Responses

The questions below are due on Monday February 18, 2019; 11:00:00 PM.

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Consider a system described by the following difference equations:

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

z[n] = 8\cdot w[n]

y[n] = y[n-1] + z[n-1]

where x[n] and y[n] represent the n^{\text{th}} samples of the input and output signals, respectively.

1) Difference Equation

Determine a difference equation for this system in terms only of x[\cdot ] and y[\cdot ].

Hint: You may find it helpful to convert the above to operator equations (in {\cal R}) before attempting to combine them.

Recall that a difference equation is in the form:

y[n] = c_0 y[n-1] + c_1 y[n-2] + \ldots + c_{k-1} y[n-k] + d_0 x[n] + d_1 x[n-1] + \ldots + d_j x[n-j]

Specify the dCoeffs: d_0 \ldots d_j and the cCoeffs: c_0 \ldots c_{k-1} for this system. In the box below, enter a Python list containing the numbers representing the coefficients. If one set of coefficients is empty, enter the empty list, [].

Difference Equation:
dCoeffs (input):
cCoeffs (output):

2) Response 1

Assume that the input to the system is:

x[n] = \begin{cases} 1 & \text{if}~n\geq 0\\ 0 & \text{otherwise} \end{cases}

Use your difference equation from above to calculate the first 5 samples of Y under this condition (i.e., calculate y[n] for n=0,1,2,3,4). Assume the system starts "at rest" (i.e., y[n]=0 for all n\lt 0).

Enter a Python list containing 5 numbers representing [y[0], y[1], y[2], y[3], y[4]].

3) Response 2

Calculate the response if the input were instead:

x[n] = \begin{cases} 2 & \text{if}~n\geq 0\\ 0 & \text{otherwise} \end{cases}

Hint! You shouldn't need to calculate this new response step-by-step!

Enter a Python list containing 5 numbers representing [y[0], y[1], y[2], y[3], y[4]].

4) Response 3

Calculate the response if the input were instead:

x[n] = \begin{cases} 2 & \text{if}~n\geq 1\\ 0 & \text{otherwise} \end{cases}

Enter a Python list containing 5 numbers representing [y[0], y[1], y[2], y[3], y[4]].

5) Response 4

Consider the system that results from modifying the original difference equations such that the one constant term is doubled:

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

z[n] = 16\cdot w[n]

y[n] = y[n-1] + z[n-1]

Calculate this new system's response to the unit step signal:

x[n] = \begin{cases} 1 & \text{if}~n\geq 0\\ 0 & \text{otherwise} \end{cases}

Enter a Python list containing 5 numbers representing [y[0], y[1], y[2], y[3], y[4]].

Check Yourself 1:

How does this result compare to "Response 2?" Why?

Why does doubling the coefficient in the second difference equation not result in the same output as doubling the input?