# Patent

## Title
Method for testing the error ratio of a device

## Abstract

A method for testing the (Bit) Error Ratio BER of a device against a maximal allowable (Bit) Error Ratio BERlimit with a early pass and/or early fail criterion, whereby the early pass and/or early fail criterion is allowed to be wrong only by a small probability D. ns bits of the output of the device are measured, thereby ne erroneous bits of the ns bits are detected. PDhigh and/or PDlow are obtained, whereby PDhigh is the worst possible likelihood distribution and PDlow is the best possible likelihood distribution containing the measured ne erroneous bits with the probablility D. The average numbers of erroneous bits NEhigh and NElow for PDhigh and PDlow are obtained. NEhigh and NElow are compared with NElimit=BERlimit×ns. If NElimit is higher than NEhigh or NElimit is lower than NElow the test is stopped.

## Background
The application is related to testing the (Bit) Error Ratio of a device, such as a digital receiver for mobile phones.
Concerning the state of the art, reference is made to U.S. Pat. No. 5,606,563. This document discloses a method of determining an error level of a data channel comprised of receiving channel parity error data indicating when bit errors occur within a set of data carried on the channel (channel error events), successively integrating the channel error events data over successive accumulation periods, comparing the integrated channel error events data with a threshold, and indicating an alarm in the event the integrated channel error events data exceeds the threshold.
Testing digital receivers is two fold: The receiver is offered a signal, usually accompanied with certain, stress conditions. The receiver has to demodulate and to decode this signal. Due to stress conditions the result can be partly erroneous. The ratio of erroneously demodulated to correctly demodulated bits is measured in the Error Ratio test. The Bit Error Ratio BER test or more general any error rate test, for example Frame Error Ratio FER, Block Error Ratio BLER each comprising a Bit Error test, is subject of this application.
BER testing is time consuming. This is illustrated by the following example: a frequently used BER limit is 0.001. The bitrate frequently used for this test is approx. 10 kbit/s. Due to statistical relevance it is not enough to observe 1 error in 1000 bits. It is usual to observe approx. 200 errors in 200 000 bits. This single BER test lasts 20 seconds. There are combined tests which contain this single BER test several times, e.g. the receiver blocking test. Within this test the single BER test is repeated 12750 times with different stress conditions. Total test time here is more than 70 h. It is known in the state of the art to fail the DUT (Device Under Test) early only, when a fixed number of errors as 200 errors are observed before 200 000 bits are applied. (Numbers from the example above).
It is the object of the present application to propose a method for testing the Bit Error Ratio of a device with reduced test time, preserving the statistical relevance.
The object is solved by the features of claim 1 or claim 4.
According to the present invention, the worst possible likelihood distribution PDhigh and/or the best possible likelihood distribution PDlow are obtained by the formulas given in claim 1 and 4, respectively. From these likelihood distributions, the average number NEhigh or NElow is obtained and compared with the limit NElimit, respectively. If the limit NElimit is higher than the average number NEhigh or smaller than the average number NElow, the test is stopped and it is decided that the device has early passed or early failed, respectively.
The dependent claims contain further developments of the invention.
The Poisson distribution preferably used for the present invention, is best adapted to the BER problem. The nature of the Bit Error occurrence is ideally described by the binomial distribution. This distribution, however, can only be handled for small number of bits and errors. For a high number of bits and a low Bit Error Ratio the binomial distribution is approximated by the Poisson distribution. This prerequisite (high number of bits, low BER) is ideally fulfilled for normal Bit Error Rate testing (BER limit 0.001) as long as the DUT is not totally broken (BER 0.5). The application proposes to overcome problems with the discrete nature of the Poisson distribution. With the same method an early pass condition as well as an early fail condition is derived.



An embodiment of the invention is described hereafter. In the drawings
FIG. 1 shows a diagram to illustrate the inventive method,
FIG. 2 shows the referenced Bit Error Ratio bernorm as a function of the measured errors ne,
FIG. 3 shows a diagram to illustrate a measurement using a first embodiment of the inventive method,
FIG. 4 shows a diagram to illustrate a measurement using a second embodiment of the inventive method and
FIG. 5 shows a diagram illustrating the position at the end of the test using the first embodiment of the inventive method as a function of probability.



