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 $y(n)$.)
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 $y(n)$.)
Show Hint
- Components in series} correspond to convolution} in the time domain or multiplication} in the Z/frequency domain.
- Components in parallel} with a summer (adder/subtractor) correspond to addition/subtraction} in both time and Z/frequency domains.
- \( h[n] = ((h_1[n]-h_2[n])*h_3[n])-h_5[n])*h_4[n] \)
- \( h[n] = ((h_1[n]-h_2[n])*h_3[n])-h_5[n])*h_4[n] \)
- \( h[n] = ((h_1[n]-h_2[n])*h_3[n])+h_5[n])*h_4[n] \)
- \( h[n] = (h_1[n]-h_2[n])*h_3[n]+h_5[n]]*h_4[n] \)
The Correct Option is A
Solution and Explanation
To solve this problem, we need to understand the system's overall impulse response based on the given block diagram description.
1. Understanding the Concepts:
- 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.
2. Given Values:
The given system has the following steps in its processing:
- Input \( X(n) \) splits into two paths.
- The first path goes through \( h_1[n] \), and the second path goes through \( h_2[n] \), where their outputs are subtracted.
- The result of the subtraction is convolved with \( h_3[n] \).
- Simultaneously, \( X(n) \) is passed through \( h_5[n] \), and the output of \( h_3[n] \) and \( h_5[n] \) are subtracted.
- This result is then convolved with \( h_4[n] \) to produce the final output \( y(n) \).
3. Finding the Overall Impulse Response:
The overall system impulse response can be described by combining all the operations. Breaking down the steps:
- The first operation involves subtracting the outputs of \( h_1[n] \) and \( h_2[n] \), followed by convolution with \( h_3[n] \):
- Then, \( h_5[n] \) is subtracted from the result of the previous operation:
- Finally, the result is convolved with \( h_4[n] \) to produce the overall impulse response \( h[n] \):
\( h[n] = ((h_1[n] - h_2[n]) * h_3[n] - h_5[n]) * h_4[n] \)
Final Answer:
The correct relation is \( h[n] = ((h_1[n] - h_2[n]) * h_3[n] - h_5[n]) * h_4[n] \).
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