Impulse Response Decomposition Methods

• Impulse response decomposition is a set of techniques for breaking down a system’s response into interpretable components using modal, subspace, and operator-theoretic approaches.
• Modal and pole-residue decompositions isolate dominant modes through spectral analysis, while subspace and GSVD methods address high-dimensional and spatially structured responses.
• Advanced algebraic, spectral, and statistical tools enable efficient decomposition with practical applications in acoustics, econometrics, control systems, and system identification.

Impulse response decomposition refers to the set of methodologies for representing, isolating, or analyzing constituent components of a system’s impulse response. For linear, time-invariant (LTI) systems, the impulse response uniquely determines the system and admits diverse analytical decompositions—from modal (pole/residue) expansions to direct/residual or orthogonal/structural splits. In time-varying, nonlinear, function-valued, or stochastic systems, more sophisticated approaches generalize decomposition concepts to functional operators or higher-order kernels. Contemporary research leverages advanced algebraic, spectral, and statistical tools to enable interpretable or computationally efficient decompositions in fields spanning signal processing, econometrics, acoustics, control, and system identification.

1. Modal and Pole-Residue Decomposition

Classical LTI systems admit a modal (or spectral) decomposition, expressing the impulse response as a sum of modes (exponentials, damped sinusoids) associated with the system’s poles and residues. For rational transfer functions $H(z)$, partial fraction expansion yields

$H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$

where $\{p_i\}$ are the poles (roots of the system denominator), and $\{r_i\}$ are residues determined by the input/output structure. In feedback delay networks (FDNs), Schlecht and Habets introduced an efficient computation of modal decomposition for large-scale systems by direct root-finding of the characteristic matrix polynomial $P(z) = D^{-1} - A$ using the Ehrlich–Aberth iteration, avoiding explicit expansion to high-degree scalar polynomials. They demonstrated that energy is concentrated in a small fraction of the total modes, with mode frequencies highly uniformly distributed and residue magnitudes exhibiting heavy-tailed distributions. This explicit modal knowledge informs artificial reverberation design and hybrid time-frequency rendering (Schlecht et al., 2019).

2. Subspace and Blockwise Decomposition

In systems with high-dimensional or spatially structured impulse responses—such as spatial room impulse responses (SRIRs)—subspace methods enable decomposition into interpretable parts. Schüldt and Fels propose a direct plus residual decomposition using a blockwise generalized singular value decomposition (GSVD) of time-local SRIR blocks against a dynamically-updated residual model. At each frame, the decomposition

$H = H_d + H_r$

(“Editor’s term”: SubDec) splits energy into a low-rank direct component (direct sound plus salient reflections) and a full-rank residual (late reverberation, noise). The GSVD singular values quantify directness; adaptive rank selection allows the algorithm to track transient reflection arrivals. This robustly outperforms geometric and plane-wave subtraction methods, delivering lower spatial-spectral reconstruction errors in both simulated and measured array recordings (Deppisch et al., 2022).

3. Impulse Response Decomposition for Statistical and Econometric Systems

In vector autoregression (VAR) frameworks, impulse-response functions (IRFs) quantify shock propagation. When error covariance is time-varying, orthogonal IRF decomposition requires adjustments. Patilea and Raïssi show that averaging pointwise Cholesky factors of nonparametric covariance estimators yields an averaged OIRF

$$\bar{\Psi}_h = \Phi_h \bar{L},\quad \bar{L} = \frac{1}{T}\sum_{t=1}^{T} L_t,$$

which differs from the OIRF constructed with a single Cholesky factor of the average covariance. They introduce an explicit heteroscedasticity index to quantify the bias of ignoring time variation. Empirically, they demonstrate that naive aggregation can overstate responses and advocate for decomposition using time-varying or averaged Cholesky factors (Patilea et al., 2020).

Instrumental-variable (IV) impulse response analysis further reveals that, for composite shocks, the local-projection-IV (LP-IV) estimand decomposes as an affine combination of constituent component IRFs: $$\beta_h = \sum_{s=1}^{S} w_s \theta_{h,ys}, \quad w_s = \frac{\mathbb{E}[z_t \varepsilon_{s,t}]}{\sum_{r=1}^S \mathbb{E}[z_t \varepsilon_{r,t}]},$$ where negative weights are possible when the instrument moves constituent shocks in different directions. This necessitates set-identification using sign restrictions or additional segmentation (e.g., sectoral data) to recover individual components (Koo et al., 2022).

