**Trust NOT fake chips!**

I'd like to share some recent experiences in using fake chips and perhaps point out the obvious:

DON'T TRUST CHEAP FAKE CHINESE CLONE CHIPS!

Especially in combination with other, not-so-cheap hardware. In my case, having no backup boards meant a sure stalling of my work for a few days until the newly ordered boards arrive, plus two fists of cash poured down the bin.

*Spartan survived the battle with China with only two pin casualties*

You might be wondering: what's his problem with using non-genuine clone chips, as long as they in fact do work? Well, I like them too, and I've used plenty of these in the past. However, as we shall see further, my problem is reliability and undocumented "features". Especially the nasty ones.

__Background of the story:__

I had to interface UART to a Spartan 6 LX150 chip, which I use for setting up some of the internals of my FPGA design, which then also sets the SPI module of my testchips. While constructing the digital system, I decided to use some of the USB-UART adapter boards out there on ebay, typically employing the notorious FT232 chip. So I bought a bunch of these (the price of $1 shipping included is very appealing), and because I know that there are some supereal fake FT232 chips out there being blackedout in software by the original FTDI driver, I bought a handful of different types from various vendors.

*Cheap USB - UART dongles*

After the parcels arrived, I discovered the CH340G chip, which is manufactured by the Big semiconductor player "Nanjing QinHeng Electronics Co.,Ltd.", offering a cheap alternative to the FT232. So I decided to pick and use that chip instead. The rest of the boards definitely used fake FT232s, which made me take the decision to use at least a genuine part from a Chinese vendor: it works, right?

And it did! I had been developing my system with this IC for about a month. Well, not for long: in a few cases, I would end up discovering that at certain conditions, the CH340G enters in some kind of latch-up state, and heats up to an unbearable temperature of over 80 deg C. It turns out that while doing this, its on-board 5 V to 3.3 V LDO fails as well (?), feeding out the deadly 5V out to my costly Spartan FPGA. Crabs! Crabs! Crabs! So I ended up popping two pins (PMOS of the LVCMOS33 driver popped, stuck to weak high, NMOS pulling down just about a few tens of milivolts) out of BANK 2. Here:

*NFET pulling dowm, but the PFET has popped...*

Very little is known about what's there inside the chip that manages the output voltages, although I wouldn't be extremely surprised if these guys have put just a huge resistive divider, ensuring a warm winter for all the Swedish, Norwegian and Finnish population. I really hope Zeptobars or JohnDMcMaster at siliconpr0n open the lid of that sucker one day. Reversing the board yields a wiring scheme which suggests that there is some kind of regulation.

*Regulation in CH340G*

Note the jumper bridging the 3V3 LDO output (low bandwidth LDO with external cap) with the 5V supply. For those who can read Chinese, here's its rich datasheet.

Oddly enough, if you turn off the power of the CH340G, let it cool down and turn it on again, the chip preserves its normal operation... until next time. I figured out it goes into latchup even when its power supply spikes and/or is unclean. Also perhaps a simple ground bounce glitch might have triggered it while connecting up the Tx/Rx lines to my board while being powered. And this is definitely not just an odd defective chip, I can reproduce the problem in all of the CH340G dongles I have here. Perhaps a design problem? Or in the best case a defective wafer lot which slipped in?

Finally, although it's normal that sh@t happens sometimes, next time please double check the vendors and quality of the parts you use... also, when buying an adapter, try finding one with opto-isolation.

And here's my final word - Crabs!

*Crabs! Crabs! Crabs!*

**Ramp kickback noise in column-parallel single-slope ADCs**

This post presents a quick investigation on the kickback noise problem in CIS ramp ADCs, and its assimilation with mass-spring-damper systems in theoretical mechanics.

In column-parallel ADC systems, the reference voltages are typically shared between all ADC slices. In the case of cyclic, SAR, and/or Sigma-Delta architectures, which use fixed DC reference voltages, it kind-of makes sense to try and minimize the ohmic resistance of the horizontally routed reference lines, so as to decrease the reference's settling time as much as possible.

