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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P31: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. | 136 DISCRETE-SIGNAL ANALYSIS AND DESIGN equation v t L let i t ej-t a phasor or a sum of phasors dt L L - L j -e j rni t dt 8-1 v t Lj ai t j Li t Vac j -l-Ic V ac and I ac are sinusoidal voltage and current at frequency 2nf. The phasor ej-t is the transformer. This is the ac circuit analysis method pioneered by Charles Proteus Steinmetz and others in the 1890s as a way to avoid having to find the steady-state solution to the linear differential equation. If the LaPlace transform is used to define a linear network with zero initial conditions on the S -plane we can replace S with j which also results in an ac circuit with sinusoidal voltages and currents. We can also start at time zero and wait for all of the transients to disappear leaving only the steady-state ac response. The Appendix of this book looks into this subject briefly. These methods are today very popular and useful. If dc voltage and or current are present the dc and ac solutions can be superimposed. A sum or difference of two phasors creates the cosine wave or sine wave excitation Iac. These can be plugged into Eq. 8-1 . ej t e-j t ej t - - e j t 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 consecutive performances of the HT of a function followed by a polarity reversal restore the starting function. In order to simplify the Hilbert operations we will use the phase shift method of Fig. 8-1c combined with filtering. But first we look at the basic definition to get further understanding. Consider the impulse response function h t 1 t which becomes infinite at t 0. The HT is defined as THE HILBERT TRANSFORM 137 the convolution of h t and the signal .s t as described in Eq. 5-4 for the discrete sequences x m and h m . The same fold and slide procedure is used in Eq. 8-3 where the symbol H means Hilbert and not the same as asterisk is the convolution operator 1 f œ s t H