In the following, the early fail condition and the early pass condition is derived with mathematical methods.
Due to the nature of the test, namely discrete error events, the early stop conditions are declared not valid, when fractional errors<1 are used to calculate the early stop limits. The application contains proposals, how to conduct the test at this undefined areas. The proposals are conservative (not risky). A DUT on the limit does not achieve any early stop condition. The application proposes to stop the test unconditioned at a specific number K (for example 200 bit) errors. As this proposal contains a pseudo paradox, an additional proposal to resolve this pseudo paradox is appended.
Based on a single measurement, a confidence range CR around this measurement is derived. It has the property that with high probability the final result can be found in this range.
The confidence range CR is compared with the specified BER limit. From the result a diagram is derived containing the early fail and the early pass condition.
With a finite number of samples ns of measured bits, the final Bit Error Ratio BER cannot be determined exactly. Applying a finite number of samples ns, a number of errors ne is measured. ne/ns=ber is the preliminary Bit Error Ratio.
In a single test a finite number of measured bits ns is applied, and a number of errors ne is measured. ne is connected with a certain differential probability in the Poisson distribution. The probability and the position in the distribution conducting just one single test is not known.
Repeating this test infinite times, applying repeatedly the same ns, the complete Poisson distribution is obtained. The average number (mean value) of errors is NE. NE/ns is the final BER. The Poisson distribution has the variable ne and is characterised by the parameter NE, the average or mean value. Real probabilities to find ne between two limits are calculated by integrating between such limits. The width of the Poisson distribution increases proportional to SQR(NE), that means, it increases absolutely, but decreases relatively.
In a single test ns samples are applied and ne errors are measured. The result can be a member of different Poisson distributions each characterized by another parameter NE. Two of them are given as follows:
The worst possible distribution NEhigh, containing the measured ne with the probability D1, is given by




  
    
      
        
          D
          1
        
        =
        
          
            ∫
            0
            
              n
              ⁢
              
                
              
              ⁢
              e
            
          
          ⁢
          
            
              
                PD
                high
              
              ⁡
              
                (
                
                  
                    NE
                    high
                  
                  ,
                  ni
                
                )
              
            
            ⁢
            
              d
              ni
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




In the example D1 is 0.002=0.2% PDhigh is the wanted Poisson distribution with the variable ni. ne is the measured number of errors.
The best possible distributions NElow, containing the measured ne with the probability D2 is given by




  
    
      
        
          D
          2
        
        =
        
          
            ∫
            ne
            ∞
          
          ⁢
          
            
              
                PD
                low
              
              ⁡
              
                (
                
                  
                    NE
                    low
                  
                  ,
                  ni
                
                )
              
            
            ⁢
            
              d
              ni
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




In the example D2 is equal D1 and it is D=D1=D2=0.002=0.2%.
To illustrate the meaning of the range between NElow and NEhigh refer to FIG. 1. FIG. 1 shows the likelihood density PD as a function of the measured number of errors ne. In the example, the actual detected number of errors ne within the measured sample of ns bits is 10. The likelihood distribution of the errors is not known. The worst possible likelihood distribution PDhigh under all possible likelihood distributions as well as the best possible likelihood distribution PDlow under all possible likelihood distributions are shown. The worst possible likelihood distribution PDhigh is characterized in that the integral from 0 to ne=10 gives a total probability of D1=0.002. The best possible likelihood distribution PDlow is characterized in that the integral from ne=10 to ∞ gives a total probability of D2=0.002. In the preferred embodiment D1 is equal to D2, i.e. D1=D2=0.002=0.2%. After having obtained the likelihood distribution PDhigh and PDlow from formulas (1) and (2), the average values or mean values NEhigh for the likelihood distribution PDhigh and NElow for the likelihood distribution PDlow can be obtained. The range between the mean value NElow and NEhigh is the confidence range CR indicated in FIG. 1.
In the case the measured value ne is a rather untypical result (in the example just 0.2% probability) nevertheless the final result NE can still be found in this range, called confidence range CR.
The probabilities D1 and D2 in (1) and (2) can be independent, but preferable they are dependent and equal (D=D1=D2).
For the Poisson distribution NElow and NEhigh can be obtained. With the formulas (3) and (4) respectively the inputs are the number of errors ne, measured in this test and the probabilities D and C=1−D. The Output is NE, the parameter describing the average of the Poisson distribution.
The following example is illustrated in FIG. 1 (D=D1=D2):