State of-the-art Ramp ADCs, however, typically employ a local storage capacitor per each column which is initially reset to a reference starting point and then the latter (lumped capacitor) is slowly discharged via a global reference current source. The ramp ADCs are a special breed of data converters which distinguish themselves with two semi-discrete — semi-continuous-time modes of operation. The latter use continuous-time comparators which sense the slewing reference voltage (ramp) and toggle after it crosses the sampling point.

Now here's my motivation — no matter how much one tries to optimize the column comparator's input-pair for performance (see below), one normally ends up with some amount of charge, kicked back onto the ramp.

*Ramp comparator kickback tradeoffs*

Kickback charge is almost linearly linked to the size of the diff pair and its load. The more we want to suppress 1/f noise, increase gain, the more kickback noise we end up absorbing on the reference voltage line. Normally! There are different types of active kickback noise reduction techniques for continuous-time comparators, usually implemented by load cut-off after comparison, passive cross-coupled bulks of the differential pair, active bias compensation, etc... Here we are focusing on the ramp reference voltage distribution line and what could we do with it to reduce the impact of kickback on the ramp.

Typically, the mitigation of kickback noise problems consists of designing a well-compensated comparator , and I do prefer these methods of root-cause elimination. However, exploring other approaches does not hurt either.

The following figure shows a typical column-parallel comparator configuration and their reference voltage line.

*Column comparator chain and reference voltage line*

Imagine that all comparators have zero offset and they all have to convert the same voltage. Alternatively, imagine that there is just a slight variation between the voltages to be converted.

It appears that when one comparator kicks charge on the reference line, it is coupled next to the neighbouring comparators which may lead to an avalanche chain reaction. If the voltages to be converted differ just slightly (as is the case with a blank image and/or if we add natural comparator offset) then a kick in the center comparator, leads to a false trigger of the neighbour. This effect is also known as crosstalk.

Here is a simulation assuming an RC line with 1 mili Ohm column-to-column segments and 120 fF column capacitors. The kicked charge is highly exaggerated to amplify the effect, using 20 fF kick capacitor and almost infinitely steep step response on the cap of 1 Volt. Vramp_col1-7 shows the reference voltage line voltage at each column, and c1-c4 are the outputs of each column comparator (ideal and 4-total in this case), we can see that all toggle at the same time, meaning that the kicked charge will affect a high number of columns, in fact, all of them in this case.

*One mili-Ohm per column-column reference line resistance*

If one increases the resistance of the reference voltage line to 1 Ohm per column, which is not unrealistic at all for a standard aluminium FSI CMOS process, we get a more dampened column-to-column variation.

*One Ohm per column-column reference line resistance (column decoupling / damping effect)*

By the use of high-resistance reference voltage line the columns are less coupled between each other. This, however, comes at the cost of bandwidth, which also alters the ramp magnitude per each column itself. You may notice that the ramps are no longer following the same voltage magnitude, but have a voltage offset instead. This offset would typically translate to ADC DN-offset, which may actually be cancelled by digital correlated double sampling. The ramp itself however, typically, for a fast ADC runs at about 1-2 us full-swing (e.g. 1.5 V). It means that the voltage line resistance cannot be increased very much as it violates the total ramp bandwidth. By using the open-circuit time constant method, the settling time of a ramp reference line would be:

$$\tau_{1} + \tau_{2} + \tau_{n} = C_{n}(R_{n-1} + R_{n}) + ... + C_{2}(R_{1} + R_{2}) + R_{n}C_{n} + ... + R_{1}C_{1}$$

Apart from the obvuous bandwidth problem, increasing the resistance of the reference voltage, would also mean that the reference current-to-voltage noise translation is also amplified. Which will be roughly proportional to the MOSFET current noise power (spot) times the sink resistance (i.e. refererence line):

$$V_{nref}^{2} \propto i_{nsource}^{2} r_{sink}$$

A good tip for checking the ramp's linearity and/or noise (assuming you run transient noise) is to actually plot its first derrivative and check for spikes (shown below).

