**Attacking a!b!=a!+b!+c! the brute-force engineering and analytic mathematical ways.**

Not very long ago I had an argue about whether there is a solution to the equation a!b!=a!+b!+c!. It was an argue between a mathematician and engineer, so while my math friend was trying to find the solution to the problem the scientific (analytical mathematic) way I decided to give it a try and brute-force it using some nested loops in Matlab. Initially I somehow made a simple uncatchable syntax mistake and my program seemed to be looping forever. The very same evening my math soul continued with the analytical solution and finally came with a statement that the equation does not have a solution. We both agreed and assumed that we are all right.

Well, the day after we found out that there indeed is at least one solution. This is a=3, b=3, c=4. The conclusion I want to bring here is that no matter how good one is in his profession, his self-confidence should stay at maximum 0.999999999999999999.

Interested in the analytical solution? Read here. Interested in the brute-force method? Find the nested loop Matlab script here.

**Offset measurement of latched comparators.**

During comparator design one is often (but not always) interested in the systematic and mismatch-related offset. When dealing with continuous-time open-loop comparators the offset measurement methodology is similar conventional OTA offset measurement e.g. DC sweep, while latched comparators require a somewhat more discrete-like manner of measuring offset.

One possible approach is to use a very precise (long sloped) linear ramp, and using massive oversampling and comparator decision storage. This approach seems to be a bit tedious as its precision is solely dependent on the ramp's slope.

An alternative method for measuring offset of latched comparators would be the binary search method. One can possibly use an ideal DAC to apply "search" voltages and depending on the comparator's decision reach a settling value with a constant binary-weighted precision. E.g. to achieve 16 bits of search accuracy only 16 cycles would be required. Here are some basic illustrative examples of the two methods.

The binary search method for measurement can be applied in various ways. One practical implementation was to implement the algorithm using Verilog-A and directly add a "search" instance where you can test your comparator directly in your existing SPICE simulation. Here is a very simple example of a possible implementation.

*Hi,
I changed some of your code because I found the array method quite confusing. What is the reason for using a large lookup table? This way, you are calculating all the values 2^16 values instead of just the 16 ones needed...
I used something like:*
if (CompDecision > vTransComp) begin
// Comparator output is high, inp>inn, ->make inp smaller
iTop = iCompinp;
iCompinp = iBot + (iTop-iBot)/2;
end
else begin
// Comparator output is low, inp < inn, make inp larger
iBot = iCompinp;
iCompinp = iBot + (iTop-iBot)/2;
end
*thus directly calculating the values if needed. The iteration stops after 16 cycles, which is the same as having a resolution of 16. So I just tried to keep it easy so I can understand it.
Just my 2¢
regards,
Stefi *

Indeed there is no need to calculate all 2^N code steps and later on comare them in a look-up table fashion. This is some code inefficiency which has not been realized back in the day, my lord... Nevertheless, the code (either way) works and it is the concept that matters. For this reason I have left the old code untouched, as it has been tested. I am however encouraging you to explore and write your own implementaton. Nevertheless, here's the original version and an aka inefficient implementation:

// VerilogA for daisyCycAd, daisyCycAdBinSchSAR, veriloga // // A component handy in comparator offset measurement. Uses a binary search algorithm with a "dive" coefficient of 2. See comments for more information. // // Initial version P1A - Deyan Dimitrov didolevski@gmail.com // `include "constants.vams" `include "disciplines.vams" module daisyCycAdBinSchSAR(vCompIn, vCompOut, vClk, vdd, gnd, vCompRef); input vCompIn, vClk; output vCompOut, vCompRef; inout vdd, gnd; electrical vCompIn, vCompOut, vClk, vdd, gnd, vCompRef; parameter real vTransClk = 1.65; parameter real vTransComp = 1.65; parameter real vSchTop = 3.3; parameter real vSchBot = 0; parameter real vCompReference = 1.5; //parameter integer Resolution = 16; parameter integer NrOfCodes = 65535; //parameter real tCompSpeed = 100e-12; real vRefInReal; real vBinSch; real vComp; real scharray[0:NrOfCodes]; //real next; integer imid; integer imin; integer imax; integer i; integer codes; analog begin @(initial_step) begin vBinSch = vSchBot + (vSchTop-vSchBot)/2; // Mid point as a start of the search imin = 0; imax = NrOfCodes-2; imid = (NrOfCodes-2)/2; for (i = 0; i < NrOfCodes-1; i = i + 1) begin scharray[i] = (((vSchTop-vSchBot)/NrOfCodes)*i)+vSchBot; // Creating reference array end $strobe("Array Max %g", scharray[imax]); $strobe("Array Min %g", scharray[imin]); end @(cross(V(vClk) - vTransClk, 1)) begin // next = $abstime + tCompSpeed; // Possible internal compensation for comparator's delay // end // @(timer(next)) begin // $strobe("Imax: %d", imax); // $strobe("Imid: %d", imid); // $strobe("Imin: %d", imin); vComp = V(vCompIn); // Strobe comparator decision // vRefInReal = V(vRefIn, gnd); if (imax >= imin) begin // Continue searching if imax >= imin if (V(vCompIn) > vTransComp) begin // Find-out which sub-array to search $strobe("%g",vComp); // vBinSch = vBinSch + (vSchTop-vBinSch)/2; imin = imid + 1; // Change min index for the upper sub-array imid = imin + ((imax-imin)/2); // Update imid to be used for strobing-out end else begin $strobe("%g",vComp); // vBinSch = vBinSch - (vBinSch-vSchBot)/2; imax = imid - 1; // Change max index for the upper sub-array imid = imin + ((imax-imin)/2); // Update imid to be used for strobing-out end end vBinSch = scharray[imid]; // Look-up at the reference array and assign to vBinSch (Votage to be strobed-out) end V(vCompOut) <+ vBinSch; // Update search voltage V(vCompRef) <+ vCompReference; // Update comparator reference end endmodule

The whole principle is simple and self-explanatory. Some delay between the latch clock and S/H clock of the search component is required to compensate for the comparator's speed. Here are some practical usage illustrations.

If you find this useful the component's symbol for Virtuoso 6 as well as the *.va code can be found here.

**A basic "seed" for plotting csv data files with**

**matplotlib.**

Well, basically all the usage guide and information about the matplotlib library is available online, however I am writing this as an emphasis on how good matplotlib actually is, moreover I can again use this short snippet as a quick "seed/template" in the future without having to write it from scratch. Here's a short example on plotting line charts from a *.csv file.

## Plotting from a csv and writing down a pdf using matplotlib. Look after the transpose function for the column plotting settings. ## ## Initial verison A Deyan Levski - didolevski@gmail.com ## ## from numpy import array import csv import matplotlib.pyplot as plt #def plotit(filename,title): filename='Slew-Rate-Limit-Errors-Capture.csv'; title='OTA residue voltage'; cs=csv.reader(file(filename,"r"),delimiter=",")# seperator of csv data is a tab ("\t") # Have to convert comma decimal to dot decimal delimiter # newlist=[[float(e.replace(".",".")) for e in x] for x in cs] #If no conversion needed paste in: newlist=[[float(e) for e in x] for x in cs] newlist=[[float(e) for e in x] for x in cs] # convert from pairs for [[x1,y1],[x2,y2]...etc] to [x1,x2,..] [y1,y2,..] ar=array(newlist).transpose() fig=plt.figure() ax=fig.add_subplot(1,1,1) ax.plot(ar[0],ar[1]) ax.plot(ar[0],ar[3]) ax.set_xlabel(r"$T_{p}$") ax.set_ylabel(r"Voltage") ax.grid(True) # Default grid settings #ax.grid(color='b', alpha=0.5, linestyle='dashed', linewidth=0.5) #ax.axis('tight') # Tight auto axis ax.set_ylim([-0.5, 4]) # Custom Y axis #axes[2].set_xlim([2, 5]) # Custom X axis plt.title(title) # Plot plt.savefig(filename+".pdf") # Save figure

One can also use it for plotting histograms, e.g. if you are running some MonteCarlo simulations directly from the SPICE engine and you do not have any GUI available to use.

