals elementwise bounded by the real number e _u > 0, the DC is operated with a finite clock period T, and

Note that A, G, and B in Equations 12-14 might be time varying.

For a particular strucure the general conditions above can typically be simplified. As an example, in the case of the chain of integrators from Fig. 3, when and the input is bounded by state bound e _x , having are sufficient conditions for an effective control. 4 Digital Estimation

As described in [9], a digital estimate u(t) of (a filtered version of) u(t) is obtained from such that where n ² > 0 is a scalar parameter and T ₀ < T ₁ describes a time window. Note that s(i), A, B, G, and G may all depend on the digital control. Continuous-time dynamically constrained least squares problem of the form in Equation 17, have been studied extensively in [10] and [11]. Additionally, [9] presented an efficient algorithm for solving the given optimization problem, at discrete-time instances. In particular, for uniformly spaced samples , where T _s > 0 is a time period and an index, the algorithm in [9] reduces to N discrete-time filters. Applying these filters to their corresponding control signa ] and combining (by addition) the result, gives a discrete-time signal ) that represents a filtered and sampled version of the voltage(s) and/or current(s) at the input of the converter device. Furthermore, as this is a discrete-time operation, the filtering is done in a fully digital unit such as a digital-signal-processing (DSP) chip. Additionally, the discrete-time estimate can be combined with additional post processing filters to further enhance the estimate.

5 Device Examples

In the following sections, we describe several examples that illustrate specific advantages of the disclosed converter devices.

5.1 Braided-Chain Converter Device

The braided chain converter device is a control-bounded converter device where the physical states are not directly the nodes of the chain. Instead, the signals associated with each node of the chain are represented on different signal vectors These signal vectors are unit vectors and zero otherwise). The idea is that, since we separated the signal vectors from the state vector, robustness and additional design flexibility can be achieved. The system description of a braided-chain converter device follows as where represents the amplification into a given signal dimension (span of signal vector). Also represents feedback within a signal dimension. Format Definition

Such a braided-chain converter device is a control-bounded converter device wherein wherein are unit vectors, and are real numbers.

Advantages

The braided-chain converter device shares all the features of the chain of integrators. However, it stands out in its ability to suppresses the impact of physical noise and mismatch by converting in a subspace other than the canonical base where each individual physical state represents a basis. From the distributed design principle, that is the braided chain, it follows that circuit resources optimally are divided uniformly over the involved circuit components. In contrast, a physical chain, that does not have a single limiting node, requires circuit resources to be allocated according to some power law. Furthermore, due to the clear division between the digital control and the analog dynamical system, the braided chain converter device naturally enables complex control strategies (such as vector quantization), Additionally, if a known disturbance is situated in a particular subspace, the converter device can exclude this subspace and thus further enhance its suppression capabilities. This can be seen as a gener- alization of principle behind differential instead of single ended amplification configurations where the common mode voltage is suppressed as to enhance the quality of the differential voltage.

An Example

A particular example of the braided chain converter device would be i.e., the signal vectors form a Hadamard basis. The amplifications are chosen as and the feedback is p The control input and observation matrix are chosen as

As in the chain of integrators, we use four one-bit flash type A/D converter to produce the control ob- servation signal the DC computes the control signal by s and subsequently the control contribution follows from four one-bit zero-order hold D/A converters. Sometimes the feedback matrix A and the control observation matrix f share a pre-transformation. This is the case for this example as can be seen by decomposing It may be particular advantageous to implement the common pre-transformation jointly as is illustrated in Figure 5 with This follows from the fact that a linear transformation can be decomposed in numerous ways comprising multiple sequential linear transformations.

