Question

The output of two digital filters can be added or the same effect can be achieved by

• A. adding their coefficients
• B. subtracting their coefficients
• C. Convolving their coefficients
• D. averaging their coefficients and then using Blackman window

Hint:

- Parallel connection of filters $\rightarrow$ Add impulse responses (coefficients).
- Cascade (series) connection of filters $\rightarrow$ Convolve impulse responses (multiply transfer functions).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is a conceptual question in digital signal processing (DSP) about combining linear, time-invariant (LTI) digital filters.

Step 2: Key Formula or Approach:
For two LTI digital filters connected in parallel:
- Input to both filters: $x[n]$
- Individual impulse responses: $h_1[n]$ and $h_2[n]$
- Individual outputs: $y_1[n] = x[n] * h_1[n]$ and $y_2[n] = x[n] * h_2[n]$
The overall output is the sum: $y[n] = y_1[n] + y_2[n]$.

Step 3: Detailed Explanation:

• The output is:
\[ y[n] = (x[n] * h_1[n]) + (x[n] * h_2[n]) \]

• Since convolution is distributive over addition:
\[ y[n] = x[n] * (h_1[n] + h_2[n]) \]

• This shows that the parallel combination is equivalent to a single LTI filter with impulse response:
\[ h[n] = h_1[n] + h_2[n] \]

• Since the coefficients of a digital filter (like an FIR filter) are the direct values of its impulse response, adding the outputs of the two filters is equivalent to adding their corresponding filter coefficients.

Step 4: Final Answer:
The same effect can be achieved by adding their coefficients, which corresponds to Option (A).