Undecimated Wavelet Packet Decomposition

• UWPD is a wavelet packet analysis method that decomposes both low-pass and high-pass branches without downsampling, preserving the input’s original sampling rate.
• It produces a redundant, shift-invariant representation with full-length subbands, reducing aliasing and temporal misalignment in signal processing tasks.
• UWPD enables finer frequency partitioning and enhanced stability, proving effective in applications like speech separation, power quality monitoring, and sequential recommendation.

Searching arXiv for papers on undecimated wavelet packet decomposition and related terminology. Undecimated Wavelet Packet Decomposition (UWPD), also referred to as the undecimated wavelet packet transform (UWPT) and, in some contexts, the full-tree undecimated stationary wavelet packet transform (SWPT), is a wavelet packet analysis in which both low-pass and high-pass branches are recursively decomposed without decimation. Instead of downsampling the subband signals, UWPD dilates the analysis filters by zero insertion as scale increases, so every retained subband remains at the original sampling rate and has the same temporal length as the input. Relative to decimated wavelet packet decomposition, this produces a redundant representation that is approximately shift-invariant and less affected by aliasing and temporal misalignment, while preserving the finer frequency partitioning characteristic of full packet trees. UWPD has been used as a front end or structural primitive in blind speech separation, speech enhancement, instantaneous power quality analysis, and graph-enhanced sequential recommendation (Missaoui et al., 2012, Sun et al., 2016, Yu et al., 2021, Liu et al., 23 Apr 2026).

1. Definition, terminology, and relation to other wavelet transforms

Wavelet Packet Decomposition (WPD) generalizes the discrete wavelet transform (DWT) by recursively splitting both approximation and detail branches with a two-channel filterbank. In its standard decimated form, WPD applies filtering followed by downsampling by $2$ at every stage, yielding a critically sampled representation with no redundancy but with shift variance. UWPD modifies this construction by suppressing the downsampling step after each filtering operation; the filters are instead dilated by inserting zeros between taps, a procedure described as the “algorithm à trous” (Missaoui et al., 2012).

In the wavelet literature represented here, “UWPD,” “UWPT,” and “SWPT” denote the same full-tree undecimated packet transform. This equivalence matters because UWPD is sometimes conflated with the stationary wavelet transform (SWT). That conflation is inaccurate: SWT is undecimated, but only the approximation branch is further decomposed at each level, whereas UWPD decomposes both branches at all levels and therefore yields $2^J$ equal-length subbands at level $J$ (Liu et al., 23 Apr 2026).

Relative to DWT, UWPD inherits the richer frequency partitioning of wavelet packets. Relative to decimated WPT, it preserves full-length subband sequences and mitigates aliasing and shift variance. In speech enhancement, these properties are described as reducing signal distortions caused by down sampling of WPT (Sun et al., 2016). In power quality analysis, the same undecimated structure is used because it reduces spectral leakage and is advantageous for separating closely spaced tones, especially interharmonics near the fundamental and integer harmonics (Yu et al., 2021).

2. Analysis equations and packet-tree structure

Let $x[n]$ be a discrete-time signal, and let $h[n]$ and $g[n]$ denote the low-pass and high-pass analysis filters. At scale $j$, the undecimated filters are formed by dyadic dilation, i.e., by inserting $2^{j-1}-1$ zeros between adjacent taps. Using the notation of the blind speech separation formulation, the level-dependent filters are

$$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$

With packet coefficients $c_{j,m}[n]$, the initialization and undecimated recursion are

$2^J$0

$2^J$1

Equivalent formulations appear in the recommendation and power-quality settings. In the 1-D SWPT notation used for sequential recommendation, if $2^J$2, then

$2^J$3

with no downsampling, so each $2^J$4 has the same length as $2^J$5 (Liu et al., 23 Apr 2026). In the power-quality formulation, the same operation is written with level-$2^J$6 scaling and wavelet filters $2^J$7 and $2^J$8 applied at full rate to every node (Yu et al., 2021).

