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

# 266 t hat if t he i nput is i dentical t o t he c

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: ely t o al 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:\: (2.69) 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...
View Full Document

## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online