Questions: FIR Filter Design and Realization

Question 1

An audio engineer needs a digital low-pass filter for music processing. She requires that the filter introduce no phase distortion — every frequency component must be delayed by the same amount. Which FIR filter property guarantees this, and why?

• A. The filter's stability (no feedback) ensures consistent phase across frequencies
• B. Symmetric impulse response coefficients h[k] = h[M−k] guarantee a constant group delay of M/2 samples at all frequencies
• C. Using the Parks-McClellan algorithm automatically produces zero phase shift at all frequencies
• D. Any FIR filter achieves linear phase because it has only finitely many coefficients

Answer: (B)

Linear phase — identical delay for all frequencies — is guaranteed by coefficient symmetry h[k] = h[M−k], which causes the filter's frequency response to factor into a real-valued magnitude term multiplied by e^{-jωM/2}, a pure linear phase term. This means every frequency is delayed by exactly M/2 samples: the output signal's shape is preserved, just shifted in time. Neither stability, nor the design algorithm, nor finiteness alone guarantees this. An asymmetric FIR filter can be designed that is stable and finite but introduces non-linear phase distortion.

Question 2

For a given stopband attenuation specification and transition bandwidth, how does the number of coefficients required by an FIR filter typically compare to an IIR filter?

• A. FIR requires fewer coefficients because it has no poles to track
• B. Both require the same number of coefficients — the design algorithm only affects coefficient values
• C. FIR requires more coefficients (higher order) to achieve the same roll-off specification
• D. It depends entirely on the window function chosen, not on the FIR/IIR distinction

Answer: (C)

IIR filters can achieve steep roll-off with far fewer coefficients than FIR filters for the same specification, because they use feedback (recursive structure) that creates poles, enabling resonance-based attenuation. FIR filters, with no poles, must approximate the desired frequency response using a finite sum of weighted delays, requiring much higher orders (more multiplications) to achieve equivalent roll-off. This computational cost is the primary reason FIR filters are not universally preferred despite their stability and linear-phase advantages.

Question 3

Symmetric FIR filter coefficients guarantee zero phase shift (no delay) at most frequencies.

• T. True
• F. False

Answer: False

Symmetry guarantees *linear* phase, not zero phase. A symmetric FIR filter of length M+1 introduces a constant group delay of M/2 samples at every frequency — every component is delayed by the same amount. This preserves signal shape but does not eliminate the delay. Zero phase would require non-causal filtering (the output depending on future inputs), which cannot be implemented in real time. The practical benefit of linear phase is shape preservation: no frequency component is delayed more or less than another, preventing phase distortion even though delay itself is present.

Question 4

An FIR filter is unconditionally stable regardless of the values of its coefficients, because it has no feedback path.

• T. True
• F. False

Answer: True

Stability of a discrete-time system depends on whether its poles lie inside the unit circle in the z-domain. An FIR filter's transfer function H(z) = Σ h[k] z^{-k} is a polynomial in z^{-1} — it has only zeros, plus trivial poles at z = 0 (inside the unit circle). Without feedback, there are no poles that can migrate outside the unit circle regardless of coefficient values. This is a structural guarantee, in contrast to IIR filters, where feedback introduces poles whose stability must be verified for each design.

Question 5

What is the Gibbs phenomenon in windowed FIR filter design, and how does the choice of window function trade off its effects?

Model answer: The Gibbs phenomenon is the persistent oscillatory overshoot (approximately 9% of the step size, or about 21 dB stopband attenuation) that appears when an infinite-length ideal filter impulse response is abruptly truncated — essentially multiplied by a rectangular window. The sharp truncation creates spectral leakage that manifests as ripples in the passband and stopband. Smoother windows (Hamming, Hann, Blackman) taper the impulse response to zero at the edges rather than cutting it abruptly, reducing ripple amplitude at the cost of a wider transition band. The tradeoff: more stopband attenuation requires a wider transition band for the same filter order.

This tradeoff is the central design decision in windowed FIR design. If the application requires steep transition from passband to stopband (narrow transition band), a rectangular window (or Parks-McClellan) is preferred, accepting some stopband ripple. If the application requires high stopband attenuation (e.g., rejecting interference near the signal band), a Blackman window is appropriate, accepting a wider transition region. Parks-McClellan avoids this tradeoff by directly optimizing the maximum ripple subject to a given filter order, typically achieving better performance than windowing for the same coefficient count.