The Four Fundamental Filter Shapes
Every filter you will ever design is a variation on one of four basic frequency-selection patterns. Each pattern defines which frequencies are passed (the passband) and which are blocked (the stopband). Understanding these four types — and the specifications that quantify their behaviour — is the foundation of all practical filter design.
Filter Specifications
Real filters cannot achieve a perfect brick-wall response. Instead, we define a set of tolerances that a practical filter must meet. These specifications map directly onto the frequency response and drive every design decision.
For a low-pass filter, the key parameters are:
| Parameter | Symbol | Meaning |
|---|---|---|
| Passband edge | ω_p | Highest frequency that must be passed with at most R_p dB of attenuation. |
| Stopband edge | ω_s | Lowest frequency that must be attenuated by at least A_s dB. |
| Passband ripple | R_p (dB) | Maximum allowed variation in gain within the passband (e.g., ±0.1 dB). |
| Stopband attenuation | A_s (dB) | Minimum rejection in the stopband (e.g., 60 dB means gain < 0.001). |
| Transition band | Δω = ω_s − ω_p | The gap between passband and stopband. Narrower Δω → higher filter order. |
Low-Pass Filter
The low-pass filter is the reference design from which all other types are derived. Its ideal frequency response is a rectangle: unity gain from ω = 0 to ω = ω_c, and zero gain above. In a windowed-sinc design, ω_c is set at the midpoint of the transition band: ω_c = (ω_p + ω_s) / 2.
Low-pass filters appear in virtually every DSP chain. Before ADC sampling, an analog low-pass anti-aliasing filter removes frequencies above f_s/2. After DAC reconstruction, a low-pass smoothing filter removes spectral images. In audio, low-pass shelving filters cut the high end. In wireless receivers, low-pass filters limit the baseband signal bandwidth to match the channel.
High-Pass Filter
A high-pass filter passes high frequencies and blocks low ones. The simplest FIR high-pass design uses spectral inversion: subtract a low-pass filter's impulse response from a unit impulse (a Dirac at n = 0). If h_LP[n] is a symmetric low-pass filter, then:
This works because the spectrum of a unit impulse is 1 for all frequencies, so H_HP(e^jω) = 1 − H_LP(e^jω). Wherever the LP has gain 1, the HP has gain 0, and vice versa. The only constraint is that the filter length must be odd to preserve symmetry after the subtraction.
Band-Pass Filter
A band-pass filter passes a contiguous range of frequencies, defined by a lower passband edge ω_L and an upper passband edge ω_H. There are two standard FIR construction routes:
The modulation method is especially elegant: by multiplying the low-pass prototype's coefficients by a cosine at the centre frequency, the LP passband is copied to both ±ω_0 in the spectrum, giving a band-pass response centred at ω_0. This is the principle behind the analysis filters in digital radio receivers.
Band-Stop (Notch) Filter
A band-stop filter — also called a notch filter when the rejected band is narrow — is the complement of a band-pass filter. It passes everything except a specific frequency band. Construction follows the same principle as the high-pass filter: a band-stop can be obtained by subtracting a band-pass filter from a unit-impulse (all-pass).
The classic application is power-line interference rejection: a 50 Hz or 60 Hz notch filter removes mains hum from biomedical recordings without affecting the signal on either side. Narrow notches require high filter orders — or IIR designs — to achieve deep attenuation across a very small bandwidth.
All four filter types are related by simple arithmetic operations on the low-pass prototype. LP is the base case; HP = impulse − LP; BP = LP_wide − LP_narrow; BS = impulse − BP. Designing one LP filter well gives you the other three for free, at the same filter order and with the same linear-phase guarantee.
LP → HP → BP → BS: one prototype, four filter types.Specifications for Each Type
While the four filter types share the same specification vocabulary (passband, stopband, ripple, attenuation), each type has slightly different parametrisation:
| Filter Type | Passband | Stopband | Key Parameter |
|---|---|---|---|
| Low-Pass | 0 … ω_p | ω_s … π | Cutoff ω_c, transition Δω |
| High-Pass | ω_p … π | 0 … ω_s | Cutoff ω_c, transition Δω |
| Band-Pass | ω_L … ω_H | 0…ω_s1, ω_s2…π | Centre ω_0, bandwidth BW |
| Band-Stop | 0…ω_p1, ω_p2…π | ω_s1 … ω_s2 | Notch centre, notch width |
Converting Between Types via Spectral Transformation
Beyond arithmetic combination of LP filters, there is a formal theory of spectral transformations that maps a prototype LP filter to any other type by substituting a new function of z for z^(−1) in the transfer function. For IIR filters especially, this approach allows a well-understood LP design (e.g., Butterworth) to be transformed into a HP, BP, or BS design analytically, preserving the filter's magnitude response shape while shifting and scaling the frequency axis.
For FIR filters, the arithmetic combination approach (difference of LP filters, modulation) is typically preferred because it keeps the design simple and preserves the linear-phase property. For IIR filters, frequency transformation in the z-domain is the standard route.
Practical Applications
- The four fundamental filter types are: low-pass, high-pass, band-pass, and band-stop (notch).
- Filter specifications use passband edge ω_p, stopband edge ω_s, passband ripple R_p (dB), and stopband attenuation A_s (dB).
- The transition band Δω = ω_s − ω_p drives filter order: narrower transition → longer filter.
- High-pass FIR filters are obtained by spectral inversion: h_HP[n] = δ[n − (N−1)/2] − h_LP[n].
- Band-pass filters are designed by subtracting two LP filters (LP_wide − LP_narrow) or by modulating a LP prototype to the centre frequency.
- Band-stop filters are obtained by subtracting a band-pass filter from an all-pass (unit impulse).
- All four types inherit the linear-phase property from the LP prototype when built using FIR arithmetic combination.
- For IIR filters, formal z-domain spectral transformation maps LP prototypes (Butterworth, Chebyshev) to HP, BP, or BS designs.