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## Measures of Performance

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**Measures of Performance**• In communications there are two measures of transmission; bit rate and baud rate. • Bit rate; this is the basic rate of transmition and indicates the number of bits • transmitted in the unit of time. Since the unit of time is the second the bit rate is • expressed as xx Bit/s. The units used are Kbit/s (103 bits) Mbit/s ( 106bits), • Gbit/s ( 109 bits), Tbit/s ( 1012bits), Pbit/s(1015bits),…. • Baud rate; when the data stream is encoded a “bit” can carry more information than the basic bit. Consider a data stream encoded in the phase of phase shift • keying signal. Now, two initial bits are encoded into one symbol. Consequently, • the new rate called baud rate is half that of the rate with which the bits arrive at the • encoder. Another way to look at baud is that the baud rate is the number of time • per second the signal changes state or varies. Therefore, • Bits per second = Bauds per second X bits per baud. • In conventional transmission one change is required to transmit a bit so the baud • rate equals the bit rate. But, a 3000 baud/s transmission where one baud • represents 4 bits the number of bits is 3000x4 = 12000 bits/s.**“1”**Transmitted stream “0” time “1” Received stream Bits in error “0” Measures of Performance • In a digital communication system the information is transported encoded in “1” and • “0” which correspond to the presence or absence of photons. The channel and the • associated equipment are not perfect so at the receiver there will be some errors in • that some “1” will be detected as “0” and vice versa. The simple diagram below • illustrates the concept. time The bit error rate, (BER), is defined as**Measures of Performance**• The system impairments that give rise to the errors are independent processes and as • as result the occurrence of the errors is random along the stream of data and since the • data themselves are random a format definition of BER that can be predicted through • the system design process and estimated through measurements required a • probabilistic definition. Following from the definition of the BER the probabilistic • definition is Since the process of errors is random in order to get a statistically robust prediction the number of bits transmitted must be large and in the limit the bit error probability is obtained. Then, Therefore, for a reasonable limit on measurement time, the minimum number of bits that yields a statistically valid test is required.**Measures of Performance**• In order to arrive at some distribution one has the address the details of the error • process. Firstly, from general information theory considerations the input alphabet • consists of two symbols, “1“ and “0” , with equal probability of occurrence. Secondly, • the probability of an error, p, and the probability of a non-error, q, equals unity. • Thus, Now, the search is for a probability distribution that predicts out of N – trials (bits sent) the probability of k events (errors). From the definition of the BER the N-trials is the number of bits transmitted and the k – events is the number of bits in error. A distribution that models exactly this process is the binomial distribution function. The binomial density distribution is defined as;**Q (x)**Integrand e x Measures of Performance • Mathematically, the BER is calculated through the use of the Q system parameter. • In the context of communications the Q is defined as; There is a number of approximations in calculating the Q - function. For x≥3 a popular approximation is There are also better approximations if required and the plot of the Q - function in the next slide is based on the Nick Kingsbury approximation.**Measures of Performance**• From the definition of the Q one can draw a number of observations about the nature • of Q. BER Q The Q - function based on the Nick Kingsbury approximation.**Threshold**G(1,σ1) G(0,σ0) “0” “1” Errors Measures of Performance • Restating the definition for convenience It is clear that A0-worst and A1-worst correspond to the higher 0 - level and the lowest 1 - level. Therefore, the Q is a measure of how close at a given time=t0 the levels corresponding to “1” and “0” are in the receiver threshold detector. Therefore, a big value of Q is a sign that the system will perform satisfactory and the big question is ; how is the Q measured?**Measures of Performance**• The objective of system design is to obtained the best performance under the design • constrains. Achieving a value of Q that satisfies the requirements seems to be the • end of search for performance and consider the conditions under which the Q was • obtained one may think that this is the best possible Q. Such a sweeping statement • would have been true for a strictly linear system. Unfortunately, optical fibre systems • operate in a quasy nonlinear regime and all the beets about global optimum operation • are off. The design has to be tested therefore against a rang eof values number of • parameters. Such an approach can be based on the idea of constant - Q contours. • The idea is illustrated in the next slide. The approach is powerful in illustrating if the • maximum Q is a global or local maximum and also illustrating the sensitivity of • performance to the parameters under study. Clearly, the process of deriving the • constant – Q contours is very consuming on computer time( simulation) and • experimental time on the test bed. • Therefore any constant – Q system assessment should be very well thought out in • terms of objectives and the importance of the objectives.