$f1 = $f2, where $f1 is the AR part and $f2 is the MA part. In DSP this is usually written
y(n) = a1*y(n-1) + ... aN*y(n-N) + b0*x(n) + ... + bM*x(n-M).
In the vector context, x, y, z are standard variables. So I remove them, add the variable n, and define dummy functions x(n), y(n). The problem is that the equation comes outyn = and not y(n) =, The x(n) seems fine. I've tried various modifications with various errors, but nothing works. I can change y(n) to q(n) and that works, but I really need yno to agree with standard texts.
code below
DOCUMENT(); # This should be the first executable line in the problem.
loadMacros(
"PGstandard.pl", # Standard macros for PG language
"MathObjects.pl",
"AnswerFormatHelp.pl",
"parserFunction.pl",
"PGcourse.pl",
);
TEXT(&beginproblem);
$showPartialCorrectAnswers = 1;
Context("Vector");
Context()->variables->remove(x); # only variables should be n
Context()->variables->add(n=> 'Real');
# how to remove y and z from Vector context?
# Component values here
parserFunction("q(n)" => "n"); # dummy function
parserFunction("x(n)" => "n"); # dummy function
$Na1 = random(3,6,1);
do {$Nb1 = random(3,6,1);} until ($Nb1 != $Na1);
$Nmax = 6;
for ($k =1;$k <= $Nmax;$k++) {$a1[$k] =0};
for ($k =1;$k <= $Nmax;$k++) {$b1[$k] =0};
$a1[0] = non_zero_random(-9,9,1); # a0
for ($k =1;$k <= $Na1;$k++) {$a1[$k] = random(-5,5,1)};
$b1[0] = non_zero_random(-9,9,1); # b0
for ($k =1;$k <= $Nb1;$k++) {$b1[$k] = non_zero_random(-5,5,1)};
for ($k =1;$k <= $Nmax;$k++) {$a1ans[$k] =0};
for ($k =1;$k <= $Nmax;$k++) {$b1ans[$k] =0};
$a1ans[0] = 1;
for ($k =1;$k <= $Na1;$k++) {$a1ans[$k] = -$a1[$k]/$a1[0]};
for ($k =0;$k <= $Nb1;$k++) {$b1ans[$k] = $b1[$k]/$a1[0]};
$f1 = Formula("$a1[0]*y(n) + $a1[1]*y(n-1) + $a1[2]*y(n-2) + $a1[3]*y(n-3) +
$a1[4]*y(n-4) + $a1[5]*y(n-5) + $a1[6]*y(n-6) ")->reduce;
$f2 = Formula("$b1[0]*x(n) + $b1[1]*x(n-1) + $b1[2]*x(n-2) + $b1[3]*x(n-3) +
$b1[4]*x(n-4) + $b1[5]*x(n-5) + $b1[6]*x(n-6) ")->reduce;
$Na2 = 1;
$Nb2 = random(4,6,1);
for ($k =1;$k <= $Nmax;$k++) {$a2[$k] =0};
for ($k =1;$k <= $Nmax;$k++) {$b2[$k] =0};
$b2[0] = non_zero_random(-9,9,1); # b0
for ($k =1;$k <= $Nb2;$k++) {$b2[$k] = random(-4,4,1)};
$Nmax = 6;
$f3 = Formula("y(n)")->reduce;
$f4 = Formula("$b2[0]*x(n) + $b2[1]*x(n-1) + $b2[2]*x(n-2) + $b2[3]*x(n-3) +
$b2[4]*x(n-4) + $b2[5]*x(n-5) + $b2[6]*x(n-6) ")->reduce;
$a2ans[0] = 1;
for ($k =1;$k <= $Na1;$k++) {$a2ans[$k] = -$a2[$k]};
for ($k =0;$k <= $Nb1;$k++) {$b2ans[$k] = $b1[$k]};
Context()->texStrings;
TEXT(EV2(<<EOT));
\{image("direct_form_2.jpg")\} $BR
This Problem is related to Problem 2.46 in the text.
$PAR
The problem is to determine the coefficients for the Direct Form
II (figure above). For this problem, we will assume the maximum
delay is $Nmax. You will need to find all coefficients. There are
likely to be several zero coefficients, if the orders of the AR and
MA parts are not equal.
$PAR
Give the coefficients of Direct Form II for the difference equation ,
$PAR
a) \( $f1 = \) $BR
\($f2\)
$PAR
For the coefficients, enter as vectors, \( {\bf a} = <a_0, a_1, ..., a_6>\) $BR
and \( {\bf b} = <b_0, b_1, ..., b_6>\) $BR
Note that some of the coefficients may be zero$BR
\({\bf a} =\) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \} $BR
\({\bf b} = \) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \}
$PAR
b) \( $f3 = $f4\)
$PAR
For the coefficients, enter as vectors, \( {\bf a} = <a_0, a_1, ..., a_6>\) $BR
and \( {\bf b} = <b_0, b_1, ..., b_6>\) $BR
Note that some of the coefficients may be zero$BR
\({\bf a} =\) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \} $BR
\({\bf b} = \) \{ans_rule(40)\} \{ AnswerFormatHelp("vectors") \}
EOT
ANS(Vector(@a1ans)->cmp());
ANS(Vector(@b1ans)->cmp());
ANS(Vector(@a2ans)->cmp());
ANS(Vector(@b2ans)->cmp());
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$BBOLD SOLUTION $EBOLD
$PAR
tmp holder
END_SOLUTION
ENDDOCUMENT(); # This should be the last executable line in the problem.