DISCRETE-SIGNAL ANALYSIS AND DESIGN- P33

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P33: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- P33146 DISCRETE-SIGNAL ANALYSIS AND DESIGN 22K 22K − + 100K 0.0025μF j⋅ω − a N := 128 ω := 0, 1..N a := 20 T(ω) := j⋅ω + a 90 deg [ 180 φ(ω) := atan2 Re(T(ω)), Im(T(ω)) ⋅ π [ Mag 1 Imag Real 0 −1 0 650 1300 1950 Hz 2600 3250 3900 200 Degrees φ(f) 100 0 0 650 1300 1950 Hz 2600 3250 3900 Figure 8-5 Elementary all-pass active RC network.is at +180◦ , according to the usual conventions [Dorf, 1990, Figs. 7-15band 7-16]. jω − a s−z T (j ω) = , T (s) = jω + a s+p THE HILBERT TRANSFORM 147Note the use of the Mathcad function atan2(x, y) that measures phaseout to ±180◦ (see also Chapter 2). The values 0.0025 μF and 100 Kare modiÞed in each usage of this circuit. Metal Þlm resistors and stableNP0 capacitors are used. The op-amp is of high quality because severalof them in cascade are usually dc coupled. Figure 8-6 shows how these basic networks can be combined to producea wideband −90◦ phase shift with small phase error and almost constantamplitude over a baseband frequency range. Each of the two all-pass net-works (I and Q) is derived from a computer program that minimizesthe phase error between the I and Q channels on two separate “wires.”[Bedrosian, 1963] is the original and deÞnitive IRE article on this subject.Examples of the circuit design and component values of RC op-amp net-works are in [Williams and Taylor, 1995, Chap. 7] and numerous articles.A simulation of this circuit from 300 to 3000 Hz using Multisim and thevalues from the book of Williams and Taylor (p. 7.36) shows a maxi-mum phase error of 0.4◦ . The 6 capacitors are 1000 pF within 1.0%. Theinput and output of each channel may require voltage-follower op-ampsto assure minimal external loading by adjacent circuitry. Copying R andC values from a handbook in this manner is sometimes quite sensiblewhen the alternatives can be unreasonably labor-intensive. A high-speedPC could possibly be used to Þne-tune the phase error in a particular appli-cation (see, for example, [Cuthbert, 1987], and also Mathcad’s optimizingalgorithms). R R R R R R − − − - + + + − + + Qout C 16.2k C 118k C 511k +IN R = 10k 1% C = 1000pF 1% 90° − R R R R R R − − − − + + + − + + Iout C 54.9k 267k C 17.4Meg CFigure 8-6 Two sets of basic all-pass networks create I and Q outputswith a 90◦ phase difference across the frequency range 300 to 3000 Hz.148 DISCRETE-SIGNAL ANALYSIS AND DESIGN The following brief discussion provides some examples regarding theusage of the Hilbert transform and its mathematical equivalent in radioequipment. Analog methods are used for visual convenience.SSB TRANSMITTERWe illustrate in Fig. 8-7 the analog design of an SSB transmitter sig-nal using the phase-shift method. It uses the −90◦ lowpass (positive-frequency) Þlter of Fig. 8-6, two double-balanced mixers, and an HFlocal oscillator [Krauss et al., 1980, Chap. 8]. The mixers create twodouble-sideband suppressed carrier (DSBSC) signals. The combiner atthe output uses the sum of these two inputs to create at the local oscil-lator frequency ω0 an LSB or the difference of the two inputs to createan USB. The BPF restricts the output to some desired frequency band.The end result is equivalent mathematically to a synthesis of the Hilberttransform and the analytic signal translated to RF that we have consideredin this chapter. There is an interesting artifact of this circuit that we should look at. 1. Start at the input, where the baseband signal is cos ωm t at 0◦ refer- ence. 2. The I -channel output (a) has a phase shift ∠θ◦ , relative to the 0◦ reference input, that varies from +64◦ at 300 Hz to −154◦ at 3 kHz. The I -channel output (a) is cosωm t + θ◦ . This effect is inherent in th ...