Question

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 $y(n)$.)

• A. \( h[n] = ((h_1[n]-h_2[n])*h_3[n])-h_5[n])*h_4[n] \)
• B. \( h[n] = ((h_1[n]-h_2[n])*h_3[n])-h_5[n])*h_4[n] \)
• C. \( h[n] = ((h_1[n]-h_2[n])*h_3[n])+h_5[n])*h_4[n] \)
• D. \( h[n] = (h_1[n]-h_2[n])*h_3[n]+h_5[n]]*h_4[n] \)

Hint:

When analyzing block diagrams of LTI systems:

• 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.

It's often easiest to write the system function in the Z-domain first, and then convert back to the time-domain impulse response by replacing multiplication with convolution. Pay close attention to the signs at the summing junctions.

Answer 1

The Correct Option is A

The problem requires determining the overall impulse response of a given system based on a block diagram configuration. Each signal path and convolution process needs to be analyzed to formulate the correct response equation:

1. Input signal \( X(n) \) is split into multiple paths.
2. One path goes through \( h_1[n] \), and another path goes through \( h_2[n] \). These outputs are then subtracted, yielding: \((h_1[n]-h_2[n])\).
3. The result is convolved with \( h_3[n] \), achieving: \(((h_1[n]-h_2[n])*h_3[n])\).
4. Concurrently, another direct path leads \( X(n) \) through \( h_5[n] \).
5. The output from the \( h_3[n] \) convolution and the output from \( h_5[n] \) are subtracted: \(((h_1[n]-h_2[n])*h_3[n])-h_5[n]\).
6. The result is then convolved with \( h_4[n] \), producing the final output: \((((h_1[n]-h_2[n])*h_3[n])-h_5[n])*h_4[n])\).

The correct expression for the overall impulse response \( h[n] \) of the system is:

\( h[n] = (((h_1[n]-h_2[n])*h_3[n])-h_5[n])*h_4[n] \)