Polyphase Implementation and Noble Identities

Sample Rate Conversion by Rational Factor $L/M$

Cascade: Interpolate by $L$ → Decimate by $M$

x[n] ──► [↑L] ──► [LPF: cutoff = min(π/L, π/M)] ──► [↓M] ──► y[n]

Single filter combines anti-imaging and anti-aliasing. Cutoff = $\min(\pi/L, \pi/M)$.

Example: Convert 44.1 kHz → 48 kHz

Ratio: $48000/44100 = 160/147$ → $L=160, M=147$

• Up-sample by 160
• Filter at $\pi/160$
• Down-sample by 147

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Polyphase Implementation (Efficient)

Instead of filtering then decimating (wasting computations), filter in polyphase branches:

Decimation Polyphase

Decompose filter $H(z) = \sum_{k=0}^{M-1} z^{-k} E_k(z^M)$

x[n] ──► E_0(z^M) ──► ↓M ──► y[n]
       E_1(z^M) ──► ↓M ──►
       ...
       E_{M-1}(z^M) ──► ↓M ──►

Only compute outputs that are kept! $M$-fold reduction in multiplies.

Interpolation Polyphase

Decompose $H(z) = \sum_{k=0}^{L-1} z^{-k} R_k(z^L)$

        R_0(z^L) ──►
        R_1(z^L) ──►     → y[n]
        ...          ↑L
        R_{L-1}(z^L) ──►

Filter branches run at lower rate → efficient!

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Noble Identities

Identity	Description
Noble 1	Filter → Decimate = Decimate → Polyphase filter
Noble 2	Interpolate → Filter = Polyphase filter → Interpolate

These allow moving rate changes across filters for efficiency.

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