## Plotting from a csv and writing down a pdf using matplotlib. Look after the transpose function for the column plotting settings. ## ## Initial verison A Deyan Levski - didolevski@gmail.com ## ## from matplotlib import rc rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']}) ## for Palatino and other serif fonts use: #rc('font',**{'family':'serif','serif':['Palatino']}) rc('text', usetex=True) from numpy import array import csv import math import numpy as np import matplotlib.pyplot as plt import matplotlib.mlab as mlab #def plotit(filename,title): filename='OTA-InputOffset-MC-4000.hash'; title='OTA Input Offset Distribution'; cs=csv.reader(file(filename,"r"),delimiter=",")# seperator of csv data is a tab ("\t") # Have to convert comma decimal to dot decimal delimiter # newlist=[[float(e.replace(".",".")) for e in x] for x in cs] #If no conversion needed paste in: newlist=[[float(e) for e in x] for x in cs] newlist=[[float(e) for e in x] for x in cs] # convert from pairs for [[x1,y1],[x2,y2]...etc] to [x1,x2,..] [y1,y2,..] ar=array(newlist).transpose() fig=plt.figure() ax = fig.add_subplot(111) # the histogram of the data n, bins, patches = ax.hist(ar[1], 30, normed=1, facecolor='green', alpha=0.75) # hist uses np.histogram under the hood to create 'n' and 'bins'. # np.histogram returns the bin edges, so there will be 50 probability # density values in n, 51 bin edges in bins and 50 patches. To get # everything lined up, we'll compute the bin centers bincenters = 0.5*(bins[1:]+bins[:-1]) acc=0 nr=0 for i in ar[1]: acc += i nr += 1 mu=acc/nr print mu acc2=0 nr=0 for i in ar[1]: acc += ((i-mu)**2) nr += 1 sd=acc/nr sd=math.sqrt(abs(sd)) print sd # add a 'best fit' line for the normal PDF y = mlab.normpdf( bincenters, mu, sd) l = ax.plot(bincenters, y, 'r--', linewidth=1) ax.set_xlabel('Input offset voltage') ax.set_ylabel('Run') ax.set_xlim(-0.05,0.05) #ax.set_ylim(0, 0.03) ax.grid(True) plt.title(r'Input offset voltage variation $\mu$=%.3f, $\sigma$=%.3f' %(mu, sd)) plt.savefig(filename+".ps") # Save figure plt.show() #ax=fig.add_subplot(1,1,1) #ax.plot(ar[0],ar[1]) #ax.plot(ar[0],ar[3]) #ax.set_xlabel(r"$T_{p}$") #ax.set_ylabel(r"Voltage") #ax.grid(True) # Default grid settings #ax.grid(color='b', alpha=0.5, linestyle='dashed', linewidth=0.5) #ax.axis('tight') # Tight auto axis #ax.set_ylim([-0.5, 4]) # Custom Y axis #axes[2].set_xlim([2, 5]) # Custom X axis #plt.title(title) # Plot #plt.savefig(filename+".pdf") # Save figure

Here is a link to the "seed" folder. Where generated pdf graps can also be found.

**INL and CFPN image artifact estimation script.**

A short function aiming to give a rough overview of INL and column mismatch effects on images in a column-parallel ADCs in CMOS Image Sensors. For now it works only with 8 bit images. Some bugs in the file read functions, not supporting all image formats could be existent.

Here are some examples of images with various INL and column mismatch:

An attempt to model randomly distributed INL was also made, however in practice ADC INL is correlated and will not necessarily be so random. Basically random INL would appear as random noise on the image. Anyway it is a good example of what would happen if theoretically we have randomly distributed INL.

You can try-out the script it is available here. It is also available at Matlab Central.

**Releasing the C source of the**

**AWS.**

Lately some nostalgy on the old plenty-of-time days has caught me. I was randomly looking through the pages, being in the process of updating and adding-up old work when I came to one of the not-very-latest versions of the source code of the AWS. I realize that I have not posted it, it might be that someone could actually benefit from it...or?

Anyway, you can find it here. You can also find the board and schematics here.