--- title: Short-Time Linear Canonical Transform url: https://www.emergentmind.com/topics/short-time-linear-canonical-transform-stlct type: topic --- # Short-Time Linear Canonical Transform Searching arXiv for recent and foundational STLCT papers to ground the article. The **short-time linear canonical transform (STLCT)** is a windowed, localized version of the **linear canonical transform (LCT)**, introduced as a time-frequency analysis tool that generalizes the **short-time Fourier transform (STFT)** by replacing the Fourier kernel with the more general LCT kernel [1910.00499]. In this formulation, a signal is first localized by a shifted window and then analyzed in an LCT domain determined by a unimodular parameter matrix \(A=(a,b,c,d)\) with \(ad-bc=1\). The resulting representation extends the STFT and short-time fractional Fourier transform settings, supports spectrogram-based analysis, and has been studied in connection with uncertainty principles, phase retrieval, sampling, and generalizations such as the short-time offset linear canonical transform [1910.00499], [2508.18973], [2605.07842], [1802.03784]. ## 1. Definition and canonical-transform setting The LCT used in STLCT theory is parameterized by \[ A=(a,b,c,d), \qquad a,b,c,d\in\mathbb R,\qquad ad-bc=1. \] For \(b\neq 0\), the LCT of a signal \(f(t)\) is \[ L_A[f](u)=\int_{-\infty}^{+\infty} f(t)\,K_A(t,u)\,dt, \] with kernel \[ K_A(t,u)=\frac{1}{\sqrt{i2\pi b}} \exp\!\left(i\frac{d}{2b}u^2-i\frac{u}{b}t+i\frac{a}{2b}t^2\right), \] and inverse matrix \[ A^{-1}=(d,-b,-c,a). \] The main STLCT developments focus on the case \(b\neq 0\), since when \(b=0\) the LCT degenerates to a chirp multiplication [1910.00499]. Given a window \(g\in L^2(\mathbb R)\) and a complex signal \[ f(\tau)=x(\tau)e^{i\phi(\tau)}\in L^2(\mathbb R), \] the STLCT is defined by \[ S_g^A f(t,u)=\int_{-\infty}^{+\infty} f(\tau)\,g(\tau-t)\,K_A(\tau,u)\,d\tau. \] This is the localized LCT analogue of the STFT: \(g(\tau-t)\) localizes the signal around time \(t\), and the LCT kernel analyzes that localized component in the \(u\)-domain [1910.00499]. An equivalent notation used in later work is \[ V_g^{(A)}f(x,\mu) :=\int_{\mathbb{R}^{d'}} K_A(t,\mu)\,\overline{g(t-x)}\,f(t)\,dt = L_A\!\big(f\,\overline{T_x g}\big)(\mu), \] where \(T_xg(t)=g(t-x)\) [2508.18973]. A closely related formulation writes \[ \mathcal S_\omega^{\mathbf A}f(x,\xi) = \int_{\mathbb R} f(t)\,\overline{\omega(t-x)}\,K_{\mathbf A}(\xi,t)\,dt, \] with \[ K_{\mathbf A}(\xi,t) = \frac{1}{\sqrt{i b}}\exp\!\left(\frac{i\pi}{b}\bigl(a t^2-2\xi t+d\xi^2\bigr)\right), \] under the standing assumption \(b>0\) [2605.07842]. These are notational variants of the same windowed LCT principle. Two special cases organize much of the theory. If \[ A=(0,1,-1,0), \] the LCT becomes the Fourier transform and STLCT reduces to the STFT [1910.00499]. If \[ A=(\cos\alpha,\sin\alpha,-\sin\alpha,\cos\alpha), \] the transform becomes a short-time fractional Fourier transform-type setting [1910.00499]. ## 2. Structural identities and relation to the STFT A central structural fact is that STLCT is reducible to an STFT after chirp modulation. One formulation states that \[ L_A f(\mu)=\frac{1}{\sqrt{b}}e^{-i\frac{d}{2b}\mu^2}\, \mathcal{F}\!\left(f(t)e^{i\frac{a}{2b}t^2}\right)\!