*First derrivative of ramp reference voltage line, kick (non-linearity) easily visible*

You can see the kick on the ramp as a spike in the center, as well as the total ramp capacitor charge alternation resulting as a change in the mean magnitude of the derrivative (left-right handside of the bottom plot). Ideally, we expect to see a flat line. This method is well-known and very efficient in the analysis of linear systems.

Finally, the ramp crosstalk effect and damping can be transformed into a theoretical mechanics problem, comprised of mass-spring-damper elements:

*Mass-Spring-Damper equivalent of the reference ramp crosstalk effect*

The weights m1-m3 are coupled to the ramp in-column slew capacitors via a second small damper (the diff-pair parasitic capacitance). The coupling point is then additionally coupled columnwise via a spring which causes the chain crosstalk reaction. Anyone daring to solve the differential equations? How about making things more complex by modelling the kick-kick avalanche of all columns?

**Surface dynamics of charge carriers measured using an indirect technique**

This is a quick follow-up on a previous post related to atomic manipulations with a scanning tunneling microscope. A few days ago Kris published a part of her work in Nature Communications — "Initiating and imaging the coherent surface dynamics of charge carriers in real space", wohooh, congratulations! I paid a visit to their lab back in 2013 (the time when she had a microscope with a gigantic chamber full of "dust", sadly no picture here) and in 2015 (when her group moved into the optical spectroscopy lab with a new high-performance mini-chambered microscope). I thought I'd continue the tradition of writing blue-sky nonsense on their work and put a few words from an ordinary person's perspective.

Apparently, plugging-in and out atoms with the tip of an STM is a lot more complicated than it looks at first sight. The tip itself is a source of charge carriers, as that's how the STM works. When the tip hovers above the scanned sample it causes carriers to tunnel through to the surface of the sample. Naturally, such charges can lead to a counter-reaction and disturbance of the spatial positions not only of the atoms directly underneath the microscope's tip, but also in a quite large radius around the tip itself. This paper describes the initial (at the start of injection) charge carrier dynamics on the scanning surface, which have been deduced by looking at the scattering patterns of toulene molecules. Oh well, or at least that is what I understood :)

The first figure in the paper shows images of Toulene molecules scattered on the surface of a Si(111) sample. A comparison between images before and after (direct?) atomic manipulations is shown. If one takes a bird's eye view on the Toulene molecules before and after the manipulation attempt, it seems like after the tip's bias has settled and reached a steady state, the Toulene molecules move in a systematic direction, that forces them to lump to each other. I.e. my interpretation is that in the before images Toulene is a lot more scattered, as compared to its position after manipulations. This change of position of the Toulene molecules on the surface of the sample is a source of information about the charge carriers' path (?) and hence behaviour. Right after the charge carriers reach the surface of the sample, they encounter a number of scattering events, before reaching a steady state (steady state = very vague expression). During that transitional time the carriers transfer their energy to the atoms on the surface, which leads to their movement and bond breaking. This movement gives indirect information about the dynamics of electrons/holes themselves.

Surely this paper is a lot more involved and is waay beyond my level of understanding of quantum physics to be able to summarize it in my own words which make sense. But, what really fascinates me — and that's why I started writing this ambitious post — is mankind's achievements in material science to this date. I mean, these things are roughly ~1 nanometer scale, and the hole/electron dynamics described occur in the order of femto seconds... i.e. that's easily deep inside the THz range. And when you add ambient temperature effects the picture becomes a complete mess, yet, it is quite well understood. On the other hand, carrier dynamics in the order of ~100s femtoseconds is not that fast, maybe that's why modern-day electronics is still struggling to conquer the THz gap?

