378 15 Causality and the IIilbert Tri2rlsform for ~ ( tto) be recovered As the input signal of both complex hand-pass filters is purely real, it is clear that in one branch only the real part of the impulse response must be considered, and in the other only the imaginary part Compared to critical sampling of real valued band-pass sigrials (see Chapter 11.3.4), the sampling rate is reductd by a factor of two Howcver, consider that in oiir example, each sampling value has a real arid i.tii imaginary part, while in Chapter 11.3.4 all sampling values were real The required number of real sampling values per time i s thus the same Howevei, we inay use analog-to-digital converters with half the sampling rate aiid we need not pay attmtion to the relation between the band edges w1 and w2 for critical sanipling Exercise 15.1 W'hich of the following discrete systems are causal? a) y[k]= ClZ[k i11 i- cgz[k] b) ~ [ k=]u [ k } ~ [ k ] c) ylk] = a [ k + 1]x[k] d) y[k] = sin(n s [ k ] ) ej y [ k ]= k ] Exercise 15.2 For w g > 0, calculate the Hilbcrt transform of a)eJwof,b) sin U , $ , c) cos q t and d) cos 2wot in the frecluency-dornairi Note: see Figure15.3 For each part, give the phase shift between the input arid output signal caused by the Hilbert transform Exercise 15.3 Consider the real signal z ( t ) with spectrum X(.jw) Some properties of the Hilbert ttransform y ( t ) = X ( x ( t ) }= - *:c(t) are to be investigatcd The irripiilse response ?t of the Hilbert, txansformator is indicated by h(t) a) Give H ( j w ) e h(t) arid sketch the magnitude arid phme of H ( j w ) How does the Hilbert transform affect / X (jw) and arg{X(jw]}? b) Is g(t) real, imaginary or complex? What symmetry does the Hilttert, txansform of the even part x e ( t )and the odd part z o ( t )have?