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

# 72a t he a mplitude response is c onstant unity for

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Unformatted text preview: ction of any transfer by properly [ adding these b asic responses. Note t hat if a particular term is in t he n umerator, its phase is a dded, b ut if t he t erm is in t he d enominator, its phase is s ubtracted. This makes it easy t o p lot t he p hase function 1. H (jw) as a function of w. C omputation of IH (jw )1, u nlike t hat of t he p hase function, however, involves t he m ultiplication a nd division of various terms. This is a formidable task, especially when we have t o p lot this funct ion for t he e ntire range of w (0 t o 00). We know t hat a log operation converts multiplication a nd division t o a ddition a nd s ubtraction. So, i nstead of plotting IH(jw)1, why not plot log IH(jw)1 t o simplify our t ask? We can take advantage of the fact t hat logarithmic units are desirable in several applications, where t he variables considered have a very large range of variation. This is p articularly t rue in frequency response plots, where we m ay have t o p lot frequency response over a frequency range s tarting from a very low frequency near 0 t o a very high frequency in t he r ange of lO lD or higher. A linear plot for such a large r ange will bury much of t he useful information. Also, t he a mplitude 6 response may h ave a very large dynamic range from a low of 1 0- 6 t o a high o f 10 . A linear plot would be unsuitable for such a situation. Therefore, logarithmic plots not only simplify o ur t ask of plotting, b ut, fortunately, they are also desirable in this situation. T he logarithmic u nit is t he d ecibel a nd is equal t o t wenty times t he l ogarithm of t he q uantity (log t o t he base 10). Therefore, 20log lO IH(jw)1 is simply the log a mplitude in decibels (dB). Thus, instead of plotting IH (jw) I, we shall plot 20l0g lO IH(jw)1 a s a function of w. T hese plots (log amplitude a nd phase) are called B ode p lots. F or t he t ransfer function in Eq. (7.12a), the l og a mplitude is This function can be p lotted as a function of w. However, we c an effect further simplification b y using t he logarithmic scale for t he variable w itself. Let us define a new variable U such t hat (jw)2\ +T (7.15a) The log-amplitude function - 20u is p lotted as a function of u in Fig. 7.3a. This is a s traight line with a slope of - 20. I t crosses t he u-axis a t u=O. T he w-scale (u = logw) also appears in Fig. 7.3a. Semilog g raphs can be conveniently used for plotting, a nd we can directly plot w on semilog paper. A ratio of 10 is a d ecade a nd a ratio of 2 is known as a n o ctave. F urthermore, a decade along t he w-scale is equivalent t o 1 u nit along t he u-scale. We can also show t hat a r atio of 2 (an octave) along t he w-scale equals t o 0.3010 (which is loglO 2) along t he u-scale.t J 30 t., . !I ~[ [ ~-. I ~~ 1--I i II~r---.... I I ! I! - _. 20 10 1 --.. ~ 0 :t: .2' 2010gIH(jw)1 = 20 log K aIa2 + 2010g 11 + j w 1+ 2010 g 11 + jw 1 - 2010gliwl bIb3 al a2 \ jb2w 20 log 1 + b ;- (7.14) - 2010gw = - 20u 0 - = logw U Hence N jW\ - 20 log 1 + ~ \ 479 7.2 Bode P lots 7 Frequency Response a nd Analog Filters 478 I II -10 .I j!...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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