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

L et t he impulse response ht be a rectangular pulse

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Unformatted text preview: stant and filtering. sinusoidal inputs of two different frequencies. T he sinusoid in Fig. 2.20b has a relatively high frequency, while t he frequency of t he sinusoid in Fig. 2.20c is low. Recall t hat t he convolution of f (t) a nd h(t) is equal to the area under t he p roduct f (r)h(t - T). T his area is shown shaded in Figs. 2.20b a nd 2.20c for the two cases. For t he high-frequency sinusoid, i t is clear from Fig. 2.20b t hat t he a rea under f (T)h(t - T) is very small because its positive a nd negative areas nearly cancel each other out. In this case t he o utput y(t) r emains periodic b ut has a rather small amplitude. This happens when t he p eriod of t he sinusoid is much smaller t han t he s ystem time-constant Th. I n contrast, for t he low-frequency sinusoid, t he period of t he sinusoid is larger t han Th, so t hat t he p artial cancellation of area under f (T)h(t - T) is less effective. Consequently, t he o utput y(t) is much larger, as depicted in Fig. 2.20c. Between these two possible extremes in system behavior, a transition point occurs when t he p eriod of the sinusoid is e qual t o t he s ystem time constant Th. T he frequency a t which this transition occurs is known as t he c utoff f requency Fe o f t he system. Because Th is t he p eriod of cutoff frequency Fe, 1 Th Fe = - (2.70) T he frequency Fe is also known as t he b andwidth of t he s ystem because t he s ystem transmits or passes sinusoidal components with frequencies below Fe while attenuating components with frequencies above Fe. O f course, t he t ransition in system behavior is g radual. T here is no d ramatic change in system behavior a t Fe = 11Th. Moreover, these results are based on a n idealized (rectangular pulse) impulse response; in practice these results will vary somewhat, depending on the exact shape ------~----------------------~ 158 2 Time-Domain Analysis of Continuous-Time Systems of h (t). R emember t hat t he "feel" of general system behavior is more i mportant t han e xact s ystem response for this qualitative discussion. Since t he s ystem time constant is equal t o its rise time, we have Tr 1 = -e F or Fe 1 = -r T (2.7la) Thus, a s ystem's b andwidth is inversely proportional to its rise time. Although Eq. (2.7la) w as d erived for an idealized (rectangular) impulse response, its implications are valid for lowpass LTIC systems in general. For a general c ase, we can show t hat l k Fe = Tr (2.7lb) where t he e xact value of k depends on t he n ature of h(t). An experienced engineer often can e stimate quickly t he b andwidth of a n unknown system by simply observing t he s ystem response to a step i nput on a n oscilloscope. 2 .7-5 Time Constant and Pulse Dispersion (Spreading) 2.7-7 The Resonance Phenomenon Finally, we come to t he fascinating phenomenon of resonance. As we have mentioned already several times, this phenomenon is observed when t he i nput signal is identical or is very similar t o a c haracteristic mode of t he system. For t he sake of simplicity a...
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