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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,
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
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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