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

# 75a it follows t hat for a causal sinusoidal input

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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&quot; 45&quot; 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 :\&quot;,,-2010g 1=0 Hence, the log-amplitude function --&gt; 0 a symptotically for w (b) For the o ther extreme case, where w » a , - 2010 g \1 + j: \&quot;&quot; = - 2010gw = - 20u « + j: \ F ig. 7 .4 (7.17a) + 2010ga (7.17b) + 20 log a = - 20 log - 45&quot; 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-' '&quot; I i 2 I I : -2. 5 3 I O.2a O .la , 1-- -1- i+-------I !i &quot;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) !:! &quot;t:l ..: 0 t: 0) ~ .&lt;: (7.19b) T he Log M...
View Full Document

Ask a homework question - tutors are online