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Unformatted text preview: he Bilinear Transformation Method e ntire frequency band (0 t o 00) in IHa(jw)1 is compressed within t he r ange (0, if)
in H[e JwT ]. Such warping of t he frequency scale is p eculiar t o t his transformation.
T o u nderstand this behavior, consider Eq. (12.55a) with s = j w / 1< / T
H[e Jw T
. = Ha (2
T 1) = (2
eiwT + 1 Ha 745 e ¥ - e=ifE.)
e ¥ + eqL = Ha ( jjtan w i) (a) Therefore, response of t he resulting digital filter a t some frequency Wd is o H [e iWdT ] = H a ( j j t an ~) = H a(jw a ) (12.60) w here 0, (12.61a)
( b) T hus, in t he resulting digital filter, t he behavior of t he desired response H a ( jw) a t
s ome frequency Wa a ppears n ot a t Wa b ut a t frequency Wd, where [from Eq. (12.61a)] O2
03 Wd = 2 T tan -1 Wa T - 2- (12.61b) Figure 12.13a shows t he p lot of Wd as a function o f Wa. For small W a , t he curve
in Fig. 12.13a is practically linear, so Wd ~ Wa. A t higher values of Wa , t here is
considerable diversion in t he values of Wa a nd Wd. T hus, t he digital filter imitates
t he desired analog filter a t low frequencies, b ut a t higher frequencies there is considerable distortion. Using this method, if we are trying t o synthesize a filter t o realize
H a (jw) d epicted in Fig. 12.13b, t he resulting digital filter frequency response will
b e, as illustrated in Fig. 12.13c. T he analog filter behavior in t he e ntire range of
Wa from 0 t o 00 is compressed in t he digital filter in t he r ange of Wd from 0 t o t r/T.
T his is as if a promising 20 y ear old man, who, after learning t hat he has only a
year to live, tries to crowd his last year with every possible adventure, passion, and
sensation t hat a normal human being would have experienced in a n e ntire lifetime.
T his compression a nd frequency warping effect is t he p eculiarity of t he bilinear
There are two ways of overcoming frequency warping. T he first is t o reduce
T (increase t he sampling rate) so t hat t he signal b andwidth is kept well below if
a nd Wa ~ Wd over t he desired frequency band. This s tep is easy t o execute, b ut
i t requires a higher sampling r ate (lower T ) t han necessary. T he second approach,
known as p rewarping, solves t he problem without unduly reducing T . 1 2.6-1 Bilinear Transformation Method with Prewarping I n prewarping, we s tart n ot with t he desired H a(jw) b ut w ith a prewarped
H a (jw) in such a way t hat t he warping because o f bilinear transformation will
compensate for the prewarping exactly. T he idea here is t o begin with a distorted
analog filter (prewarping) so t hat t he d istortion caused by bilinear transformation
will be canceled by the built-in (prewarping) distortion. T he idea is similar t o oj~1 !COa2 C O_ . Ma3 [ jOlT] ~e 0, (c) O2
COd' COd2 COd3 1< T C O_ F ig. 12.13 Frequency warping in bilinear transformation: (a) mapping relationship of
analog and digital frequencies (b) analog response (c) corresponding digital response.
t he one used in prestressed concrete, in which a concrete beam is precompressed
initially. W hen loaded, t he b eam experiences tension, which is canceled b y t he
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