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In [[Dynamical system|mathematical dynamics]], '''discrete time''' and '''continuous time''' are two alternative frameworks within which to model [[Variable (mathematics)|variables]] that evolve over time.
In [[Dynamical system|mathematical dynamics]], '''discrete time''' and '''continuous time''' are two alternative frameworks within which to model [[Variable (mathematics)|variables]] that evolve over time.
在[[数学动力学]] '''Mathematical dynamics'''中，'''离散时间 Discrete time'''和'''连续时间 Continuous time'''是两种可供选择的框架，这两种框架可以对随时间变化的'''变量 Variables'''进行建模。
[[Image:Sampled.signal.svg|right|thumb|Discrete sampled signal|链接=Special:FilePath/Sampled.signal.svg]]
Discrete sampled signal
==离散时间 Discrete time==
[[Image:Sampled.signal.svg|thumb|离散采样信号 Discrete sampled signal|链接=Special:FilePath/Sampled.signal.svg|替代=]]
'''Discrete time''' 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 [[discrete variable]]. 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 [[integer]] values of the variable "time".
'''Discrete time''' 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 [[discrete variable]]. 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 [[integer]] values of the variable "time".
Discrete time 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 discrete variable. 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 integer values of the variable "time".
'''离散时间 Discrete time'''将变量的值视为发生在不同的、分离的”时间点” ，或者等价于在整个非零时间区域(“时间段”)中没有变化，也就是说，时间被视为一个[[离散变量]] '''Discrete variable'''。因此当时间从一个时间周期移动到下一个时间周期时，非时间变量从一个值跳到另一个值。这种对时间的观测相当于一个数字时钟，如在一段时间内给出一个固定的读数10:37，然后跳到一个新的固定读数10:38，等等。在这个框架中，每个感兴趣的变量在每个时间段都被测量一次。任何两个时间周期之间的测量数量都是有限的。测量通常按照“时间”变量的连续[[整数]] '''Integer'''值进行。
A '''discrete signal''' or '''discrete-time signal''' is a [[time series]] consisting of a [[sequence]] of quantities.
A '''discrete signal''' or '''discrete-time signal''' is a [[time series]] consisting of a [[sequence]] of quantities.
'''离散信号'''或'''离散时间信号'''是由一[[序列]] '''Sequence'''量组成的[[时间序列]] '''Time series''' 。
Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by [[Sampling (signal processing)|sampling]] from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated [[sampling rate]].
Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by [[Sampling (signal processing)|sampling]] from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated [[sampling rate]].
与连续时间信号不同，离散时间信号不是连续变量的函数，然而，它可以是从连续时间信号中[[采样]] '''Sampling'''得到的。当一个离散时间信号是通过在均匀间隔时间对序列进行采样的，它就有一个关于[[采样率]] '''Sampling rate'''的问题。
Discrete-time signals may have several origins, but can usually be classified into one of two groups:<ref name=":0">"Digital Signal Processing" Prentice Hall - Pages 11-12</ref>
离散时间信号可能有几种来源，但通常可分为两类:<ref name=":0" />
*By acquiring values of an [[analog signal]] at constant or variable rate. This process is called [[Sampling (signal processing)|sampling]].<ref>"Digital Signal Processing: Instant access." Butterworth-Heinemann - Page 8</ref>
*通过以恒定或可变速率获取[[模拟信号]] '''Analog signal'''的值。 这个过程称为[[采样]] '''Sampling。'''
*By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.
*通过观察固有的离散时间过程，例如特定经济指标的每周峰值。
* By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.
== 连续时间 Continuous time==
In contrast, '''continuous time''' views variables as having a particular value for potentially only an [[infinitesimal]]ly short amount of time. Between any two points in time there are an [[infinity|infinite]] number of other points in time. The variable "time" ranges over the entire [[real number line]], or depending on the context, over some subset of it such as the non-negative reals. Thus time is viewed as a [[continuous variable]].
In contrast, '''continuous time''' views variables as having a particular value for potentially only an [[infinitesimal]]ly short amount of time. Between any two points in time there are an [[infinity|infinite]] number of other points in time. The variable "time" ranges over the entire [[real number line]], or depending on the context, over some subset of it such as the non-negative reals. Thus time is viewed as a [[continuous variable]].
