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class="vector-toc-numb">2</span> <span>Continuous time</span> </div> </a> <ul id="toc-Continuous_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relevant_contexts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relevant_contexts"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relevant contexts</span> </div> </a> <ul id="toc-Relevant_contexts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types_of_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types of equations</span> </div> </a> <button aria-controls="toc-Types_of_equations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Types of equations subsection</span> </button> <ul id="toc-Types_of_equations-sublist" class="vector-toc-list"> <li id="toc-Discrete_time_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_time_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Discrete time</span> </div> </a> <ul id="toc-Discrete_time_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_time_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_time_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Continuous time</span> </div> </a> <ul id="toc-Continuous_time_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graphical_depiction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graphical_depiction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> 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Not to be confused with <a href="/wiki/Discrete_variable" class="mw-redirect" title="Discrete variable">Discrete variable</a>.</div> <p>In mathematical dynamics, <b>discrete time</b> and <b>continuous time</b> are two alternative frameworks within which <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> that evolve over time are modeled. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Discrete_time">Discrete time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=1" title="Edit section: Discrete time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sampled.signal.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Sampled.signal.svg/220px-Sampled.signal.svg.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Sampled.signal.svg/330px-Sampled.signal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Sampled.signal.svg/440px-Sampled.signal.svg.png 2x" data-file-width="585" data-file-height="330" /></a><figcaption>Discrete sampled signal</figcaption></figure> <p><b>Discrete time</b> views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a <a href="/wiki/Discrete_variable" class="mw-redirect" title="Discrete variable">discrete variable</a>. Thus a non-time variable jumps from one value to another as time moves from one time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential <a href="/wiki/Integer" title="Integer">integer</a> values of the variable "time". </p><p>A <b>discrete signal</b> or <b>discrete-time signal</b> is a <a href="/wiki/Time_series" title="Time series">time series</a> consisting of a <a href="/wiki/Sequence" title="Sequence">sequence</a> of quantities. </p><p>Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by <a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a> from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated <a href="/wiki/Sampling_rate" class="mw-redirect" title="Sampling rate">sampling rate</a>. </p><p>Discrete-time signals may have several origins, but can usually be classified into one of two groups:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li>By acquiring values of an <a href="/wiki/Analog_signal" title="Analog signal">analog signal</a> at constant or variable rate. This process is called <a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Continuous_time">Continuous time</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=2" title="Edit section: Continuous time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In contrast, <b>continuous time</b> views variables as having a particular value only for an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimally</a> short amount of time. Between any two points in time there are an <a href="/wiki/Infinity" title="Infinity">infinite</a> number of other points in time. The variable "time" ranges over the entire <a href="/wiki/Real_number_line" class="mw-redirect" title="Real number line">real number line</a>, or depending on the context, over some subset of it such as the non-negative reals. Thus time is viewed as a <a href="/wiki/Continuous_variable" class="mw-redirect" title="Continuous variable">continuous variable</a>. </p><p>A <b>continuous signal</b> or a <b>continuous-time signal</b> is a varying <a href="/wiki/Quantity" title="Quantity">quantity</a> (a <a href="/wiki/Signal_(information_theory)" class="mw-redirect" title="Signal (information theory)">signal</a>) whose domain, which is often time, is a <a href="/wiki/Continuum_(set_theory)" title="Continuum (set theory)">continuum</a> (e.g., a <a href="/wiki/Connected_space" title="Connected space">connected</a> interval of the <a href="/wiki/Real_number" title="Real number">reals</a>). That is, the function's domain is an <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable set</a>. The function itself need not to be <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>. To contrast, a <a href="/wiki/Discrete_time" class="mw-redirect" title="Discrete time">discrete-time</a> signal has a <a href="/wiki/Countable_set" title="Countable set">countable</a> domain, like the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. </p><p>A signal of continuous amplitude and time is known as a continuous-time signal or an <a href="/wiki/Analog_signal" title="Analog signal">analog signal</a>. This (a <a href="/wiki/Signal_(electrical_engineering)" class="mw-redirect" title="Signal (electrical engineering)">signal</a>) will have some value at every instant of time. The electrical signals derived in proportion with the physical quantities such as temperature, pressure, sound etc. are generally continuous signals. Other examples of continuous signals are sine wave, cosine wave, triangular wave etc. </p><p>The signal is defined over a domain, which may or may not be finite, and there is a functional mapping from the domain to the value of the signal. The continuity of the time variable, in connection with the law of density of <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, means that the signal value can be found at any arbitrary point in time. </p><p>A typical example of an infinite duration signal is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e37cfe091bd31d8093ce6b397ceb3bca01319e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.245ex; height:2.843ex;" alt="{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }"></span></dd></dl> <p>A finite duration counterpart of the above signal could be: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=\sin(t),\quad t\in [-\pi ,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=\sin(t),\quad t\in [-\pi ,\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5a62642ad4f66510bdde0918bab7caf79c64e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.367ex; height:2.843ex;" alt="{\displaystyle f(t)=\sin(t),\quad t\in [-\pi ,\pi ]}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1b63ab4c491516b6e1444f97dac7ab5e439d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.188ex; height:2.843ex;" alt="{\displaystyle f(t)=0}"></span> otherwise.