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lockin_low_pass: compute magnitude noise analytically

master
Matt Huszagh 2020-11-23 16:16:21 -08:00
parent 8ae20009d7
commit 8806feb423
1 changed files with 48 additions and 39 deletions
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87
dsp/tests/lockin_low_pass.rs
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@ -332,30 +332,30 @@ fn sampled_noise_amplitude(
///
/// | sqrt((I+n*sin(x))**2 + (Q+n*cos(x))**2) - sqrt(I**2 + Q**2) |
///
/// * I is the in-phase component of the part of the input signal we
/// care about (component of the input signal with the same frequency
/// as the demodulation signal).
/// * I is the in-phase component of the portion of the input signal
/// with the same frequency as the demodulation signal.
/// * Q is the quadrature component.
/// * n is the total noise amplitude (from all contributions, after
/// attenuation from filtering).
/// * x is the phase of the demodulation signal and can be chosen to
/// be anywhere in the range [0, 2pi) to maximize this expression.
/// * x is the phase of the demodulation signal.
///
/// We need to find the demodulation phase (x) that maximizes this
/// expression. We could compute this, because we know I, Q, and n,
/// but that's a fairly expensive computation and probably
/// overkill. Instead, we can employ the heuristic that when |I|>>|Q|,
/// sin(x)=+-1 (+- denotes plus or minus) will maximize the error,
/// when |Q|>>|I|, cos(x)=+-1 will maximize the error and when
/// |I|~|Q|, max,min(sin(x)+cos(x)) will maximize the error (this
/// occurs when sin(x)=cos(x)=+-1/sqrt(2)). Whether a positive or
/// negative noise term maximizes the error depends on the values and
/// signs of I and Q (for instance, when I,Q>0, negative noise terms
/// will maximize the error since the sqrt function is concave down),
/// but the difference should be modest in each case so we should be
/// able to get a reasonably good approximation by using the positive
/// noise case. We can use the maximum of all 3 cases as a rough
/// approximation of the real maximum.
/// expression. We can ignore the absolute value operation by also
/// considering the expression minimum. The locations of the minimum
/// and maximum can be computed analytically by finding the value of x
/// when the derivative of this expression with respect to x is
/// 0. When we solve this equation, we find:
///
/// x = atan(I/Q)
///
/// It's worth noting that this solution is technically only valid
/// when cos(x)!=0 (i.e., x!=pi/2,-pi/2). However, this is not a
/// problem because we only get these values when Q=0. Rust correctly
/// computes atan(inf)=pi/2, which is precisely what we want because
/// x=pi/2 maximizes sin(x) and therefore also the noise effect.
///
/// The other maximum or minimum is pi radians away from this
/// value.
///
/// # Arguments
///
@ -388,21 +388,17 @@ fn magnitude_noise(
.abs()
};
let mut max_noise: f64 = 0.;
for (in_phase_delta, quadrature_delta) in [
(total_noise_amplitude, 0.),
(0., total_noise_amplitude),
(
total_noise_amplitude / 2_f64.sqrt(),
total_noise_amplitude / 2_f64.sqrt(),
),
]
.iter()
{
max_noise = max_noise.max(noise(*in_phase_delta, *quadrature_delta));
}
let phase = (in_phase_actual / quadrature_actual).atan();
let max_noise_1 = noise(
total_noise_amplitude * phase.sin(),
total_noise_amplitude * phase.cos(),
);
let max_noise_2 = noise(
total_noise_amplitude * (phase + PI).sin(),
total_noise_amplitude * (phase + PI).cos(),
);
max_noise
max_noise_1.max(max_noise_2)
}
/// Compute the maximum phase deviation from the correct value due to
@ -415,12 +411,25 @@ fn magnitude_noise(
/// See `magnitude_noise` for an explanation of the terms in this
/// mathematical expression.
///
/// Similar to the heuristic used when computing the error in
/// `magnitude_noise`, we can use (sin(x)=+-1,cos(x)=0),
/// (sin(x)=0,cos(x)=+-1), and the value of x that maximizes
/// |sin(x)-cos(x)| (when sin(x)=1/sqrt(2) and cos(x)=-1/sqrt(2), or
/// when the signs are flipped) as cases to test as an approximation
/// for the actual maximum value of this expression.
/// This expression is harder to compute analytically than the
/// expression in `magnitude_noise`. We could compute it numerically,
/// but that's expensive. However, we can use heuristics to try to
/// guess the values of x that will maximize the noise
/// effect. Intuitively, the difference will be largest when the
/// Y-argument of the atan2 function (Q+n*cos(x)) is pushed in the
/// opposite direction of the noise effect on the X-argument (i.e.,
/// cos(x) and sin(x) have different signs). We can use:
///
/// * sin(x)=+-1 (+- denotes plus or minus), cos(x)=0,
/// * sin(x)=0, cos(x)=+-1, and
/// * the value of x that maximizes |sin(x)-cos(x)| (when
/// sin(x)=1/sqrt(2) and cos(x)=-1/sqrt(2), or when the signs are
/// flipped)
///
/// The first choice addresses cases in which |I|>>|Q|, the second
/// choice addresses cases in which |Q|>>|I|, and the third choice
/// addresses cases in which |I|~|Q|. We can test all of these cases
/// as an approximation for the real maximum.
///
/// # Arguments
///