相反，连续时间将变量视为可能仅在[[极短|无限]] '''Infinitesimally'''小的时间内具有的特定值，在任意两个时间点之间有[[无限]] '''Infinite'''个其他时间点。变量“时间”的范围是整个[[实数线]] '''Real number line'''，或者取决于上下文，取决于它的某个子集，如非负实数。因此时间被看作是一个[[连续变量]] '''Continuous variable'''。
A '''continuous signal''' or a '''continuous-time signal''' is a varying [[quantity]] (a [[signal (information theory)|signal]])whose domain, which is often time, is a [[Continuum (set theory)|continuum]] (e.g., a [[connected space|connected]] interval of the [[real number|reals]]). That is, the function's domain is an [[uncountable set]]. The function itself need not be [[continuous function|continuous]]. To contrast, a [[discrete time]] signal has a [[countable set|countable]] domain, like the [[natural number]]s.
'''连续信号 Continuous signal''' 或'''连续时间信号''' '''Continuous-time signal'''是[[可变的量]] Quantity ([[信号]] '''Signal''')，它的域，通常是一个[[连续体]] '''Continuum'''(例如，[[实数]] '''Connected'''的[[连通]] '''Connected'''区间)。也就是说，函数的域是一个[[不可数的集合]] '''Uncountable set'''。函数本身不需要是[[连续]] '''Continuous'''的。相比之下，[[离散时间]] '''Discrete time'''信号像[[自然数]] '''Natural numbers'''一样有一个[[可数]] '''Countable'''域。
A signal of continuous amplitude and time is known as a continuous-time signal or an [[analog signal]]. This (a [[Signal (electrical engineering)|signal]]) 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.
A signal of continuous amplitude and time is known as a continuous-time signal or an [[analog signal]]. This (a [[Signal (electrical engineering)|signal]]) 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.
具有连续振幅和时间的信号称为连续时间信号或[[模拟信号]] '''Analog signal'''。这(一个[[信号]] '''Signal''')在任何时刻都有一定的值。电信号与温度、压力、声音等物理量成正比，且通常是连续信号。连续信号的其他例子，有如正弦波、余弦波、三角波等。
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 [[real numbers]], means that the signal value can be found at any arbitrary point in time.
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 [[real numbers]], means that the signal value can be found at any arbitrary point in time.
信号是定义在一个域上，这个域可能是有限的，也可能不是有限的，并且存在一个从该域到信号值的函数映射。时间变量的连续性，与[[实数]] '''Real numbers'''密度定律有关，意味着信号值可以在任意时间点找到。
A typical example of an infinite duration signal is:
A typical example of an infinite duration signal is:
无限持续信号的一个典型例子是:
无限持续信号的一个典型例子是:
<math>f(t) = \sin(t), \quad t \in \mathbb{R}</math>
<math>f(t) = \sin(t), \quad t \in \mathbb{R}</math>
A finite duration counterpart of the above signal could be:
A finite duration counterpart of the above signal could be:
上述信号的有限持续时间对应的可以是:
<math>f(t) = \sin(t), \quad t \in [-\pi,\pi]</math> and <math>f(t) = 0</math> otherwise.
<math>f(t) = \sin(t), \quad t \in [-\pi,\pi]</math> and <math>f(t) = 0</math> otherwise.
<math>f(t) = \sin(t), \quad t \in [-\pi,\pi]</math> and <math>f(t) = 0</math> otherwise.
<math>f(t) = \sin(t), \quad t \in [-\pi,\pi]</math> 和 <math>f(t) = 0</math> 否则.
