Model for a Distributed Radio Telesope Patrik Flekenstein Department of Mathematis, Rohester Institute of Tehnology, Rohester, NY 14623, U.S.A. E-mail address: pat@sh.rit.edu
Abstrat. In this paper, a stohasti model for a single radio telesope is generated. The model takes into aount interferene from other radio soures (inluding the sky), errors in the measurement of the detetor s position, errors in the measurement of time at the detetor station, noise due to the quantum nature of light, and noise in the detetor. This model is then used as the basis for a omputer simulation of a distributed array of radio telesopes. The omputer simulation suessfully resolves a target signal using a few hundred very bad detetors. The simulation is further employed to analyze the model s sensitivity to various parameters suh as the number of detetors and the magnitude of errors in time measurements.
Contents Chapter 1. Summary and onlusions 5 Chapter 2. Glossary of variables 7 Chapter 3. Bakground information 9 Chapter 4. Formulation of the model 11 1. Measuring time 12 2. Measuring signals 12 Chapter 5. Analysis of the model 15 Chapter 6. Interpretation of the model 17 Chapter 7. Further work 21 Appendix A. Distribution of ɛ h / + ɛ t 23 Appendix B. Distribution of sum of Poisson distributed variables 25 Appendix. Bibliography 27 3
CHAPTER 1 Summary and onlusions This paper develops a stohasti model of the behaviour of a single radio telesope subjet to inoming signals from various radio soures (inluding the sky). In the model, the time is expressed in terms of the time when the signal one is attempting to measure would pass the enter of the Earth. The resulting stohasti model expresses the probability that the detetor will detet z photons at a partiular time t: (1) p d (z, t) = x=0 p i x (x, t) x+ 1 2 e 1 y=x 1 2 2 σ d 2π ( z y σ d ) 2 dy This expresses the probability that z photons are deteted by the i-th detetor at the time when the i-th detetor should be reeiving the signal from the primary radio soure (the soure being observed) whih will pass the enter of the Earth at time t. Here σ d is the standard deviation of the noise in the detetor. The integral in this equation arises from the tenuous assumption that noise in the detetor an be adequately modeled with a normal distribution. In the above equation, the term p i x (x, t) gives the probability that x photons from the various radio soures will reah the detetor at time t. That probability is: (2) p i x (x, t) = ɛ T = ( ( )) x j s j t + hij hi1 ɛ T e 1 2 e ( ɛ T (σ h /) 2 +σ 2 t ) 2 x! ((σh /) 2 + σt 2 ) 2π dɛ T j (t+ sj h ) ij h i1 ɛ T Here, s j (t) is the strength of the j-th signal whih would pass the enter of the Earth during the sampling interval entered at time t. The h ij terms are the height of the i-th reeiver measured parallel to a ray whih starts at the enter of the Earth and extends toward the j-th radio soure. The ɛ T term is the total error in the station s time measurements due to errors in its positional measurement h ij and errors in its diret time measurement. The standard deviation of the errors in positional measurements is σ h, and the standard deviation of the errors in diret time measurements is σ t. The speed of light is denoted. Beause equations 1 and 2 are quite unwieldy, a omputer simulation was developed based upon this single detetor model to determine the performane of an array of these telesopes. The omputer simulation shows that even faint signals an be deteted with very poor detetors when the auray of positional and time measurements is within urrently attainable limits. 5
6 1. SUMMARY AND CONCLUSIONS Chapter 2 gives a summary of the variables used in the development of this model. Chapter 3 provides bakground information about radio telesope arrays and forms the base upon whih this model is ast. In hapter 4, the goals for this model are established. It goes on to highlight the problems in time and position measurements assoiated with distributing the telesopes in this array, and it mentions the problems of deteting signals whih are inurred at eah detetor. Chapter 5 goes on from there to derive equations 1 and 2. The analysis in this hapter depends heavily upon some proofs about the sums of normally distributed variables and the sums of Poisson distributed variables. Those proofs are in appendix A and appendix B. Chapter 6 employs a omputer simulation based upon the equations derived in hapter 5. It shows empirial results of running the simulation with various soures of radio interferene, various magnitudes of time errors, and various numbers of telesopes in the array. Chapter 7 goes on to suggest plaes for further exploration before one attempts to reate this type of telesope array.