The packet tree admits a nominal frequency assignment analogous to decimated WPD. For sampling frequency $2^J$9, node $J$0 approximately spans

$J$1

In the undecimated setting, the absence of downsampling does not change these nominal passbands; it changes redundancy and shift behavior instead (Missaoui et al., 2012). For $J$2 kHz, the blind speech separation paper reports the nominal widths of a full packet tree as $J$3 kHz at level $J$4, $J$5 kHz at level $J$6, $J$7 kHz at level $J$8, $J$9 kHz at level $x[n]$0, and selected perceptually adjusted nodes of about $x[n]$1 kHz and $x[n]$2 kHz at levels $x[n]$3 and $x[n]$4 (Missaoui et al., 2012).

3. Shift invariance, redundancy, reconstruction, and boundary treatment

Because UWPD eliminates decimation, small time shifts in the input cause corresponding shifts in the coefficients without the aliasing artifacts associated with downsampling. In the SWPT derivation, if $x[n]$5, then

$x[n]$6

which formalizes the shift-invariant behavior of the undecimated packet coefficients (Liu et al., 23 Apr 2026). This property is central in applications where temporal alignment across bands matters, such as subband-wise graph propagation or speech transient analysis.

The price of shift invariance is redundancy. At level $x[n]$7, UWPD yields $x[n]$8 subbands, each with the same length as the original signal; if a full level-$x[n]$9 packet is retained, the redundancy factor is approximately $h[n]$0 (Missaoui et al., 2012). In the dual-tree complex speech-enhancement construction, the first three levels are undecimated and the dual-tree doubles the redundancy, so the stage-1 redundancy is $h[n]$1 (Sun et al., 2016). This redundancy is exploited for stability, but it increases coefficient storage and convolution cost.

A standard misunderstanding is that undecimation by itself guarantees perfect reconstruction for any packet system. The standard property stated in the blind speech separation and SWPT explanations is narrower: perfect reconstruction in an undecimated two-channel packet framework is obtained when analysis and synthesis filter pairs are chosen as biorthogonal or orthonormal wavelets with appropriate duals, and the same level-dependent dilations are used in synthesis (Missaoui et al., 2012, Liu et al., 23 Apr 2026). One general synthesis relation is

$h[n]$2

iterated upward to recover $h[n]$3 (Missaoui et al., 2012). In some applications, however, UWPD is used only for analysis and parameter estimation. The blind speech separation method does not reconstruct from UWPD coefficients; it estimates the ICA unmixing from selected UWPD coefficients and then separates directly in the time domain (Missaoui et al., 2012).

Boundary handling becomes consequential because undecimated convolutions preserve the original index grid at every level. In WPGRec, boundary artifacts are reduced by symmetric extension plus learnable boundary tokens,

$h[n]$4

with $h[n]$5 selected by grid search (Liu et al., 23 Apr 2026). In power quality monitoring, overlapped sliding windows are used to reduce Hilbert-transform end effects (Yu et al., 2021).

4. Filterbank design and frequency tiling strategies

UWPD is not tied to a single filterbank design; its behavior depends strongly on the chosen wavelet family, tree depth, and node-retention policy. In blind speech separation, the base transform is a five-level UWPD using Daubechies-4 ($h[n]$6) filters on $h[n]$7 kHz speech. The full tree is then adjusted to match the Bark critical bands within the $h[n]$8–$h[n]$9 kHz Nyquist range by selecting and merging nodes. The resulting perceptually adjusted tree is termed the critical bands–undecimated wavelet packet decomposition (CB-UWPD) tree and is intended to realize approximately $g[n]$0 perceptual bands under $g[n]$1 kHz. The paper states that no spectral weighting is applied; perceptual characteristics are achieved by node selection according to Bark band edges (Missaoui et al., 2012).

In instantaneous power quality analysis, filter design is treated as the central technical issue. The proposed method constructs new scaling and wavelet filters with narrow transition bands for UWPT using a conjugate quadrature mirror filter bank relationship. The design starts from a half-band low-pass prototype $g[n]$2, forms a non-negative half-band filter $g[n]$3, enforces $g[n]$4 by spectral factorization, and obtains the high-pass as

$g[n]$5

The reported design chooses $g[n]$6 of odd order $g[n]$7, which yields scaling and wavelet filters of order $g[n]$8, and sets the passband edge $g[n]$9. At level $j$0 with $j$1 Hz, the resulting UWPT achieves an effective transition bandwidth of about $j$2 Hz, whereas with conventional wavelets such as “DB45” the level-5 transition bandwidth is about $j$3 Hz (Yu et al., 2021).

