We consider a transmission of a sequence of binary elements independent, that can take values 0 or 1 with the same proba

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We Consider A Transmission Of A Sequence Of Binary Elements Independent That Can Take Values 0 Or 1 With The Same Proba 1 (174.84 KiB) Viewed 66 times

We consider a transmission of a sequence of binary elements independent, that can take values 0 or 1 with the same probability. We use a modulation with an alphabet of two signals so(t) and sı(t) of period T, defined over the interval [0, T[. Over the interval [KT, (k+1)T[, we transmit soft-kT) or sı(t - KT) according to the value of the binary element to transmit. We focus on the time interval [0, T[. We assume that the receiver is formed by a filter of transfer function G(f), impulse response h(t) followed by a threshold detector and a sampler at instant to. On the interval [0, T[, the signal r(t) at the input of the receiver is equal to : r(t) = so(t) + b(t) or r(t) = si(t) + b(t) where b(t) represents a white Gaussian noise, centered, independent of the signal and that has a power spectral density No/2. 1. Calculate the value of the signal at the sampling instant, in the absence of noise, respectively when the binary element 0 or 1 is transmitted. We denote up and ui these values that we express in integral forms function of G(f) and respectively So(f) and Si(f). 2. Compute the error probability and show that is dependent of the ratio p, quotient of the difference (u - U1) by the deviation o of the noise at the output of the sampler. Express 02 as function of No and G(f). 3. By using Schwarz inequality, show that the ratio p is maximal if the filter of response g(t) is adapted to the signal (81(t) - so(t)], which means : g(t) = K[sı(to - t) - so(to - t)], where K is a constant. 4. Derive the error probability and show that it can be written as : 1 Pe=1 / orie Q d2 4N I (s() – so(6) de What does d physically represent ? By using this result, find the error probability for the code NRZ and the code RZ.