CHAPTER 2 Glossary of variables This table summarizes the variables used in the development of the distributed radio-telesope model. Throughout this paper, variables whih are randomly distributed will be written in bold. 1 Desription Variable Units the strength of the signal from the j-th soure that will s j (t) photons/s pass the enter of the Earth at time t the height of i-th observation point in the diretion toward h ij m the j-th soure with respet to the Earth s enter the error in height measurement ɛ h m the standard deviation of height measurements σ h m the error in time measurement ɛ t s the standard deviation of time measurements σ t s the total error in time measurement aused by ɛ h and ɛ t ɛ T s the error in the detetor ɛ d photons/s the standard deviation of the error in the detetor σ d photons/s the speed of light m/s the radius of the Earth R m 1 Unfortunately, a bold epsilon ɛ is indistinguishable from a normal epsilon ɛ in L ATEX. However, the subsripts on them will still be bold. 7
CHAPTER 3 Bakground information Many of the best radio observatories use multiple radio telesopes in tandem for an observation. The outputs of the individual telesopes are ombined to form a learer image of the target objet than any one of the telesopes ould provide. There is a tradeoff between the number of radio telesopes required and the sensitivity of eah telesope. The Very Large Array 1 uses 27 25m-telesopes, the Very Long Baseline Array 2 uses 10 25m-telesopes, and the proposed Ataama Large Millimeter Array 3 will use 64 12m-telesopes. If a muh larger array were possible with muh smaller telesopes (ones that individuals ould afford), one would hope to be able to harness some of the 400 members of the Soiety of Amateur Radio Astronomers 4 and some of the two million users who have donated omputer time to the SETI@Home 5 projet. With wide enough support, one may be able to reate an array large enough to make useful observations. Radio telesope arrays link several telesopes together. Signals from the radio soure hit the different telesopes at slightly different times beause the signals have to travel different distanes to eah of the telesopes. The signals from these telesopes are then reombined. The position of eah telesope with respet to the radio soure is taken into aount in order to synhronize the signals. The reombined signals reinfore eah other. Noise in the telesopes and from other stellar soures have the same probability of being in-phase as out of phase while the synhronized signals will be in-phase. By arefully plaing multiple telesopes, sientists an selet partiular frequenies from partiular soures despite large amounts of bakground interferene. The Very Long Baseline Array takes a different approah than most other arrays. The 27 telesopes of the Very Large Array, for example, are all on the same plot of land in New Mexio. The 10 telesopes of the Very Long Baseline Array are not in lose proximity. One telesope is in St. Croix, another in Hawaii, others in California, et. Rather than preisely positioning the telesopes to some prealulated formation, the sientist adjusts for the atual positions of the radio telesopes to synhronize their data streams. This ompensation an be aomplished so long as the loation of eah telesope is preisely known and the data stream for eah telesope is preisely timestamped. Global Positioning Satellite (GPS) 6 reeivers are beoming ommonplae. Amateurs an preisely know their latitude, longitude, and elevation. The Network 1 http://www.ao.nrao.edu/vla/html/vlahome.shtml 2 http://www.ao.nrao.edu/vlba/html/vlba.html 3 http://www.alma.nrao.edu/ 4 http://www.bambi.net/sara.html 5 http://setiathome.ssl.berkeley.edu/ 6 http://www.aero.org/publiations/gpsprimer/index.html 9
10 3. BACKGROUND INFORMATION Time Protool 7 is nearing nanoseond auray on new omputers. Computer users will be able to aurately measure time intervals and synhronize time with standard loks. The tehnology to synhronize data streams from amateur telesopes will soon be widespread. The only missing tehnology is heap radio telesopes. This, however, is largely due to demand. There has never been a use for heap radio telesopes. If a large, distributed array of heap radio telesopes were possible, heap radio telesopes ould readily be onstruted. 7 http://www.eeis.udel.edu/~mills/preise.htm