The value of a finite (or infinite) duration signal may or may not be finite. For example,
The value of a finite (or infinite) duration signal may or may not be finite. For example,
有限(或无限)持续时间信号的值可能是有限的，也可能不是。比如说,
有限(或无限)持续时间信号的值可能是有限的，也可能不是。比如说,
<math>f(t) = \frac{1}{t}, \quad t \in [0,1]</math> and <math>f(t) = 0</math> otherwise,
<math>f(t) = \frac{1}{t}, \quad t \in [0,1]</math> and <math>f(t) = 0</math> otherwise,
[0,1] </math > 和 < math > f (t) = 0 </math > 否则,
<math>f(t) = \frac{1}{t}, \quad t \in [0,1]</math> 和 <math>f(t) = 0</math> 否则,
is a finite duration signal but it takes an infinite value for <math>t = 0\,</math>.
is a finite duration signal but it takes an infinite value for <math>t = 0\,</math>.
是一个有限持续时间的信号，但它取一个无限的值 <math>t = 0\,</math>.
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.
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.
在许多学科中，惯例是连续信号必须总是有一个有限值，这在物理信号的情况下更有意义。
在许多学科中，惯例是连续信号必须总是有一个有限值，这在物理信号的情况下更有意义。
For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example，the <math>t^{-1}</math> signal is not integrable at infinity, but <math>t^{-2}</math> is).
在某些情况下，只要信号在任何有限区间上可积，无限奇点是可以接受的(例如, <math>t^{-1}</math> 该信号在无穷远处不可积，但 <math>t^{-2}</math> 是可积的).
Any analog signal is continuous by nature. [[Discrete-time signal]]s, used in [[digital signal processing]], can be obtained by [[Sampling (signal processing)|sampling]] and [[Quantization (signal processing)|quantization]] of continuous signals.
Any analog signal is continuous by nature. [[Discrete-time signal]]s, used in [[digital signal processing]], can be obtained by [[Sampling (signal processing)|sampling]] and [[Quantization (signal processing)|quantization]] of continuous signals.
任何模拟信号本质上都是连续的。[[数字信号处理]] '''Digital signal processing'''中使用的是[[离散时间信号]] '''Discrete-time signals'''，可以通过对连续信号的[[采样]] '''Sampling'''和[[量化]] '''Quantization'''来获得。
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 [[image processing]], where two space dimensions are used.
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 [[image processing]], where two space dimensions are used.
连续信号也可以定义在时间以外的独立变量上。另一个非常常见的独立变量是空间，它在[[图像处理]] '''Image processing'''中特别有用，在图像处理中使用两个空间维度。
==相关背景 Relevant contexts==
Discrete time is often employed when [[empirical]] [[measurement]]s are involved, because normally it is only possible to measure variables sequentially. For example, while [[economic activity]] 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, [[gross domestic product]] will show a sequence of [[Calendar year#Quarters|quarterly]] values.
Discrete time is often employed when [[empirical]] [[measurement]]s are involved, because normally it is only possible to measure variables sequentially. For example, while [[economic activity]] 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, [[gross domestic product]] will show a sequence of [[Calendar year#Quarters|quarterly]] values.
当涉及到[[经验测量]] '''Empirical measurements'''时，通常采用离散时间，因为通常只能按顺序测量变量。例如，虽然[[经济活动]] '''Economic activity'''实际上是连续发生的，但是没有经济完全停顿的时刻，只能对经济活动进行分散的测量。出于这个原因，例如，公布的[[国内生产总值]] '''Gross domestic product''' 数据将显示一系列的[[季度]] '''Quarterly'''值。
When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses [[time series]] or [[regression analysis|regression]] methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, ''y''<sub>''t''</sub> might refer to the value of [[income]] observed in unspecified time period ''t'', ''y''<sub>''3''</sub> to the value of income observed in the third time period, etc.
When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses [[time series]] or [[regression analysis|regression]] methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, ''y''<sub>''t''</sub> might refer to the value of [[income]] observed in unspecified time period ''t'', ''y''<sub>''3''</sub> to the value of income observed in the third time period, etc.
当人们试图根据其他变量和/或他们自己先前的值对这些变量进行实证解释时，他们使用[[时间序列]] '''Time series'''或[[回归]] '''Regression'''方法，在这些方法中，变量被索引，下标表示观测发生的时间周期。例如，''y''<sub>''t''</sub> 可能是指在非特定时间段 t 观察到的[[收入]] '''Income'''值，''y''<sub>''3''</sub> 是指在第三时间段观察到的收入值，等等。
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.