For the given setup the circuit implementation can efficiently realized using differential op-amps and unit resistors. The positive and negative multiplications then follow from crossing or alternatively not crossing wires between inputs and outputs. Such an implementation is shown in Fig. 6. The resistor network, which is marked by the H(R) box, is described in Fig. 7. To simplify the figure, the connect- ing wires are not drawn specifically instead the box is written as a matrix of resistor pairs where each (k, f)-th element of the matrix represents the connection between the Ath differential input and the fc-th differential output. Furthermore, the feedback capacitor C and unit resistor R are selected such that 5.2 Leap Frog Converter Device

The leap frog converter device is control-bounded converter device with a particular feedback struc- ture. The feedback enables complex conjugate pole pairs and can therefore provide both amplification as well as sharper transition between passband and stopband in the resulting amplitude response of the linear dynamical system. An example of the difference is shown in Fig. 9 where the amplitude response of a braided chain (dashed line) with and a leap frog system (solid line) of same complexity are illustrated. From the figure we see that the leap frog system can have a relatively larger cutoff frequency as well as sharper passband stopband transition. In full generality this means that any transfer function where are filter coefficients can be realized by the dynamical system

Since, the dynamical system of a leap frog converter device essentially is a ladder filter with amplifi- cation, there exists filter-synthesis tools that will determine for given filter coefficients.

Formal Definition

A leap frog converter device is a control-bounded converter device wherein and wherein c are unit vectors, and are real numbers.

Advantages

The leap frog converter device inherits the advantages of the braided-chain converter device. In addi- tion, it enables further flexibility as the transfer function G(s) can have larger bandwidth in comparison to the braided chain for the same number of physical states. Furthermore, the transfer function can be designed without concern for stability as the control and estimation unit ensures stability regardless of poles and zero placements. Finally, feedback paths provide additional noise suppression due to noise shaping.

An Example A particular example of the leap frog converter device follows from choosing , , ; as Hadamard basis vectors with Furthermore, by choos- ing where a local control policy s with N one-bit flash A/D and N zero-order hold D/A converters. The resulting transfer function is plotted in Fig. 9. Since the SNR associated with the estimate of the control and estimation unit is proportional to the integral of the transfer function, this configuration gives high performance for frequencies up to roughly / = 600 kHz. For comparison, the chain of integrator converter device from Fig 3 with the same b , b values is also shown in Fig. 9. As seen from the figure, the leap frog implementation (solid) outperforms the chain of integrator converter device (dashed) with a bandwidth of 600 kHz (in terms of bandwidth). 5.3 Oscillator Converter Device

An oscillator converter device enables analog-to-digital conversion for signals that reside at frequen- cies bands not centered around the zero frequency. In contrast to a conventional modulation scheme this technique only modulates the control observation and control contribution s(t) but not the signal path directly. This enables the oscillator converter device structure to be part of a larger control-bounded converter device structure, for example the braided-chain and leap frog converter device. The oscillator converter device is realized by the dynamical system where and β , and θ are real numbers and . Note that in contrast to the previous structures the oscillator converter device has a two dimensional input vector (in-phase and quadrature signal). Further- more, the oscillator converter device has a time varying control input and observation matrix where are real numbers and The control input and observation matrices can be seen as modulating and demodulating respectively the control signal and state vector. The control observation matrix in Equation (43) suggests analog multiplications. This can alternatively be implemented with analog multipliers i.e., changing the R(·) in Equation 43 to followed by M low-pass anti-aliasing filters

Formal Definition

An oscillator converter device is a control-bounded converter device wherein Advantages

The main advantage of the oscillator converter device, compared to bandpass Δ∑ converter devices or systems that down-modulates the signal before analog-to-digital conversion, is that the modulation is done in the control path and not the signal path. Subsequently, the digital control can be operated without requiring excessively fast or precise sampling operations in the included analog-to-digital converter. Additionally, the analog multipliers (the control observation matrix) do not need to be implemented at the same level of precision as the rest of the dynamical system as they are not part of the signal path and their output is only used by the low complexity analog-to-digital converter. In contrast, the digital multiplication (the control input matrix) can more easily be implemented as shown in the example below. Furthermore, the signal path is not degraded by the modulation and can be a part of a larger control- bounded converter device structure. Additionally, as both modulation and demodulation is done with the same oscillator, synchronization and phase alignment of the oscillator do not impact the performance of the converter device, i.e. a free running oscillator can be used for modulation.