**Measures of Performance**• The concept of constant - Q contours system assessment. Direction of steepest descent – maximum sensitivity. Parameter B Q2 Q1 Q3 Q4 Parameter A**Measures of Performance**The penalty in Q in terms of DresIL and DresTOT where the term mean: DresIL = Lspan x D – Dline where D ; dispersion of the fibre and the Dline the dispersion of the compensating fibre. DresTOT= Nspan x Lspan x D - D lineTOT whereNspan the number of spans, L the length per span D the dispersion of the fibre and DlineTOT the total in line compensation. • The penalty in Q - dB for residual dispersion in a system with 5 spans of 100 km each, NRZ • pulses with 7 channels at 10Gbit/s each. High penalty at total balanced dispersion indicates strong XPM effects. Possible global minimum!**Measures of Performance**• In the context of BER we have addressed the following: • (1) Required number of bits for a given BER, that is, how long the test will take. • (2) Given a set of BER measurements how much confidence we can place on the readings. • (3) The Q is a measure of the system performance but apart from measuring it it seems that the performance measures are understood. So, • Have we finish with measures of performance? • Unfortunately no, but why? • In order to appreciate this statement one has to assess the nature of the BER. • The BER is global measure of performance and it is opaque as to the causes of • the performance; good, bad or indifferent. One needs more information regarding the • performance of the system so that the performance can be improved if necessary. • Such information can be provided by using the information embedded in the pulse • stream .**Measures of Performance**• A stream of data consisting of “1” and “0” is shown in the figure below. Notice the differences between nominally similar pulses and that the features of each pulse depend on the previous sequence of pulses. All the impairments highlighted in this figure will affect the BER of the system. So, the question now arises; if the features of a pulse embedded in such a stream depends on its position in the stream how bad can such a pulse be?**Amplitude decision**space Threshold Time reference (clock) Measures of Performance • The reason the attention is focussed on an individual pulses is that the BER is based • on individual pulses be in error. • A digital receiver makes a bit by bit decision on the presence of “1” or “0”. To do that a • threshold is established that bisects the amplitude decision space and a time • reference is used that defines the sampling time when the receiver inspect the result • of the sampling and makes the decision. The optimum combination of time and • amplitude is when the distance between “0” and “1” is the maximum ( maximum pulse • opening). The ideal situation is shown below. However, if the adjacent pulses affect • the pulse under study then the decision of the threshold detector is compromised. • But by how much is compromised?**“1”**“0” 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 time 0 1 0 Measures of Performance • To assess the impact of the adjacent pulses on the performance of the pulse under • study the concept of the “eye diagram” was born. Conceder all the combinations of • three digits taking values 1 and 0. Then, there are eight triplets that are shown below. If the basic pulse shape is as shown below then, the eye diagram ( pattern) for three bits can be constructed as shown in the next slide. Since the pattern has three bits then there is one pulse in the middle to be detected and one on each side.**Measures of Performance**• The number of blocks of lines in the eye diagram is given by where a and b the number of interfering bits before and after the bit under study. So, for a=b=1 N blocks of lines = 8. Clearly, this is the same number that gives the number of the sampling with replacement of two states (1 and 0) taken three at a time. If more bits participate , say a = b = 2, then 32 blocks are required to construct the eye diagram. Continuing with the example if the eight blocks are superimposed the ideal eye diagram is shown in the next slide. The diagram also includes the key features of the eye that will be of use later.**Measures of Performance**Maximum eye opening Inner eye – receiver operating space Ideal threshold Amplitude window Ideal sampling time The ideal eye diagram of three bits and its key features. From all the impairments a telecommunication system is subject to noise is the irreducible minimum; noise is always present. From the eye diagram above it is not easy to ascertain the value of diagram. So, let us add noise!