Math & Units

Curve Fitting

Q: What is the difference between curve fitting and interpolation?

Interpolation forces the model to pass exactly through every measured point, so a 50-point S-parameter sweep yields a function that reproduces all 50 samples but can oscillate wildly between them. Curve fitting instead chooses a model with fewer free parameters than data points and minimizes the aggregate residual error, so the resulting curve does not generally hit each sample but captures the underlying trend and rejects measurement noise. For broadband RF data with random instrument noise of 0.05 to 0.1 dB, a least-squares polynomial or rational fit produces a smoother, more physically meaningful model than spline interpolation, which would chase that noise. Use interpolation when the data are essentially noise-free and you only need values between grid points; use fitting when you must extract a compact, noise-robust model.

Q: How do you choose the model order for fitting S-parameters?

Model order is the number of poles in a rational or vector-fitting model, or the polynomial degree in a least-squares fit. Start low and increase order while monitoring the root-mean-square residual error. The error drops sharply at first, then flattens once the model captures every real resonance; the knee of that curve sets the appropriate order. For a single-resonance bandpass response you typically need 2 to 4 poles, for a multi-cavity filter 8 to 20 poles, and for a long transmission structure many more. Adding poles beyond the knee fits measurement noise rather than physics, a condition called overfitting that produces unstable extrapolation and non-passive models. Vector fitting with a passivity-enforcement step is the industry standard for generating SPICE-compatible macromodels from measured or simulated S-parameters.

Q: What does R-squared tell you about an RF curve fit?

The coefficient of determination R-squared reports the fraction of the variance in the measured data explained by the fitted model, ranging from 0 to 1. A value of 0.999 means 99.9 percent of the data variation is captured, leaving 0.1 percent as residual. For RF magnitude data an R-squared above 0.99 is usually adequate, but R-squared alone can be misleading: it always rises as you add parameters, and it weights large values more heavily, so it can look excellent while the fit is poor near nulls or stopband notches. Engineers therefore also inspect the residual plot for structure, report root-mean-square error in dB, and for complex S-parameters check that the fitted model remains causal and passive across the full frequency band rather than trusting a single scalar metric.

/kurv FIT-ing/

A numerical procedure that builds a continuous mathematical function to approximate a set of discrete measurement points, usually by minimizing the sum of squared residuals between the model and the data. In RF and microwave work it converts swept measurements, such as S-parameters, gain, and noise figure, into compact analytic models that support interpolation between frequency points, smooth extrapolation, and broadband de-embedding. Common methods range from low-order least-squares polynomials for magnitude trends to rational and vector-fitting models with 2 to 20 poles for resonant networks, and the engineer balances residual error against model order to avoid fitting instrument noise rather than device physics.

Category: Math & Units

Common Method: Least Squares / Vector Fit

Target RMS Error: < 0.1 dB

From Discrete Samples to Continuous RF Models

A vector network analyzer reports a device under test only at the discrete frequencies it sweeps, perhaps 201 or 1601 points across a band. Many downstream tasks need values everywhere in between or a closed-form expression that a circuit simulator can evaluate: time-domain convolution, statistical yield analysis, and behavioral macromodeling all require a continuous description. Curve fitting supplies that description by selecting a parameterized function family and solving for the parameter set that minimizes the deviation from the measured samples. The defining feature is that the model has fewer free parameters than data points, so it cannot pass through every sample; instead it captures the dominant behavior while averaging out random measurement noise on the order of 0.05 dB in magnitude and a few hundredths of a degree in phase.

The most common formulation is linear least squares, where the model is linear in its coefficients even if it is a high-degree polynomial in frequency. The normal equations then yield a closed-form solution. Resonant RF structures, however, are poorly represented by polynomials because their sharp poles and zeros demand a ratio of polynomials. Rational fitting and the closely related vector-fitting algorithm address this by placing complex-conjugate pole pairs that correspond directly to physical resonances, which is why vector fitting has become the standard route from measured S-parameters to passive, causal SPICE macromodels.

Across both formulations the central engineering judgment is model order. Too few parameters underfit and leave structured residual error; too many overfit, chasing noise and producing models that ring or go non-passive when extrapolated outside the measured band. Plotting root-mean-square residual versus order reveals a knee where added parameters stop reducing real error, and that knee sets the order an experienced engineer accepts.

Least Squares and Goodness of Fit

Least-Squares Objective:
minimize S = ∑_i=1^N w_i × |y_i − f(x_i, β)|²

Coefficient of Determination:
R² = 1 − (SS_res / SS_tot)

Root-Mean-Square Residual:
RMSE = √( (1/N) × ∑_i=1^N |y_i − f(x_i, β)|² )

Rational / Vector-Fit Model:
f(s) ≈ ∑_k=1^P r_k / (s − p_k) + d + s·e

Where y_i = measured data, f = model, β = parameter vector, w_i = weights, N = sample count, SS_res/SS_tot = residual/total sum of squares, P = pole count, p_k/r_k = poles/residues, s = jω. Example: a 4-pole vector fit of a cavity S₂₁ reaches RMSE ≈ 0.03 dB.

Choosing a Fitting Method

Method	Model Form	Typical Order	Best For	Main Risk
Linear least squares (polynomial)	Polynomial in f	2 to 6 degree	Smooth magnitude / gain slope	Oscillation at high degree
Rational fitting	Ratio of polynomials	4 to 16 poles	Resonant S-parameters	Spurious poles
Vector fitting	Pole-residue sum	8 to 20 poles	Passive SPICE macromodels	Non-passivity if unconstrained
Nonlinear least squares	Arbitrary f(x, β)	3 to 8 params	Equivalent-circuit extraction	Local minima, needs seed
Cubic spline (interpolation)	Piecewise cubic	N segments	Noise-free dense sweeps	Amplifies measurement noise

Common Questions

Frequently Asked Questions

What is the difference between curve fitting and interpolation?

Interpolation forces the model through every measured point, so it reproduces all samples but can oscillate between them and amplify instrument noise. Curve fitting uses fewer parameters than data points and minimizes aggregate residual error, capturing the underlying trend while rejecting random noise on the order of 0.05 to 0.1 dB. Use interpolation for noise-free dense sweeps; use fitting to extract a compact, noise-robust model.

How do you choose the model order for fitting S-parameters?

Order is the pole count in a rational or vector fit, or polynomial degree in least squares. Increase order while watching the RMS residual: it drops then flattens, and the knee sets the right order. A single resonance needs 2 to 4 poles, a multi-cavity filter 8 to 20. Beyond the knee you fit noise, not physics, which causes overfitting and unstable extrapolation. Vector fitting with passivity enforcement is the standard for SPICE macromodels.

What does R-squared tell you about an RF curve fit?

R² reports the fraction of data variance the model explains, from 0 to 1; above 0.99 is usually adequate for magnitude data. It is not sufficient alone: it always rises with more parameters and underweights nulls, so a fit can look excellent yet miss a stopband notch. Engineers also inspect the residual plot for structure, report RMSE in dB, and verify the complex model stays causal and passive across the band.

Related Terms

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