\left(\frac{\mu}{b}\right), \] so the LCT differs from the Fourier transform by chirp factors and a rescaling [2508.18973]. Correspondingly, STLCT magnitudes can be re-expressed through an STFT-like object after chirp modulation and an LCT change of variables: \[ V_g^{(A)}f(x,\mu) = e^{\frac{i\mu(d\mu-x)}{b}} \,V^{(B)}_{L_C(g)}L_A(f)(\mu,d\mu-x), \] with appropriate matrices \(B,C\) [2508.18973]. Another formulation gives the identity \[ \mathcal S_\omega^{\mathbf A}f(x,\xi) = \frac{1}{\sqrt{ib}}\,e^{i\pi\frac{d}{b}\xi^2} \,\mathcal V_\omega \check f\!\left(x,\frac{\xi}{b}\right) = \frac{1}{\sqrt{ib}}\,e^{i\pi\frac{d}{b}\xi^2} \,\mathcal F\bigl(f^{\omega,x}\bigr)\!\left(\frac{\xi}{b}\right), \] where \[ \check f(t)=e^{i\pi \frac{a}{b}t^2}f(t), \qquad f^{\omega,x}(t)=f(t)\overline{\omega(t-x)}e^{i\pi\frac{a}{b}t^2}, \] and \(\mathcal V_\omega\) denotes the STFT [2605.07842]. This identifies STLCT as an STFT of a chirp-modulated signal, up to a known chirp factor and the rescaling \(\xi\mapsto \xi/b\). The 2019 uncertainty paper also establishes a frequency-domain representation: \[ S_g^A f(t,u)=\int_{-\infty}^{+\infty} L_A(\xi)\,G(\xi\mid u,t)\,d\xi, \] where \[ G(\xi\mid u,t)= -i2\pi b\,e^{i2b(\xi-u)}\,L_{A_1}^g(\xi-u)\,K_A(t,u)\,K_A^*(t,\xi), \qquad A_1=(0,b,-1/b,d). \] This identity expresses the STLCT through an interaction between the signal’s LCT and the window’s LCT [1910.00499]. These formulas explain why STLCT inherits much of the analytic structure of Gabor analysis while extending it to broader canonical domains. This suggests that many STFT techniques can be transported to the STLCT setting after chirp compensation and coordinate rescaling, a point made explicit in later phase-retrieval work [2508.18973], [2605.07842]. ## 3. Local signal models, spectrograms, and moment decompositions The STLCT literature introduces localized signal and spectral quantities to quantify concentration. For a fixed time \(t\), the local signal is \[ f_t(\tau)=f(\tau)g(\tau-t), \] with normalization factor \[ Q(t)=\int_{-\infty}^{+\infty}|f(\tau)g(\tau-t)|\,d\tau, \] and normalized local signal \[ q_t(\tau)=\frac{1}{Q(t)}f(\tau)g(\tau-t), \qquad \int |q_t(\tau)|\,d\tau=1. \] The corresponding local spectrum is defined by \[ L_u^A(\mu)=L_A^f(\mu)\,L_A^g(u-\mu), \] with normalization \[ P(u)=\int_{-\infty}^{+\infty} |L_A^f(\mu)L_A^g(u-\mu)|\,d\mu, \] and normalized local spectrum \[ p_u(\mu)=\frac{1}{P(u)}L_A^f(\mu)L_A^g(u-\mu). \] These quantities formalize localization in both time and LCT-frequency variables [1910.00499]. The paper further defines \[ s_g^A f(t,u)=\int_{-\infty}^{+\infty} L_A^f(\mu)\,L_A^g(u-\mu)\,K_A^*(t,\mu)\,d\mu, \] and proves the identity \[ |S_g^A f(t,u)|^2=|s_g^A f(t,u)|^2. \] Accordingly, the magnitude squared of the STLCT acts like a spectrogram [1910.00499]. Moment quantities are then defined from \(|S_g^A f(t,u)|\). The mean time and mean frequency are \[ t_{A,S}=\frac{1}{S_g^A f(t,u)^2}\int\!\!\int t\,|S_g^A f(t,u)|\,dt\,du, \qquad u_{A,S}=\frac{1}{S_g^A f(t,u)^2}\int\!\!\int u\,|S_g^A f(t,u)|\,dt\,du, \] and the time and frequency spreads are \[ T_{A,S}^2=\frac{1}{S_g^A f(t,u)^2}\int\!\!\int (t-t_{A,S})^2\,|S_g^A f(t,u)|\,dt\,du, \] \[ F_{A,S}^2=\frac{1}{S_g^A f(t,u)^2}\int\!\!