**Maximum eye**opening Inner eye - receiver operating space Threshold Amplitude window Sampling time Measures of Performance • A three bit eye diagram with Gaussian noise added. Notice the serious reduction of • the receiver operating space. It is the reduction and shape of the receiver operating • space that makes possible to identify the various impairments that lead to the global • BER.**Measures of Performance**• Two measured eye diagrams that display dispersion and noise effects. Inner eye Inner eye (a) Measured eye; transmitted eye. (b) Measured eye; received eye. The worst case diagram( pattern) defines the open inner part of the eye. The absolute and normalised eye opening (aperture) is a key parameter because it can be used to calculate the BER of the system from experimental data.**Measures of Performance**• The absolute eye opening is defined • by The normalised eye opening is defined by The eye opening ( aperture).**Peak ISI**Noise margin Slope; sensitivity to timing errors Time jitter Window over which the signal can be sampled. Optimum sampling time Measures of Performance • The eye above highlights some additional performance metrics that can be measured • such as ; intersymbol interference (ISI), time jitter, timing error sensitivity, and noise • margin. The worst eye diagrams are obtained for the following pattern conditions; • worst case “1”; …000010000…; worst case “0”; …111101111…**Measures of Performance**There is a number of measurements that can be made and extract information from the eye diagram. They are summarised below • The interpretation of some measurement are summarise below.**Measures of Performance**• In th development of the eye diagram it was tacitly assumed that there will be • detectors that can handle the signal bandwidth. • One of the fastest detector today is the u2t XPDV4120R with an electrical bandwidth • of 90 GHz and around 10 ps response. Frequency response of the u2t XPDV4120R detector. Impulse response of the u2t XPDV4120R detector.**Measures of Performance**• It is therefore expected that the bandwidth of a detection system for measuring the eye • diagram will be of the order of 80 to100 GHz. For system operating above that range it • will not be possible to measure the eye diagram. The solution lies in using optical • sampling. Conceptually optical sampling is similar to electronic sampling but at much • high speeds. The principle of optical sampling is summarised in the next slide. • The bandwidth of the sampling system ( temporal resolution) depends on the sampling • window (gating). For a Gaussian shaped gating window the sampling system • bandwidth is given by Using a soliton pulse as the gating window a window duration of 210 fs was achieved leading to a sampling system bandwidth of 2000 GHz. The next plus one slide shows waveforms acquired through an optical sampling system and the pulses of a 640 Gbit/s optical transmission experiment.**Optical**window Optical sample ≈ ≈ ≈ time Low bandwidth O/E detection time Reconstructed pulse amplitude. time Measures of Performance • The principle of optical sampling.**Data**Sampling gate Sampling pulse source O/E Detector Signal processing Clock recovery Clock processing Optical signal Electronic signal Measures of Performance Optically sampled waveform. 640-Gb/s optical signal. Synchronous sampling configuration.**Complex plane**Imaginary axis “1” (1, angle )V Real axis ● ● 0+ j 0 1+ j 0 “0” (0, angle 0)V (a) Conventional representation. (b) The “constellation” representation Measures of Performance • Until now the symbols of “1” and “0” for binary transmission have been defined as level • of, say, voltage or current. There is however an alternative representation that • conveys the same amount of information. Consider again a binary signal of “1” and “0” • and let us say that they correspond to voltages1 V and 0V that change with time. • Then, the complete description is one that contains also the phase, that is, phasors • are used for the complete description. The conventional representation is shown • below on the left. On the right there is the description using the complex plane. • Clearly, both representation contain the same about of information .The representation • on the right is called for reason that will become apparent very soon, the • “constellation”**Imaginary**● ● v2 01 Imaginary v1 00 ● ● ● ● v3 Real 10 Real ● ● v4 11 Measures of Performance • Perhaps, this example does not demonstrate the power of the new representation. • Consider now four voltages corresponding to four signal level represented by ; • v1=1+j 0, v2= 0+ j1, v3= -1, j 0 and v4= 0 – j. The constellation is as shown below and • it should be clear now the advantages of the representation. In fact that constellation • represents a four level phase shift keying, (PSK) format. Now, let us farther assume • that the PSK four level format encodes bits according the following rule v1=00, • v2=01,v3=10 and v4=11. Then, instead of depicting the voltages the symbols can be • directly represented in the constellation diagram.