DSP 101
M11 · L04
Module 11: Multirate Signal Processing

Filter Banks & Polyphase Structures

Split a wideband signal into M subbands, process each independently, then reassemble — the architecture behind MP3, OFDM, and wavelet compression.

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DSP 101
M11 · L04
Concept

What is a Filter Bank?

An analysis bank decomposes the input into M subbands. A synthesis bank reconstructs the original. When designed correctly, x̂[n] = x[n − k] — perfect reconstruction.

M
subbands
π/M
bandwidth each
1:1
data rate ratio
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DSP 101
M11 · L04
Analysis Bank

Filtering & Decimating

Apply M bandpass filters then downsample each by M. Total data rate stays the same: M subbands × fs/M = fs.

DFT Modulated Bank
H_k(z)=H_0\!\left(z\,e^{-j2\pi k/M}\right)
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DSP 101
M11 · L04
Synthesis Bank

Upsampling & Summing

Upsample each subband by M, filter with synthesis filters Gk, then sum all M branches. Joint design of Hk & Gk achieves perfect reconstruction.

PR condition (simplified)
Alias terms cancel + distortion = pure delay z−k
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DSP 101
M11 · L04
Efficiency

Polyphase Decomposition

Split H(z) into M shorter branches Ek(zM). All arithmetic runs at the subband rate fs/M. Combine branches with an FFT — computation drops by a factor of M.

Polyphase Form
H(z)=\sum_{k=0}^{M-1}z^{-k}E_k(z^M)
05 / 11
DSP 101
M11 · L04
Two-Channel Case

Quadrature Mirror Filters

  • M = 2: one lowpass H0, one highpass H1
  • H1(z) = H0(−z)  →  h1[n] = (−1)n h0[n]
  • Aliasing from the two decimators cancels exactly
  • Johnston & paraunitary (CQF) designs achieve exact PR
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DSP 101
Interactive
Try It

M-Channel Bank

Adjust M to see how the subbands tile the spectrum. Each colored band is one subband of width π/M.

M 4 bands
07 / 11
DSP 101
M11 · L04
Application: Audio Codecs

MP3 & AAC

  • MP3: 32-channel polyphase bank + MDCT subbands
  • AAC: pure MDCT filter bank, better resolution
  • Allocate bits by perceptual importance per subband
  • Masked subbands receive fewer (or zero) bits
  • Wavelet (JPEG 2000): recursive QMF tree → multi-resolution
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DSP 101
M11 · L04
Application: OFDM

OFDM as a Filter Bank

The OFDM transmitter is a polyphase synthesis bank — each subcarrier is one channel. The IFFT combines M data streams into one wideband signal. The receiver's FFT is the analysis bank.

Why cyclic prefix?
Prevents inter-subband aliasing from the channel — same role as the anti-aliasing filter in a classical filter bank.
09 / 11
DSP 101
Filter Banks

Split, process, reconstruct

Four checks on analysis/synthesis banks, critical sampling, QMF alias cancellation, and OFDM.

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DSP 101
M11 Complete
Module 11 Complete

Multirate DSP Done

  • Filter banks split & recombine signals via analysis/synthesis pairs
  • Polyphase decomposition cuts cost by factor M — all work at fs/M
  • QMF banks (M=2) cancel aliasing; PR conditions ensure exact recovery
  • Powers MP3, AAC, JPEG 2000, OFDM, hearing aids, wavelets
Up Next — Module 12
Capstone & Real-World DSP →
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