Z-transform - Wikiparency

# Z-transform  ## Microsims — p5.js ### Z-transform (p5.js) · `Σ x[n]z⁻ⁿ` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/4HJeXX2SZ" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Z-transform — p5.js microsim"></iframe> </div> *The discrete-time answer to Laplace; pole-zero positions in the z-plane govern a digital filter's behavior.* **Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/4HJeXX2SZ) · movement *II · Transforms & the frequency domain* · library `p5js` ### Related microsims Live sims on neighbouring articles — 6 of them inside this article's own Wikipedia link tree: - [[Control_theory]] *(in tree)* - [[Convolution]] *(in tree)* - [[Detection_theory]] *(in tree)* - [[Digital_image_processing]] *(in tree)* - [[Digital_signal_processing]] *(in tree)* - [[Discrete_cosine_transform]] *(in tree)* *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  ## Links (Wikipedia order)  `Abraham_de_Moivre` · `Absolute_convergence` · `Advanced_z-transform` · `Aliasing` · `Anti-aliasing_filter` · `Anticausal_system` · `Audio_signal_processing` · `Bilinear_transform` · `Causal_system` · `Complex_number` · `Constant-Q_transform` · [[Control_theory]] · [[Convolution]] · `Cross-correlation` · [[Detection_theory]] · `Digital_filter` · [[Digital_image_processing]] · [[Digital_signal_processing]] · `Dirac_delta_function` · `Dirichlet_series` · `Discrete-time_Fourier_transform` · `Discrete_Fourier_transform` · [[Discrete_cosine_transform]] · [[Discrete_time_and_continuous_time]] · `Downsampling_(signal_processing)` · `Eliahu_I._Jury` · `Encyclopedia_of_Mathematics` · [[Estimation_theory]] · `Final_value_theorem` · `Finite_impulse_response` · `Formal_power_series` · `Fourier_series` · [[Frequency_response]] · `Fundamental_theorem_of_algebra` · `Generating_function` · `Generating_function_transformation` · `Geometric_series` · `Heaviside_step_function` · `Hertz` · `Imaginary_unit` · `Impulse_invariance` · [[Impulse_response]] · `Initial_value_theorem` · [[Integral_transform]] · `John_R._Ragazzini` · `Kronecker_delta` · [[Laplace_transform]] · `Laurent_series` · [[Least-squares_spectral_analysis]] · `Linearity` · `Lotfi_A._Zadeh` · `Matched_Z-transform_method` · `Mathematics` · `Multiplication` · `Nyquist_frequency` · `Nyquist_rate` · [[Nyquist–Shannon_sampling_theorem]] · `Oversampling` · `Parseval's_theorem` · `Periodic_function` · `Periodic_summation` · `Permutation` · `Polar_coordinate_system` · `Pole–zero_plot` · `Probability_theory` · [[Quantization_(signal_processing)]] · `Radian` · `Radius_of_convergence` · `Random_variable` · `Real_number` · `Residue_theorem` · [[Sampling_(signal_processing)]] · [[Sequence]] · [[Signal_processing]] · `Speech_processing` · `Starred_transform` · `Time-scale_calculus` · [[Time_domain]] · [[Transfer_function]] · `Two-sided_Laplace_transform` · `Undersampling` · `Unit_circle` · `Upsampling` · [[Wayback_Machine]] · `Witold_Hurewicz` · `Zak_transform` · `Zeros_and_poles` > Signal Processing concept · part of the Signal Processing Portal · movement II · !31 変換 henkan.svg  !480 *Rendered from the live microsim (▶ motion).*  ## See it next [![Laplace transform|200](Laplace_transform_thumb.png)](Laplace_transform) *→ [[Laplace_transform|Laplace transform]]*  --- Back to Signal Processing Portal · the room · Semiotic gateway  ## What it is The z-transform maps a discrete-time signal x[n] to a function of the complex variable z, X(z) = Σ_{n=-∞}^{∞} x[n] z^{-n}, serving as the discrete-time analogue of the Laplace transform. ## How it works / why it matters Its region of convergence in the complex z-plane, together with the locations of poles and zeros, characterizes a discrete linear time-invariant system: convolution in time becomes multiplication of z-transforms, so difference equations turn into rational transfer functions. It is foundational for designing and analyzing IIR and FIR digital filters, and a causal system is stable exactly when all its poles lie inside the unit circle |z| = 1. The z-plane relates to the s-plane through z = e^{sT}. ## Signs & universals Instantiates: transformation, signal, frequency. ## Related [[Laplace_transform]] · Starred transform · Advanced z-transform · Matched Z-transform method · Discrete-time Fourier transform · Digital filter  ## From the Real GENERATIVE library > In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation.[1][2] ([Wikipedia](https://en.wikipedia.org/wiki/Z-transform))   *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*   **Microsim — three.js (Wikitube framework), pending deploy:** *Z-transform: |H(z)| as a rubber sheet over the z-plane* will play here once `https://wikitube-3d-microsims.netlify.app/signal/Z-transform.html` is live.  *Built from `MICROSIM_GUIDE/specs/sims/Z-transform.json`; part of the [[Signal_processing]] set ([[PORTAL_Signal_Processing]]).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Z-transform) : [Wikitube](https://en.wikitube.io/wiki/Z-transform) ## Previous hub tags Tree parent: [[Control_theory]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*