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Chapter 26

Optimal and adaptive filtering

Four lessons in Part IX, Random signals and statistical signal processing. Read them in order, or start anywhere: a prerequisite is a link, never a gate.

Start with 26.1
0 of 4 read1 on the essential pathabout 80 minutes

Lessons in this chapter

Two stacked panels. Received x[n]: 512 noisy samples, n = 0 to 511, with the 64-sample template drawn over them at the current lag and a bracket marking it. Matched-filter output: the sum of the products at each lag ℓ, from 0 to 448. Lag ℓ = 448: the template covers samples 448 to 511, and the output there is −6.7; the output is drawn for lags 0 to 448. A diamond marks the peak, 31.8 at lag 300.Two stacked panels. Received x[n]: 512 noisy samples, n = 0 to 511, with the 64-sample template drawn over them at the current lag and a bracket marking it. Matched-filter output: the sum of the products at each lag ℓ, from 0 to 448. Lag ℓ = 448: the template covers samples 448 to 511, and the output there is −6.7; the output is drawn for lags 0 to 448. A diamond marks the peak, 31.8 at lag 300.

Lesson 1 Essential16 minYou are hereRead

Matched filters and detection

To find a known pulse in white noise, correlate with it: no linear filter lifts it further above the noise. A threshold then decides.

Two stacked panels against Ω from 0 to π. PSDs, in dB from −25 to 25: the signal's PSD, a solid curve labelled signal, from 12.8 dB at 0 to −12.8 dB at π, and the noise's PSD, a flat dashed line labelled noise. Wiener gain H_opt, from 0 to 1, with a dotted level at ½. Noise at 10 dB. The gain is 0.655 at 0 and 0.005 at π, ½ at 0.032π, where a small tick marks it. Error power 0.586.Two stacked panels against Ω from 0 to π. PSDs, in dB from −25 to 25: the signal's PSD, a solid curve labelled signal, from 12.8 dB at 0 to −12.8 dB at π, and the noise's PSD, a flat dashed line labelled noise. Wiener gain H_opt, from 0 to 1, with a dotted level at ½. Noise at 10 dB. The gain is 0.655 at 0 and 0.005 at π, ½ at 0.032π, where a small tick marks it. Error power 0.586.

Lesson 220 minYou are hereRead

The Wiener filter

Recover a signal from noise with the least mean-square error: weight each frequency by the signal's share of the power, or solve the Wiener–Hopf equations.

One panel, the two taps at equal scales: h_0 across from −0.2 to 1.4, h_1 up from −1.0 to 0.6. Contours of the error J at 0.02, 0.05, 0.17: ellipses around the bottom, h_true = (0.800, −0.400), marked by a cross. The steepest-descent path is dashed; the LMS path is solid, with a mark at the current taps. Step size μ = 0.05. After 300 steps, LMS is at (0.812, −0.369), J = 0.012; steepest descent at (0.799, −0.399), J = 0.010.One panel, the two taps at equal scales: h_0 across from −0.2 to 1.4, h_1 up from −1.0 to 0.6. Contours of the error J at 0.02, 0.05, 0.17: ellipses around the bottom, h_true = (0.800, −0.400), marked by a cross. The steepest-descent path is dashed; the LMS path is solid, with a mark at the current taps. Step size μ = 0.05. After 300 steps, LMS is at (0.812, −0.369), J = 0.012; steepest descent at (0.799, −0.399), J = 0.010.

Lesson 322 minYou are hereRead

Adaptive filters: LMS

A filter that learns its taps: LMS walks down the error bowl in noisy steps, and an adaptive canceller removes mains hum.

Misalignment in dB against sample n from 1 to 2000 on a logarithmic axis, for LMS (a line ending in a square) and RLS (a line ending in a dot), identifying 8 taps from a coloured input. Both curves drawn up to sample 2000: LMS at −47.1 dB, RLS at −56.1 dB.Misalignment in dB against sample n from 1 to 2000 on a logarithmic axis, for LMS (a line ending in a square) and RLS (a line ending in a dot), identifying 8 taps from a coloured input. Both curves drawn up to sample 2000: LMS at −47.1 dB, RLS at −56.1 dB.

Lesson 422 minYou are hereRead

RLS and the Kalman filter

RLS solves least squares again at every sample, so it learns fast; the Kalman filter predicts with a model and corrects with measurements.

After this chapter

Where to go next.

The chapters either side, and the rest of Part IX in the library.

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