In WPGRec, the wavelet family $j$4 is selected by grid search over $j$5, and the decomposition depth $j$6 is selected from $j$7, with full-tree decomposition and no pruning. The stated motivation is to align multi-resolution temporal modeling with graph propagation at matching scales while preserving equal-length, shift-invariant subbands (Liu et al., 23 Apr 2026).

5. Algorithmic roles in representative systems

In blind speech separation, UWPD serves as a preprocessing stage whose purpose is to increase non-Gaussianity before independent component analysis. Two observed mixtures $j$8 and $j$9 are decomposed by the five-level CB-UWPD tree, producing coefficient sequences $2^{j-1}-1$0. For each retained node, kurtosis is computed on a zero-mean, unit-energy coefficient sequence $2^{j-1}-1$1 as

$2^{j-1}-1$2

For each mixture channel, the node with the highest kurtosis is selected, giving $2^{j-1}-1$3 and $2^{j-1}-1$4. FastICA is then run on these selected coefficients to estimate an unmixing matrix $2^{j-1}-1$5, and the final separation is performed on the original time-domain mixtures through

$2^{j-1}-1$6

with the reported mixing matrix

$2^{j-1}-1$7

The rationale is explicit: ICA exploits non-Gaussianity, and maximizing kurtosis enhances the statistical independence of components (Missaoui et al., 2012).

In speech enhancement, UWPD appears as the first stage of a two-stage dual-tree complex wavelet packet transform (DTCWPT). The method uses $2^{j-1}-1$8 levels of undecimated DTCWPT followed by $2^{j-1}-1$9 levels of decimated DTCWPT, giving $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$0 total levels at an $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$1 kHz sampling rate. The undecimated stage provides shift-invariant, low-aliasing analytic subbands, while the decimated stage refines frequency resolution with limited additional redundancy. A speech presence probability (SPP) estimator is derived in this complex packet domain under a one-sided generalized Gamma prior for speech magnitude and a complex Gaussian noise model, and a generalized MMSE magnitude estimator is then applied to the complex coefficients before inverse two-stage synthesis (Sun et al., 2016).

In instantaneous power quality monitoring, UWPT is the first stage of a two-stage decomposition method for multi-tone voltage and current signals containing interharmonics and transient disturbances. Interharmonic frequencies are first estimated via Hanning-window two-point IpDFT; then single-sideband modulation shifts each interharmonic away from subband edges, an $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$2-level UWPT with the designed narrow-transition filters isolates the component, the shift is inverted, and the extracted interharmonic is subtracted from the signal. A second-stage FS-DWT is then applied to the residual fundamental and integer harmonics, and the Hilbert transform computes instantaneous amplitudes and phases for power quality indices (Yu et al., 2021).

In sequential recommendation, UWPD is applied along the temporal axis of item-embedding sequences. WPGRec uses

$$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$3

where each subband representation $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$4 has the same temporal length as the extended sequence. Subband-wise Chebyshev graph propagation is then performed independently,

$$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$5

and the resulting subband representations are fused by energy- and spectral-flatness-aware gating (Liu et al., 23 Apr 2026).

6. Empirical results, trade-offs, and practical interpretation

Across the cited applications, UWPD is associated with improved stability under temporal shifts, better isolation of localized spectral structure, and gains in downstream estimation tasks, but those gains are coupled to greater redundancy and to more demanding filterbank design.

