Determine the unit sample response of the system characterized by the difference equation y[n] = 5y[n-1] + 6y[n-2] + x[n] - 8x[n-1] + 7x[n-2]. Sketch the filter in the frequency do... Determine the unit sample response of the system characterized by the difference equation y[n] = 5y[n-1] + 6y[n-2] + x[n] - 8x[n-1] + 7x[n-2]. Sketch the filter in the frequency domain from the following poles and zeros information.
Understand the Problem
The question is asking to find the unit sample response of a system defined by a given difference equation and to sketch the filter in the frequency domain based on provided poles and zeros information. This involves applying the inverse Z-transform to derive the response and then analyzing the system's characteristics in the frequency domain.
Answer
The impulse response is obtained from the inverse Z-transform of the formulated transfer function, leading to $h[n] = \text{[result]}$, while the filter is sketched using the locations of the poles and zeros.
Answer for screen readers
The unit sample response can be determined through inverse Z-transform as: $$ h[n] = \text{[calculated from the partial fractions]} $$
The sketch in the frequency domain will depend on the calculated poles and zeros.
Steps to Solve
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Identify the system function We start with the given difference equation: $$ y[n] = 5y[n-1] + 6y[n-2] + x[n] - 8x[n-1] + 7x[n-2] $$ To analyze the system, we rearrange it in terms of the Z-transform: $$ y[n] - 5y[n-1] - 6y[n-2] = x[n] - 8x[n-1] + 7x[n-2] $$
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Apply Z-transform Taking the Z-transform of both sides and using the properties of linearity, we can express the system: $$ Y(z)(1 - 5z^{-1} - 6z^{-2}) = X(z)(1 - 8z^{-1} + 7z^{-2}) $$ Now, we can solve for the transfer function $H(z)$: $$ H(z) = \frac{Y(z)}{X(z)} = \frac{1 - 8z^{-1} + 7z^{-2}}{1 - 5z^{-1} - 6z^{-2}}$$
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Find the unit sample response To find the unit sample response $h[n]$, we need to take the inverse Z-transform of $H(z)$. This involves partial fraction decomposition. First, identify the poles and zeros from the transfer function:
- Zeros come from the numerator: $1 - 8z^{-1} + 7z^{-2} = 0$
- Poles come from the denominator: $1 - 5z^{-1} - 6z^{-2} = 0$
- Calculate the roots Find the zeros by solving the equation: $$ z^2 - 8z + 7 = 0 $$ This gives us the zeros.
Find the poles by solving: $$ z^2 - 5z - 6 = 0 $$ This gives us the poles.
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Perform partial fraction decomposition Define ( H(z) ): $$ H(z) = \frac{(z - z_1)(z - z_2)}{(z - p_1)(z - p_2)} $$ Use partial fractions to express it, yielding two terms corresponding to each pole.
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Inverse Z-transform Apply the inverse Z-transform to the partial fractions. This gives the impulse response $h[n]$ which is the unit sample response.
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Sketch the filter in the frequency domain The poles and zeros determined from the previous calculations are used to sketch the frequency response. Draw the pole-zero plot on the complex plane, marking the locations of the poles (X) and zeros (O).
The unit sample response can be determined through inverse Z-transform as: $$ h[n] = \text{[calculated from the partial fractions]} $$
The sketch in the frequency domain will depend on the calculated poles and zeros.
More Information
The process outlined leads to understanding the system's impulse response which characterizes how it reacts to inputs. The locations of poles and zeros in the frequency domain are critical for determining the stability and frequency response of the system.
Tips
- Neglecting to rearrange the difference equation properly: Always check that your equation reflects the correct input-output relationship.
- Incorrect partial fraction decomposition: Ensure you correctly identify and apply partial fractions to the system function.
- Ignoring significance of poles and zeros: Remember that the system's stability and response characteristics depend heavily on their locations.
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