DSP 101
M7 · L3
Module 7 — FIR Filter Design
Filter Types

The four fundamental frequency-selection patterns — low-pass, high-pass, band-pass, and band-stop — their specifications, and how all four are derived from a single low-pass prototype using spectral transformation.

1 / 9
DSP 101
M7 · L3
Four Fundamental Types
Choose Your Passband
LP
Low-Pass
HP
High-Pass
BP
Band-Pass
BS
Band-Stop

Each type selects which frequencies pass and which are rejected. The low-pass filter is the reference — all others are derived from it.

2 / 9
DSP 101
M7 · L3
Filter Specifications
Passband, Stopband, Ripple

Real filters cannot achieve a perfect brick-wall response. Specifications define the tolerances:

  • Passband edge ω_p — highest frequency that must pass with ≤ R_p dB attenuation
  • Stopband edge ω_s — lowest frequency that must be rejected by ≥ A_s dB
  • Transition band Δω = ω_s − ω_p — narrower Δω means higher filter order
  • Passband ripple R_p — variation allowed within the passband (e.g., ±0.1 dB)
  • Stopband attenuation A_s — minimum rejection (e.g., 60 dB → gain < 0.001)
3 / 9
DSP 101
M7 · L3
LP and HP
Invert the Spectrum

A high-pass FIR filter is obtained by spectral inversion of a low-pass design:

HP via Spectral Inversion
h_{\text{HP}}[n] = \delta\!\left[n - \tfrac{N-1}{2}\right] - h_{\text{LP}}[n]
Key Rule
Subtract the LP impulse response from a unit impulse at its centre. Passband and stopband swap. Linear phase is preserved.
4 / 9
DSP 101
M7 · L3
Band-Pass Filter
Two Routes to a Band
  • Difference method: design LP_wide (cutoff ω_H) and LP_narrow (cutoff ω_L), subtract — only the band between ω_L and ω_H remains
  • Modulation method: design LP prototype with cutoff BW/2, multiply coefficients by 2cos(ω_0 n) — shifts the LP band to centre frequency ω_0
  • Both methods inherit linear phase from the LP prototype
  • Both filters must have the same length for coherent combination
Radio Receivers
The modulation method is the principle behind digital channel filters in SDR and 5G baseband chips.
5 / 9
DSP 101
M7 · L3
Band-Stop / Notch
Block a Frequency Band

A band-stop filter subtracts a band-pass filter from an all-pass — passing everything except the rejected band. The narrow-band notch removes a single interference frequency.

Band-Stop from Band-Pass
h_{\text{BS}}[n] = \delta\!\left[n - \tfrac{N-1}{2}\right] - h_{\text{BP}}[n]
Classic Application
50/60 Hz notch filters remove mains hum from ECG, EEG, and audio recordings without affecting adjacent frequencies.
6 / 9
DSP 101
M7 · L3
Spectral Transformation
One Prototype, Four Types
  • LP — the base design, windowed-sinc method
  • HP = impulse − LP — spectral inversion
  • BP = LP_wide − LP_narrow — or modulation of LP prototype
  • BS = impulse − BP — complement of band-pass
  • All inherit linear phase and the same order as the LP prototype
  • IIR filters use z-domain frequency transformation (not arithmetic)
7 / 9
DSP 101
Quick check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

8 / 9
DSP 101
M7 · L3
Key Takeaways
What You Learned
  • Four filter types: LP, HP, BP, BS — each selects a different frequency region
  • Specs: passband edge ω_p, stopband edge ω_s, ripple R_p, attenuation A_s, transition Δω
  • HP = impulse − LP (spectral inversion, same length, same order)
  • BP = LP_wide − LP_narrow, or LP prototype modulated to ω_0
  • BS = impulse − BP (complement of band-pass)
  • Narrow notch → high order or IIR design for deep rejection over small bandwidth
Read full article →
Next lesson All lessons
9 / 9
1 / 8 muted