Signal Processing and Linear Systems-B.P.Lathi copy

75a it follows t hat for a causal sinusoidal input

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Unformatted text preview: 7.3b. For a zero a t t he o rigin, the phase is Ljw = 90°. T his is a mirror image of the phase plot for a pole a t t he origin and is shown d otted in Fig. 7.3b. 3. First-Order Pole (or Zero) 100a 90" 45" T he Log Magnit:ude T he log a mplitude because of a first-order pole a t - a is - 20 log 11 + j: I· Let us investigate t he a symptotic behavior of this function for extreme values of w (w « a a nd w » a). (a) For w « a, - 2010 g \l+ j :\",,-2010g 1=0 Hence, the log-amplitude function --> 0 a symptotically for w (b) For the o ther extreme case, where w » a , - 2010 g \1 + j: \"" = - 2010gw = - 20u « + j: \ F ig. 7 .4 (7.17a) + 2010ga (7.17b) + 20 log a = - 20 log - 45" a (Fig. 7.4a). This represents a straight line (when plotted as a function of u , t he log of w) w ith a slope of - 20 d B/decade (or - 6 d B/octave). When w = a, t he log amplitude is zero [Eq. (7.17b)]. Hence, this line crosses t he w-axis a t w = a, as illustrated in Fig. 7.4a. Note t hat t he a symptotes in (a) and (b) meet a t w = a. T he e xact log amplitude for t his pole is - 2010+ 00_ (7.16) ~) - 2010 g ( o (1 + :~) ~ A mplitude a nd p hase response o f a f irst-order pole o r zero. This exact log magnitude function also appears in Fig. 7.4a. Observe t hat t he actual plot and the asymptotic plots are very close. A maximum error of 3 dB occurs a t w = a. T his frequency is known as t he c orner f requency or b reak f requency. T he error everywhere else is less t han 3 dB. A plot of the error as a function of w is shown in Fig. 7.5a. This figure shows t hat t he e rror a t one octave above or below the corner frequency is 1 dB and the error a t two octaves above or below the corner frequency is 0.3 dB. T he a ctual plot can be obtained by adding the error to the asymptotic plot. T he a mplitude response for a zero a t - a (shown dotted in Fig. 7.4a) is identical t o t hat of the pole a t - a with a sign change, and therefore is t he mirror image (about the O-dB line) of the amplitude plot for a pole a t - a. Phase (7.18) T he phase for t he first-order pole a t - a is 482 o :§ 7 -- --D.S I-- ~ 7.2 Bode Plots 7 Frequency Response and Analog Filters I - I. S ,~d-' '" I i 2 I I : -2. 5 3 I O.2a O .la , 1-- -1- i+-------I !i "I i '.. L / I . '- ..- 1\ i L i I NI I, 1 4. Second-Order Pole (or Zero) ! Let us consider t he second-order pole in Eq. (7.l1a). T he d enominator term is 1---4--+-- 1-(a)1 i I s2+b2 8 + b3. We shall introduce the often-used standard form s2+2(wn8+W~ instead 2 of 8 + b28 + b3. W ith this form, the log amplitude function for t he second-order 1 term in Eq. (7.13) becomes I O)=a O.Sa 7.4b) is identical to t hat of the pole a t - a with a sign change, a nd therefore is t he mirror image (about the 0 0 line) of the phase plot for a pole a t - a. b+=1-+-I I 2a (0--+ IOa Sa - 20Io g I1+2 j 6 l (:n (~:rl + (7.19a) and the phase function is 4 0) 483 2 iOJ) !:! "t:l ..: 0 t: 0) ~ .<: (7.19b) T he Log M...
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