\int (u-u_{A,S})^2\,|S_g^A f(t,u)|\,dt\,du. \] With the auxiliary matrix \[ A_1=(0,b,-1/b,d), \] the corresponding means and spreads of the window \(g\) are denoted \(t_g,u_{A_1,g},T_g,F_{A_1,g}\) [1910.00499]. A key decomposition lemma yields \[ t_{A,S}=t_f-t_g, \qquad T_{A,S}=T_f\,T_g, \] \[ u_{A,S}=u_{A,f}-u_{A_1,g}, \qquad F_{A,S}^2=F_{A,f}^2+F_{A_1,g}^2. \] These relations separate signal and window contributions to the STLCT centers and spreads [1910.00499]. A plausible implication is that window design directly controls the attainable localization geometry in the STLCT plane, not merely the smoothness of the analysis operator. ## 4. Uncertainty principles in STLCT domains The foundational uncertainty paper generalizes several classical principles from Fourier and LCT analysis to STLCT for complex signals [1910.00499]. The starting point is the known LCT uncertainty principle \[ T_f^2\,F_{A,f}^2 \ge |b|. \] Using the moment decomposition above, the paper proves the STLCT time-frequency uncertainty relation \[ T_{A,S}^2\,F_{A,S}^2 \ge |b|. \] This is Theorem 1 and is presented as the STLCT analogue of the classical time-frequency uncertainty relation [1910.00499]. The same paper establishes a two-domain uncertainty relation for two different LCT parameter matrices \[ A=(a_1,b_1,c_1,d_1),\qquad B=(a_2,b_2,c_2,d_2): \] \[ F_{A,S}^2\,F_{B,S}^2 \ge \frac{(a_1b_2-a_2b_1)^2}{4}. \] This is Theorem 2 and quantifies the incompatibility of concentration in two distinct STLCT frequency-like domains [1910.00499]. For the fractional Fourier special case, \[ A=(\cos\alpha,\sin\alpha,-\sin\alpha,\cos\alpha),\qquad B=(\cos\beta,\sin\beta,-\sin\beta,\cos\beta), \] the bound becomes \[ F_{\alpha,S}^2\,F_{\beta,S}^2 \ge \frac{\sin^2(\alpha-\beta)}{4}. \] The dependence on \(\alpha-\beta\) shows that separation of fractional angles strengthens the lower bound [1910.00499]. The spectrogram itself also satisfies a conditional uncertainty principle. For fixed \(t\), the conditional mean of \(u\) is \[ u_t=\frac{1}{Q(t)}\int_{-\infty}^{+\infty} u\,|S_g^A f(t,u)|\,du, \] and for fixed \(u\), the conditional mean of \(t\) is \[ t_u=\frac{1}{P(u)}\int_{-\infty}^{+\infty} t\,|s_g^A f(t,u)|\,dt. \] The conditional standard deviations are \[ \sigma_{u|t}^2=\frac{1}{Q(t)}\int_{-\infty}^{+\infty}(u-u_t)^2\,|S_g^A f(t,u)|\,du, \] \[ \sigma_{t|u}^2=\frac{1}{P(u)}\int_{-\infty}^{+\infty}(t-t_u)^2\,|s_g^A f(t,u)|\,dt. \] Theorem 3 states \[ \sigma_{u|t}^2\,\sigma_{t|u}^2 \ge \frac{1}{Q(t)P(u)} \int_{-\infty}^{+\infty} \left(S_g^A f(t,u)\right)^* \left(\frac{1}{2}[A,B]+\frac{i}{2}\right) f(\tau)\,d\tau, \] where \(A,B\) are Hermitian operators, specifically \[ A=\tau-t,\qquad B=\frac{b}{i}\frac{d}{d\tau}-u_t,\qquad [A,B]=i. \] This is an operator-based uncertainty principle for the shape of the STLCT spectrogram itself [1910.00499]. The broader short-time canonical-transform literature extends this perspective. The short-time OLCT, which reduces to STLCT when \(T=0\) and \(n=0\), satisfies a Lieb-type uncertainty principle: \[ \int_{\mathbb{R}^2} |V_{g,A}f(x,u)|^p\,dx\,du \le (2\pi b)^{\,1-\frac{p}{2}} \,\|f\|_2^p\,\|g\|_2^p, \qquad p>2, \] and an essential support lower bound \[ |\Omega| \ge (2\pi b)\,(1-\varepsilon)^{\frac{p-2}{p}} \] under the stated normalization and concentration hypotheses [1802.03784]. Since short-time OLCT is explicitly described as a generalization of STLCT, these results situate STLCT within a wider uncertainty-theoretic hierarchy [1802.03784]. ## 5. Phase retrieval, sampling geometry, and reconstruction theory Recent work has shifted STLCT research from concentration inequalities toward inverse problems. A 2025 study considers phase retrieval from intensity measurements \[ \big|V_\varphi^{(A)} f(\lambda)\big|,\qquad \lambda\in\Lambda, \] with