An Example

A particular example of an oscillator converter device is shown in Fig. 8. The linear dynamical system has two states that are interconnected such that the states resonate at the angular frequency w. Furthermore, both the in-phase and quadrature signal part (where one could be a dummy signal) are equally amplified by b. The system input and dynamics can therefore be summarized as

The control uses a free running oscillator sin(ωt + Φ) of the same frequency as the resonance fre- quency of the dynamical system. Furthermore, as both the control input and observation matrix is imple- mented with the same oscillator, the unknown phase f is immaterial for the control. Another key point of this implementation is that the analog multiplication in the control observation is not required to be implemented with the same precision as the rest of the analog part of the circuit. On the other hand the digital multiplication in the control input matrix, which should be implemented with precision, can be realized by switching between different phase shifted versions of the oscillators output. The proposed design can be summarized by the control input and observation matrix

With two one-bit A/D and D/A converters converting each of their inputs and outputs separately and a control policy of

S.4 Oscillator Chain Converter Device

The oscillator chain converter device is a control-bounded converter device that makes us of multiple oscillator converter devices. This leads to a converter device structure that achieves high performance conversion at frequency bands not centered around zero. As a single oscillator converter device requires two states, the length of the state vector of oscillator chain converter device will be 27V instead of TV. The oscillator chain converter device’s dynamical system can be written as where 0 ₂ x ₂ is a two-by-two zero matrix. As in the leap frog converter device case the /¾, , / _/ v represents the forward gain between signal dimensions, and _j the feedback gain. is the angular resonance frequency of the fc-th oscillator node of the chain. represent the connection angles between pairs of oscillator states. Furthermore, the control input and observation matrix can be written as and where are such that they have the same row rank and column rank respectively as the matrix and diag (C ₁ , . . . , C _N ) constructs a block diagonal matrix with C ₁ , . . . , C _N on the diagonal.

Formal Definition

An oscillator chain is a control-bounded converter device wherein

Advantages The oscillator-chain converter device, much like the braided-chain converter device, provides an exponential resolution increase in the order of chain nodes N. However, unlike the chain of integrator converter device, the oscillator chain converts a signal centered at one or several carrier frequencies. The main advantage of the oscillator chain compared to first demodulating and then applying conventional A/D conversion at baseband, is that the modulation does not appear in the signal path in the oscillator chain. Subsequently, the modulation (control observation matrix) does not need to be implemented with the same level of precision as the overall converter device. Note that the same does not apply for the control input matrix as the resulting signal enters the signal path directly. However, this can be done more simply as illustrated in the oscillator converter device example since it requires a digital multiplication instead of an analog one. Additionally, in comparison with an undersampling approach, the sampling at the control interface needs not be as precise as its input is effectively down converted (demodulated) before feed to the low complexity A/D converter.

An Example

5.5 Configurable Converter Device

A configurable converter device does not only maintain a bounded state vector through the control contribution s(t) but can additionally adapt the dynamical system to redistribute analog circuit resources on demand or by time varying constraints. Subsequently, the dynamical system of a configurable con- verter device is time changing (in synchronization with a global clock) and at a given time t, where

least one combination fulfill the conditions of Section 2.1. This means that the converter device may also temporarily operate in a configuration that is not covered by these conditions.

The configurable converter device may redistribute the circuit resource between conversion pro- cesses, i.e. the scalar inputs of the vector u(t), by changing the elements of any of A, B, G, or G. In particular, this can be done to change the amount of power consumed by a particular conversion process or to change the overall power consumption of the whole converter device. A particular advantageous scenario would be a mobile application where power is a scarce resource, and the converter device can be adapted to operate in different power consumption modes as a specified effective conversion resolution might vary over time. Subsequently, in such a scenario, substantial power savings can be achieved while still maintaining a continuous conversion process.