**Point Spread-Function, Optical and Modulation Transfer Functions**

In solid-state imaging, the obtained image contrast and spatial resolution, apart from the electrical circuit performance with respect to crosstalk and linearity, vastly depends on the spatial organization of the light sensing pixels. An image sensor can essentially be looked into as if it was a non-ideal "electrical" transducer, distorting the fed through it light information. In the general case, if we exclude all electrical complexities, an imaging system's performance can be extremely complicated for evaluation also due to the numerous second order effects accumulated along the optical signal path. This implies that we need to apply some restrictions to the degrees of freedom in our analyses, and hence, apply a suitable to the particular application divide and conquer approach. This post aims to give an introduction to the Modulation Transfer Function parameter and its implication in image sensor optical array design.

With that respect, two fundamental properties in imaging systems can be identified:

__ 1. Linearity__ — just as any other black-box system, it implies that the output corresponding to a sum of inputs should also be equal to the sum of the inputs processed individually.

__ 2. Invariance__ — the projected image in the spatial domain remains the same even after the imaged object is moved to another location in "space".

Linearity is usually taken care of accurate readout electronics and photodetector design. Invariance, or spatial resolution, however, greatly depends on the chosen geometrical shape of the pixels, and their arrangement in space. A common benchmark parameter in imaging systems is their Modulation Transfer Function (MTF), which is to a large extent linked with the arrangement of the array elements. In order to continue with the definition and implications of MTF, we must first have a look at what a Point Spread-Function (PSF) in optics is.

Imagine you can find a point light source (e.g. a torch) with an infinitely small aperture, which you then point to the image sensor in your mobile phone camera. What you would expect to see on the display is the same point at the same location where you pointed the beam:

*Beam with an infinitely small aperture projected on the image sensor plane*

Real optical systems, however, suffer from optical imperfections, which result in smearing out of the energy around the infinitely small aperture beam you pointed towards the system, hence, yielding a loss of sharpness. The *Point Spread Function* provides a measure of the imaging system's smearing out at a single, infinitely small, physical point in the imaging plane. In the case of a real pixel in an image sensor, its PSF can be defined as a function of its effective aperture:

*Effective spatial aperture definition of a pixel/photodetector*

The PSF for the region within the photodiode is equal to some constant (modulated by the quantum efficiency of the photodiode), and for the region where the pixel is covered with the metal we can assume that the PSF is zero. Although, metal layers in modern integrated circuits are so thin, that photons can still tunnel through them. Then formally:

$$ S(x) = \begin{cases} s_{o} & x_{o} - L/2 \le x \le x_{o} + L/2 \\ 0 & \text{for } x \text{ is all others} \end{cases} $$Which in simple words represents a boxcar function.

*Point Spread Function for the above pixel is similar to a boxcar function*

By knowing the Point Spread Function of the array, we can estimate the global Optical Transfer Function of the whole imaging system by using a simple mathematical exercise. Note that the PSF is a spatial domain function. Just as in electronics, apart from looking at signals in the time domain, frequency domain analyses prove to be extremely representative too. This is also the case with 2D imaging and their spatial and frequency domains. The Optical and Modulation Transfer Functions are usefully represented in the frequency domain. But what exactly is OTF and MTF? Let's have a look at the physical effects they describe, by examining the following pattern:

If we feed in an ideal optical pattern as the one shown at the top, due to the limited single photodetector aperture (opening/fill factor), the reconstructed at the output image would be smeared out. Just as the torch test with an infinitely small aperture. Depending on the effective aperture size (pixel fill factor) as well as the geometrical arrangement of individual pixels, the smearing would have a different magnitude for different spatial sizes. To simplify the optical system's evaluation, this smearing can be expressed in the frequency domain, which is done by the OTF and MTF. An imaging system can be viewed of as a low-pass filter in the frequency domain, thus OTF and MTF represent the system's amplitude-frequency characteristics.