equivalence relation \[ f\sim h \quad \Longleftrightarrow \quad f=e^{i\alpha}h \ \text{for some } \alpha\in\mathbb{R}. \] The central question is whether \[ \big|V_\varphi^{(A)} f(\lambda)\big| = \big|V_\varphi^{(A)} h(\lambda)\big| \quad \forall \lambda\in\Lambda \] implies \(f\sim h\) [2508.18973]. The main positive result holds on a **rectangular square-root lattice** \[ \Lambda=A(\sqrt{\mathbb{Z}})^{2d'}, \] with \[ A=\operatorname{diag}(\tau_1,\dots,\tau_{d'},\,v_1,\dots,v_{d'}), \] so that \(\Lambda=\Psi\times\Gamma\), where \[ \Psi=\prod_{j=1}^{d'} \tau_j\sqrt{\mathbb{Z}},\qquad \Gamma=\prod_{j=1}^{d'} v_j\sqrt{\mathbb{Z}}. \] If the window \(\varphi\) belongs to \[ \mathcal{O}_m^n(\mathbb{C}^{d'}), \] and \[ \tau_j<\frac{1}{\sqrt{2n_j e}} \quad\text{and}\quad v_j0\) such that \[ |f(p_j)|\ge \gamma,\qquad 1\le j\le J. \] With \[ r:=\max_{1\le j\le J-1}(p_{j+1}-p_j), \qquad I=[p_1-r,p_J+r], \] the global stability constant depends on the maximal spacing between adjacent anchor points rather than the full interval size [2605.07842]. This suggests a phase-propagation mechanism localized by nonvanishing samples. The same paper further gives a finite-data noisy model \[ Y_{n,k} = M_f\!\left(\frac{\beta n}{2},hk\right)+\eta_{n,k}, \qquad M_f(x,t)=\bigl|\mathcal S_{\check\varphi}^{\mathbf A}f(x,t)\bigr|^2, \qquad \|\eta\|_\infty\le \delta, \] and develops an explicit reconstruction algorithm with quantitative robustness guarantees [2605.07842]. ## 6. Generalizations, higher-dimensional variants, and comparative frameworks STLCT has been embedded into several larger transform families. The most immediate is the **short-time offset linear canonical transform**. In the OLCT setting, the parameter set is \[ A=\begin{bmatrix} a & b & T\\ c & d & n \end{bmatrix}, \qquad ad-bc=1, \] where \(T\) is a time-shift parameter and \(n\) is a frequency-modulation / frequency-offset parameter [1802.03784]. The short-time OLCT is defined by \[ V_{g,A}f(x,u) = \int_{-\infty}^{+\infty} f(t)\,g(t-x)\,K_A(t,u)\,dt, \] and the paper explicitly states that when \[ T=0,\qquad n=0, \] the short-time OLCT reduces to STLCT [1802.03784]. This places STLCT as the zero-offset member of a six-parameter localized canonical-transform family. A 2026 study introduces a convolution-based **short time offset linear canonical transform (STOLCT)** and treats STLCT as the benchmark transform introduced in earlier work. In that paper, STLCT is described as a short-time analysis transform built from the convolution structure of the LCT, and its reconstruction/sampling formula is quoted as \[ f(x)\overline{g(x-u)}= e^{\frac{ia}{2b}x^2}\sum_{n\in \mathbb{Z}} e^{\frac{ia}{2b}\frac{n\pi}{\eta}^2} f\!\left(\frac{n\pi}{\eta}\right)\overline{g\!\left(\frac{n\pi}{\eta}-u\right)}\operatorname{sinc}\!\left(\frac{\eta x- n\pi}{b \pi}\right). \] The authors use this only for numerical comparison against their STOLCT sampling theorem and report that reconstruction errors decrease as \(\eta\) increases for both methods, while STOLCT “produces smaller errors, demonstrating better reconstruction performance” than STLCT [2606.14102]. Since the paper’s discussion of STLCT is explicitly indirect, this should be read as a comparative statement within the STOLCT framework rather than a redevelopment of STLCT theory itself. Higher-dimensional and hypercomplex extensions also use STLCT as a reference model. In three dimensions, a paper on octonion