CHAPTER 4 Formulation of the model Radio signals from pulsars and other soures are onstantly hitting the Earth. Figure 1 shows a signal from one pulsar approahing several detetors on Earth. The signal will hit the detetors at different times beause the detetors are different distanes away from the soure (the pulsar). For the purposes of this model, it is assumed that the Earth and the soures of radio signals are all motionless. The length of time for any single point of observation will be under 1/100-th of a seond for this array. This model assumes that any motion of the soure or the Earth during that 1/100-th of a seond will be negligible and that the oordinate transformations to adjust for position over a series of observation points an easily be added later. The oordinate transformations h 3 h 2 h 1 Figure 1. Signal hitting sensors on the Earth 11
12 4. FORMULATION OF THE MODEL would just obfusate the main line of this development if they were to be inluded from the outset. 1. Measuring time The relative distanes of the detetors from the soures are ritial fators in this model. This model uses h ij to represent the height of the i-th detetor with respet to the j-th soure. The quantity h ij is the height of the detetor measured parallel to a ray originating at the enter of the Earth and extending toward the j-th radio soure. For example, if the j-th radio soure were diretly over the north pole, then h ij would be h ij = R sin θ i where R is the radius of the Earth and θ i is the degrees latitude north of the equator for the i-th detetor. An impliit assumption here is that eah radio soure is far enough away that one an assume that the inoming radio waves are parallel without inurring signifiant error. The enter of the Earth is the point of referene for all measurements in this model. In order to determine the strength of the signal from the j-th soure that would have hit the enter of the Earth at time t, one must ompensate for the fat that the signal hit the i-th detetor sooner beause the i-th detetor was h ij meters loser to the j-th soure than the enter of the Earth is. Beause light travels at meters per seond, one must hek the output of the i-th detetor from time t ij where t ij = t h ij However, beause there is some error in the measurement of position of the detetor and in the measurement of time at the detetor station, t ij is randomly distributed. (3) t ij = t ( hij + ɛ ) h + ɛ t where ɛ h is the error in the measurement of position and ɛ t is the error in the measurement of time. This model assumes that the error in the measurement of position ɛ h and the error in the measurement of time ɛ t are eah normally distributed with zero mean. The standard deviation of the error in position measurement is denoted σ h, and the standard deviation of the error in time measurement is denoted σ t. In pratie, these errors may not be normally distributed. But, hopefully the number of telesopes involved will be large enough to make it reasonable to use normal distributions as approximations of the true distributions. Also, these errors may not have zero means. But, beause the model only depends on the relative positions and times of the detetors, a bias in these distributions would not affet the results. The key here is to synhronize the signals from the different detetors not to determine the exat time the signal would pass the enter of the Earth. 2. Measuring signals The atual signal that reahes a detetor has noise in it already. Radio waves are not emitted at a onstant rate from a soure. Due to the quantum nature of light, a onstant soure has a onstant probability of emitting photons as opposed to a onstant emission of photons. Thus, the number of photons atually released
2. MEASURING SIGNALS 13 by a soure follows a Poisson distribution whose parameter is the ideal emission rate for the soure. 1. Note: Radio waves are eletromagneti waves. They are of a lower frequeny than visible light waves, but they are fundamentally the same thing. Radio waves are omposed of photons just as visible light waves are. If the soure would ideally emit s j (t) photons for time t, then the probability of x photons from the soure arriving at the i-th detetor at time t ij is given by: (4) p x (x) = (s j(t)) x e sj(t) x! Various soures ontribute to noise in the detetor thermal noise, g-r noise and 1/f noise. 2 Rather than model eah of these individually, this model assumes that the net effet will be additive noise that is normally distributed with zero mean and standard deviation σ d. This is the most tenuous assumption thus far and must ertainly be taken up in future work. 1 [Kithin] p. 41 2 [Kithin] p. 40