  
    
      
        
          NE
          low
        
        =
        
          
            qchisq
            ⁡
            
              (
              
                D
                ,
                
                  2
                  ·
                  ne
                
              
              )
            
          
          2
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          NE
          high
        
        =
        
          
            qchisq
            ⁡
            
              (
              
                C
                ,
                
                  2
                  ⁢
                  
                    (
                    
                      ne
                      +
                      1
                    
                    )
                  
                
              
              )
            
          
          2
        
      
    
    
      
        (
        4
        )
      
    
  




Example

Number of errors: ne=10
Probability: D=0.002 C=0.998
Result
NElow=3.26
NEhigh=22.98
Interpretation
Having measured ne=10 errors in a single test, then with a low probability D=0.002 the average number of errors NE in this test is outside the range from 3.26 to 22.98 or with a high probability C=0.998 inside this range from 3.26 to 22.98.
Such as the width of the Poisson distribution, the confidence range CR increases proportional to SQR(ne), that means, it increases absolutely, but decreases relatively.
If the entire confidence range CR, calculated from a single result ne, is found on the good side (NElimit>NEhigh) of the specified limit NElimit we can state: With high probability C, the final result NE is better than the limit NElimit. Whereby NElimit is given by

NElimit=BERlimit·ns  (5)

and BERlimit is the Bit Error Rate allowable for the device and obtained by an ideal long test with an infinite high number of bit samples ns.

If the entire confidence range CR, calculated from a single result ne, is found on the bad side (NElimit<NElow) of the specified limit NElimit we can state: With high probability C, the final result NE is worse than the limit.
With each new sample and/or error a new test is considered, reusing all former results. With each new test the preliminary data for ns, ne and ber is updated. For each new test the confidence range CR is calculated and checked against the test limit NElimit.
Once the entire confidence range CR is found on the good side of the specified limit (NElimit>NEhigh), an early pass is allowed. Once the entire confidence range CR is found on the bad side of the specified limit (NElimit<NElow) an early fail is allowed. If the confidence range CR is found on both sides of the specified limit (NElow<NElimit<NEhigh) it is evident neither to pass nor to fail the DUT early.
FIG. 1 illustrates the above conditions. Of course, NElimit is a fixed value not altering during the test, but NElow and NEhigh as well as the confidence range CR are altering during the test. For reasons of illustration, however, the three possibilities of the possible positions of the confidence range CR with respect to the constant limit NElimit are drawn for the same example in FIG. 1.
The above can be described by the following formulas:
The current number of samples ns is calculated from the preliminary Bit Error Ratio ber and the preliminary number of errors ne

ber=ne/ns  (6)

After a full test the final Bit Error Ratio is

BERlimit=NElimit/ns  (7)

for abbreviation in the formula:

bernorm=ber/BERlimit=ne/NElimit (normalised ber)  (8)

Early pass stipulates:

NEhigh<NElimit  (9)

Early fail stipulates:

NElow>NElimit  (10)

Formula for the early pass limit:




  
    
      
        
          ber
          norm
        
        =
        
          ne
          
            NE
            high
          
        
      
    
    
      
        (
        11
        )
      
    
  




This is the lower curve (bernormpass (ne, C)) in FIG. 2, which shows bernorm as a function of ne.
Formula for the early fail limit:




  
    
      
        
          ber
          norm
        
        =
        
          ne
          
            NE
            low
          
        
      
    
    
      
        (
        12
        )
      
    
  