analysis defines the 3D-STLCT by \[ \mathcal V_{A_1,A_2,A_3}\{f\}(w,u) = \frac{1}{(2\pi)^{3/2}\sqrt{|b_1b_2b_3|}} \int_{\mathbb R^3} f(x)\,\phi(x-u)\, e^{i\theta_1}\,e^{i\theta_2}\,e^{i\theta_3}\,dx, \] with three coordinate-wise LCT parameter matrices \(A_k=(a_k,b_k,c_k,d_k)\), \(k=1,2,3\) [2202.00551]. The paper’s main object is a short-time octonion linear canonical transform, but it explicitly presents 3D-STLCT as the classical comparison object and gives a structural relation between them [2202.00551]. A more speculative extension appears in work on the **linear canonical space-time transform** for \(C\ell_{3,1}\)-valued signals. That paper does not define a short-time LCST explicitly, but it states that its results form the theoretical backbone for any STLCT or short-time linear canonical space-time transform built in the \(C\ell_{3,1}\) setting [2405.10990]. This suggests a possible path from one-dimensional STLCT to Clifford-valued space-time canonical analysis, though the short-time construction itself is not supplied there. ## 7. Conceptual significance and recurring themes Across the literature, STLCT occupies a specific position: it is the short-time, windowed form of the LCT, just as the STFT is the short-time form of the Fourier transform [1910.00499]. Its chief role is to provide localized analysis in canonical transform domains broader than ordinary frequency, including Fourier and fractional Fourier cases [1910.00499]. Three themes recur. First, **localization is constrained but structured**. The uncertainty relations show that STLCT inherits fundamental lower bounds on simultaneous concentration in time and canonical-frequency variables, in two different STLCT domains, and in the conditional geometry of its spectrogram [1910.00499]. This indicates that generalized canonical flexibility does not remove the localization trade-offs already familiar from Fourier analysis. Second, **the transform is tightly linked to Gabor analysis**. The chirp-modulation identities reduce STLCT to STFT-like objects, enabling transfer of uniqueness, analyticity, and sampling arguments between the two settings [2508.18973], [2605.07842]. In the Fourier case \[ \mathbf A=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}, \] the STLCT becomes the Gabor transform exactly [2605.07842]. Third, **sampling geometry matters as much as transform geometry**. For general \(L^2\) signals, phaseless STLCT recovery is possible on carefully designed rectangular square-root lattices, fails on uniform lattices in \(L^2(\mathbb R)\), and becomes possible again on uniform lattices for band-limited spaces or Gaussian shift-invariant spaces under explicit structural conditions [2508.18973], [2605.07842]. A plausible implication is that the main obstruction in STLCT phase retrieval is not the canonical kernel itself but the interaction between window analyticity, function class, and sampling set. In this sense, STLCT has evolved from a generalized time-frequency representation into a platform for inverse problems and sampling theory. The 2019 uncertainty-principle results established its concentration geometry [1910.00499]; subsequent work on phase retrieval, explicit reconstruction, and stability has shown that this geometry can also support rigorous recovery theory in non-Fourier canonical domains [2508.18973], [2605.07842]. Source: https://www.emergentmind.com/topics/short-time-linear-canonical-transform-stlct