CHAPTER 5 Analysis of the model In this model, the terms ɛ h and ɛ t are ombined into a total error in time due to the errors in positional measurement and time measurement. This total error in time is denoted ɛ T. Thus, equation 3 from page 12 beomes: ( ) hij (5) t ij = t + ɛ T Given that ɛ h and ɛ t are normally distributed with zero means and standard deviations of σ h and σ t respetively, one would like to know the distribution of the total error in time ɛ T. The total error in time ɛ T is normally distributed with zero mean and a standard deviation of (σ h /) 2 + σt 2. The full derivation of this is given in appendix A. There will be more than one soure whose radio signal will strike the detetor. Beause of this, one must modify equation 4 to take into aount multiple soures. Taking s 1 (t) to be the signal one wishes to observe, one will synhronize the time signals of the i-th detetor using t i1 from equation 5. In other words, the output of the i-th detetor should be heked at time t hi1. One must take into aount the strength of the other signals whih would hit that detetor at the same time. To do this, one must hek the strength of the j-th signal at time t + hij hi1. One an readily verify that if one is heking the strength of the signal from the primary soure, one would be heking the strength of the signal that would reah the enter of the Earth at time t. Given multiple variables whih are eah Poisson distributed, their sum is Poisson distributed. The full derivation of this is found in appendix B. Beause of this equation 4 beomes: (6) p i x (x, t) = ( j s j ( t + hij hi1 )) x e j (t+ sj h ) ij h i1 for multiple radio signal soures. In order to take into aount the errors in time measurement, one must sum the probability from equation 6 times the probability of a partiular error in time measurement over all possible values of the error in time measurement. That is: (7) p i x (x, t) = ɛ T = x! p i x (x, t ɛ T ) p ɛt (ɛ T ) dɛ T Beause the error in time measurement ɛ T is normally distributed with zero mean and a standard deviation of (σ h /) 2 + σ 2 t (see page 15), the funtion p ɛt (ɛ T ) 15
16 5. ANALYSIS OF THE MODEL is given by: 1 ɛ T (8) p ɛt (ɛ T ) = e 2 (σ h /) 2 +σ t 2 ((σh /) 2 + σt 2 ) 2π ( It simplifies things greatly if one assumes that that the total error in time measurement ɛ T is the same for eah soure. In reality, the error in time due to the error in position measurement ɛ h will be different for soures whih are in different diretions. However, beause the speed of light is very large ompared to the standard deviation of the error in position measurement σ h, the whole value ɛ h has a very minor effet on ɛ T anyway. With this assumption, equation 7 an be written out ompletely: (9) p i x (x, t) = ɛ T = ) 2 ( ( )) x j s j t + hij hi1 ɛ T e 1 2 e ( ɛ T (σ h /) 2 +σ 2 t ) 2 x! ((σh /) 2 + σt 2 ) 2π dɛ T j (t+ sj h ) ij h i1 ɛ T This expresses the probability that x photons reah the i-th detetor at the time when the i-th detetor should be reeiving the signal from the primary radio soure (the soure being observed) whih will pass the enter of the Earth at time t. The equation for the probability that the i-th detetor detets z photons at that time must also take into aount the noise in the detetor. In the previous hapter, this was assumed to be additive noise whih was normally distributed with zero mean and a standard deviation σ d. Thus, the probability that the i-th detetor detets z photons is obtained by multiplying the probability that there were x photons present and z x noise in the detetor summed over all possible values of x and z x. This is: (10) p d (z, t) = x=0 p i x (x, t) x+ 1 2 e 1 y=x 1 2 2 σ d 2π ( z y σ d ) 2 dy The integral in the above equation raises immediate warning flags about the hoie of a normal distribution to model the detetor noise. The integral serves to mesh the ontinuous normal distribution with the disrete summation by rounding off the noise in the detetor to the nearest whole photon. However, the hopes of simplifying equation 10 are slim. On the brighter side, it is lear that when all of the various random variables obtain their mean values and when the primary soure is the only soure, the deteted quantity is preisely the one sought. So, while unwieldy, the model is on trak. Unfortunately, this equation only models a single detetor. Taking into aount all of the detetors in the array involves determining how the average of variables distributed as in equation 10 is distributed. Rather than takle this diretly, the author wrote a omputer simulation employing the above model. This simulation is used in the following hapter to make preditions with this model and to explore its sensitivities.