**Amplitude –**Random variable Amplitude 01 ● ● Phase 00 Phase – Random variable φ 01 00 ● ● 11 ● ● φ ● ● 10 11 Measures of Performance • The constellation diagram in the previous slide showed very clearly the position of the • symbols in the plane. In order to see the impact of transport consider the 4-symbol • PSK again but now rotated by 45o andusing the unit circle for reference. Notice, the • defined amplitude and phase of each symbol. This is the transmitted constellation. (a) Transmitted constellation. (b) Received constellation. During transmission the constellation has been subjected to random amplitude and phase variations so the receiver has to estimate what was transmitted.**Actual signal**point Amplitude error ● Error vector ● Imaginary axis φ Ideal signal point Phase error Real axis Measures of Performance • The receiver has to make a decision regarding the received signal. This is • accomplished by using the concept of the minimum Euclidean distance. Consider the • details of one of constellation points , say, “00”. Because of transmission impairments • the actual signal point is not the same as that of the transmitted signal. The error is • expressed as an “error vector magnitude”, (EVM), and it is an expression of how far • from the ideal signal the received signal is. The receiver decides what signal has been sendby using the shortest distance to the ideal point. Since the space is 2 - D Euclidean the distance is termed as “Euclidian”. In this particular example the receiver detects the correct signal point but in worst channels or with more signal points in the constellation errors are made. In the literature instead of the terms real and imaginary axes in-phase, (I), and quadrature ,(Q), terms are used.**Measures of Performance**• The estimation of EVM is based on the following metric; where N the number of symbols in the constellation plane. The number of levels along the in-phase or quadrature axis for the ideal constellation is The measurement of EVM is not simple but there are instruments that address the measurement issues. Intuitively, one expects that and calculations presented in the next slide support this.**Measures of Performance**• These two graphs demonstrate the interaction between EVM and S/N ratio. The reason EVM has relevance to optical communications is that the modulation formats are in general multilevel and as a result the use of constellation diagrams is very useful in depicting the key features of the system.**Appendix III - A**The BER using the Eye Diagram**The Calculation of BER using the Eye Diagram**• The system performance information embedded in the eye diagram can be used to • calculate the BER. The probability density functions of detected signals and the • regions of errors for “1” and “0” are presented in the figure below.**A 1 - worst**A 0 - worst The Calculation of BER using the Eye Diagram • The key assumption is that the noise associated with the “1” and “0” is additive white • Gaussian noise ( AWGN) of zero mean and variance σn2 provided there is no phase • noise.. The implication of this assumption is that any sample and (signal + noise) can • be written mathematically as Also, the statistically strong condition of statistical indolence is imposed on the signal and noise samples. The (signal +noise) model is shown below.**The Calculation of BER using the Eye Diagram**• The BER can be stated as, where (1) and P(0) the up-priory probabilities of transmitting “1” and “0” and P(0/1) and P(1/0) the conditional probabilities of sending a “1” and detecting a “0” and sending a “0” and detecting a “1”. For maximum information transport P(1) = P(0) = 0.5. Then, where σ0 and σ1 correspond to the noise standard deviation at “0” and ”1” respectively.**The Calculation of BER using the Eye Diagram**• Making the following substitutions, we obtain, where erfc (x) is defined as the complementary error function and given by A good approximation for x > 3 is**The Calculation of BER using the Eye Diagram**It can be shown that the BER is minimised if the threshold is located where the errors for 1 and 0 are equal. This is achieved when • When σ0 ≈ σ1 then the last equation simplifies to A key system parameter is defined by the following equation, Using the definition of Q the BER is now given by**The Calculation of BER using the Eye Diagram**This is the worst BER because in calculating it the effect of ISI was taken into account. Clearly, the system performance is impaired and the impact of the worst case BER calculation can be ascertain from the following equation; Since A Norm ≤ 1 then δ(dB) worst case ≥ 0. The system design is based on a value of Q that satisfies the requirements. Therefore It is important to compare the theoretical expected Qth with the achieved Qeye. The Q – impairment can be assessed through the equation, There is a number of measurements one can make and extract information from the eye diagram. They are summarised below. There are instruments that can estimate the quoted parameters through the use of efficient algorithms.**The Calculation of BER using the Eye Diagram**The interpretation of some measurement are summarise below.