PowerPoint PresentationSignals in NoiseAnalog Signals in Noise: ExampleAnalog Signals in NoiseSlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Gaussian Random ProcessesGaussian Random Processes (cont’d)Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19ExampleSlide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Matched FilteringSlide 30Slide 31Slide 32Copyright 2001, S.D. Personick. All Rights Reserved. 1Telecommunications Networking ITopic 3Quantifying the Performance of Communication Systems Carrying Analog Information Dr. Stewart D. PersonickDrexel UniversityCopyright 2001, S.D. Personick. All Rights Reserved. 2Signals in NoiseThe basic model:data or informationsignalsignal + noisedata or informationnoise+Copyright 2001, S.D. Personick. All Rights Reserved. 3Analog Signals in Noise: ExampleEngineTemperature Sensors= ca (volts)where a= temperature (C)+noiser=s+n =ca+nc= .01 volt/degree-C n(av) =0, var(n) = .0001 volt**2Copyright 2001, S.D. Personick. All Rights Reserved. 4Analog Signals in NoiseExample (continued)a= a number representing information to be communicated. apriori, a is a Gaussian random variable with variance A, and zero average values = a signal in the form of a voltage proportional to a …. = ca (volts), where c is a known constantr = a received signal with additive noise = s + n (volts)n = a Gaussian random variable with variance N (volts**2)How can we estimate “a”, if we receive “r”?Copyright 2001, S.D. Personick. All Rights Reserved. 5Signals in Noise•What is the criterion for determining whether we’ve done a good job in estimating “a” from the received signal “r”?•It would have something to do with the difference between our estimated value for “a” and the true value for “a”•Example: minimize E{(a-a)**2}, where a is the estimated value of “a”, given rCopyright 2001, S.D. Personick. All Rights Reserved. 6Signals in Noise•We will give a proof, on the blackboard, that a, the estimate of “a” that minimizes the average value of (a-a)**2 isa(r) = E (a|r) = the “expected value” of “a”, given “r”•The above is true for any probability distributions of “a” and “n”; and for the specific cases given, a(r) = r/c [Ac**2/(Ac**2 +N)]Copyright 2001, S.D. Personick. All Rights Reserved. 7Signals in NoiseHarder example:a = a Gaussian random variable with variance A, representing informations(t) = a signal of duration T (seconds) where s(t) = a c(t), and c(t) is a known waveformr(t) = a received signal = s(t) + n(t), where n(t) is a “random process” having a set of known statistical characteristicsHow do we estimate a, given r(t)?Copyright 2001, S.D. Personick. All Rights Reserved. 8Signals in Noise•What is the criterion for evaluating how good an estimate of “a” we have derived from r(t)?•How do we describe the noise n(t) in a way that is useful in determining how to estimate “a” from r(t)?Copyright 2001, S.D. Personick. All Rights Reserved. 9Signals in Noise•Suppose n(t) = nc(t) + x(t) where:n is a Gaussian random variable of variance N; and where, in some loosely defined sense:•x(t) is a random process that is statistically independent of the random variable “n”, and where …. (continued on next slide)Copyright 2001, S.D. Personick. All Rights Reserved. 10Signals in Noise•...x(t) is also “orthogonal” to the known waveform c(t), then we can construct a “hand waiving” argument that suggests that we can ignore x(t), and concentrate on the noise nc(t), as we attempt to estimate the underlying information variable “a”.•To make this argument more precisely requires a deep understanding of the theory of random processesCopyright 2001, S.D. Personick. All Rights Reserved. 11Signals in Noise•We will show (using the blackboard) that we can convert this problem into the earlier problem of estimating an information parameter “a” from a received signal of the form r= ca + n.•While doing so, we will introduce the concept of a “matched filter”Copyright 2001, S.D. Personick. All Rights Reserved. 12Gaussian Random Processes•If we look at (“sample”) the random process n(t) at times: t1,t2,t3,…,tj, then we get a set of random variables: n(t1), n(t2), n(t3), …,n(tj).•If the set of random variables {n(tj)} has a joint probability density that is Gaussian, then n(t) is called a Gaussian random processCopyright 2001, S.D. Personick. All Rights Reserved. 13Gaussian Random Processes (cont’d)•Any linear combination of samples of a Gaussian random process is a Gaussian random variable•Extending the above, the integral of the product n(t)c(t) over a time interval T is also a Gaussian random variable if, n(t) is a Gaussian random process, and c(t) is a known functionCopyright 2001, S.D. Personick. All Rights Reserved. 14Gaussian Random Processes (cont’d)•Let n(t) be a random process (not necessarily Gaussian)•Define “n” as follows:n= the integral over T of n(t)c(t)/W, whereW= the integral over T of c(t)c(t)•Then, we can write n(t) as follows:n(t)= nc(t) + “the rest of n(t)”Copyright 2001, S.D. Personick. All Rights Reserved. 15Gaussian Random Processes (cont’d)•If n(t) is a “white, Gaussian random process”, then:-n is a Gaussian random variable, and- “rest of n(t)” is statistically independent of “n”…I.e., “the rest of n(t)” contains no information that can help of estimate either “n” or “a”Copyright 2001, S.D. Personick. All Rights Reserved. 16Gaussian Random Processes (cont’d)•Furthermore, we can build a correlator that works as follows. It takes the received waveform, r(t), multiplies it by the known waveform c(t), integrates over the time interval T, and finally divides by Wz= {the integral over T of r(t)c(t)}/WCopyright 2001, S.D. Personick. All Rights Reserved. 17Gaussian Random Processes (cont’d)•Going back to the definitions and equations above, we find thatz= a + n, where “a” is the original information variable we wish to estimate, and n is a Gaussian random variable•Thus by introducing the correlator, we convert the new problem to (continued)Copyright 2001, S.D. Personick. All Rights Reserved. 18Gaussian Random Processes (cont’d)•…the old problem of estimating a scalar (“a”) from another scalar (“r”); where r= a+n, and where n is a Gaussian random variable•The correlation operation is also known as “matched filtering”, because it can be