Question

Linear phase is easily achievable in:

• A. IIR filters
• B. FIR filters
• C. Analog filters
• D. Band pass filters

Hint:

Perfect linear phase responses are easily achieved in FIR filters simply by making their filter coefficients symmetric around the center point ($h[n] = h[M-n]$).

Answer 1

The Correct Option is B

Concept: A digital filter provides a linear phase response if its phase shift varies linearly with frequency. This means the phase response satisfies the equation: \[ \angle H(e^{j\omega}) = -\alpha\omega \] This property is important because the derivative of phase with respect to frequency, known as the group delay ($\tau_g = -\frac{d\angle H(e^{j\omega})}{d\omega} = \alpha$), remains constant across all frequencies. A constant group delay ensures that all frequency components of an input signal experience the exact same time delay through the filter, preventing phase distortion. This is critical for preserving the shape of biomedical signals, such as the waves in an ECG trace.

Step 1: Reviewing the linear phase condition for FIR filters.
A Finite Impulse Response (FIR) filter is governed by a finite convolution sum: \[ y[n] = \sum_{k=0}^{M} b_k x[n-k] \] An FIR filter is guaranteed to have a linear phase response if its impulse response coefficients, $h[n]$, exhibit symmetry or anti-symmetry around their midpoint: \[ h[n] = h[M-n] \quad \text{(Symmetric)} \] \[ h[n] = -h[M-n] \quad \text{(Anti-symmetric)} \] Designing this symmetry into an FIR filter is simple: you choose matching coefficient pairs on both sides of the center tap (e.g., $b_0 = b_M$, $b_1 = b_{M-1}$, etc.).

Step 2: Comparing with alternative filter types.

• IIR and Analog Filters: Infinite Impulse Response (IIR) filters and analog filters utilize feedback loops (poles). The presence of poles makes it mathematically impossible to achieve a perfectly linear phase response across the entire passband. While their phase responses can be linearized using additional all-pass equalization stages, this adds complexity and cannot achieve the perfect linearity inherent to symmetric FIR filters. Thus, a linear phase response is easily and naturally achieved in FIR filters, matching Option (B).