This is the upper curve (bernormfail (ne, D)) in FIG. 2.
As the early pass limit is not defined for ne=0 (normally the case at the very beginning of the test for a good DUT), an artificial error event with the first sample can be introduced. When the first real error event occurs, the artificial error is replaced by this real one. This gives the shortest possible measurement time for an ideal good DUT. For example ns=5000 for BERlimit=0.001 and probability D=D1=D2=0.2%.
As the early fail limit uses NElow<1 for small ne<k (in the example below k=5) due to a decision problem at a fractional error, the early fail limit at ne=k is extended with a vertical line upwards. This ensures that a broken DUT hits the early fail limit in any case after a few samples, approx. 10 in the example. In other words, the test is not stopped as long as ne is smaller than k.
With each new sample and/or error a new test is considered, reusing all former results. With each new test the preliminary data for ns, ne and ber and bernorm are updated and a bernorm/ne coordinate is entered into the bernorm-diagram. This is shown in FIG. 3. Once the trajectory crosses the early fail limit (bernorm (ne,D)) or the early pass limit (bernorm (ne,C)) the test may be stopped and the conclusion of early fail or early pass may be drawn based on this instant.
FIG. 3 shows the curves for early fail and early pass. bernorm is shown as a function of the number of errors ne.
For the simple example demonstrated in FIG. 3, it is BERlimit=0.2=1/5 and the final Bit Error Ratio BER=0.25 (1/4). The test starts with the first bit sample, for which no error is detected. For the second sample, a first error is detected and the preliminary Bit Error Ratio ber=ne/ns=1/2 and bernorm=ber/BERlimit becomes 1/2:1/5=5/2. bernorm after the second sample is marked with a cross a in FIG. 3. For the third, fourth and fifth sample, no further error occurs and bernorm subsequently becomes 5/3, 5/4 and 5/5, respectively, which is marked with the crosses b, c and d in FIG. 3, respectively. The sixth sample brings a new error and ne becomes 2. Consequently, ber=ne/ns becomes 2/6 and bernorm becomes 10/6. This situation is marked with cross e in FIG. 3. For the seventh; eighth and ninth sample, no further error occurs and the situation after the seventh, eighth and ninth sample is marked with crosses f, g, h in FIG. 3, respectively. The tenth sample brings a third error. Consequently, ber becomes 3/10 and bernorm becomes 15/10. This situation is marked with cross i in FIG. 3. As can be seen from FIG. 3, the trajectory is between the early fail curve and the early pass curve at the beginning of the test, but converges to a line Z, which crosses the early fail curve after about forty errors. After forty errors, it can thus be decided that the tested DUT early fails the test.
If no early stop occurs the BER test may be stopped, after the following condition is valid:

ne>=K  (13)

and the DUT shall be passed, if ns is sufficiently high. K is a maximum number of errors. For example K can be 200.

If the trajectory neither crosses the early fail curve nor the early pass curve after K (for example 200) errors have occurred, the DUT can be finally passed. If the DUT, however, is rather good or rather bad, the tests can be stopped much earlier, long before the K=200 errors have occurred. This significantly shortens the total test time.
In the above embodiment early fail means: a DUT is failed and a probability of 0.2% that it is actually better than the limit is accepted. Further early pass means: the DUT is passed and a probability of 0.2% that it is actually worse than the limit is accepted. If the test is stopped at 200 errors the DUT is passed without any early fail or early pass condition arbitrarily. It can cross the vertical 200 error line in FIG. 2 at different heights, each height is connected with a certain statistical interpretation: The probability to have a DUT better (worse) than the limit is indicated in FIG. 5. The vertical in FIG. 5 shows the position in FIG. 2 at the end of the test. The horizontal in FIG. 5 shows the respective probability.
This embodiment contains a pseudo paradox, due to statistical nature of the test and limited test time, demonstrated with the following example: A DUT, which is early failed, would pass if the probability for a wrong decision has been reduced by widening the early stop limits, or vice versa, if the test is stopped at 200 errors and the DUT is arbitrarily failed. A DUT, which is early passed, would fail if the probability for a wrong decision has been reduced.
The following embodiment resolves this pseudo paradox and additionally accelerates the test. This is done by a meaningful redefinition of the early pass limit maintaining the early fail limit. Early pass means now: A DUT is passed and a probability of 0.2% that it is actually worse than M times the specified limit (M>1) is accepted. This is a worse DUT limit. This shifts the early pass limit upwards in FIG. 2 as shown in FIG. 4. Bernormpass (ne, C) in FIG. 2 becomes bernormbadpass (ne, C) in FIG. 4. Bernormfail (ne, D) remains unchanged. Now it is

NElimit,M=BERlimit·M·ns  (14)

and an early pass is allowed, if NElimit·M=NElimit,M>NEhigh.

