# Re: [Boost-bugs] [Boost C++ Libraries] #9672: PDF and CDF of a laplace distribution throwing domain_error

Subject: Re: [Boost-bugs] [Boost C++ Libraries] #9672: PDF and CDF of a laplace distribution throwing domain_error
Date: 2017-12-05 18:41:47

#9672: PDF and CDF of a laplace distribution throwing domain_error
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Reporter: HS <tan@â€¦> | Owner: Paul A. Bristow
Type: Feature Requests | Status: closed
Milestone: Boost 1.56.0 | Component: math
Version: Boost Development Trunk | Severity: Showstopper
Resolution: fixed | Keywords: laplace
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Comment (by merciu):

SIGNAL THEORY & APPLICATIONS

Examiners: Dr. S.S. Singh/Dr. D.H. Lawrence

Attempt FOUR questions only Time allowed: 3 hours
Total number of questions = 6

Not more than 2 questions from either section A or B

All questions carry equal marks.
The figures in brackets indicate the relative weightings of parts of a
question.

Special requirements: Graph paper.
Mathematical Formulae (B& C)

1) The circuit in Figure Q1 is made up of a low-pass filter connected
via an operational amplifier to a high-pass filter. The amplifier is
configured as a unity gain buffer.

a) Comment on the role of the buffer and explain what the
loading effect of the high-pass filter network will be on the low-pass
filter network. Give reasons for your explanation, which includes an
understanding of the current drawn by the operational amplifier inputs
from the low-pass filter network and the current drawn by the high-pass
filter from the operational amplifier output. (3)

b) Derive from first principles and using Laplace transforms
an expression for the Laplace transform Vc1(s) developed across the
capacitor C1 in terms of the Laplace transform Vin(s) of the time varying
input voltage Vin. Ensure that the expression includes component symbols
C1 and R1, and that the Laplace transform VR1(s) of the resistor voltage
VR1 has been eliminated from the expression. (10)

c) Derive an expression for the Laplace transform VR2(s)
developed across the resistor R2 in terms of the Laplace transform Vin(s)
of the time varying input voltage Vin. Assume that the initial voltage of
both capacitors is zero for the purpose of your derivation. (12)

QUESTION 1 CONTINUED ON PAGE 3/10

Note: Laplace Transform of dx/dt is -x(t=0) +
sX(s)

FIGURE Q1

2) In the circuit of Figure Q2 the switch S is used to divert the
current from the resistor r to the resistor R and capacitor C which are
themselves connected in series. When, the switch is operated in this
manner a constant current source IS supplies current I to the RC network.

a) Write down an expression for the relationship between the
constant current I from the current source and the current IR flowing in
the resistor R after the switch is operated as described above. (1)

b) Write down an expression for the relationship between the
currents IR and IC shown flowing in the resistor R and capacitor C
respectively. (1)

c) Write down an expression that relates the current flow IC
in the capacitor to the rate of change of the output voltage. (2)

d) Derive an expression for the relationship between the
output voltage and the constant current input for the circuit in Figure
Q2. Assume that at the instant the switch is operated to divert the
current in the RC network, the voltage at the output is not zero. (10)

e) If a constant current load was connected across the output
terminals how would this affect the output voltage, assuming that the
constant load current was half that of the input current IS? Ensure that
your explanation is supported by mathematical justification. Sketch a
graph of the output voltage against time with no load current and also the
output voltage against time with a constant current load. The value of
the constant current load must be half that of the input current.
Indicate any relationship that exists between both graphs. (11)

QUESTION 2 CONTINUED ON PAGE 5/10

Note: Laplace Transform of dx/dt is -x(t=0) + sX(s)
Inverse Laplace transform of 1/s2 is t
Inverse Laplace transform of 1/s is 1

3) Examine the large-signal bipolar triangular voltage source and
full-wave-rectifier circuit in Figure Q3. The c

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Ticket URL: <https://svn.boost.org/trac10/boost/ticket/9672#comment:10>
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