Find out the transfer function. The image depicts a signal processing block diagram featuring a feedback loop with summation, scaling, and delay elements. The layout includes the following components:1. **Inputs and Outputs**: - **x[n]** is the main input to the system. - **y[n]** is the output of the system.2. **Summation Blocks**: - There are two summation nodes, represented by circles with Σ. The top summation block takes inputs from the main input **x[n]** and the feedback loop, and it delivers the processed signal onwards. - The bottom summation block combines inputs from the output of the feedback process and the feedback loop itself to produce the final output **y[n]**.3. **Scaling Multipliers**: - Two identical triangular scaling blocks are present, each labeled with a gain of **k/2**. These scaling elements are in the feedback path and adjust the signal amplitude coming from the delay element and returning to the summation node.4. **Feedback Loop**: - The signal from the output of the top summation node enters a scaling block, then passes through a **z⁻¹** delay element. This element delays the signal by one sample period. - The delayed output, **q[n]**, is then scaled again by another **k/2** multiplier before being summed into both the top and bottom summation blocks.This configuration represents a discrete-time feedback system commonly used in digital control systems and digital filters. The delay element indicates discrete time processing, and the scaling factors manage the feedback intensity, affecting the system’s stability and response characteristics.

Answered: Find out the transfer function

Introductory Circuit Analysis (13th Edition)

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Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

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Find out the transfer function.

The image depicts a signal processing block diagram featuring a feedback loop with summation, scaling, and delay elements. The layout includes the following components:

1. **Inputs and Outputs**:
- **x[n]** is the main input to the system.
- **y[n]** is the output of the system.

2. **Summation Blocks**:
- There are two summation nodes, represented by circles with Σ. The top summation block takes inputs from the main input **x[n]** and the feedback loop, and it delivers the processed signal onwards.
- The bottom summation block combines inputs from the output of the feedback process and the feedback loop itself to produce the final output **y[n]**.

3. **Scaling Multipliers**:
- Two identical triangular scaling blocks are present, each labeled with a gain of **k/2**. These scaling elements are in the feedback path and adjust the signal amplitude coming from the delay element and returning to the summation node.

4. **Feedback Loop**:
- The signal from the output of the top summation node enters a scaling block, then passes through a **z⁻¹** delay element. This element delays the signal by one sample period.
- The delayed output, **q[n]**, is then scaled again by another **k/2** multiplier before being summed into both the top and bottom summation blocks.

This configuration represents a discrete-time feedback system commonly used in digital control systems and digital filters. The delay element indicates discrete time processing, and the scaling factors manage the feedback intensity, affecting the system’s stability and response characteristics.

Expert Solution

Step 1

For this system, the input is $x [n]$ and the output is $y [n]$ . Since $z^{- 1}$ is equivalent to unit time delay, the system equations will be

$x [n] = q [n] - \frac{k}{2} q [n - 1] - - - > (1) y [n] = q [n] + \frac{k}{2} q [n - 1] - - - > (2)$

Taking Z transform on both sides of equation (1) and (2), the following can be obtained:

$X (z) = \{1 - \frac{k}{2} z^{- 1}\} Q (z) - - - > (3) Y (z) = \{1 + \frac{k}{2} z^{- 1}\} Q (z) - - - > (4)$

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