Multirate Polyphase Filters and Filter Banks, (GREEN Technology, also known as DSP Magic)

Recently, someone posted a question on a DSP blog I visit occasionally. How does one design a very narrow bandwidth low pass filter? One version of the problem is a filter with 10 Hz wide pass band, a 10 Hz wide transition band, and a 1 kHz sample rate. Stopband attenuation >80 dB with passband ripple <0.01 dB. This a very bad combination: low transition bandwidth with high sample rate! I think students post their homework problems on the blog so I seldom volunteer to do their homework. I did however read the many suggestions posted on the blog submitted by regular subscribers to the blog. They were interesting to read but nothing clever and of limited value. Some were just plain silly, but to quote a famous line, “who am I to judge?” The consensus was that some problems are hard and require lots of resources, this is one of them! All it takes is lots of filter coefficients and lots of multiply and adds. 405 taps seemed to be about the right number. When I read one suggestion from someone I know at Westminster University in London, I simply had to throw my hat in the ring. It then became a game: how small could you make the filter and still satisfy the specifications? For a week I submitted daily solutions requiring fewer and fewer coefficients. I started at 38 M&A per input sample and I stopped when I reached 6 M&A per input sample!

The presentation will show how to build narrowband filters with more than an order of magnitude reduction of workload. The only requirement is that there be a large ratio of sample rate to bandwidth. Once we learn the simple trick to accomplish this reduction we pose the question, Can we achieve similar reduction in workload when there is not a large ratio of sample rate to bandwidth? The answer surprisingly is yes! We will share the recipe for the secret sauce so you too will know how wideband filters can also be implement with more than an order of magnitude workload reduction. How about an I-Q filter pair with 1400 taps per arm replaced with a resampling filter requiring only 100 real multiplies?

What this presentation is about and why it matters

This talk by Fred Harris explains how multirate polyphase filters and filter banks let you implement narrowband and wideband filters much more efficiently than a single, long FIR. The practical punchline: if you can change sample rate and rearrange where filtering happens, you often turn hundreds of multiplies-per-sample into a few dozen (or fewer). For engineers building radios, channelizers, software-defined radios, or any system that must isolate many narrow bands inside a wide spectrum, these techniques reduce cost, power, and time-to-market while sometimes enabling designs that were previously impractical.

Who will benefit the most from this presentation

• DSP engineers and system designers working on receivers, channelizers, or software-defined radio.
• Students learning multirate processing, FIR design, or filter-bank concepts who want engineering intuition and practical recipes.
• Anyone optimizing computational load for real-time filtering, resampling, or multi-channel extraction.

What you need to know

The talk builds on a few core ideas. You don’t need to be an expert, but familiarity with the items below will help you get more from the explanations and examples.

• Sampling and aliasing: Understand that sampling repeats the spectrum and that decimation (downsampling) aliases those repeats into lower bands. Harris uses aliasing intentionally as a tool rather than an enemy.
• FIR filtering basics: Convolution, impulse response, and the cost of long FIRs (multiplications per sample) — why a 400–600 tap filter can be expensive in real-time systems.
• IQ representation: Why complex baseband (I/Q) removes image ambiguity and how analog IQ mismatch motivates digital solutions.
• Equivalence theorem (mix–filter order): Mixing (heterodyning) before filtering is equivalent to modulating the filter coefficients and filtering at the carrier — this lets you move filters to the signal instead of moving the signal to the filter.
• Polyphase decomposition: A long FIR can be partitioned into M subfilters (polyphase components) when you plan to decimate by M. That lets most work be done at the lower rate because each input sample only traverses one subfilter.
• Noble identities and twiddle factors: Noble identities permit reordering resampler and filter blocks. Twiddle factors (phasors) reinsert the phase lost when reorganizing delays; they connect the polyphase structure to DFT/FFT operations.
• Filter banks and the (I)DFT: The polyphase outputs can be combined with an IDFT/FFT (or DFT) to give multiple channel outputs simultaneously (analysis/synthesis filter banks, perfect-reconstruction under proper design).

Small inline math reminders you may see in the talk: phasors and rotation use terms like $e^{j\omega n}$; decimation by $M$ creates $M$ aliased Nyquist zones; roots of unity are $e^{j2\pi k/M}$ which play into the IDFT.

Glossary

• Decimation — Downsampling by an integer factor $M$; discards samples and causes spectrum aliasing.
• Interpolation — Upsampling (zero-stuffing) followed by filtering to reconstruct a higher-rate signal.
• Polyphase decomposition — Rewriting an FIR as $M$ subfilters that process staggered input samples; key to efficient decimation/ interpolation.
• Noble identity — Equality that allows swapping certain sampling-rate change operations with filtering without changing overall behavior.
• Twiddle factors — Complex phase multipliers (roots of unity) that appear when reorganizing delays and are closely related to FFT coefficients.
• Filter bank — A set of analysis filters (and their dual synthesis filters) that split a signal into frequency subbands; can be implemented with polyphase+DFT.
• Channelizer — Practical filter bank that extracts many narrow channels from a wideband input (often using polyphase + FFT).
• Equivalence theorem — The principle that mixing then filtering is equivalent to modulating filter coefficients and filtering before mixing.
• Perfect reconstruction — Property of some filter banks that allows exact (or near-exact) recovery of the input after analysis and synthesis.
• Roots of unity — The complex numbers $e^{j2\pi k/M}$ used in DFT/IDFT and in derotation of aliased components.

Final note

Fred Harris’s presentation mixes solid engineering intuition, historical perspective, and practical tricks (equivalence theorem, noble identities, polyphase partitioning) that let you do “applied magic” — achieving dramatic reductions in computation while preserving filter performance. If you work on radios, channelizers, or any system with expensive FIR work, this talk is a compact, example-rich guide to techniques you can apply right away. Bring a notebook: the talk rewards careful listening and will give you concrete ideas for moving filter work into lower-rate domains and leveraging FFTs to share a single long filter across many channels.

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