Similar to electronics, the OTF can be derived by computing the Fourier transform of the Point Spread Function with is the equivalent to the impulse response in electronic linear time-invariant systems:

$$ S(f) = \int_{-\infty}^{\infty} S(x) e^{j 2 \pi f_{x}}~dx $$Substituting with the PSF we get the following definition:

$$ S(f) = s_{o} \int_{x_{o}-L/2}^{x_{o}+L/2} e^{j 2 \pi f_{x}}~dx \propto \frac{sin(\pi f L)}{\pi f L} $$*Simple overview of spatial vs frequency domain in optical systems*

What is the difference between OTF and MTF? OTF contains a negative and imaginary part from the fourier transformation and carries also phase information. MTF is defined as the ratio of the output modulation to the input modulation as a function of the spatial frequency, but normalized. It is typically expressed as [strips/cycles]/mm. Thus, to normalize, MTF is derrived from the absolute ratio of the OTF, normalized with the OTF at zero frequency (or DC in analogy with electronics).

$$ MTF(f) = \frac{|S(f)|}{|S(f=0)|} $$

Hence, the MTF is a sinc function:

$$ MTF(f) = \frac{sin(\pi f L)}{\pi f L} $$

Various pixel array configurations exist, which directly affect the MTF coming from the sensor's aperture. It is very important to note the aperture MTF is not the only source, nor a global MTF degradation parameter. In imaging of moving objects, such as Time-Delay-Integration imaging mode, discrete (temporal) MTF from object synchronization and misalignment also occurs. In addition, the lens also add-up to the total camera system's MTF. The good news is that as the MTF is normalized and dimensionless, we can easily multiply all MTFs of all sources of degradation to identify the global MTF. Again, just as in electronics, the MTF degradation order, as it progresses is important. Identical to electrical amplifiers and the Friis formula, we should always place the highest (if possible) MTF optical part to come first along the signal chain. Unfortunately, in optics swapping parts along the signal chain is a rather more difficult task, than it is with amplifier chains.

References:

[1] K. Rossmann et al., *Tools for the study of imaging systems*, Technical Report, Department of Radiology, The University of Chicago and the Argonne Cancer Research Hospital, August 1969.

**A quick review of Time-to-Digital Converter architectures**

*I have been recently looking at various ADC architectures employing coarse-fine interpolation methods using TDCs, so I thought I'd share a quick list with brief explanation of the most common TDC architectures.*

__ 1. Counter based TDC__ – Probably the simplest method to digitize a time interval is by using a stopcounter. Counter based TDCs use a reference clock oscillator and asynchronous binary counters controlled by a start and a stop pulse. A simple timing diagram depicting the counter method's principle of operation is shown below:

*Counter based TDC principle of operation*

The time resolution of counter based TDCs is determined by the oscillator frequency and its quantization error is generally dependent on the clock period. Counter method TDCs offer a virtually unlimited dynamic range, as with the addition of a single bit the counter's capacity is doubled. This comes at the cost of an increased quantization error. The start signal can be either synchronous or asynchronous with the clock. In the case of an asynchronous start and stop pulses its quantization error can be totally random. Similarly if the start pulse is synchronous with the clock, this TDCs' quantization error is fully deterministic, if we exclude second order effects from clock and start pulse jitter respectively. With current CMOS process nodes 45nm or so, the highest achievable resolutions vary in the order of a few nanoseconds to 200 picoseconds. The counter based TDC's resolution is generally limited by the reference clock frequency stability and the metastability and gate delay of the stop latches. To increase the counting speed an often reffered to as Double Data Rate (DDR) counting scheme can be employed, which uses a counter incrementing its value on both rising and falling edges of the count clock.

