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In a digital computer simulation of a bandpass filter, the complex envelope of the impulse response is used, where as shown in Fig. 4–3. The complex impulse response can be expressed in terms of quadrature components as k(t) = 2hx(t) + j2hy(t) where and The complex envelopes of the input and output are denoted, respectively, by g1(t) = x1(t) + jy1(t) and g2(t) = x2(t) + jy2(t). The bandpass filter simulation can be carried out by using four real baseband filters (i.e., filters having real impulse responses), as shown in Fig. P4–17. Note that although there are four filters, there are only two different impulse responses: hx(t) and hy(t).
(a) Using Eq. (4–22), show that Fig. P4–17 is correct.
(b) Show that hy(t) 0 (i.e., no filter is needed) if the bandpass filter has a transfer function with Hermitian symmetry about is the bounded spectral bandwidth of the bandpass filter. This Hermitian symmetry implies that the magnitude frequency response of the bandpass filter is even about fc and the phase response is odd about fc.
Figure P4–17

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