CHAPTER 6 Interpretation of the model Figure 1 shows the mean squared error of a distributed array of radio telesopes as a funtion of the number of telesopes. In this plot, the only radio soures were the primary target and the sky. The primary soure signal was taken to be a sine wave of frequeny 100 Hz. Note: Young nasent pulsars rotate at about 100 Hz, and they slow down as they age. 1 The soure was taken to be 1/1000-th the brightness of the sky. The standard deviation in position measurement ɛ h was taken to be 0.8 meters (reasonable under urrent GPS performane speifiations). And, the standard deviation of time measurement ɛ t was taken to be 0.01 seonds (well within the urrent laims of the Network Time Protool). The detetors were sampled at 1024 Hz. The detetor was taken to be 20 times as noisy as the sky. By noise in the sky, one means the noise aused by the Poisson distribution of photons emitted by the sky. The standard deviation of the Poisson distribution is the square root of its mean. For this simulation, the sky s mean brightness was taken to be 10000 photons per seond. Thus, the standard deviation of the sky noise is 100 photons 1 [UCSC] seond paragraph 0.24 0.22 0.2 0.18 0.16 Mean squared-error 0.14 0.12 0.1 0.08 0.06 0.04 100 200 300 400 500 600 700 800 900 1000 Number of detetors Figure 1. Mean squared-error as a funtion of the number of detetors 17
18 6. INTERPRETATION OF THE MODEL 0.6 0.5 0.4 Mean squared-error 0.3 0.2 0.1 0 0-0.5-1 -1.5-2 -2.5-3 -3.5-4 Log of standard deviation of time errors -4.5-5 Figure 2. Mean squared-error as a funtion of the log 10 of the standard deviation of the time error σ t per seond. The detetor was taken to have a standard deviation of 2000 photons per seond. From figure 1, it is lear that if only a single radio soure (plus sky noise) is being deteted by more than 350 telesopes, the mean squared-error easily stays below 0.02 photons per seond on a signal whose amplitude is 10 photons per seond. That is a mean squared-error less than 1/500-th of the signal against a noisy sky with a very poor detetor. Figure 2 shows the model s sensitivity to errors in time measurements. It plots the mean squared-error as a funtion of the log 10 of the standard deviation of time measurements ɛ t for a system of 500 detetors sattered randomly over the surfae of the Earth. In this plot, the only radio soures were the primary target and the sky. The primary soure signal was taken to be a sine wave of frequeny 100 Hz as it was for figure 1. The soure was taken to be 1/1000-th the brightness of the sky. The standard deviation in position measurement ɛ h was again taken to be 0.8 meters. The detetors were again taken to be 20 times as noisy as the sky as they were for figure 1. Beause the frequeny of the soure was taken to be 100 Hz, it is unsurpising that the mean squared-error of the array drops off dramatially one the standard deviation of the time error drops below 1/100-th of a seond. For multiple radio soures, the results are still promising. Assuming that eah detetor is an omni-diretional radio antenna (that is, it has a good response to radio signals oming from any diretion), the mean squared-error still drops below 2 photons per seond with around 150 detetors. Figure 3 shows the situation where there are fifty radio soures in addition to the sky. The mean squared-error here is with respet to the primary soure whih has an amplitude of 10 photons per seond. As with the earlier plots, the detetors are taken to be 20 times as
6. INTERPRETATION OF THE MODEL 19 0.7 0.6 0.5 0.4 Mean squared-error 0.3 0.2 0.1 0 100 150 200 250 300 350 400 450 500 Number of detetors Figure 3. Mean squared-error as a funtion of the number of detetors with multiple soures of interferene noisy as the sky. The amplitude of the primary soure is taken to be 1/1000-th the brightness of the sky. The primary soure has a frequeny of 100 Hz. The other radio soures are sattered about the sky and have similar amplitudes and wavelengths as the primary soure. Beause of this, one need not point the individual telesopes in any diretion. The array an image the entire sky at one time. One an hoose a diretion from whih to analyze the data by seleting height values for the detetors h i1 s appropriate to that diretion. Thus, the array of telesopes is quite robust. Even with really poor telesopes, the array an distinguish faint soures against a bright sky amid interferene from other soures.
