Multirate processing
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| decimation | LẤY MẪU GIẢM |
| interpolation/ɪnˌtɜːpəˈleɪʃn/ | NỘI SƯP |
| polyphase | PHÂN TÁCH ĐA PHA |
Filter first, then drop
- Decimation by $M$ keeps every $M$th sample — after a low-pass below $f_s/2M$, or the folded spectrum aliases. Interpolation by $L$ stuffs $L-1$ zeros — then a low-pass below the old Nyquist erases the images.
8 kHz speech to 48 kHz: interpolate by 6, low-pass at 4 kHz. Run naively the filter works at 48 kHz — six times the cost; polyphase runs it as 6 subfilters at 8 kHz: the same arithmetic, $1/6$ the multiplies ✓.
{exercise:uni-ee1206-order}
Decimating by M is safe only if —
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Keeping 1 sample in M folds M spectral copies together; whatever sits above the new Nyquist aliases permanently. The fence goes up first.
One converter, not two
- Rational ratios cascade: 48 → 72 kHz is $L = 3$, $M = 2$ — upsample to 144 kHz, one low-pass at 24 kHz (stops the $\uparrow 3$ images and the $\downarrow 2$ aliases at once), keep every second sample.
{exercise:uni-ee1206-ratio} {exercise:uni-ee1206-polyphase}
Resampling 48 kHz audio to 72 kHz uses —
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48 × 3/2 = 72: upsample by 3 to 144 kHz, one 24 kHz low-pass, then keep every second sample.
Polyphase decomposition of the 8-to-48 kHz interpolator —
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Polyphase is exact algebra: 6 phase subfilters each running at 8 kHz replace one filter at 48 kHz — identical arithmetic, one-sixth the multiplies.