sidFreqBTFDR¶

Python equivalent: sid.freq_btfdr

Blackman-Tukey spectral analysis with frequency-dependent resolution.

result = sidFreqBTFDR(y, u)
result = sidFreqBTFDR(y, [])
result = sidFreqBTFDR(y, u, 'Resolution', R)
result = sidFreqBTFDR(y, u, 'Resolution', R, 'Frequencies', w, 'SampleTime', Ts)

Like sidFreqBT, but the window size varies across frequencies. The user specifies a resolution parameter (in rad/sample) instead of a fixed window size. Finer resolution (smaller R) uses a larger window and gives lower variance but coarser frequency detail, while coarser resolution (larger R) uses a smaller window.

This is an open-source replacement for the System Identification Toolbox function 'spafdr'.

Inputs¶

Name	Description
`y`	Output data, (N x n_y) matrix. Column vector for SISO. For multiple trajectories: (N x n_y x L) array or cell array {y1, y2, ...} for variable-length data. Spectral estimates are ensemble-averaged across trajectories.
`u`	Input data, (N x n_u) matrix. Column vector for SISO. For multiple trajectories: (N x n_u x L) or cell array. Use [] for time series (output spectrum only).

Name-value options¶

Name	Description
`'Resolution'`	Frequency resolution in rad/sample. Scalar (uniform) or vector of same length as frequency grid (per-freq). Default: 2*pi / min(floor(N/10), 30).
`'Frequencies'`	Frequency vector in rad/sample, in (0, pi]. Default: 128 linearly spaced values.
`'SampleTime'`	Sample time in seconds. Default: 1.0.

Outputs¶

Name	Description
`result`	Struct with fields:
`.Frequency`	(n_f x 1) frequency vector, rad/sample
`.FrequencyHz`	(n_f x 1) frequency vector, Hz
`.Response`	(n_f x n_y x n_u) complex frequency response
`.ResponseStd`	(n_f x n_y x n_u) standard deviation of Response
`.NoiseSpectrum`	(n_f x n_y x n_y) noise spectrum
`.NoiseSpectrumStd`	(n_f x n_y x n_y) standard deviation
`.Coherence`	(n_f x 1) squared coherence (SISO only)
`.SampleTime`	sample time in seconds
`.WindowSize`	(n_f x 1) vector of window sizes M_k
`.DataLength`	number of samples N
`.NumTrajectories`	number of trajectories L
`.Method`	'sidFreqBTFDR'

Examples¶

N = 1000; u = randn(N, 1);
y = filter([1 0.5], [1 -0.8], u) + 0.1*randn(N, 1);
result = sidFreqBTFDR(y, u, 'Resolution', 0.3);
sidBodePlot(result);

Algorithm¶

For each frequency w_k: 1. Determine local window size M_k = ceil(2*pi / R_k). 2. Compute Hann window W_{M_k} and biased covariances up to lag M_k. 3. Compute windowed spectral estimates via direct DFT. 4. Form \(G(w_k)\) and \(Phi_v(w_k)\) as in sidFreqBT. 5. Compute asymptotic uncertainty using local window norm.

References¶

Ljung, L. "System Identification: Theory for the User", 2nd ed., Prentice Hall, 1999. Sections 6.3-6.4.

Specification¶

SPEC.md §5 — Frequency-Dependent Resolution

Changelog¶

2026-03-24: First version by Pedro Lourenço.