For function-valued shocks, Seo and Seong formalize the impulse response as a bounded linear operator (functional IRF) in separable Hilbert-space models, constructing regularized operator estimates via the Schur complement and spectral truncation (Seo et al., 11 Mar 2025).

4. Nonlinear and Higher-Order Decomposition

In nonlinear or bilinear systems, linear decomposition via convolution no longer suffices. The Volterra series framework expands the output as a sum of multidimensional convolutions. Hartmann and Beattie characterize the bilinear system impulse response as

$$h(t) = \sum_{k=1}^{\infty} h_k(t), \quad h_k(t) = \frac{1}{k!} c^{\top} e^{A t} N^{k-1} b,$$

aligning the time-diagonal (“impulse”) values of the Volterra kernels with factorial-weighted contributions. Proper assignment of those values on lower-dimensional simplex faces ensures compatibility between the impulse response and higher-order kernels (Varona et al., 2018).

In data-driven nonlinear time series, as in VANAR (vector autoencoder nonlinear autoregression), the impulse response is operationally defined as the difference between shocked and unshocked model simulations. No further structural decomposition, such as orthogonalization, is embedded; thus, any mechanism for partitioning direct/indirect or component-wise shocks must be implemented on top (Cabanilla et al., 2019).

5. Product-Convolution, Wavelet, and Operator-Theoretic Decomposition

For systems with space- or parameter-varying impulse responses, Sarti et al. describe a product-convolution expansion, representing the operator as a sum of convolution-weighted terms built from representative impulse responses and smooth spatial weights. This expansion admits conversion to a wavelet domain—facilitating sparse, quasi-linear time implementations for large-scale imaging deconvolution: $$K[f](x) \approx \sum_{k=1}^{m} u_k * (v_k \odot f)(x),$$ where $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$0 are sampled PSFs and $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$1 are spatial weights. The induced wavelet matrix can be constructed sparsely and exploited for fast inversion with substantial speed-ups for image restoration tasks (Escande et al., 2020).

6. Application-Driven Decompositions: Acoustic Dereverberation and Phase-Based Identification

Direct manipulation of the measured impulse response frequently forms the basis for practical filtering and system estimation problems. Ciba introduces an algorithm for dereverberation based on cepstral domain impulse response adaptation, where the estimated impulse response $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$2 is reshaped by frequency-bin-specific exponential fading derived from blind $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$3 estimates: $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$4 yielding a “direct+tail” decomposition in the time-frequency domain. The reconstructed $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$5 prioritizes direct path and early reflections by suppressing late decay, improving dereverberation filter efficacy for speech and music signals (Ciba, 10 Jan 2026).

Flax (in “Allpass impulse response modelling”) presents allpass phase removal for system identification. Here, a single known allpass phase (chirp) is imposed on the excitation and exactly removed from the output spectrum—allowing recovery of the unknown system’s band-limited impulse response. However, this is not a decomposition of $H(z) = \sum_{i} \frac{r_i}{1 - p_i z^{-1}},$6 into a sum or cascade of allpass components in the usual sense (Flax, 2023).

7. Significance, Limitations, and Future Perspectives

Impulse response decomposition underpins diverse analysis and synthesis tasks—ranging from physical modeling and acoustic analysis to macroeconomic shock identification and large-scale inverse problems. Modal and subspace decompositions facilitate interpretable rendering, selective modification, or computational acceleration. In statistical settings, explicit decomposition of shocks and their effects has revealed both limitations of naive structural assumptions and practical remedies for identification problems.

Limitations arise in generalizing linear decomposition to nonlinear, nonstationary, composite, or functional settings. In nonlinear and black-box models, meaningful constituent separation of impulse responses remains an open challenge, often reduced to a difference between counterfactual predictions rather than algebraic or spectral partition.

Research continues to advance operator-theoretic and data-driven decomposition, pushing the boundaries of what aspects of an impulse response can be isolated and manipulated for physical interpretability, controllability, or computational tractability. This ongoing progress is exemplified by developments in operator-valued function estimation, block-Schur regularization, and the integration of domain-specific knowledge (e.g., decay statistics, subspace priors) into decomposition frameworks.