Additionally or alternatively thereto, the effective conversion resolution can be adapted for each con- version processes without changing the joint effective conversion resolution or the power consumption of the converter device. A particularly advantageous scenario would be a multi-input conversion task where the scalar input signals are not all active at the same time. In that case, a larger average effective conversion resolution can be achieved (for the same power budget) by constantly reallocating the cir- cuit resources to the signals that are active at any given time. Furthermore, the signal activation can be evaluated by the digital control and estimation unit via the estimate ύ(ί).

Formal Definition

A configurable converter device is a control-bounded converter device wherein A, B, G, and G are configurable by said control signal such that at any given time t where kT ≤ t < (k + 1 )T, they can be written as and estimation unit is adapted to bring said converter device into at least two different configurations wherein said two different configurations have different elements in at least one of A, B, G, and G.

In particular, said at least two different configurations may have different zero elements in at least one of A, B, G In particular, said at least two different configurations may have a power consumption differing by a factor F > 1,2, in particular F > 1.5, in particular F > 2, and in particular F > 10.

In particular, said at least two different configurations may differ in effective conversion resolution for at least one scalar input signal.

Advantages

The fully configurable converter device enables us to do adaptive conversion where the analog net- work can be updated to further enhance the signal characteristics that are observed by the control and estimation unit. It can also be used for adaptive power management where only a subset of the involved circuit resources are utilized until the control and estimation unit detects the presence of an input signal and subsequently additionally resources are activated. Furthermore, an adaptive analog network can be used to share the circuit resources between multiple conversion processes (multiple input signals) and adapt the allocation of signal dimension used in each process in accordance with each input dimension’s activity, signal conditioning demand, or power and performance constraints.

An Example

Fig. 11 shows an example of a configurable converter device where the general analog network can be adaptively changed by the DC via the analog multiplexer (MUX) in the feedback path. Practically this means that the DC can alter the feedback matrix A of the dynamical system within a given set of configurations. Additionally, the configurable converter device is a multi-input converter device with two input signals The key feature of this design is that inter- nal signal dimensions can be allocated adaptively to these two conversion processes, i.e. the available configurations are and a static input matrix

Furthermore, the control contribution is generated with the same control as in Section 5.1 with input matrix and output matrix as in (24) and (25).

In this example the DC chooses the configuration by evaluating the signal activity of the estimates u. Namely, in case one of them falls below a threshold, and is thereby classified as not active, the DC switches to the configuration where the other input uses three signal dimensions instead of two. This yields a substantial increase in conversion performance for the latter input. Note that even though less resources are used for the signal that was classified as not active, it is still being converted and contin- uously monitored (using one signal dimension). In the case the signal becomes active again, the DC switches into a configuration where two or three signal dimension are utilized for this input depending on the classification of the other input at that time. 5.6 Effective Control Converter Device

Most DS modulators do not have guaranteed stable operations. That means that the modulator might occasionally stall and hang in a particular state when a particular input signal is fed into the system. Both excessive simulation at the design stage, to make sure that signals of interest do not exhibit this be- haviour, and continuous monitoring and resets during operations are standard techniques to manage such instabilities. These techniques can be extended and used for the same purpose in all the previously pro- posed converter devices Additionally, the control-bounded converter device makes it possible to ensure an effective control as is described in Section 3.

Formal Definition An effective control converter device is a control -bounded converter device wherein for wherein is a set of all state vectors elementwise bounded by the real number is provided by the DC given the observation ( ) { || ( )|| } is a set of all input vector signals elementwise bounded by the real number , and the digital control operates with a clock period T. Advantages

The main advantage of having an effective control is the fact that converter device is guaranteed to not stall or hang during its normal operations. This means that the design can be made without the addition of excessive simulations and the uncertainty of not having simulated enough relevant signals of interest, Examples

All the necessary conditions for an effective control can been explicitly stated in the context of each previously presented converter device example as follows:

- the control of the braided-chain converter device example from Section 5.1 can be made effective by

2 β T ≤ 1 (78) given that the elements of the state vector and input signal are bounded by the same bound. - The control of the leap-frog converter device example from Section 5.2 can be made effective by setting T = 0.476m seconds where T is the time period between control updates.