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.
此外，当研究人员试图发展一个理论，以解释什么是观察下的离散时间，往往理论本身是表达在离散时间，以促进发展的时间序列或回归模型。
On the other hand, it is often more mathematically [[Closed form solution|tractable]] to construct [[Scientific theory|theoretical model]]s in continuous time, and often in areas such as [[physics]] an exact description requires the use of continuous time. In a continuous time context, the value of a variable ''y'' at an unspecified point in time is denoted as ''y''(''t'') or, when the meaning is clear, simply as ''y''.
On the other hand, it is often more mathematically [[Closed form solution|tractable]] to construct [[Scientific theory|theoretical model]]s in continuous time, and often in areas such as [[physics]] an exact description requires the use of continuous time. In a continuous time context, the value of a variable ''y'' at an unspecified point in time is denoted as ''y''(''t'') or, when the meaning is clear, simply as ''y''.
另一方面，数学上[[更容易]] '''Tractable'''于处理在连续时间上建立[[理论模型]] '''Theoretical models'''，而在[[物理学]] '''Physics'''等领域，精确的描述往往需要使用连续时间。在连续时间上下文中，未指定时间点上变量 y 的值通常表示为 y (t) ，或者，如果含义清楚，则简单地表示为 y。
==方程的类型 Types of equations==
===离散时间 Discrete time===
Discrete time makes use of [[difference equation]]s, also known as recurrence relations. An example, known as the [[logistic map]] or logistic equation, is
Discrete time makes use of [[difference equation]]s, also known as recurrence relations. An example, known as the [[logistic map]] or logistic equation, is
离散时间利用[[差分方程]] '''Difference equations'''，也称为递推关系。一个被称为[[逻辑图]] '''Logistic map'''或逻辑方程的例子是
<math> x_{t+1} = rx_t(1-x_t),</math>
<math> x_{t+1} = rx_t(1-x_t),</math>
in which ''r'' is a [[Parameter#Mathematical functions|parameter]] in the range from 2 to 4 inclusive, and ''x'' is a variable in the range from 0 to 1 inclusive whose value in period ''t'' [[nonlinearity|nonlinearly]] affects its value in the next period, ''t''+1. For example, if <math>r=4</math> and <math>x_1 = 1/3</math>, then for ''t''=1 we have <math>x_2=4(1/3)(2/3)=8/9</math>, and for ''t''=2 we have <math>x_3=4(8/9)(1/9)=32/81</math>.
in which ''r'' is a [[Parameter#Mathematical functions|parameter]] in the range from 2 to 4 inclusive, and ''x'' is a variable in the range from 0 to 1 inclusive whose value in period ''t'' [[nonlinearity|nonlinearly]] affects its value in the next period, ''t''+1. For example, if <math>r=4</math> and <math>x_1 = 1/3</math>, then for ''t''=1 we have <math>x_2=4(1/3)(2/3)=8/9</math>, and for ''t''=2 we have <math>x_3=4(8/9)(1/9)=32/81</math>.
其中 r 是2到4包含的参数，x 是0到1包含的变量，其周期t的值对下一周期t+1的值产生非线性影响。例如，如果 <math>r=4</math> 和 <math>x_1 = 1/3</math>，那么对于 t=1，我们有 <math>x_2=4(1/3)(2/3)=8/9</math>，对于t=2,我们有<math>x_3=4(8/9)(1/9)=32/81</math>。
其中 r 是2到4包含的参数，x 是0到1包含的变量，其周期 t 的值对下一周期 t + 1的值产生非线性影响。例如，如果 < math > r = 4 </math > 和 < math > x1 = 1/3 </math > ，那么对于 t = 1，我们有 < math > x2 = 4(1/3)(2/3) = 8/9 </math > ，对于 t = 2，我们有 < math > x3 = 4(8/9)(1/9) = 32/81 </math > 。
Another example models the adjustment of a price P in response to non-zero excess demand for a product as
Another example models the adjustment of a price P in response to non-zero excess demand for a product as
另一个例子模型的[[价格]] '''Price''' ''P'' 的调整，以响应非零[[过剩需求|超额需求]] '''Excess demand'''的产品如下<math>P_{t+1} = P_t + \delta \cdot f(P_t,...)</math>
where <math>\delta</math> is the positive speed-of-adjustment parameter which is less than or equal to 1, and where <math>f</math> is the [[excess demand function]].
where <math>\delta</math> is the positive speed-of-adjustment parameter which is less than or equal to 1, and where <math>f</math> is the [[excess demand function]].
