TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P32

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P32:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | THE HILBERT TRANSFORM 141 N-1 xh n k 0 k XH n -exp -j-2-n- -n xa n Re x n - j Re xh n h 50 0 20 30 60 n 10 40 i N-1 XA k -1 xa n exp -j 2 n n 0 j n Nk Re XA k k Figure 8-3 continued 142 DISCRETE-SIGNAL ANALYSIS AND DESIGN f The spectrum XH k is converted to the time domain xh n using the IDFT. This is the Hilbert transform HT of the input signal x n . g The HT xh n of the input signal sequence is plotted. Note that xh n is a real sequence as is x n . h The formula xa n for the complex analytic signal in the time domain. i There are two time-domain plot sequences one dashed for the imaginary part of xa n and one solid for the real part of xa n . These I and Q sequences are in phase quadrature. j The spectrum XA k of the analytic signal is calculated. k The spectrum of the analytic signal is plotted. Only the negativefrequency real components 2 same as 62 and 8 same as 56 appear because the minus sine was used in part H . If the plus sign were used in part h only the positive-frequency real components at 2 and 8 would appear in part k . Note that the amplitudes of the frequency components are twice those of the original spectrum in part c . All of this behavior can be understood by comparing parts c and e where the components at 2 and 8 cancel and those at 2 and 8 add but only after the equation in part h is used. The j operator in part h aligns the components in the correct phase either to augment or to cancel. This is the baseband analytic signal also known as the lowpass equivalent spectrum Carlson 1986 pp. 198-199 that is centered at zero frequency. To use this signal for example in radio communication it must be frequency-translated. It then becomes a true single-sideband signal at positive SSB frequencies with suppressed carrier. If this SSB RF signal is represented as phasors it is a two-sided SSB phasor spectrum one SSB sideband at positive RF frequencies and the other SSB sideband at negative RF frequencies. The value of the positive suppressed carrier .

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