TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P33

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P33: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. | 146 DISCRETE-SIGNAL ANALYSIS AND DESIGN j ffl - a N 128 ffl 0 a 20 T ffl . a Figure 8-5 Elementary all-pass active RC network. is at 180 according to the usual conventions Dorf 1990 Figs. 7-15b and 7-16 . T j œ j œ - a s - z ------ T s ------- j œ a s p THE HILBERT TRANSFORM 147 Note the use of the Mathcad function atan2 x y that measures phase out to 180 see also Chapter 2 . The values and 100 K are modihed in each usage of this circuit. Metal him resistors and stable NP0 capacitors are used. The op-amp is of high quality because several of them in cascade are usually dc coupled. Figure 8-6 shows how these basic networks can be combined to produce a wideband -90 phase shift with small phase error and almost constant amplitude over a baseband frequency range. Each of the two all-pass networks I and Q is derived from a computer program that minimizes the phase error between the I and Q channels on two separate wires. Bedrosian 1963 is the original and dehnitive IRE article on this subject. Examples of the circuit design and component values of RC op-amp networks are in Williams and Taylor 1995 Chap. 7 and numerous articles. A simulation of this circuit from 300 to 3000 Hz using Multisim and the values from the book of Williams and Taylor p. shows a maximum phase error of . The 6 capacitors are 1000 pF within . The input and output of each channel may require voltage-follower op-amps to assure minimal external loading by adjacent circuitry. Copying R and C values from a handbook in this manner is sometimes quite sensible when the alternatives can be unreasonably labor-intensive. A high-speed PC could possibly be used to hne-tune the phase error in a particular application see for example Cuthbert 1987 and also Mathcad s optimizing algorithms . Figure 8-6 Two sets of basic all-pass networks create I and Q outputs with a 90 phase difference across the frequency range 300 to 3000 Hz. 148 DISCRETE-SIGNAL ANALYSIS AND DESIGN The following brief

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