Sensor fusion — complementary filter (simple), Kalman filter (optimal): Question bank

Before we start, a quick reminder of the four numbers everything below turns on, so no symbol appears un-defined:

True or false — justify

Two sensors are only worth fusing if one is more accurate than the other

False — they are worth fusing when their errors live in different frequency bands (slow drift vs fast noise); even two equally-bad sensors help if their weaknesses don't overlap.

The complementary weights

α

and

1 - α

must sum to exactly 1

True — this is the discrete echo of

H_{lp} + H_{hp} = 1

, which guarantees unity gain on a constant (DC) input; otherwise a steady angle would be scaled wrongly and leave a bias.

A larger measurement-noise variance

R

tells the Kalman filter to trust the sensor more

False —

R

is noise, so bigger

R

means a jitterier sensor; the gain

K = \frac{P ^{-}}{P ^{-} + R}

shrinks, so the filter trusts the measurement less.

At steady state the Kalman filter behaves like a complementary filter

True — once

P

converges to a constant,

K

stops changing, so the Kalman update becomes a fixed-weight blend, exactly a complementary filter with frozen gain.

The Kalman gain

K

and the complementary weight

α

play the same role

Roughly —

K

corresponds to

(1 - α)

, the weight on the measurement; the difference is

K

adapts each step while

α

is frozen.

Integrating the gyro gives a drift-free absolute angle

False — integrating a tiny constant bias grows without bound (drift); the gyro is only trustworthy short-term (high frequency).

The accelerometer angle is instantaneous and reliable moment-to-moment

False — it gives an absolute reference with no drift, but every vibration adds fast noise, so it is only trustworthy long-term (low frequency).

Precisions (inverse variances) add when two Gaussian estimates are fused

True —

\frac{1}{P _{k}} = \frac{1}{P _{k}^{-}} + \frac{1}{R}

; information adds, which is why the fused uncertainty is smaller than either input's.

The Kalman filter can make the posterior variance larger than both inputs

False — fusing two independent Gaussians always yields a variance smaller than either, since precisions add and every precision is positive.

The complementary filter needs an explicit numerical derivative of the angle

False — the high-pass factor contains an

s

, but the gyro already supplies the rate

\dot{θ}

, so no differentiation of noisy data is ever performed.

Spot the error

"Set

Q = 0

; the model is exact so we save computation."

With

Q = 0

the predicted

P^{-} = P_{k - 1}

never grows, so

P \to 0

K \to 0

, and the filter stops listening to sensors — this is filter divergence when reality inevitably deviates from the model.

"Choose

α

near 1 to kill accelerometer noise completely."

α

near 1 does suppress accel noise, but it also barely pulls back toward the absolute reference, so gyro drift returns unbounded;

α

is a trade-off, not a noise switch.

"Because a good measurement arrived, raise

R

so we lean into it."

Backwards — a good (low-noise) measurement means small

R

, which raises

K

; you don't inflate

R

to trust a reading.

"After update, carry

P

forward unchanged into the next predict."

Wrong — predict must add process noise,

P_{k}^{-} = P_{k - 1} + Q

; skipping the

+ Q

makes the filter falsely certain and deaf to sensors.

"The blend weight

α

has units of seconds."

α = \frac{τ}{τ + Δ t}

is a ratio of two times, so it is dimensionless and lives in

(0, 1)

; the seconds cancel.

"Kalman needs a fixed gain because complementary filters use one."

The whole point of Kalman is a time-varying gain that tracks uncertainty; a fixed gain is only optimal at steady state.

"A big Kalman gain always means a better estimate."

A big

K

just means "lean toward the measurement"; it's appropriate only when the prior is more uncertain than the sensor — during transients or after a reset — not universally good.

Why questions

Why must the two transfer functions sum to 1 rather than to any constant?

Summing to 1 forces unity DC gain: a constant input passes through at full scale, so a truly steady angle is neither amplified nor shrunk — any other sum introduces steady-state bias.

Why does the Kalman gain shrink as

P

shrinks?

K = \frac{P ^{-}}{P ^{-} + R}

: as we grow confident (small

P^{-}

), the numerator falls, so we weight new measurements less — we already know a lot, so each reading matters less.

Why does the predict step grow uncertainty instead of leaving it fixed?

Between measurements the true state can wander by process noise

Q

; adding independent variances gives

P^{-} = P + Q

, honestly widening our belief so the next measurement is still heard.

Why does fusing two imperfect sensors beat the better one alone?

Because their errors occupy different bands — the accel covers the gyro's slow drift and the gyro covers the accel's fast noise — so each patches exactly the other's weak spot.

Why is the high-pass path applied to the gyro and the low-pass to the accelerometer, not the reverse?

The gyro is trustworthy at high frequency (short-term smoothness) and the accel at low frequency (long-term truth); the filters route each signal to the band where it is reliable.

Why is minimizing the posterior variance the right criterion for choosing

K

It directly minimizes the expected squared error of the estimate; setting

\frac{d P}{d K} = 0

P = (1 - K)^{2} P^{-} + K^{2} R

yields the unique variance-minimizing, hence optimal-linear, gain.

Edge cases

What happens to

α

as the sample time

Δ t \to 0

with

τ

fixed?

α = \frac{τ}{τ + Δ t} \to 1

— infinitely fast sampling means each tick trusts the gyro almost fully, since barely any drift accumulates between ticks.

What happens as

τ \to 0

α \to 0

, so the estimate collapses onto the raw accelerometer angle — the low-pass cutoff moves so high that no gyro smoothing survives.

What is

K

when the sensor is noiseless,

R \to 0

K = \frac{P ^{-}}{P ^{-} + 0} = 1

— a perfect measurement is believed completely, and the posterior variance

P = (1 - 1) P^{-} = 0

collapses to certainty.

What is

K

when the sensor is useless,

R \to \infty

K \to 0

— an infinitely noisy reading is ignored, and

P

stays at

P^{-}

; the update reduces to pure prediction.

What does the Kalman filter do on the very first step with a huge initial

P_{0}

A large

P_{0}

makes

K \approx 1

, so it snaps almost fully onto the first measurement — sensible, because we admitted we knew nothing at start.

Q

is set very large, how does the filter respond to each new reading?

P^{-} = P + Q

becomes large, pushing

K

toward

1

, so the filter chases every measurement (fast but jittery) — modelling the state as highly unpredictable.

What is the posterior variance if both inputs have equal variance

P^{-} = R = σ^{2}

\frac{1}{P} = \frac{1}{σ ^{2}} + \frac{1}{σ ^{2}}

gives

P = \frac{σ ^{2}}{2}

— two equally-trusted sources halve the variance, the clearest sign that information adds.

Recall One-line self-test

If someone says "just freeze the Kalman gain, it's simpler," what's the one-sentence rebuttal? ::: That's only optimal at steady state; during transients a frozen gain either over-trusts a stale prior or ignores a fresh measurement, so you'd re-invent the complementary filter and lose the adaptivity that made Kalman worth the effort.