Domain	UWPD role	Reported outcome
Blind speech separation	CB-UWPD preprocessing plus kurtosis-based node selection before FastICA	Proposed method outperforms SOBI, JADE, and FastICA in SIR/SDR, segmental SNR, and PESQ (Missaoui et al., 2012)
Power quality monitoring	First-stage UWPT with newly designed narrow-transition filters	Maximum relative error across all single-phase PQIs $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$6; proposed runtime faster than FS-WPT and slower than FS-DWT (Yu et al., 2021)
Speech enhancement	Three undecimated dual-tree levels before four decimated levels	At low input SNRs with nonstationary noise, average $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$7 PESQ gain and more than $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$8 dB SegSNR improvement over OM-LSA, SMPO, and MMSE-SPP (Sun et al., 2016)

In blind speech separation, experiments were carried out on instantaneous mixtures of two speech sources using two sensors, with TIMIT speech sampled at $$h^{(j)}[n] = \uparrow 2^{\,j-1}\{h\}[n], \qquad g^{(j)}[n] = \uparrow 2^{\,j-1}\{g\}[n].$$9 kHz and three gender-pairing cases: Female+Male, Female+Female, and Male+Male. The proposed CB-UWPD plus FastICA system reported average SIR/SDR values of $c_{j,m}[n]$0, $c_{j,m}[n]$1, and $c_{j,m}[n]$2, compared with FastICA values of $c_{j,m}[n]$3, $c_{j,m}[n]$4, and $c_{j,m}[n]$5, respectively. Segmental SNR and PESQ also improved in the reported tables, with particularly large gains in the Male+Male mixtures, where the average SIR/SDR improvement over FastICA was approximately $c_{j,m}[n]$6 dB (Missaoui et al., 2012).

In power quality monitoring, the single-phase tests used $c_{j,m}[n]$7 Hz, an analysis window of approximately $c_{j,m}[n]$8 s, overlapped sliding windows, and additive white Gaussian noise with SNR approximately $c_{j,m}[n]$9 dB. The methods compared were the proposed UWPT + FS-DWT + HT pipeline, FS-DWT, FS-WPT, and STFT with a $2^J$00 s window. For single-phase PQIs, the proposed method reported maximum relative error across all PQIs of at most $2^J$01, whereas STFT reached up to $2^J$02, FS-DWT up to $2^J$03, and FS-WPT up to $2^J$04. For three-phase stationary stages, most PQIs were within $2^J$05 error, while $2^J$06 remained at approximately $2^J$07–$2^J$08 error. The computational time per $2^J$09 s window was $2^J$10 s for the proposed single-phase method, $2^J$11 s for FS-DWT, and $2^J$12 s for FS-WPT; in the three-phase case, the corresponding runtimes were $2^J$13 s, $2^J$14 s, and $2^J$15 s (Yu et al., 2021).

In sequential recommendation, the evidence is qualitative rather than numerical in the supplied material. The reported finding is that WPGRec consistently outperforms sequential and graph-based baselines on four public benchmarks, with particularly clear gains on sparse and behaviorally complex datasets. An ablation identified “No Wavelet” as producing the most noticeable drop among model variants, while removing boundary handling caused smaller but consistent declines (Liu et al., 23 Apr 2026). This suggests that, in this setting, the equal-length and shift-invariant packetization is not merely a preprocessing convenience but part of the model’s scale-alignment mechanism.

The principal trade-offs recur across domains. UWPD is redundant, so computational load and memory usage are higher than in decimated WPD or DWT pipelines (Missaoui et al., 2012, Yu et al., 2021). Filter and tree design are application-dependent: Bark-band selection in speech separation, CQMFB-derived narrow-transition filters in power-quality analysis, and shallow full-tree SWPT with Coiflets or Symlets in recommendation (Missaoui et al., 2012, Yu et al., 2021, Liu et al., 23 Apr 2026). Gains also depend on signal content. In speech separation, the largest reported improvements occurred for Male+Male mixtures (Missaoui et al., 2012). In power-quality analysis, narrow transitions were obtained at the cost of higher stopband ripple, and the two-stage UWPT then FS-DWT construction was introduced to manage that trade-off (Yu et al., 2021).

Taken together, these results position UWPD as a structurally distinctive wavelet packet formalism rather than a minor implementation variant of WPT. Its defining features—full-tree decomposition without decimation, equal-length subbands, approximate shift invariance, and redundancy—make it especially suitable when temporal alignment, closely spaced spectral components, or robust subband statistics are central to the downstream task (Missaoui et al., 2012, Liu et al., 23 Apr 2026, Yu et al., 2021, Sun et al., 2016).