__ 2. Nutt interpolated TDC (Time Intervalometer)__ – initially proposed by Ronald Nutt [1], the Nutt architecture is based on measuring the quantization errors formed in the counter method and subtracting these to form the final counter value. The sketch below depicts the basic principle of the Nutt method:

*Principle of Nutt interpolation method*

Typically a short-range TDC is used for synchronous fine quantization error measurement between the counter's clock and the stop pulse. The used short-range TDC as a fine quantizer can be of any type as long as its input range matches with the largest expected quantization error to be measured, which is the counter's clock period $T$. In case of the use of a Double Data Rate (DDR) counter the input range is reduced to $T/2$. The global TDC presicion of a TDC employing the counter scheme with Nutt interpolation is improved by a factor of $2^{M}$ where M is the resolution of the fine TDC in bits.

The challenges the design of Nutt interpolator based TDCs are generally linked with the difficulty of matching the gain of the fine TDC with the clock period of the coarse counter. Both INL and DNL of the fine interpolation TDC is translated as a static DNL error at the combined output. Moreover, any noise in the fine TDC also translates as a DNL error in the final TDC value, which creates a non-deterministinc DNL error performance which cannot be corrected for.

__ 3. Flash TDC__ – This is probably one of the simplest short-range TDC architectures. It uses a clock delay line and a set of flip flops controlled by the stop pulse for strobing the phase-delayed start pulse.

*Basic Flash TDC core architecture*

It can be employed in a standalone asynchronous scheme, where the start pulse is fed to the delay line and the stop pulse gates the flip flops. Alternatively it can be used as an interpolator to the counter scheme, in which case it's start pulse is controlled by the low-frequency count clock. In the case of the last configuration it is important that the sum of the delays in the delay line matches with the clock period $T$, or $T/2$ in the case of a DDR counting scheme.

Typically one way of delay synchronization is the deployment of a phase locked loop, which keeps the last delayed clock in-phase with the main count clock period. Usually the choice of delay number is based on binary sets of $2^{N}$. The strobed value in the flip flops is thermometer coded, which is consecutively converted to binary and added to the final value. Alternatively a scheme employing a ring oscillator incrementing the binary counter can also be used. In such schemes the complexity is moved from a PLL design to a constant-gain ring oscillator challenge.

An important aspect within Flash TDCs is that their time resolution is still limited to a single gate delay of the CMOS process.

__ 4. Vernier TDC__ – The Vernier TDC, compared to Flash TDCs aims to improve the converter resolution beyond the gate delay limit. A principle diagram of a classic Vernier architecture is shown below:

*Basic Vernier TDC core architecture*

Two sets of delay chains with a phase difference are used, where the stop signal delays use a slightly smaller time delay. This causes the stop signal to slightly catch-up the start signal as it propagates through the delay line. The time resolution of Vernier TDCs is practically determined by the time difference between the delay lines $t_{res}=\tau_{1}-\tau_{2}$. If no gain calibration of the digitized value is intended, the choice of time delay and number of delay stages in Vernier TDCs should again be carefully considered. The chosen delay for the start pulse divided by the delay time difference should equal to the number of used delay stages, which to retain the binary coding, whould be $2^{N}$.

$$\frac{t_{ds}}{t_{d}}=N$$

The Vernier TDC principle is inspired by the secondary scale in calipers and micrometers for fine resolution measurements. It was invented by the French mathematician Pierre Vernier [2].

__ 5. Time-to-Voltage Converter + ADC__ – the TVC architecture converts time interval into votlage. It is difficult to achieve a high time resolution with such schemes as traditionally the only reasonable way of converting time into voltage is by using a current integrator in which a capacitor is charged with constant current for the duration of the measured time interval.

*Basic principle of TVC TDCs*

After the time measurement is complete, a traditional ADC is used to quantize the integrated voltage on the capacitor.

These architectures could be used in applications where a mid- or large time measurement ranges are required. For practical reasons, the TVC type TDCs do not suit well as a time interpolator combined with counter based TDCs. The noise and linearity difficulties in current integrators for high speeds are setting a high lower bound for the time dynamic range of TVCs.

