The overall impulse response of the system shown in figure is given by (Block diagram provided: Input $X(n)$ splits. One path goes to $h_1[n]$, another to $h_2[n]$. The outputs of $h_1[n]$ and $h_2[n]$ are subtracted. This result is convolved with $h_3[n]$. Separately, $X(n)$ also goes to $h_5[n]$. The output of $h_3[n]$ and $h_5[n]$ are subtracted. This result is convolved with $h_4[n]$ to produce $y(n)$.)
To solve this problem, we need to understand the system's overall impulse response based on the given block diagram description.
- Impulse Response: The impulse response of a system describes its output when an impulse function \( \delta(n) \) is applied as input. It characterizes the system's behavior.
- Convolution: Convolution of two sequences \( x(n) \) and \( h(n) \) is defined as:
\[ y(n) = (x * h)(n) = \sum_{k=-\infty}^{\infty} x(k) h(n-k) \]
- System Description: According to the problem, the system has multiple blocks, where the input \( X(n) \) is processed in different paths and then combined via subtraction or addition, followed by convolution with other impulse responses. The final output \( y(n) \) results from these operations.
The given system has the following steps in its processing:
The overall system impulse response can be described by combining all the operations. Breaking down the steps:
\( h[n] = ((h_1[n] - h_2[n]) * h_3[n] - h_5[n]) * h_4[n] \)
The correct relation is \( h[n] = ((h_1[n] - h_2[n]) * h_3[n] - h_5[n]) * h_4[n] \).