Interpolated FIR (IFIR) Digital Filters

This presentation delves into the theory, operation, and design of very efficient linear-phase interpolated FIR (IFIR) digital filters.

What this presentation is about and why it matters

This talk introduces Interpolated FIR (IFIR) digital filters — a two-stage FIR structure invented to implement very narrowband, linear-phase FIR filters with much lower computational cost than a single long tapped-delay-line FIR. IFIR filters are especially powerful when you need a narrow transition band (small difference between passband and stopband) at an ordinary sample rate: instead of designing one long FIR with hundreds of taps, you split the work into a "shaping" stage that uses an expanded delay structure and a second "image‑reject" stage to remove unwanted repeated passbands (images).

Why this matters: in embedded and real‑time DSP systems (wireless front ends, sensors, audio subbands, instrumentation, etc.), computation, power, and memory are limited. IFIR can reduce the number of multiplies/adds dramatically for narrowband filters: the talk’s design example reduces a 230‑tap design to a 59‑tap cascade — roughly a 75% cut in workload. Knowing IFIR gives you another efficient tool for practical filter implementation.

Who will benefit the most from this presentation

• DSP engineers implementing narrowband filters on constrained hardware (microcontrollers, FPGAs, low‑power ASICs).
• Students and practitioners who want a practical alternative to very long linear‑phase FIR filters.
• Anyone designing multistage filter banks, communicaton receivers, or precision measurement systems where passband/stopband specs are tight and computational cost matters.

What you need to know

To get the most from the talk, you should be comfortable with basic FIR filter concepts and frequency‑domain reasoning. Key ideas used in the presentation:

• Tapped‑delay‑line FIR: an FIR filter implemented as a set of taps (coefficients) on successive sample delays. A single long FIR directly implements a desired impulse response but can be costly when many taps are required.
• Linear phase: symmetric coefficients give constant group delay. IFIR structures preserve linear phase because each subfilter is an FIR with symmetric coefficients.
• Impulse‑response interpolation: the shaping stage is created by taking a prototype FIR and inserting extra delay elements between taps (i.e., zero coefficients in between). This is an interpolation of the prototype impulse response — note: it is not changing the signal sample rate.
• Images and spectral repetition: inserting M−1 delays between taps in the shaping stage compresses its frequency response so the original passband appears M times across the spectrum (these are called images). These images are centered at multiples of fs/M.
• Cascade and multiplication: the overall IFIR response is the product of the shaping filter and the image‑reject filter. The image‑reject stage is designed to remove the undesired images while leaving the desired passband intact.
• Key relationships: the prototype filter is designed at scaled frequencies. For a chosen expansion factor m, the prototype passband and stopband scale as
  F_p,proto = m · F_p
  F_s,proto = m · F_s
  and the transition width is F_trans = F_s − F_p.
• Choice of m (expansion factor): selecting m is critical — it controls how many images appear and how expensive each subfilter must be. The talk emphasizes that an optimum m exists (and can be obtained from a simple formula involving F_p, F_s, and F_trans), and that you should round to an integer value for implementation. In the example, plugging the specs yields a non‑integer m which is rounded to 7.
• Practical design flow: (1) pick desired F_p, F_s, attenuation; (2) compute/choose optimum m; (3) design prototype at scaled frequencies; (4) insert M−1 zeros between prototype taps to get shaping filter; (5) design image‑reject filter at original sample rate; (6) cascade and verify.

Glossary

• IFIR — Interpolated FIR: a two‑stage FIR approach using a shaping stage with expanded tap delays and an image‑reject stage.
• Shaping filter — the first IFIR subfilter formed by inserting extra delays between prototype taps; it shapes the desired baseband response but creates images.
• Image‑reject filter — the second subfilter that removes unwanted image passbands created by the shaping filter.
• Expansion factor (m) — the integer number of delay elements inserted between prototype taps; the design parameter that controls efficiency.
• Prototype filter — the original FIR (before inserting extra delays) whose impulse response is interpolated to build the shaping filter.
• Images — repeated passbands in the shaping filter response that appear at frequencies spaced by fs/m.
• Transition band — the frequency interval between the passband edge and the stopband start (F_trans=F_s−F_p).
• Linear phase — FIR symmetry that preserves waveform shape by producing constant group delay across the passband.
• Normalized frequency — frequency expressed as a fraction of the sampling rate (e.g., 0.02 of fs).
• Cascade — placing two filters in series; the overall frequency response is the product of the individual magnitude responses.

Final notes — why watch this talk

Rick Lyons presents IFIR filters with practical clarity: he explains the structure, walks through a complete design example (including numerical values and the resulting computational savings), and dispels a common misunderstanding — the word "interpolated" refers to the impulse response, not to resampling the signal. If you design filters for constrained targets or want an efficient alternative to very long FIRs, this concise, example‑driven presentation is a useful, actionable resource. Expect clear visuals, a practical design recipe, and an approachable explanation of why IFIR deserves to be better known in everyday DSP work.

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