Question E7F12:
Which of the following is generally true of Finite Impulse Response (FIR) filters?  

Choices:  
A: FIR filters can delay all frequency components of the signal by the same amount  
B: FIR filters are easier to implement for a given set of passband rolloff requirements  
C: FIR filters can respond faster to impulses  
D: All these choices are correct    

Correct Answer is  A: FIR filters can delay all frequency components of the signal by the same amount  

Correct answer: A — FIR filters can delay all frequency components of the signal by the same amount.

1) Why A is correct
- Many FIR filters can be designed to have exactly linear phase. Linear phase means constant group delay, so all frequency components are delayed equally (no phase distortion).
- For an FIR of length N with symmetric or antisymmetric coefficients, $h[n] = \pm h[N-1-n]$, the phase is linear:
  $$\phi(\omega) = -\omega \frac{N-1}{2}$$
  Hence the group delay is constant:
  $$\tau_g(\omega) = -\frac{d\phi(\omega)}{d\omega} = \frac{N-1}{2} \text{ samples}$$
- This uniform delay property is a hallmark advantage of FIR filters.

2) Why the other choices are incorrect
- B: “FIR filters are easier to implement for a given set of passband rolloff requirements” — Not generally. For the same transition width and stopband attenuation, IIR filters usually achieve the specs with a much lower order (fewer multipliers), so FIR is not “easier” in that sense. FIRs are simpler structurally (no feedback) and always stable, but they often need many more taps for sharp rolloff.
- C: “FIR filters can respond faster to impulses” — Not generally. Although FIR impulse responses are finite, their length N sets both settling time and latency; sharper FIR filters require larger N, increasing delay. IIR filters can meet similar selectivity with fewer poles/zeros and often have lower latency. So FIR does not inherently “respond faster.”
- D: “All these choices are correct” — False because B and C are not generally true.

3) Useful formulas and rules of thumb
- FIR output:
  $$y[n] = \sum_{k=0}^{N-1} h[k]\,x[n-k]$$
- Linear-phase condition (FIR):
  $$h[n] = h[N-1-n] \text{ or } h[n] = -h[N-1-n] \Rightarrow \text{linear phase}$$
- Group delay (linear-phase FIR):
  $$\tau_g = \frac{N-1}{2} \text{ samples},\quad \tau = \frac{\tau_g}{f_s} \text{ seconds}$$
- Rule of thumb: FIR = exact linear phase and guaranteed stability; IIR = sharper transitions for a given order but nonlinear phase (frequency-dependent delay).