e All of the cont,inuous and discrete signals that, we have considered so far in the timeclomain arid freqitcnry-doincri W P L signals ~ that could be described by xiiatliematical luncIions 1% could calculate the valoes of the signal, add and subtract signals, clday signals and torrn derivatives ancl integrals We found integration vcry uscful, foi convolution, for Fourier arid Laplace tiansfornis, ~ n t wc~ l also u s t d complex iiitegrrttion for the inverse Laplace and inverse z-transfornris For discrete signals summation is iised in d of irittegration Shis was all possiblc bcc*ause we l m l assiimcd that every s I had one definite value at avenj’ poiiit in time, md that every signal coiilil he described by a mathematical formula, however complicated that formula might be M M Lsignals ~ that occur in practise not conform to this assumption It ~voulclbe theoretically possible to desrribe the sprcch signal from Figure 1.1 wit 2.1 the properties of the human vocal tract by superiiiiposirig various waves, but this would not lead to a tecliiiically realistic solution It is cornpletely impossible to assign functions to iioise signals, or Sigriiils made np of c ascillations A i i ~ w coiiccpt nnist be found to represent such irregular pro Just unclerstwding that, a signal waveform can have aii uiiprcdictablc val is t h c b r efore rtid[)IT1, does not actually help much To deal with system inputs and outputs in the way we are nsed to, random signals must he described by non-random or ‘deterministic’ quantities This can be prrtorrn.d by the so-called expected values, which arc iritrodncc~din the next section Then we will cleal with stationary and ergo&; raiidoni processes, for w2iich a significantly simpler calculation i s possi blt with cxpertetl valuts, An important clabs of exp tl values arc correlation functions which will also be disc-ussad All of tlicsc forms for describing random signals will be introduced for continuous signals, a i d the chapter concludes by cxteudiiig t h t concepts to dibcrete rantloni signals .1.1 at Are Ran The signals that we have been working with 60 far are called ddernimzstzr sz.gnala, which means that a signal has a ksiowri unambiguous value at every poiiit ill time A signal ctui also be deterministic when it cannot, he described by siinple mathematiral functions but i i i s t e d for example, by an iiifiiiite Fourier series,