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
From: Boost C++ Libraries (noreply_at_[hidden])
Date: 2017-12-05 18:41:47

#9672: PDF and CDF of a laplace distribution throwing domain_error
--------------------------------------+-----------------------------
  Reporter:  HS <tan@…>               |      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
--------------------------------------+-----------------------------

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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