DISCRETE-SIGNAL ANALYSIS AND DESIGN- P31

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P31: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- P31136 DISCRETE-SIGNAL ANALYSIS AND DESIGNequation di(t) v(t) = L ; let i(t) = ej ωt (a phasor or a sum of phasors) dt di(t) = j ωej ωt = j ωi(t) (8-1) dt v(t) = Lj ωi(t) = j ωLi(t) Vac = j ωLIacV ac and I ac are sinusoidal voltage and current at frequency ω = 2πf. Thephasor e jωt is the “transformer.” This is the ac circuit analysis methodpioneered by Charles Proteus Steinmetz and others in the 1890s as a wayto avoid having to Þnd the steady-state solution to the linear differentialequation. If the LaPlace transform is used to deÞne a linear network (withzero initial conditions) on the S -plane, we can replace “S ” with “j ω, whichalso results in an ac circuit with sinusoidal voltages and currents. We canalso start at time = zero and wait for all of the transients to disappear,leaving only the steady-state ac response. The Appendix of this booklooks into this subject brießy. These methods are today very popular and useful. If dc voltage and/orcurrent are present, the dc and ac solutions can be superimposed. A sum or difference of two phasors creates the cosine wave or sinewave excitation I ac . These can be plugged into Eq. (8-1): ej ωt − e−j ωt ej ωt + e−j ωt j sin ωt = , cos ωt = (8-2) 2 2 The HT always starts and ends in the time domain, as shown in Figs.8-1 and 8-2. The HT of a ( + sine) wave is a ( − cosine) wave (as in Fig.8-1) and the ( − cosine) wave produces a ( − sine) wave. Two consecutiveperformances of the HT of a function followed by a polarity reversalrestore the starting function. In order to simplify the Hilbert operations we will use the phase shiftmethod of Fig. 8-1c combined with Þltering. But Þrst we look at the basicdeÞnition to get further understanding. Consider the impulse responsefunction h(t) = 1/t, which becomes inÞnite at t = 0. The HT is deÞned as THE HILBERT TRANSFORM 137the convolution of h(t) and the signal s(t) as described in Eq. (5-4) for thediscrete sequences x (m) and h(m). The same “fold and slide” procedureis used in Eq. (8-3), where the symbol H means “Hilbert” and ∗ (not thesame as asterisk *) is the convolution operator: +∞ 1 s(τ) H [s(t)] = s (t) = h(t) ∗ s(t) = ˆ dτ (8-3) π −∞ t −τ In this equation τ is the “dummy” variable of integration. The valueof the integral and H[s(t)] become inÞnite when t = τ and the integral iscalled “improper” for this reason. First, the problem of the “exploding”integral must be corrected. This is done by separating the integral intotwo or more integrals that avoid t = τ. H[s(t)] =ˆ (t) = h(t) ∗ s(t) s lim 1 −ε s(τ) 1 +∞ s(τ) (8-4) = dτ + dτ ε→0 π −∞ t −τ π +ε t −τThis equation is called the “principal” value, also the Cauchy principalvalue, in honor of Augustin Cauchy (1789– 1857). As the convolutionis performed, certain points and perhaps regions must be excluded. This“connects” us with Fig. 8-1, where the value of the HT became very largeat three locations. There is also a problem if s(t) has a dc component. Equations (8-3)and (8-4) can become inÞnite, and the dc region should be avoided. Thecommon practice is to reduce the low-frequency response to zero at zerofrequency.The Perfect Hilbert TransformerThe procedure in Fig. 8-1c is an all-pass network [Van Valkenburg, 1982,Chapts. 4 and 8], also known as a quadrature Þlter [Carlson, 1986, p.103]. Part (c) shows that its gain at all phasor frequencies, positive andnegative, is ± 1.0, and that it performs an exact + 90◦ or − 90◦ phase shift.This is the practical software deÞnition of the perfect Hilbert transformer. It is useful to point out at this time that the HT of a +sine wave is a(−cosine) wave and the HT of a +cosine wave is a (+sine) wave. At a138 DISCRETE-SIGNAL ANALYSIS AND DESIGNspeciÞc frequency, a ± 90◦ phase shift network can accomplish the samething, but for the true HT the wideband constant amplitude and widebandconstant ± 90◦ are much more desirable. This is a valuable improvementwhere these wideband properties are important, as they usually are. In software-deÞned DSP equipment the almost-perfect HT is fairly easy,but in hardware some compromises can creep in. Digital integrated cir-cuits that are quite accurate and stable are available from several vendors,for example the AD9786. In Chapter 2 we learned how to convert atwo-sided phasor spectrum into a positive-sided sine–cosine–θ spectrum.When we are working with actual analog signal generator outputs (pos-itive frequency), a specially designed lowpass network with an almostconstant −90◦ shift and an almost constant amplitude response over somedesired positive frequency range is a very good component in an analogHT which we will describe a little later. Please note the following: For this lowpass Þlter the relationshipbetween negative frequency phase and positive-frequency phase is notsimple. If the signal is a perfectly odd-symmetric sine wave (Fig. 2-2c),the positive- and negative-frequency sides are in opposite phase, just likethe true HT. But if the input signal is an even-symmetric cos ...