Consider the signal given as: h(t) = e-tu(t).
Introductory Circuit Analysis (13th Edition)
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Quantization and Resolution
Quantization is a methodology of carrying out signal modulation by the process of mapping input values from an infinitely long set of continuous values to a smaller set of finite values. Quantization forms the basic algorithm for lossy compression algorithms and represents a given analog signal into digital signals. In other words, these algorithms form the base of an analog-to-digital converter. Devices that process the algorithm of quantization are known as a quantizer. These devices aid in rounding off (approximation) the errors of an input function called the quantized value.
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Signals and Systems
Need a b c
Transcribed Image Text:Q.1 Consider the signal given as:
h(t) = e-lu(t).
(1)
a) Find the time constant t of h(t) and find h(t), h(2t), h(3T), h(4r), h(5t).
b) We know that the output of a continuous-time LTI system, whose input x(t) and impulse
response h(t) are nonzero for -o < t <o, is given by the convolution integral
y(t) = x(t) + h(t) = S, x(t – t)h(t)dr = x(t)h(t – t)dr.
(2)
c) Now let x(t) = 8(t), the unit – impulse signal. Show that, using (2) and one of the
properties of the impulse signal 8(t), the output y(t) for this input is found as
y(t) = h(t).
d) Now let h(t) be given as in (1). Giving the mathematical justification, make the required
change in the integral limits of the convolution integral given in (2) according to the
impulse response signal h(t) given in (1) and rewrite (2).
e) Now let
x(t) = u(t),
(3)
the unit-step signal, which implies another change in the integral limits in (2). Giving the
mathematical justification, make the change in the integral limits of (2) according to (3).
f) Now substitute (1) and (3) into (2) and find y(t) by taking the integral given in (2) with
the modified limits and show that
У (t) %3D (1 — е-")u(t).
Hint: Use u(t- T) =1 for 0<T<t.
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