A1 Simple Statistics

Fundamentals of Communications A1: Simple Statistics EE3158 Professor Ian Groves [email protected] www.ctr.kcl.ac.uk/members

Simple Statistics      

Averages Spreads Z scores Normal Distribution Finding probabilities Application to Telecommunications

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Averages  Mean, median and mode Weekly rent paid by 15 students sharing accommodation, 1998 45 35 51 45 51 40 42 46 37 42 47 49

(£) 49 36 42

 Mean (or average)  add observations and divide by number of observations 657/15 = 43.8

x ∑ x= n A1 - Simple Statistics

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Averages… 2  Median – the middle observation.  rank the observations and find the middle one (n+1)/2th observation  35 36 37 40 42 42 42 45 45 46 47 49 49 51 51  the 8th observation (15+1)/2 is 45

 Mode – the most frequent observation  in this case 42

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Spreads  Standard Deviation (SD) calculated as below  calculate residuals – individual observation minus mean  square and sum these  divide by number 2  of observations minus 1 [gives Yi − Y   Variance] SD =    take square − 1 root  for Standard Deviation

∑( n 

)



 example peoples heights (cm)  190 185 182 208 186 187 189 179 183 191 179  mean 187.18  SD 8.02 A1 - Simple Statistics

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Z Scores  used to ‘normalise’ data

X −X i Zi = SD observation − mean Z= standard deviation A1 - Simple Statistics

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Normal Distribution  A general statistical theorem, the Central Limit Theorem, states that the probability distribution of any quantity which arises as the sum of the effects of a large number of separate contributions is the Gaussian (or normal) distribution, which for unit variance and zero mean is given by:

Z(x ) =

1 −x 2 2 e 2π

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Normal Distribution…2  the probability that x is greater than some value xo is found by integrating this equation over the range xo to infinity:

P(x > xo ) =

∞

∫ Z (t)dt

xo

 also known as the Gaussian Q–function.  we cannot solve this in closed form, rather numerically integrate and tabulate.  total area under curve (integral over +/infinity) =1 A1 - Simple Statistics

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Areas Under the Normal Curve

 ‘bell’ shaped curve – symmetrical  most common observations fall within +/- one SD A1 - Simple Statistics

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Tabulated Results  Values of Q(x) for various x x 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Q(x) 0.50000 0.30854 0.15866 0.06681 0.02275 0.00621 0.00135 0.00023 3.17 x 10-5

 Some ‘rules of thumb’  68% of a population fall within =/- 1 SD  95% fall within +/- 2 SD  99% fall within +/- 3 SD  SD is Standard Deviation

 For large values of x we can approximate Q(x) as Z(x)/x with less than 10% error for x>3

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Gaussian Probabilities level Xo 1.28 2.33 3.09 3.73 4.28 4.76 5.21 5.62

20log Xo (dB) 2.1 7.3 9.8 11.4 12.6 13.6 14.3 15.0

+ 6dB 8.1 13.3 15.8 17.4 18.6 19.6 20.3 21.0

Error Rate 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8

 values for Xo taken from published tables for Xo < 3  else computed from approximation (Excel solver function)  expressed as signal–to–noise ratio (power which is voltage squared so use 20log(Xo)  +6dB column for pulse detection in presence of noise… A1 - Simple Statistics

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Error Rates  For digital telecommunication systems we are interested in low error rates when detecting pulses in the presence of noise.  An error occurs if the instantaneous noise voltage exceeds half the pulse amplitude E volts.  If error rate is 10-3, say, Xo is 3.09 from previous table.  0.5E/SD = 3.09, so E/SD = 6.18  20log(6.18) = 15.8 dB i.e. 6dB greater than first column  we can now plot an error rate curve for a rectangular pulse detected in the presence of noise as a function of signal to noise A1ratio. - Simple Statistics 12

Error Rate Curve Error Probability 0 -1

Error Rate (10^y)

-2 -3 -4 -5 -6 -7 -8 -9 8

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Signal to Noise Ratio (dB)

 Calculated curve for a rectangular pulse from Slide 11 A1 - Simple Statistics

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