Question

Which of the following defines the FIR filter for length \( M \), input \( x(n) \), and output \( y(n) \)?

• A. \( y(n) = \sum_{K=0}^{M} b_k x(n-k) \)
• B. \( y(n) = \sum_{K=0}^{M+1} b_k x(n+k) \)
• C. \( y(n) = \sum_{K=0}^{M-1} b_k x(n-k) \)
• D. \( y(n) = \sum_{K=0}^{M} b_k x(n+k) \)

Hint:

A FIR filter is implemented using convolution: \( y(n) = \sum_{k=0}^{M-1} b_k x(n-k) \). It depends only on past and current inputs.

Answer 1

The Correct Option is C

Step 1: A Finite Impulse Response (FIR) filter is defined by the convolution sum: \[ y(n) = \sum_{k=0}^{M-1} b_k x(n-k) \] where:
- \( x(n) \) is the input signal,
- \( y(n) \) is the output signal,
- \( b_k \) are the filter coefficients,
- \( M \) is the filter length. 
Step 2: FIR filters are non-recursive, meaning they only depend on current and past input values. 
Step 3: Evaluating the given options:
- (A) Incorrect: The upper limit should be \( M-1 \), not \( M \).
- (B) Incorrect: FIR filters use past values, not future values (\( x(n+k) \)).
- (C) Correct: Matches the standard FIR filter definition.
- (D) Incorrect: Uses \( x(n+k) \), which does not define a standard FIR filter.