FIR Filter Design: Window Method and Parks-McClellan

FIR Filter Design: Window Method

Principle

1. Start with ideal frequency response $H_d(e^{j\omega})$ (brick-wall)
2. Compute ideal impulse response $h_d[n]$ via IDTFT
3. Truncate to finite length $M$ using window $w[n]$
4. $h[n] = h_d[n] \cdot w[n]$

Ideal Impulse Responses

Filter Type	$h_d[n]$
Lowpass ($\omega_c$)	$\frac{\omega_c}{\pi} \text{sinc}(\omega_c n)$
Highpass ($\omega_c$)	$\delta[n] - \frac{\omega_c}{\pi} \text{sinc}(\omega_c n)$
Bandpass ($\omega_1, \omega_2$)	$\frac{\omega_2}{\pi}\text{sinc}(\omega_2 n) - \frac{\omega_1}{\pi}\text{sinc}(\omega_1 n)$
Bandstop	$\delta[n] - h_{\text{bandpass}}[n]$

Window Functions: The Trade-off Table

Window	Main Lobe Width	Peak Sidelobe	Transition Width	Stopband Atten.
Rectangular	$4\pi/M$	-13 dB	$0.9\pi/M$	21 dB
Bartlett (Triangular)	$8\pi/M$	-27 dB	$1.8\pi/M$	25 dB
Hanning	$8\pi/M$	-31 dB	$3.1\pi/M$	44 dB
Hamming	$8\pi/M$	-43 dB	$3.3\pi/M$	53 dB
Blackman	$12\pi/M$	-58 dB	$5.5\pi/M$	74 dB
Kaiser ($\beta$)	Adjustable	Adjustable	Adjustable	Adjustable

Rule of thumb: For stopband attenuation $A$ dB, choose window with sidelobes < $A$ dB. Then $M \approx \frac{\text{Transition width factor} \cdot \pi}{\Delta\omega}$.

Worked Example: Lowpass FIR via Hamming Window

Specs: $\omega_p = 0.2\pi$, $\omega_s = 0.3\pi$, $A_s > 50$ dB

1. Cutoff: $\omega_c = (\omega_p + \omega_s)/2 = 0.25\pi$
2. Transition: $\Delta\omega = \omega_s - \omega_p = 0.1\pi$
3. Window: Hamming (53 dB > 50 dB)
4. Length: $M \approx \frac{3.3\pi}{0.1\pi} = 33$ → use $M=33$ (odd for Type I linear phase)
5. Ideal response: $h_d[n] = 0.25 \text{sinc}(0.25(n-16))$ for $n=0..32$
6. Window: $w[n] = 0.54 - 0.46\cos(2\pi n / 32)$
7. Final: $h[n] = h_d[n] \cdot w[n]$

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FIR Design: Parks-McClellan (Optimal Equiripple)

The Parks-McClellan algorithm (Remez exchange) designs optimal minimax FIR filters.

Advantages over Window Method

• Directly specifies passband/stopband edges and ripples
• Optimal: Minimizes maximum error (Chebyshev criterion)
• Often requires fewer taps for same specs

Design Parameters

Passband edge:    ωp
Stopband edge:    ωs
Passband ripple:  δp (linear) or Ap (dB)
Stopband ripple:  δs (linear) or As (dB)
Weight ratio:     W = δp/δs (trade-off)

MATLAB: firpm(N, f, a, w)

• f: Frequency vector [0, ωp, ωs, 1] normalized to π
• a: Desired amplitude [1, 1, 0, 0] for lowpass
• w: Weight vector [W, 1] (higher weight = tighter constraint)

Linear Phase Types

| Type | Symmetry | Length | Zero at ω=π? | Use Case | |—|—|—|—|—| | Type I | Even ($h[n]=h[M-1-n]$) | Odd | No | General LP, HP, BP | | Type II | Even | Even | Yes | LP, BP only | | Type III | Odd ($h[n]=-h[M-1-n]$) | Odd | Yes | Differentiators, Hilbert | | Type IV | Odd | Even | No | Differentiators, Hilbert |

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