- The control of the oscillator converter device example from Section 5.3 can be made effective by given that both the elements of the state vector and input signal are bounded by the same bound.

- The control of the oscillator-chain converter device example from Section 5.4 can be made effec- tive by T= 0.5/μ seconds.

- The control of the configurable converter device example from Section 5.5 can be made effective by making sure that for any of its configurations (78) holds.

5.7 Overcomplete Control

For control-bounded converter devices, higher order quantizers may be used to improve the perfor- mance and stability of the conversion. The main drawback of a high order quantizer is the higher order DAC that is required in the feedback path. The precision that such an DAC can be implemented with is typically a limiting factor when increasing the number of bits in the quantizer.

The control-bounded converter device can also make use of higher order quantizers. More so it offers an alterative approach by controlling in overlapping subspaces of the state space rather than increasing the quantization in a single physical space (as described above). The overlapping control spaces are realized by choosing where M is larger than N and the non-zero column vectors of G form an overcomplete set respectively. Furthermore, the DC uses one-bit D/A converters to ensure an effective control. Performance and stabil- ity may be enhanced further by using hysteresis (Schmitt triggers) in the A/D conversion of the control observation. Note that introducing such hysteresis does not alter the estimation task from Section 4. Note any control-bounded converter device may benefit from an overcomplete control independently of their A and B matrix structure.

Formal Definition Advantages

A control-bounded converter device utilizing an overcomplete control enhances robustness as the control effort is distributed over multiple overlapping control inputs. Furthermore, the sensitivity to circuit imperfections, relating to the control input matrix, is reduced as the impact of each individual control input is reduced. Additional performance follows from the fact that the control discretize the state space in finer partitions.

An Example

A particular example would build upon a braided chain converter device utilized with a Hadamard basis as in (22) and (23). How'ever, to make use of an overcomplete control, the control input and observation matrix follows

With this setup sufficient conditions for an effective control are

2 βT < 1 (84) given that the elements of the state vector and input signal are bounded by the same bound.

5.8 Multi-Input Converter Device

The control-bounded analog-to-digital converter device can also be advantageous for multi-input analog-to-digital conversion. In particular, the conversion resources can be shared among the differ- ent conversion processes and thereby support both flexible resource allocation and performance gains, alternatively resource savings.

The converter devices presented in the previous sections can be transformed into a multi-input control-bounded analog-to-digital converter device by altering the A and B matrices. Specifically, by jointly considering L independent converter devices, i.e. organizing their corresponding Ae and matrices in two block diagonal matrices and

The system resources are then shared by altering the system as where is a transformation matrix such that (ai) from Claim 1 is satisfied. The new repre- sentation resulting from (87) and (88) are potentially dense matrices and therefore involving many shared physical states in the conversion process of each input channel. In summary, it is the H _N transformation matrix that enables multiple, otherwise independent conversion processes, to be combined such that they share physical circuit resources.

Formal Definition

A analog-to-digital converter device where and the input signal u(i) takes values in for U > 1 and

An Example

A particular example would use the Hadamard matrix, that was previously mentioned in the braided chain example from Section 5.1, as the transformation matrix. In this example four input channels, L = 4, are converted using four identical chain-of-integrators converter devices, i.e. and Furthermore, the transformation matrix is defined as : where the Hadamard matrix is defined recursively as and the b parameter is choosen as in the example from Section 5.1.

Notes While there are shown and described presently preferred embodiment of the invention, it is to be distinctly understood that the invention is not limited thereto but may be otherwise variously embodied and practiced within the scope of the following claims.

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