How it works
Downsampling by factor M: keep every Mth sample, x_d[n] = x[nM]. This compresses the frequency spectrum by M, causing aliasing if signal energy exists above π/M. The anti-aliasing LPF must have cutoff ωc = π/M before downsampling. Upsampling by factor L: insert L−1 zeros between samples, x_u[n] = x[n/L] for n divisible by L, 0 otherwise. This inserts L−1 images of the baseband spectrum up to 2π — an anti-imaging LPF with cutoff π/L removes them. Polyphase decomposition splits the filter H(z) into M subfilters E_k(z^M), enabling each subfilter to run at the lower rate, reducing computation by M. Noble identities allow moving downsamplers and upsamplers past filters in the signal flow graph to simplify structures.
Key points to remember
The fundamental identity: downsampling by M reduces computation to 1/M of the full-rate filter cost; without polyphase decomposition, the FIR filter runs at the high rate and wastes M−1 out of every M output samples. Sampling rate conversion by a rational factor L/M is achieved by upsampling by L first, filtering, then downsampling by M — the order is fixed because upsampling first avoids aliasing. Polyphase filter banks are the basis of OFDM, MP3 audio coding (QMF banks), and ADSL modems. Two-channel QMF (quadrature mirror filter) banks use filters H0(z) for the low band and H1(z) = H0(−z) for the high band — the mirror symmetry at π/2 is where the name comes from. Perfect reconstruction requires carefully designed analysis and synthesis filter pairs.
Exam tip
The examiner always asks you to explain why the anti-aliasing filter must come before the downsampler and not after — state that once aliasing occurs in the sampled signal it cannot be removed by any subsequent filter, making the pre-filter order mandatory.