System Functionals

In this exercise, we will examine the system functionals for the various primitive systems and combinations that we have seen in 6.01.

For each of the systems below, solve for its system functional. Specify your answer by entering the coefficient lists associated with the numerator and denominator polynomials (in ${\cal R}$) of the system functional, with the zeroth-order coefficient first.

1) Primitives

1.1) Gain

What are the system functionals for each of the systems represented below?

$y[n] = 8x[n]$

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

Gain(9)

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

1.2) R

$y[n] = x[n-1]$

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

$y[n+1] = x[n]$

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

R(7)

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

R(4)

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):

2) Combinations

3) FeedforwardAdd

If ${\cal H}_1 = \frac{n_1}{d_1}$ and ${\cal H}_2 = \frac{n_2}{d_2}$ are both system functionals (where $n_1$, $n_2$, $d_1$, and $d_2$ represent polynomials in ${\cal R}$), what is the system functional of the combination shown below? Enter your answer as a Python expression involving the variables n_1, n_2, d_1, and/or d_2.

4) Cascade

If ${\cal H}_1 = \frac{n_1}{d_1}$ and ${\cal H}_2 = \frac{n_2}{d_2}$ are both system functionals (where $n_1$, $n_2$, $d_1$, and $d_2$ represent polynomials in ${\cal R}$), what is the system functional of the combination shown below? Enter your answer as a Python expression involving the variables n_1, n_2, d_1, and/or d_2.

5) FeedbackAdd

If ${\cal H}_1 = \frac{n_1}{d_1}$ and ${\cal H}_2 = \frac{n_2}{d_2}$ are both system functionals (where $n_1$, $n_2$, $d_1$, and $d_2$ represent polynomials in ${\cal R}$), what is the system functional of the combination shown below? Enter your answer as a Python expression involving the variables n_1, n_2, d_1, and/or d_2.

6) Big System

What is the system functional of the following system?

Hint: Look for some of the patterns from above in the block diagram.

System Functional:
Numerator coeffs (in ${\cal R}$):
Denominator coeffs (in ${\cal R}$):