DISCRETE-SIGNAL ANALYSIS AND DESIGN- P32

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P32:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P32 THE HILBERT TRANSFORM 141 N−1 xh(n) : = ∑ k=0 XH(n)⋅exp −j⋅2⋅π⋅ k ⋅n N (f ) 2 RealRe(xh(n)) 0 −2 0 10 20 30 40 50 60 n (g) xa(n) : = Re(x(n)) − j⋅Re(xh(n)) (h) 2 Imaginary Real sequence sequenceIm(xa(n)) 0Re(xa(n)) −2 0 10 20 30 40 50 60 n (i ) N−1 XA(k) : = 1 ⋅ N n=0 ∑ xa(n)⋅exp −j⋅2⋅π⋅ n ⋅k N (j ) 1 Real 0.5 Re(XA(k)) 0 −0.5 0 10 20 30 40 50 60 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 imag- inary 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 negative- frequency 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 equiv-alent spectrum [Carlson, 1986, pp. 198–199] that is centered at zerofrequency. To use this signal, for example in radio communication, it mustbe frequency-translated. It then becomes a true single-sideband “signal”at positive SSB frequencies with suppressed carrier. If this SSB RF signalis represented as phasors, it is a two-sided SSB phasor, spectrum, oneSSB sideband at positive RF frequencies and the other SSB sideband atnegative RF frequencies. The value of the positive suppressed carrier fre-quency ω0 can be anything, but in the limit, as ω0 →0, the idea of anactual SSB signal disappears (in principle), as mentioned before.SINGLE-SIDEBAND RF SIGNALAt radio frequencies the single-sideband (SSB) signal contains informa-tion only on upper singlesideband (USSB) or only on lower singlesideband THE HILBERT TRANSFORM 143(LSSB). The usual “carrier” that we see in conventional AM is miss-ing. A related approach is the “vestigial” opposite sideband, which tapersoff in a special manner. Some systems (e.g., shortwave broadcast) usea reduced-level pilot carrier (−12 dB), that is used to phase lock to aninput signal. In the usual peak-power-limited system the power in the pilotcarrier reduces slightly the power in the desired single sideband. Specialmethods are used to modulate and demodulate the SSB signal, which addsomewhat to the cost and complexity of the equipment. The big plus fac-tor is that almost all of the transmitted and received signal power residein one narrow sideband, where they are most effective. Incidentally, and to digress for a moment, AM mode is criticizedbecause of all of the “wasted” power that is put into the carrier. Nothingcould be further from the truth. The basic AM receiver uses a very simplediode detector for the AM signal. The AM carrier is the “local oscillator”for this detector which demodulates (translates) the AM signal to audio(see Fig. 3-5). The t ...