CHAPTER 7 Further work Most of the interpretation of this model is based upon a omputer simulation. As suh, more analysis should be done to verify that the omputer simulation faithfully reprodues the model and orretly outputs data. All of the ode whih the author wrote for this omputer simulation is freely available at: http://www.nklein.om/produts/sope The omputer simulation assumes also that the detetors attempted to synhronize their sample times (as best they an given the total error in time measurement ɛ T ). In reality, it is more likely that eah detetor s sampling will have its own time offset and that one would like to know the output of a detetor for a time period that is partially in one sample and partially in another sample. For example, if a telesope sampled every 1/1000-th of a seond beginning at time zero, but the adjustment fator t i1 for that telesope is 1/2500-th of a seond, then one would rather the telesope had started reording 1/2500-th of a seond earlier. It would be nie to model the situation where the values from a partiular telesope must be interpolated from the values that the telesope reorded. Another assumption in the omputer simulation is that the detetors are distributed with equal probability over the entire surfae of the Earth and that the pulsar radio soures are evenly distributed over the sky. In reality, hanes are good that the detetors will be lustered in high population areas and that the pulsars will most likely be in the diretion of the galati enter. A more detailed model of the Earth and the loations and strengths of various pulsars would benefit the simulation greatly. As mention on page 13, the assumption that the noise in the detetor an be modeled with a normal distribution of zero mean is questionable. A more detailed model of the detetor noise whih takes into aount thermal noise, g-r noise, and 1/f noise is desirable. However, despite these onerns, the model paints a promising piture for a distributed observatory. It is hopeful that an array suh as this may prove viable. 21
APPENDIX A Distribution of ɛ h / + ɛ t Given that ɛ h is normally distributed with mean 0 and standard deviation H and that ɛ t is normally distributed with mean 0 and standard deviation T, we want to find the distribution of ɛ h +ɛ t. We will show that the sum is distributed normally with mean 0 and standard deviation (H/) 2 + T 2. Proof. Under the assumptions, the probability that ɛ h takes on any partiular value h is given by: (11) p ɛh (h) = e 1 2( h H ) 2 H 2π and the probability that ɛ t takes on any partiular value t is given by: (12) p ɛt (t) = e 1 2( t T ) 2 T 2π We now onern ourselves with the probability that ɛ h + ɛ t takes on any partiular value z. This probability is given by the following integral whih sums the probabilities of eah possible ombination of h and t that result in a partiular z: ( (13) p z z = h ) ( + t = p ɛh (h) p ɛt z h ) dh We shall substitute into equation 13 the values of p ɛh and p ɛt given by equations 11 and 12. (14) (15) p z (z) = = = 1 HT 2π e 1 2( h H ) 2 1 H 2π e 2( z h/ T ) 2 T dh 2π e 1 2 1 e 1 2 HT 2π ( ) h 2 H 2 + z2 2zh/+h 2 / 2 T 2 dh [ ( H 2 / 2 +T 2 ) h2 2zH 2 h/+h 2 z 2 H 2 T 2 Equation 14 looks hopeless at first glane. But, beause e 2( 1 h µ σ ) 2 dh = σ 2π it remains only to massage the integral into this form. This is easier than it may appear beause it doesn t matter what µ turns out to be. It doesn t figure into the result of the integral. 23 ] dh
24 A. DISTRIBUTION OF ɛ h / + ɛ t To massage the integral into the form of equation 15, we will omplete the square in the exponent of equation 14 without introduing any new terms ontaining h. p(z) = e 1 2 (H2 / 2 +T 2 ) h2 2H 2 zh/+ H4 z 2 H 2 +T 2 2 H 2 T 2 HT 2π e H 4 z 2 1 H 2 +T 2 2 +H2 z 2 2 H 2 T 2 ( 1 1 H 2 ) H 2 +T 2 2 z 2 h (H/) = e 2 T 1 2 +T H 2 2 z (H/) 2 2 2 +T 2 HT e HT 2π ( ) 1 1 H2 z H 2 +T 2 2 = e 2 T 2 1 h H 2 2 z [(H/) 2 +T 2 ] 2 HT (H/) e 2 +T dh HT 2π Now, the integral is in the form of equation 15. The value of σ is dh dh HT. (H/)2 +T 2 The value of µ is not very pretty, but it does not enter into the value of the integral. Now, we an replae the integral with its value. (16) p(z) = e 1 2 ( ) 2 z (H/) 2 +T 2 HT 2π ( ) 2 1 z = e 2 (H/) 2 +T 2 (H/) 2 + T 2 2π HT 2π (H/) 2 + T 2 Equation 16 shows that z = ɛ h / + ɛ t is normally distributed with mean 0 and standard deviation (H/) 2 + T 2. By extension (if one takes to be 1), one an see that the sum of normally distributed variables x 1, x 2,... x n with zero means and standard deviations X 1, X 2,... X n respetively is normally distributed with zero mean and standard deviation n i=1 X i. Partiularly, if X 1 = X 2 = = X n = X, then the sum is normally distributed with zero mean and standard deviation n X.