</dd></dl> <p>The value of a finite (or infinite) duration signal may or may not be finite. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)={\frac {1}{t}},\quad t\in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {1}{t}},\quad t\in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aeff78fa14fae46a25060d2a74afc98bae569d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.714ex; height:5.176ex;" alt="{\displaystyle f(t)={\frac {1}{t}},\quad t\in [0,1]}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1b63ab4c491516b6e1444f97dac7ab5e439d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.188ex; height:2.843ex;" alt="{\displaystyle f(t)=0}"></span> otherwise,</dd></dl> <p>is a finite duration signal but it takes an infinite value for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60270e96a3e533f6dbc68bd51f481c0d504caf41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.488ex; height:2.176ex;" alt="{\displaystyle t=0\,}"></span>. </p><p>In many disciplines, the convention is that a continuous signal must always have a finite value, which makes more sense in the case of physical signals. </p><p>For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82182729910430d9930cc0cbf96a3f43050b381c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.172ex; height:2.676ex;" alt="{\displaystyle t^{-1}}"></span> signal is not integrable at infinity, but <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{-2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e35caaad0e31891f7c2a23806c52dc6818f44e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.172ex; height:2.676ex;" alt="{\displaystyle t^{-2}}"></span> is). </p><p>Any analog signal is continuous by nature. <a href="/wiki/Discrete-time_signal" class="mw-redirect" title="Discrete-time signal">Discrete-time signals</a>, used in <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, can be obtained by <a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a> and <a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">quantization</a> of continuous signals. </p><p>Continuous signal may also be defined over an independent variable other than time. Another very common independent variable is space and is particularly useful in <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, where two space dimensions are used. </p> <div class="mw-heading mw-heading2"><h2 id="Relevant_contexts">Relevant contexts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=3" title="Edit section: Relevant contexts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Discrete time is often employed when <a href="/wiki/Empirical" class="mw-redirect" title="Empirical">empirical</a> <a href="/wiki/Measurement" title="Measurement">measurements</a> are involved, because normally it is only possible to measure variables sequentially. For example, while <a href="/wiki/Economic_activity" class="mw-redirect" title="Economic activity">economic activity</a> actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely. For this reason, published data on, for example, <a href="/wiki/Gross_domestic_product" title="Gross domestic product">gross domestic product</a> will show a sequence of <a href="/wiki/Calendar_year#Quarters" title="Calendar year">quarterly</a> values. </p><p>When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses <a href="/wiki/Time_series" title="Time series">time series</a> or <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, <i>y</i><sub><i>t</i></sub> might refer to the value of <a href="/wiki/Income" title="Income">income</a> observed in unspecified time period <i>t</i>, <i>y</i><sub><i>3</i></sub> to the value of income observed in the third time period, etc. </p><p>Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often the theory itself is expressed in discrete time in order to facilitate the development of a time series or regression model. </p><p>On the other hand, it is often more mathematically <a href="/wiki/Closed_form_solution" class="mw-redirect" title="Closed form solution">tractable</a> to construct <a href="/wiki/Scientific_theory" title="Scientific theory">theoretical models</a> in continuous time, and often in areas such as <a href="/wiki/Physics" title="Physics">physics</a> an exact description requires the use of continuous time. In a continuous time context, the value of a variable <i>y</i> at an unspecified point in time is denoted as <i>y</i>(<i>t</i>) or, when the meaning is clear, simply as <i>y</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Types_of_equations">Types of equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=4" title="Edit section: Types of equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Discrete_time_2">Discrete time</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=5" title="Edit section: Discrete time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Discrete time makes use of <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a>, also known as recurrence relations. An example, known as the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a> or logistic equation, is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c656ae2536cc6d115e9e3f0951e1c893133982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.174ex; height:2.843ex;" alt="{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}"></span></dd></dl> <p>in which <i>r</i> is a <a href="/wiki/Parameter#Mathematical_functions" title="Parameter">parameter</a> in the range from 2 to 4 inclusive, and <i>x</i> is a variable in the range from 0 to 1 inclusive whose value in period <i>t</i> <a href="/wiki/Nonlinearity" class="mw-redirect" title="Nonlinearity">nonlinearly</a> affects its value in the next period, <i>t</i>+1. