Answer:
To find the differential equation for the given system, let's analyze the diagram in Figure 1. Based on the diagram, we can determine the system function and then use it to evaluate the output for the given input.
First, let's define some variables:
x(n) - input signal
y(n) - output signal
From Figure 1, we can see that the system has two inputs:
The input signal x(n) passes through the Z^-1 block.
The delayed version of the input signal x(n) also passes through the Z^-1 block.
Now, let's write the difference equation for the system. The difference equation relates the current output y(n) to the current and past inputs x(n) and x(n-1):
y(n) = a1 * x(n) + a2 * x(n-1)
To determine the coefficients a1 and a2, we can observe the diagram:
The input x(n) passes through a gain of 2.
The delayed input x(n-1) passes through a gain of 19.
Both signals are then summed.
Based on these observations, we have:
a1 = 2
a2 = 19
Therefore, the difference equation for the system is:
y(n) = 2 * x(n) + 19 * x(n-1)
Next, let's find the system function H(z). The system function is the z-transform of the difference equation. Taking the z-transform of the differential equation yields:
Y(z) = a1 * X(z) + a2 * X(z) * z^-1
Dividing both sides by X(z), we get:
H(z) = Y(z) / X(z) = a1 + a2 * z^-1
Substituting the values of a1 and a2, we have:
H(z) = 2 + 19 * z^-1
Now, let's evaluate the output y(n) for the given input x(n):
x(n) = (n + 1) * 0.5^u(n) + 2 * cos(2nπ)
Substituting this expression into the difference equation, we get:
y(n) = 2 * [(n + 1) * 0.5^u(n) + 2 * cos(2nπ)] + 19 * [(n - 1) * 0.5^u(n-1) + 2 * cos(2(n-1)π)]
Simplifying further, we can compute y(n) for the given input values of x(n).
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