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Extra resources for Digital Communication: Principles and System Modelling
Example text
In the next section, we’ll try to discuss the reason and logic behind this assumption. 3 How We Can Predict the Future? The approach to signal prediction (estimation) is discussed here using the spirit of Taylor [2]. Let us consider, a signal m(t), which have derivatives of all order at t. Using Taylor series we can express m(t + TS ) as m(t + TS ) = m(t) + T2 TS m(t) + S m(t) + ............. 1! 2! 36a) shows that from knowledge of the signal and its derivative at instant t, we can predict a future signal at t = TS .
Now, the derivative m(kTS ) ≈ m[kTS ] − m(kTS − TS ) . TS So form Eq. 36b), we obtain m[k] − m[k − 1] TS = 2m[k] − m[k − 1]. 37) It shows that, we can find a crude prediction of the k + 1th sample from the two previous samples. The approximation improves if we take more terms in Eq. 36a). To determine the higher order derivatives in the services, we require more samples from the past. Larger the number of past samples, we can get a better approximation. 38) The right hand side is m[k] ˆ the predicted value of m[k].
Here ωs − 2πB < 2πB Or, ωs < 4πB Or, fs < 2B Fig. 3 Sampled spectra for fS
