Computing Poles

The questions below are due on Sunday February 24, 2019; 11:00:00 PM.

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This problem will talk through the process of finding the poles of a system, through two examples.

1) System Functionals and Poles

System functionals are useful because they concisely represent everything that we need to know about a system in order to determine its behavior. System functionals are also convenient to use because they are modular in the sense that one can calculate the system function of a complicated system from the system functionals of its constituent subsystems. This modularity results from a fundamental property of systems functionals: if a system is composed of adders, gains, and delays, then the system functional can always be expressed as the ratio of two polynomials of $\mathcal{R}$, where $\mathcal{R}$ represents the delay operator.

As discussed in lecture, the asymptotic (long-term) behavior of a system is determined by the system's poles, which can be obtained as follows.

Determine the system functional, which is a ratio of polynomials in \mathcal{R}, where \mathcal{R} represents the delay (right-shift) operator.
Convert the system functional to a system function, which is a ratio of polynomials in z, by substituting 1\over z for \mathcal{R}.
Find the roots of the denominator polynomial of z.

2) Example

Consider the system specified by:

\frac{Y}{X} = \frac{1}{1-{\cal R} - {\cal R}^2}

We would like to factor that denominator into the form (1-p_0{\cal R})(1-p_1{\cal R})\ldots, where the p_0, p_1, etc., are the poles of the system. However, factoring polynomials into that form is difficult! What we care about in the end is finding the poles p_0, p_1, etc. So we can construct a new polynomial that has those numbers as its roots, and use our existing knowledge to find those roots (and, thus, the poles!).

To find the poles of this system, we first substitute in {\cal R} = \frac{1}{z}:

\frac{1}{1 - \frac{1}{z} - \frac{1}{z^2}}

We then convert to a rational polynomial in z:

\frac{z^2}{z^2 - z - 1}

Now, the roots of the denominator are our poles. In this case, we can use the quadratic formula to find them:

Poles: \frac{1 \pm \sqrt{5}}{2}

3) Part 1

Consider the system described by the following block diagram:

What is this system's 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.

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

Now do the conversion using {\cal R} = z^{-1}, and simplify to a rational polynomial in z. Specify your answer by entering the coefficient lists associated with the numerator and denominator polynomials (in z), with the zeroth-order coefficient first.

Numerator coeffs (in z):
Denominator coeffs (in z):

The poles of the system are the roots of the denominator polynomial in z after this conversion. What are the poles of this system? Enter your answer as a Python list of numbers. Note that you can use Python expressions to represent fractions if need be.

Poles:

4) Part 2

Consider the system described by the following system functional:

\frac{Y}{X} = \frac{{\cal R}^2}{1-0.3{\cal R}}

Convert this system functional to a rational polynomial in z. Specify your answer by entering the coefficient lists associated with the numerator and denominator polynomials (in z), with the zeroth-order coefficient first.

Numerator coeffs (in z):
Denominator coeffs (in z):

What are the poles of this system? As above, enter your answer as a Python list.

Poles:

5) Part 3

Which of the following determines the number of poles a system will have?