There are three high level parameters for the test:

    
    
        Probability to make a wrong decision (proposal: C=0.2%)
        Final stop (proposal: K=200 errors)
        Definition of a Bad DUT (BERlimit·M)
    
    


These parameters are interdependent. It is possible to enter two of them and to calculate the third one. To make the test transparent, it is proposed to enter the wrong decision probability C and the final stop condition K and to derive the Bad DUT factor M. This is done in the following manner: The early pass limit is shifted upwards by a factor of M, such that the early fail and the early pass limit intersect at 200 errors in the example as shown in FIG. 4. FIG. 4 also shows the test limit obtained from the crossing point from the early fail curve and the easy pass curve, corresponding to Mtest.
There are infinite possibilities to resolve the above mentioned paradox.
In the example above the open end between the limits was originally declared pass, and time was saved by multiplying the early pass limit with M (M>1), shifting it upwards such that the early fail and the early pass curve intersect at 200 errors (200: example from above). Such only a DUT, bad with high probability, is failed (customer risk).
The complementary method is: The open end between the limits is declared fail, and time is saved by multiplying the early fail limit with m(0<m<1), shifting it downwards, such that the early fail and the early pass curve intersect at 200 errors (200: example from above). Such only a DUT, good with high probability, is passed (manufacturer risk).
The compromise method is: The open end between the limits is partitioned in any ratio: the upper part is declared fail and the lower part is declared pass. Time is saved by multiplying the early fail limit with m (0<m<1) and such shifting it downwards and by multiplying the early pass limit with M (M>1) and such shifting it upwards. So the early fail and the early pass curve intersect at 200 errors (200: example from above).
With given D1 and D2 the early fail curve and the early pass curves in FIG. 3 and FIG. 2 or FIG. 4 can be calculated before the test is started. During the test only bernorm=ne/NElimit has to be calculated and to be compared with the early pass limit and the early fail limit as explained with respect to FIG. 3 and FIG. 4. Thus, no intensive calculation has to be done during the test.

## Claims

1. A method for testing a bit error ratio BER of a device against a maximal allowable Error Ratio BERlimit with an early pass criterion, whereby the early pass criterion is allowed to be wrong only by a small probability D1, comprising the following steps
measuring ns bits of the output of the device, thereby detecting ne erroneous bits of these ns bits;
assuming that the likelihood distribution giving a distribution of the number of erroneous bits ni in a fixed number of samples of bits is PD(NE,ni), wherein NE is an average number of erroneous bits, obtaining PDhigh from





  
    D
    1
  
  =
  
    
      ∫
      0
      
        n
        ⁢
        
          
        
        ⁢
        e
      
    
    ⁢
    
      
        
          PD
          high
        
        ⁡
        
          (
          
            
              NE
              high
            
            ,
            ni
          
          )
        
      
      ⁢
      
        d
        ni
      
    
  



 wherein PDhigh is the worst possible likelihood distribution containing the measured ne erroneous bits with the probability D1;
obtaining the average number of erroneous bits NEhigh for the worst possible likelihood distribution PDhigh;
comparing NEhigh with NElimit=BERlimit ns;
if NElimit is higher than NEhigh stopping the test and deciding that the device has early passed the test; and
if NElimit is smaller than NEhigh continuing the test whereby increasing ns.

2. A method for testing the bit error ratio BER according to claim 1,
characterized in that
the likelihood distribution PD(NE,ni) is a Poisson distribution.

3. A method for testing the bit error ratio BER according to claim 1 or 2,
characterized in that
for avoiding a undefined situation for ne=0 starting the test with an artificial error ne=1 not incrementing ne when a first error occurs.

