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systems. 156 2 Time-Domain Analysis of Continuous-Time Systems
/ (t) h ( t) 2.7 Intuitive Insights into System Behavior 157 h ( t) y ( t) *
o t- (a) o t- o
h ( t-t) Fig. 2 .19 Rise time of a system. Thus, t he t ime c onstant in this case is simply (the negative of the) reciprocal of t he
s ystem's characteristic root. For the multimode case, h(t) is a weighted sum of t he
s ystem's characteristic modes, a nd Th is a weighted average of t he t ime constants
associated with t he n modes of t he system. 2 .7-3 y ( t) ( b) Time Constant and Rise Time o f a System T he s ystem t ime constant may also be viewed from a different perspective.
T he u nit s tep r esponse y(t) of a system is t he convolution of u (t) w ith h (t). L et
t he impulse response h(t) be a rectangular pulse of width T", as shown in Fig.
2.19. This a ssumption simplifies t he discussion, yet gives satisfactory results for
qualitative discussion. T he r esult of this convolution is i llustrated in Fig. 2.19.
Note t hat t he o utput does n ot rise from zero t o a final value instantaneously as t he
i nput rises; instead, t he o utput takes Th seconds t o accomplish this. Hence, t he rise
time Tr of t he s ystem is equal t o t he s ystem time constant:\:
This result a nd F ig. 2.19 show clearly t hat a s ystem generally does n ot r espond to
a n i nput i nstantaneously. Instead, it takes time Th for the system t o r espond fully. 2 .7-4 t- t- Time Constant and Filtering A l arger t ime c onstant implies a sluggish system because the system takes a
longer time to r espond fully t o a n i nput. Such a s ystem cannot respond effectively
to rapid variations in t he i nput. In contrast, a smaller time constant indicates t hat
a s ystem is c apable of responding to rapid variations in the input. Thus, there is a
d irect connection between a system's time constant a nd its filtering properties.
Consider a high-frequency sinusoid t hat varies rapidly with time. A system
with a large t ime c onstant will not be able to respond well t o this input. Therefore, such a s ystem will suppress rapidly varying (high-frequency) sinusoids a nd
o ther high-frequency signals, thereby acting as a lowpass filter ( a filter allowing t he
t ransmission o f low-frequency signals only). We shall now show t hat a s ystem with
a time constant T h a cts as a lowpass filter having a cutoff frequency of Fe = 11 T h
Hz, so t hat s inusoids with frequencies below Fe Hz are t ransmitted r easonably well,
while those with frequencies above Fe Hz are suppressed.
To d emonstrate t his fact, let us determine t he system response t o a sinusoidal
i nput f (t) by convolving this input with t he effective impulse response h (t) in Fig.
2.20a. F igures 2 .20b a nd 2.20c show t he process of convolution of h (t) w ith t he
:j: Because of varying definitions of rise time, the reader may find different results in the literature.
The qualitative and intuitive nature of this discussion should always be kept in mind. ( c) F ig. 2 .20 Time con...
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