By Marcelo S. Alencar

Communique structures introduces Fourier remodel, convolution and definitions of autocorrelation and gear spectral density. It additionally introduces suggestions of likelihood, random variables, and stochastic tactics and their functions to the research of linear structures. additionally, the ebook covers speech coding, amplitude modulation, quadrature modulation, perspective modulation, cellular mobile structures, propagation channels and extra. The e-book additionally comprises 5 appendices which disguise issues reminiscent of Fourier sequence and transforms, Hilbert remodel, formulation utilized in the textual content, GSM mobile approach and others.

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**Example text**

41) A signal y{t) is given by the following expression *)4/ y{t) = - I ^dr. where the signal x{t) has a Fourier transform X(a;). By using properties of the Fourier transform determine the Fourier transform of y{t)42) Given that the Fourier transform of the signal f{t) = cos{(jJot) is F{uj) = 7r[6{w + Uo) + S{w — cc;^)], determine the Fourier transform of the signal g{t) — sm{ujot — <^)^ where (/) is a phase constant. Sketch the magnitude and phase graphs of the Fourier transform of this signal.

Application 4: Finally, consider A — R and I — R. 7. 7. Analog signal. Algebra of Sets In order to construct an algebra of sets or, equivalently, to construct a field over which operations involving sets make sense, a few properties have to be obeyed. 1) li A ^ T then A ^ T. A \s the set containing desired results, or over which one wants to operate; 2) UA^Td^ndBeT then A \J B e T, The above properties guarantee the closure of the algebra with respect to finite operations over sets. , Jl G J^, because Vt = A^ A.

40 COMMUNICA TION SYSTEMS 48) Using properties of the Fourier transform determine the Fourier transform of the function |^|. Plot the corresponding magnitude and phase spectrum of that transform. 49) Show t h a t u{t) * u{t) = r ( t ) , where u{t) represents the unit step function and r{t) denotes the ramp with slope 1. 50) Determine the Fourier transform for each one of the following functions: u{t — T ) , t, te~"'^u{t)^ J, ^ . Plot the corresponding time domain diagrams and the respective magnitude and phase spectrum of the associated transforms.