4. A method for testing the bit error ratio BER of a device against a maximal allowable Error Ratio BERlimit with an early fail criterion, whereby the early fail criterion is allowed to be wrong only by a small probability D2, comprising the following steps
measuring ns bits of the output of the device, thereby detecting ne erroneous bits of these ns bits;
assuming that the likelihood distribution giving a distribution of the number of erroneous bits ni in a fixed number of samples of bits is PD(NE,ni), wherein NE is an average number of erroneous bits, obtaining PDlow from the





  
    D
    2
  
  =
  
    
      ∫
      ne
      ∞
    
    ⁢
    
      
        
          PD
          low
        
        ⁡
        
          (
          
            
              NE
              low
            
            ,
            ni
          
          )
        
      
      ⁢
      
        d
        ni
      
    
  



 wherein PDlow is the best possible likelihood distribution containing the measured ne erroneous bits with, the probability D2;
obtaining the average number of erroneous bits NElow for the best possible likelihood distribution PDlow;
comparing NElow with NElimit=BERlimit ns;
if NElimit is smaller than NElow stopping the test and deciding that the device has early failed the test; and
if NElimit is higher than NElow continuing the test whereby increasing ns.

5. A method for testing the bit error ratio BER according to claim 4,
characterized in that
the likelihood distribution PD(NE,ni) is a Poisson distribution.

6. A method for testing the bit error ratio BER according to claim 4,
characterized in that
for avoiding a undefined situation for ne<k, wherein k is a small number of errors, not stopping the test as long as ne is smaller than k.

7. A method for testing the bit error ratio BER according to claim 6,
characterized in that
k is 5.

8. A method for testing a bit error ratio BER according to any one of claims 4 to 7,
characterized by
an additional early pass criterion, whereby the early pass criterion is allowed to be wrong only by a small probability D1, comprising the following additional steps
assuming that the likelihood distribution giving the distribution of the number of erroneous bits ni in a fixed number of samples of bits is PD(NE,ni), wherein NE is an average number of erroneous bits, obtaining PDhigh from





  
    D
    1
  
  =
  
    
      ∫
      0
      
        n
        ⁢
        
          
        
        ⁢
        e
      
    
    ⁢
    
      
        
          PD
          high
        
        ⁡
        
          (
          
            
              NE
              high
            
            ,
            ni
          
          )
        
      
      ⁢
      
        d
        ni
      
    
  



 wherein PDhigh is the worst possible likelihood distribution containing the measured ne erroneous bits with the probability D1;
obtaining the, average number of erroneous bits NEhigh for the worst possible likelihood distribution PDhigh;
comparing NEhigh with NElimit=BERlimit ns;
if NElimit is higher than NEhigh stopping the test and deciding that the device has early passed the test; and
if NElimit is smaller than NEhigh continuing the test, whereby increasing ns.

9. A method for testing the bit error ratio BER according to claim 8,
characterized in that
for avoiding a undefined situation for ne=0 the test with an artificial error ne=1 not incrementing ne when a first error occurs.

10. A method for testing the bit error ratio BER according to claim 8,
characterized in that
the probability D1 for the wrong early pass criterion and the probability D2 for the wrong early fail criterion are equal (D1=D2).

11. A method for testing the bit error ratio BER according to any one of claims 4 to 7,
characterized by
an additional early pass criterion, whereby the early pass criterion is allowed to be wrong only by a small probability D1, comprising the following additional steps
assuming that the likelihood distribution giving the distribution of the number of erroneous bits ni in a fixed number of samples of bits is PD(NE,ni), wherein NE is an average number of erroneous bits, obtaining PDhigh from





  
    D
    1
  
  =
  
    
      ∫
      0
      
        n
        ⁢
        
          
        
        ⁢
        e
      
    
    ⁢
    
      
        
          PD
          high
        
        ⁡
        
          (
          
            
              NE
              high
            
            ,
            ni
          
          )
        
      
      ⁢
      
        d
        ni
      
    
  



 wherein PDhigh is the worst possible likelihood distribution containing the measured ne erroneous bits with the probability D1;
obtaining the average number of erroneous bits NEhigh for the worst possible likelihood distribution PDhigh;
comparing NEhigh with NElimit,M=BERlimit M ns, with M>1;
if NElimit,M is higher than NEhigh stopping the test and deciding that the device has early passed the test; and
if NElimit,M is smaller than NEhigh continuing the test, whereby increasing ns.