__ 6. Time Differenec Amplifier based TDC__ – one concept for time difference amplification (TDA) in the digital domain is introduced by Abbas et al. [3]. An analog time stretcher allows for resolution improvement by amplifying the input time interval and successively converting it with a lower resolution TDC. The concept is similar to the analog voltage gain stages placed at the input of ADCs. A time amplifier based on the originally reported by Abbas et at. architecture [3] is shown below:

*TDA principle and a basic TDA cell*

The circuit represents a winner-takes-it-all scheme, where the gates are forced in a metastable state performing a mutual exclusion operation, the consecutive output inverters apply regenerative gain to the MUTEX element. By using two MUTEX elements linked with the start and stop signal in parallel, with a time offset and then edge-combining their outputs one can amplify the initial edge's time difference. The difference in the output voltages of a bistable element in metastability is approximately $\Delta_{V} = \theta.\Delta_{t}.e^{1/\tau}$, where $\tau$ is the intrinsic gate time constant, $\theta$ the conversion factor from time to initial voltage change at the metastable nodes and $\Delta_{t}$ is the incoming pulse time difference [4]. We can note that the output rate of change at the metastable node is exponenitally dependent. Thus, if we intentionally delay "forwards" and "backwards" in time two MUTEX elements and combine their edges, we would acquire a linear time difference amplification caused by the logical edge combination. The output time difference in the corresponding case would be equal to $\Delta_{out} = \tau . ln(Td + \Delta_{in}) - \tau . ln(Td - \Delta_{in}) $. Several different variations of the latter circuit exist, most of which base on the logarithmic voltage dependency of latches in metastable state.

All time difference amplifier TDC approaches impose challenges related to the linearity of TDAs as well as their usually limited amplification factors.

__ 7. Successive approximation TDC__ – this architecture uses the binary search algorithm to resolve a time interval. A principle diagram of a 4-bit binary search TDC is shown, as presented by Miyashita et al. [5].

*Successive approximation TDC architecture*

Imagine that the stop pulse slightly leads the start pulse with time difference of $\Delta = 1.5\tau$. The arbiter D3 would then detect a lead and reconfigure the multiplexer delays such that they lag the start signal by $4\tau$, thus leading to a difference of $2.5\tau$. The resolved MSB in that case would be the inverted value of D3, or '0'. Further on, arbiter D2 detects a lead in the start pulse over the stop by $2.5\tau$, in which case it would reconfigure the multiplexer delays in stage two to lag the stop signal by $2\tau$. The MSB-1 value is the inverted value of D2 arbiter which would thus be '1'. The MSB-2 value would be equal to 0, respectively, as the stop signal now leads with $0.5\tau$. Finally, arbiter D0 dechipers a '1' due to the start signal now leading with $0.5\tau$.

This topology effectively utilizes a time-domain SAR scheme and has a time resolution of $\tau$. Open-loop binary search schemes as the propsed here [5] require good matching between the delay elements, which typically suffer from low PSRR. Nevertheless, this SAR scheme is relatively new to the TDC world and might provide us with promising food for future research. Compared to the Flash TDC, the SAR scheme offers a $2^{N}-N$ reduction of strobe latches.

__ 8. Stochastic TDC__ – this family of TDCs uses the same core principle of operation as Flash TDCs, however, it is formed redundantly. Here is a sketch:

*Principle diagram of a Stochastic TDC architecture*

Because the Flash latches have an intrinsic threshold mismatch, they practically introduce a natural Gaussian dither into the resolution of each Flash bit. If we read out the values of all latches and average their values we can extract digitized values with sub quantized step resolution. In order take advantage of the oversampling, however, Flash TDCs require a large set of latches and delay elements, which comes at the cost of power. They are traditionally reserveed for DLL-type applications and often require a digital back-end calibration [6].

**References:**

[1] Ronald Nutt, *Digital Time Intervalometer*, Review of Scientific Instruments, 39, 1342-1345 (1968)