Inapplied mathematics,discretization is the process of transferringcontinuous functions, models, variables, and equations intodiscrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as abinary variable (creating adichotomy formodeling purposes, as inbinary classification).

Discretization is also related todiscrete mathematics, and is an important component ofgranular computing. In this context,discretization may also refer to modification of variable or categorygranularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.

Whenever continuous data isdiscretized, there is always some amount ofdiscretization error. The goal is to reduce the amount to a level considerednegligible for themodeling purposes at hand.

The termsdiscretization andquantization often have the samedenotation but not always identicalconnotations. (Specifically, the two terms share asemantic field.) The same is true ofdiscretization error andquantization error.

Mathematical methods relating to discretization include theEuler–Maruyama method and thezero-order hold.

Discretization is also concerned with the transformation of continuousdifferential equations into discretedifference equations, suitable fornumerical computing.

The following continuous-timestate space model

wherev andw are continuous zero-meanwhite noise sources withpower spectral densities

can be discretized, assumingzero-order hold for the inputu and continuous integration for the noisev, to

with covariances

where

andT is thesample time. IfA isnonsingular,${\displaystyle {\mathbf{B}}_{\mathbf{d}}={\mathbf{A}}^{-1}({\mathbf{A}}_{\mathbf{d}}-\mathbf{I})\mathbf{B}.}$

The equation for the discretized measurement noise is a consequence of the continuous measurement noise being defined with a power spectral density.^[1]

A clever trick to computeA_d andB_d in one step is by utilizing the following property:^{[2]: p. 215}

WhereA_d andB_d are the discretized state-space matrices.

Numerical evaluation ofQ_d is a bit trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of it^[3]The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition ofG with the upper-right partition ofG:$${\displaystyle {\mathbf{Q}}_{\mathbf{d}}=({{\mathbf{A}}_{\mathbf{d}}}^{\mathrm{\top }}{)}^{\mathrm{\top }}({{\mathbf{A}}_{\mathbf{d}}}^{-1}{\mathbf{Q}}_{\mathbf{d}})={\mathbf{A}}_{\mathbf{d}}({{\mathbf{A}}_{\mathbf{d}}}^{-1}{\mathbf{Q}}_{\mathbf{d}}).}$$

Starting with the continuous model$${\displaystyle \dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)}$$we know that thematrix exponential is$${\displaystyle \frac{d}{dt}{e}^{\mathbf{A}t}=\mathbf{A}{e}^{\mathbf{A}t}=e^{\mathbf{A}t}\mathbf{A}}$$and by premultiplying the model we get$${\displaystyle e^{-\mathbf{A}t}\dot{\mathbf{x}}(t)=e^{-\mathbf{A}t}\mathbf{A}\mathbf{x}(t)+e^{-\mathbf{A}t}\mathbf{B}\mathbf{u}(t)}$$which we recognize as$${\displaystyle \frac{d}{dt}[e^{-\mathbf{A}t}\mathbf{x}(t)]=e^{-\mathbf{A}t}\mathbf{B}\mathbf{u}(t)}$$and by integrating,which is an analytical solution to the continuous model.

Now we want to discretise the above expression. We assume thatu isconstant during each timestep.We recognize the bracketed expression as${\displaystyle \mathbf{x}[k]}$, and the second term can be simplified by substituting with the function${\displaystyle v(\tau )=kT+T-\tau }$. Note that${\displaystyle d\tau =-dv}$. We also assume thatu is constant during theintegral, which in turn yields

which is an exact solution to the discretization problem.

WhenA is singular, the latter expression can still be used by replacing${\displaystyle e^{\mathbf{A}T}}$ by itsTaylor expansion,$${\displaystyle e^{\mathbf{A}T}=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}(\mathbf{A}T{)}^k.}$$This yieldswhich is the form used in practice.

Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps${\displaystyle e^{\mathbf{A}T}\approx \mathbf{I}+\mathbf{A}T}$. The approximate solution then becomes:$${\displaystyle \mathbf{x}[k+1]\approx (\mathbf{I}+\mathbf{A}T)\mathbf{x}[k]+T\mathbf{B}\mathbf{u}[k]}$$

This is also known as theEuler method, which is also known as the forward Euler method. Other possible approximations are${\displaystyle e^{\mathbf{A}T}\approx (\mathbf{I}-\mathbf{A}T{)}^{-1}}$, otherwise known as the backward Euler method and${\displaystyle e^{\mathbf{A}T}\approx (\mathbf{I}+\frac{1}{2}\mathbf{A}T)(\mathbf{I}-\frac{1}{2}\mathbf{A}T{)}^{-1}}$, which is known as thebilinear transform, or Tustin transform. Each of these approximations has different stability properties. The bilinear transform preserves the instability of the continuous-time system.

Instatistics and machine learning,discretization refers to the process of converting continuous features or variables to discretized or nominal features. This can be useful when creating probability mass functions.

Ingeneralized functions theory,discretizationarises as a particular case of theConvolution Theoremontempered distributions

${\displaystyle \mathcal{F}\{f\ast \mathrm{III}\}=\mathcal{F}\{f\}\cdot \mathrm{III}}$

${\displaystyle \mathcal{F}\{\alpha \cdot \mathrm{III}\}=\mathcal{F}\{\alpha \}\ast \mathrm{III}}$

where${\displaystyle \mathrm{III}}$ is theDirac comb,${\displaystyle \cdot \mathrm{III}}$ is discretization,${\displaystyle \ast \mathrm{III}}$ isperiodization,${\displaystyle f}$ is a rapidly decreasing tempered distribution(e.g. aDirac delta function${\displaystyle \delta }$ or any othercompactly supported function),${\displaystyle \alpha }$ is asmooth,slowly growingordinary function (e.g. the function that is constantly${\displaystyle 1}$or any otherband-limited function)and${\displaystyle \mathcal{F}}$ is the (unitary, ordinary frequency)Fourier transform.Functions${\displaystyle \alpha }$ which are not smooth can be made smooth using amollifier prior to discretization.

As an example, discretization of the function that is constantly${\displaystyle 1}$ yields thesequence${\displaystyle [..,1,1,1,..]}$ which, interpreted as the coefficients of alinear combination ofDirac delta functions, forms aDirac comb. If additionallytruncation is applied, one obtains finite sequences, e.g.${\displaystyle [1,1,1,1]}$. They are discrete in both, time and frequency.

• Discrete event simulation
• Discrete space
• Discrete time and continuous time
• Finite difference method
• Finite volume method for unsteady flow
• Interpolation
• Smoothing
• Stochastic simulation
• Time-scale calculus

• Robert Grover Brown & Patrick Y. C. Hwang (1997).Introduction to random signals and applied Kalman filtering (3rd ed.).ISBN 978-0471128397.
• Chi-Tsong Chen (1984).Linear System Theory and Design. Philadelphia, PA, USA: Saunders College Publishing.ISBN 978-0030716911.
• C. Van Loan (Jun 1978)."Computing integrals involving the matrix exponential"(PDF).IEEE Transactions on Automatic Control.23 (3):395–404.doi:10.1109/TAC.1978.1101743.hdl:1813/7095.
• R.H. Middleton & G.C. Goodwin (1990).Digital control and estimation: a unified approach. Prentice Hall. p. 33f.ISBN 978-0132116657.

• Discretization in Geometry and Dynamics: research on the discretization of differential geometry and dynamics

• Numerical analysis
• Applied mathematics
• Functional analysis
• Iterative methods
• Control theory

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