379 15.4 Exercises c) Calculate the cross-correlation fuiictioii of the (deterministic) signals z(t ) and y ( t ) for z = 0, i.e yZy(Q)= f-", s ( t ) y ( t )d t Which property of the Kilbert transforni of real signals can be derived froni this result? Note: see Chapter 9.8 aiid 9.9 d) Compute the Hilbert transform of the Fourier series [a,, cos(w0vt) + b , sin(ln/&)], with WO > Exercise 15.4 w Given the signal ~ ( t=) n 1- cosw,t and its I-Iilttert transform i ( t )= 3-I(e(t)} uJ9t a) Calculate arid sketch X ( p ) o or(t) Note: Vse the multiplication tlieorem of the Fourier tramforin, Chapter 9.7 b) Determine d ? ( ~ ~o )- o ?(t) graphically and calculate r i ( f ) Note* consider Figure 15.3 c) Let r n ( f ) x ( t ) + ~ i ( t ) Which spccial property exhibits X ( l ( ~ wo )- - o ~ ~ ( t ) ? 7= d) Calculate X , ( j w ) forrnally irt depcndciice on X ( J W ) Exercise 15.5 What is tlie connection between the energy of a signal ~ ( tand ) the Hilbcrt transformed signal Z ( t ) = f { x ( i ) ] ' ? Assume that the signal has no DC component Note: refer to thc%Parscval theorem in Chapter 9.8 Exercise 15.6 1,et X l ( j w ) be the sprc:trurn of the real signal sl(l) and X2(jw) = (1 sign(w))XI (jw)the one-sided spectrum of a signal :EZ ( t ) + a) Det,erminc z2( t )in deperideace on % ( t ) Note: Consider which transform corresponds t o the required convolution, b) Determine the relatioii of the signal ciiergies of energies directly in the frequency domain zl(l;)urid ~ ( t ) Stnt,e both c) Determine the rela,tion of the signal energies of x ~ ( t und ) zz(t)by a consideration in the time domain llse the result of Exercise 15.5 Exercise 15.7 A sketch of the real function P ( J w )is sl-lomm, which is part of the transfer function