APPENDIX B Distribution of sum of Poisson distributed variables Given two variables x and y whih are Poisson distributed with parameters X and Y respetively, we will show that the sum z = x+y is Poisson distributed with parameter X + Y. Proof. Beause x is Poisson distributed, the probability that it takes on any partiular value x is given by: (17) p x (x) = Xx e X Similarly, the probability that y takes on any partiular value is: (18) p y (y) = Y y e Y The probability that z takes on any partiular value z is the probability that x takes on a partiular value x times the probability that y takes on a value of z x. There are many ombinations of x and y that aomplish this. Thus, we must sum the probabilities over eah ombination. Sine neither x nor y an be negative, x an only range from 0 to z. The probability then that the sum x + y takes on the value z is given by: z (19) p x+y (z) = p x (x) p y (z x) x=0 where p x is given by equation 17 and p y is given by equation 18. Substituting equations 17 and 18 into equation 19, we obtain: p x+y (z) = z X x e X x=0 x! x! y! = e (X+Y ) z x=0 Y (z x) e Y (z x)! X x x! Y (z x) (z x)! We an simplify this greatly by multiplying the whole equation by z!/z!. p x+y (z) = e (X+Y ) z! z x=0 z! x!(z x)! Xx Y (z x) The expression z! x!(z x)! an be rewritten ( z x). Beause ( z x) is the oeffiient of X x Y (z x) in the binomial expansion of (X + Y ) z, we an get rid of the summation 25
26 B. DISTRIBUTION OF SUM OF POISSON DISTRIBUTED VARIABLES in the above equation. (20) p x+y (z) = e (X+Y ) z! z x=0 ( ) z X x Y (z x) x = (X + Y )z e (X+Y ) z! This shows that x + y is Poisson distributed with parameter X + Y. By extension, the sum of Poisson distributed variables x 1, x 2,... x n with respetive parameters X 1, X 2,... X n is Poisson distributed with parameter n i=1 X i.
Bibliography [Carr] Carr, Joseph J. Serets of RF Ciruit Design, Seond Edition. TAB Books; New York; 1997. ISBN 0-0701-1673-3. [Christiansen] Christiansen, W. N. and J. A. Högbom. Radiotelesopes. University Press, Cambridge, 1969. [Kithin] Kithin, C. R. Astrophysial Tehniques, Third Edition. Institute of Physis Publishing; Bristol; 1998. ISBN 0-7503-0498-7. [SPIE98] Phillips, Thomas G. Advaned Tehnology MMW, Radio, and Terahertz Telesopes; Proeedings of the SPIE; Volume 3357. SPIE The International Soiety for Optial Engineering; Bellingham, Washington; 1998. ISBN 0-8194-2804-3. [Shannon] Shannon, Claude E. A Mathematial Theory of Communiation. Bell Systems Tehnial Journal, vol. 27; July and Otober; 1948. Downloaded from http://m.bell-labs.om/m/ms/what/shannonday/paper.html on 2000-10-19. [Spiegel] Spiegel, Murray R. Shaum s Outline of Theory and Problems of Statistis, Seond Edition. MGraw-Hill, In.; New York; 1988. ISBN 0-0706-0234-4. [Swenson] Swenson, Jr. G. W. An Amateur Radio Telesope. Pahart Publishing House; Tuson; 1980. ISBN 0-9129-1806-3. [UCSC] UCSC Compat Objet Group. Radio Pulsars: An Introdution. http://pulsar.uolik.org/og/pulsars/intro.html viewed 2000-11-05. 27