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b0f4d0814f95038bbe2568d00fdea9a5324284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=4}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=1/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=1/3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faba5be07c7f75a75394e2182073049d34a71247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.97ex; height:2.843ex;" alt="{\displaystyle x_{1}=1/3}"></span>, then for <i>t</i>=1 we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=4(1/3)(2/3)=8/9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}=4(1/3)(2/3)=8/9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ad453baa721363b8bc562243cb92d09f547309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.824ex; height:2.843ex;" alt="{\displaystyle x_{2}=4(1/3)(2/3)=8/9}"></span>, and for <i>t</i>=2 we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}=4(8/9)(1/9)=32/81}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>32</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>81</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{3}=4(8/9)(1/9)=32/81}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d3802ec387160e2b430f0f50e3f62af0aa03e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.149ex; height:2.843ex;" alt="{\displaystyle x_{3}=4(8/9)(1/9)=32/81}"></span>. </p><p>Another example models the adjustment of a <a href="/wiki/Price" title="Price">price</a> <i>P</i> in response to non-zero <a href="/wiki/Excess_demand" class="mw-redirect" title="Excess demand">excess demand</a> for a product as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>δ</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83c9dc3a944668bfd47384fa8a10a2d9a30c83dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.945ex; height:2.843ex;" alt="{\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...)}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> is the positive speed-of-adjustment parameter which is less than or equal to 1, and where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is the <a href="/wiki/Excess_demand_function" title="Excess demand function">excess demand function</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Continuous_time_2">Continuous time</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=6" title="Edit section: Continuous time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Continuous time makes use of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. For example, the adjustment of a price <i>P</i> in response to non-zero excess demand for a product can be modeled in continuous time as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>λ</mi> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b326719c232f275382643e1763afccfe0ccced9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.899ex; height:5.509ex;" alt="{\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...)}"></span></dd></dl> <p>where the left side is the <a href="/wiki/First_derivative" class="mw-redirect" title="First derivative">first derivative</a> of the price with respect to time (that is, the rate of change of the price), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the speed-of-adjustment parameter which can be any positive finite number, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is again the excess demand function. </p> <div class="mw-heading mw-heading2"><h2 id="Graphical_depiction">Graphical depiction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=7" title="Edit section: Graphical depiction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A variable measured in discrete time can be plotted as a <a href="/wiki/Step_function" title="Step function">step function</a>, in which each time period is given a region on the <a href="/wiki/Horizontal_axis" class="mw-redirect" title="Horizontal axis">horizontal axis</a> of the same length as every other time period, and the measured variable is plotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graph appears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-axis point. In this technique, the graph appears as a set of dots. </p><p>The values of a variable measured in continuous time are plotted as a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a>, since the domain of time is considered to be the entire real axis or at least some connected portion of it. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Aliasing" title="Aliasing">Aliasing</a></li> <li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Digital_data" title="Digital data">Digital data</a></li> <li><a href="/wiki/Discrete_calculus" title="Discrete calculus">Discrete calculus</a></li> <li><a href="/wiki/Discrete_system" title="Discrete system">Discrete system</a></li> <li><a href="/wiki/Discretization" title="Discretization">Discretization</a></li> <li><a href="/wiki/Normalized_frequency_(digital_signal_processing)" class="mw-redirect" title="Normalized frequency (digital signal processing)">Normalized frequency</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></li> <li><a href="/wiki/Time-scale_calculus" title="Time-scale calculus">Time-scale calculus</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Discrete_time_and_continuous_time&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">"Digital Signal Processing", Prentice Hall - pages 11–12</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">"Digital Signal Processing: Instant access", Butterworth-Heinemann - page 8</span> </li> </ol></div></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGershenfeld,_Neil_A.1999" class="citation book cs1">Gershenfeld, Neil A. (1999). <i>The Nature of mathematical Modeling</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-57095-6" title="Special:BookSources/0-521-57095-6"><bdi>0-521-57095-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nature+of+mathematical+Modeling&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=0-521-57095-6&rft.au=Gershenfeld%2C+Neil+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+time+and+continuous+time" class="Z3988"></span></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWagner,_Thomas_Charles_Gordon1959" class="citation book cs1">Wagner, Thomas Charles Gordon (1959). <i>Analytical transients</i>. Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+transients&rft.pub=Wiley&rft.date=1959&rft.au=Wagner%2C+Thomas+Charles+Gordon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiscrete+time+and+continuous+time" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Discrete_time_and_continuous_time&oldid=1252434412">https://en.wikipedia.org/w/index.php?title=Discrete_time_and_continuous_time&oldid=1252434412</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Time_in_science" title="Category:Time in science">Time in science</a></li><li><a href="/wiki/Category:Dynamical_systems" title="Category:Dynamical systems">Dynamical systems</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 October 2024, at 10:02<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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