Why is the cutoff tied to min(π/L, π/M)?

After **↑L**, spectral copies appear every **2π/L** in normalized high-rate frequency; a low-pass must keep below roughly **π/L** to suppress images before they fold. Before **↓M**, energy above about **π/M** would alias when you keep every **M**th sample. Using the **stricter** of the two keeps both effects under control in this simplified full-band sine test.

Is this the same as a polyphase implementation computationally?

Mathematically the **outputs match** (up to numerical detail and boundary handling). A true polyphase structure avoids explicitly forming the long zero-stuffed sequence and runs **L** cheap convolutions at the **input** rate; this page emphasizes the **signal-flow** picture first.

What goes wrong when I skip the FIR?

Images from zero-stuffing remain, and decimation **folds** them into the output band, producing **ghost tones** and a **broken** spectrum—classic missing **anti-alias / anti-image** filtering.

Why is output length close to, but not exactly, N_in·L/M?

The simulator emits output samples whose high-rate timestamps fall on the original input window after FIR group-delay compensation. Because the ratio **L/M** is rational and sample counts are integers, the count is **floor((N_in − 1)L/M) + 1**, which is the number of decimated instants that land from the first through the last input sample.

Polyphase L/M Resampling

Sampling-rate conversion by integer factors L (upsample) and M (downsample) is usually drawn as zero insertion by L, a linear-phase FIR low-pass at the intermediate rate L·F_s, then decimation by M. The FIR must simultaneously attenuate spectral images created by stuffing zeros (anti-imaging) and prevent aliasing when the high-rate signal is decimated by M (anti-alias), which in the idealized full-band setting leads to a cutoff on the order of min(π/L, π/M) radians per high-rate sample. The polyphase decomposition rewrites the same FIR as L shorter subfilters E_r(z^L) that operate only on input-rate samples—this is the computational form behind efficient interpolators and rational resamplers (Noble identities). The page implements the direct high-rate convolution path for clarity, aligns output samples to the original finite input window after compensating FIR group delay, lets you toggle a no-filter path to see aliasing/imaging, and compares windowed-sinc spectra before and after resampling. It does not implement coefficient sharing, multistage decimation, or half-band shortcuts.

Who it's for: Undergraduate DSP or audio students learning multirate systems, interpolation, decimation, image rejection, and why generic FIRs are oversized when you exploit polyphase structure.

Key terms

Polyphase decomposition
Noble identities
Upsampling
Decimation
Anti-aliasing filter
Anti-imaging filter
Rational resampling
Windowed sinc

Live graphs

How it works

Integer L/M resampling in canonical form: zero-insert by L, FIR low-pass at the high rate with cutoff tied to min(π/L, π/M) to suppress imaging (after upsampling) and aliasing (before downsampling), then keep every M-th sample. The same FIR can be rearranged by the Noble identities into a polyphase bank of L shorter filters running at the input rate—identical arithmetic, lower rate inside each branch. Toggle skip FIR to see what breaks when the anti-alias / anti-image step is removed.

Key equations

u[k] = Σ x[i] δ[k − iL] (zero stuffing), y[m] = (u * h)[m₀ + mM]

H(z) = Σ_{r=0}^{L−1} z^{−r} E_r(z^L) — L polyphase components E_r

Frequently asked questions

Why is the cutoff tied to min(π/L, π/M)?: After ↑L, spectral copies appear every 2π/L in normalized high-rate frequency; a low-pass must keep below roughly π/L to suppress images before they fold. Before ↓M, energy above about π/M would alias when you keep every Mth sample. Using the stricter of the two keeps both effects under control in this simplified full-band sine test.
Is this the same as a polyphase implementation computationally?: Mathematically the outputs match (up to numerical detail and boundary handling). A true polyphase structure avoids explicitly forming the long zero-stuffed sequence and runs L cheap convolutions at the input rate; this page emphasizes the signal-flow picture first.
What goes wrong when I skip the FIR?: Images from zero-stuffing remain, and decimation folds them into the output band, producing ghost tones and a broken spectrum—classic missing anti-alias / anti-image filtering.
Why is output length close to, but not exactly, N_in·L/M?: The simulator emits output samples whose high-rate timestamps fall on the original input window after FIR group-delay compensation. Because the ratio L/M is rational and sample counts are integers, the count is floor((N_in − 1)L/M) + 1, which is the number of decimated instants that land from the first through the last input sample.

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