Which of the options is an equivalent representation of the signal flow graph shown here?

Correct Answer:

.

Step-by-Step Explanation:

To find the equivalent representation of the given signal flow graph (SFG), let us analyze the node relations and reduce the graph step-by-step.

1. Analyzing the Original Graph:

Let the nodes in the main graph be labeled sequentially from left to right as $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$.

From the given image, the branch gains between the nodes are defined as follows:

- Branch from $x_1$ to $x_2$: gain = 1
- Branch from $x_2$ to $x_3$: gain = a
- Branch from $x_3$ to $x_4$: gain = d
- Feedback branch from $x_4$ to $x_3$: gain = c
- Feedback branch from $x_4$ to $x_2$: gain = e
- Branch from $x_4$ to $x_5$: gain = 1

2. Eliminating the Middle Node ($x_3$):

The signal at node $x_3$ receives inputs from $x_2$ (via gain a) and from $x_4$ (via feedback gain c):

$x_3=a{x}_2+c{x}_4$

The signal at node $x_4$ is driven by node $x_3$ via gain d:

$x_4=d{x}_3$

Substituting the expression for $x_3$ into $x_4$:

$x_4=d(a{x}_2+c{x}_4)=ad{x}_2+cd{x}_4$

3. Solving for $x_4$ in terms of $x_2$:

Rearranging the terms containing $x_4$:

$x_4-cd{x}_4=ad{x}_2$

$x_4(1-cd)=ad{x}_2$

$x_4=\left(\frac{ad}{1-cd}\right)x_2=a\left(\frac{d}{1-c\mathrm{d<!--\; mrow-->}}\right)x_2$

4. Conclusion:

The forward branch gain connecting node $x_2$ directly to node $x_4$ becomes:

$G=a\left(\frac{d}{1-cd}\right)$

While the feedback loop of gain e from node $x_4$ back to node $x_2$ remains unchanged.

Thus, the reduced signal flow graph matches the image in the first option.