Constant Level Stopband Sidelobes: Tsk-Tsk-Tsk, Don’t do it!

This presentation will show you how to design and implement FIR filters with an interesting and desired variation of its spectrum. The Remez (or FIRPM) filter algorithm designs FIR filters with selectable and different constant level pass band ripple and constant level stop band ripple. This is what the recursive elliptic filter does. In an alternate design we modify the Remez to replace the constant level stop band attenuation ripple with a 1/f spectral decay. Why would we do this? We do it to protect our signal passband from stopband aliasing levels when we reduce sample rate after reducing signal bandwidth. Come by and listen, I will show you why and how and demonstrate the advantages of using this option.

What this presentation is about and why it matters

This talk shows a practical but often-overlooked design choice: stopband sidelobes that are constant in level (equal-ripple) versus stopbands that are intentionally tilted so their sidelobes fall with frequency (for example, ~1/f). Fred Harris explains why constant-level stopbands are risky in real systems — especially when you reduce the sample rate after filtering — and demonstrates two engineering fixes: a simple modification of the Parks–McClellan/Remez penalty function to produce tilted sidelobes, and a pragmatic "snip the impulse edge" trick when you must retrofit a design.

Why it matters: when you downsample or decimate, energy that lived outside the new Nyquist interval aliases back into your passband. If the stopband energy is flat, those many aliased copies can add up and raise the in-band noise (roughly by 10·log10(number_of_images) dB). Designing the stopband to decay (1/f, 1/f^2, ...) pushes the high-frequency stopband energy down so that its aliases no longer dominate the passband.

Who will benefit the most from this presentation

• DSP engineers designing filters for multirate systems (decimators/interpolators, channelizers).
• Communications and radar engineers who care about in-band noise, EVM, or intersymbol interference after resampling.
• Students learning FIR/IIR design trade-offs and the Parks–McClellan (Remez) algorithm.
• FPGA/ASIC implementers concerned about coefficient quantization and finite-precision effects.

What you need to know

• Nyquist and aliasing: if you reduce sample rate (decimate), spectral content outside the new Nyquist band folds back (aliases) into the passband. Many small out-of-band terms can sum to a significant in-band level.
• FIR vs IIR behavior: recursive (IIR) designs like elliptic or Chebyshev often show equal-ripple stopbands; FIR designs using Remez/Parks–McClellan can also produce equal-ripple stopbands by default.
• Parks–McClellan / Remez: a minimax (equal-ripple) algorithm that lets you weight passband and stopband errors independently. Harris shows how to replace the flat stopband weight with a frequency-dependent penalty so sidelobes tilt like 1/f.
• Filter length rule of thumb: Harris gives a practical approximation for FIR tap count. In words: filter length scales with sample rate divided by transition bandwidth and proportional to required attenuation. One compact form used in the talk is

  N ≈ (attenuation_dB / 22) · (Fs / ΔF)

  (written in MathJax as $N \approx (A_{dB}/22)\cdot (F_s/\Delta F)$).
• Sidelobe tilt and penalty functions: using a stopband weight that increases with frequency produces sidelobes that fall approximately as 1/f. A quadratic weight gives ~1/f^2, etc. Implemented by supplying a custom FRF/penalty vector to Remez.
• Snipping method: for an existing Remez FIR, trimming (zeroing or tapering) the last sample(s) of the impulse response removes the edge impulse that creates flat stopband components and can produce a 1/f-like decay — at the cost of slightly increased passband ripple and a small rise near the band edge.
• Finite precision and quantization: coefficient quantization and arithmetic word-length limit how far the 1/f tail can keep falling. A practical rule Harris cites is ~5 dB of stopband improvement per extra bit of coefficient precision (not the ideal 6 dB), so budget bits accordingly.
• Practical trade-offs: tilting the stopband reduces aliased in-band energy but can increase in-band ripple (causing pre/post-echoes). The engineer must balance alias suppression, passband fidelity, filter length, and implementation cost.

Glossary

• Passband ripple — Variation of gain inside the passband (allowed deviation from unity gain).
• Stopband ripple — Residual gain level in the stopband (design spec usually in dB down from 0 dB).
• Remez / Parks–McClellan — An algorithm for designing minimax (equal-ripple) FIR filters using weighted error and alternation theorem.
• Sidelobes — The out-of-band lobes of a filter's frequency response that decay away from the main lobe.
• 1/f tilt — A stopband envelope that decays roughly proportional to 1/f (frequency), reducing high-frequency stopband energy.
• Decimation / Resampling — Reducing sample rate (e.g., keep every Nth sample); causes spectral aliases if content exists beyond the new Nyquist.
• Impulse response snipping — Trimming/tapering edge coefficients to remove edge impulses that create flat stopbands.
• Penalty / weight function (FRF) — Frequency-dependent weights used by Remez to penalize error in different bands; making it grow with frequency produces tilted sidelobes.
• Finite-precision effects — Quantization of coefficients and arithmetic that can lift zeros and raise sidelobes; practical rule: ~5 dB improvement per extra bit.
• Pre-/post-echo — Time-domain echoes produced by passband ripple; frequency-domain modulation (ripple) maps to time shifts via Fourier duality.

This overview is AI-generated from the session transcript. Spot an issue? Let us know.