Signals and Systems - Electrical Engineering

Problems 163

2.17. Response to unbounded inputs versus BIBO stability
The BIBO stability assumes that the input is always bounded, limited in amplitude. If that is not the case,
even a stable system would provide an unbounded output. Consider the analog averager, with an input–
output relationship of

y(t)=

1

T

∫t

t−T

x(τ)dτ

(a) Suppose that the input to the averager is a bounded signalx(t)(i.e., there is a finite valueMsuch that

|x(t)|<M). Find the value for the bound of the outputy(t)and determine whether the averager is BIBO

stable or not.

(b)Let the input to the averager bex(t)=tu(t)(i.e., a ramp signal). Compute the outputy(t)and determine

if it is bounded or not. Ify(t)is not bounded, does that mean that the averager is an unstable system?

Explain.

2.18. Sampler and hold circuit
In an analog-to-digital converter (ADC), the analog signal is first sampled and then each of its samples is
converted into a digital value. Since each of the samples is obtained momentarily, there is the need for a
circuit that holds the value long enough for the ADC to convert it into a binary number. The circuit having
the sampler and the hold circuit is called thesampler and holdcircuit, an example of which is shown in
Figure 2.24. The input is the sampled signalxs(t), which we are considering a train of rectangular pulses of
duration 1 and periodicityTsand different magnitudes corresponding tox(nTs).
The valuerC<< 1, where 1 is the duration of the pulse andRC>>TswhereTs>> 1, is the sampling
period. The first condition allows the capacitor to be charged fast in 1 seconds, and the second condition
allows slow discharge inTsseconds.
(a) Consider the first sample conversion. Let the input to the hold circuit be a pulse of duration 1 and
amplitudex( 0 ). The switch has been opened beforet= 0 so that the capacitor is discharged. The
switch closes att= 0 and remains closed untilt= 1 and then it opens. Carefully draw the voltage in
the capacitor fromt= 0 toTs.
(b)Since the RC circuit is a linear time-invariant system, the output corresponding to the other samples
can be found from the result of the first sample. Suppose the analog signal is a rampx(t)=tu(t),
sampled withTs= 1 and 1 =0.1. Plot the voltage in the capacitor fromt= 0 tot=4 sec.

2.19. AM envelope detector—MATLAB
Consider an envelope detector that is used to detect the message sent in the AM system shown in the
examples. The envelope detector as a system is composed of two cascaded systems: one that computes
the absolute value of the input (implemented with ideal diodes), and a second that low-pass filters its input
(implemented with an RC circuit). The following is an implementation of these operations in the discrete
time so we can use numeric MATLAB.

FIGURE 2.24

Problem 2.18: sample and hold circuit.

C R

+

−

+

−

xs(t)

r

x∗(t)