其中 <math>\delta</math> 是小于或等于1的正调整速度参数，而 <math>f</math> 是[[过剩需求|超额需求函数]] '''Excess demand function'''。
===连续时间 Continuous time ===
Continuous time makes use of [[differential equation]]s. For example, the adjustment of a price ''P'' in response to non-zero excess demand for a product can be modeled in continuous time as
Continuous time makes use of [[differential equation]]s. For example, the adjustment of a price ''P'' in response to non-zero excess demand for a product can be modeled in continuous time as
连续时间是利用的[[差分方程]] '''Difference equations'''。例如，对产品非零[[过剩需求|超额需求]] '''Excess demand'''价格 ''P'' 的调整可以在连续时间内建模为<math>\frac{dP}{dt}=\lambda \cdot f(P,...)</math>
where the left side is the [[first derivative]] of the price with respect to time (that is, the rate of change of the price), <math>\lambda</math> is the speed-of-adjustment parameter which can be any positive finite number, and <math>f</math> is again the excess demand function.
where the left side is the [[first derivative]] of the price with respect to time (that is, the rate of change of the price), <math>\lambda</math> is the speed-of-adjustment parameter which can be any positive finite number, and <math>f</math> is again the excess demand function.
其中左边是价格对时间的[[一阶导数]] '''First derivative'''(即价格变化率) ，<math>\lambda</math> 是调整速度参数，可以是任意正的有限数字，<math>f</math> 又是超额需求函数。
==图形描绘 Graphical depiction==
A variable measured in discrete time can be plotted as a [[step function]], in which each time period is given a region on the [[horizontal axis]] 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.
A variable measured in discrete time can be plotted as a [[step function]], in which each time period is given a region on the [[horizontal axis]] 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.
在离散时间中测量的一个变量可以绘制为一个[[阶跃函数]] '''Step function'''，其中每个时间周期在[[水平轴]] '''Horizontal axis'''上给定一个与每个其他时间周期相同长度的区域，测量的变量绘制为在整个时间周期区域内保持不变的高度。在这种图形技术中，图表显示为一系列水平步骤。或者，可以将每个时间段视为时间上的一个分离点，通常位于水平轴上的一个整数值处，测量变量绘制为该时间轴点之上的一个高度。在这种技术中，图表显示为一组点。
The values of a variable measured in continuous time are plotted as a [[continuous function]], since the domain of time is considered to be the entire real axis or at least some connected portion of it.
The values of a variable measured in continuous time are plotted as a [[continuous function]], since the domain of time is considered to be the entire real axis or at least some connected portion of it.
以连续时间测量的变量值被绘制为[[连续函数]] '''Continuous function'''，因为时间域被认为是整个实轴或至少它的一些连接部分。
==另请参阅 See also==
==See also==
{{div col|colwidth=30em}}
{{div col|colwidth=30em}}
*[[Aliasing]]
*[[别名 Aliasing]]
*[[Bernoulli process]]
*[[伯努利过程 Bernoulli process]]
*[[Digital data]]
*[[数字数据 Digital data]]
*[[Discrete calculus]]
*[[离散微积分 Discrete calculus]]
*[[Discrete system]]
*[[离散系统 Discrete system]]
*[[Discretization]]
*[[离散化 Discretization]]
*[[Normalized frequency (digital signal processing)|Normalized frequency]]
*[[归一化频率（数字信号处理）|归一化频率 Normalized frequency (digital signal processing)|Normalized frequency]]
*[[Time-scale calculus]]
*[[时间尺度演算 Time-scale calculus]]
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==参考文献 Reference==
== References ==
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{{reflist}}
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*{{cite book
*{{cite book
[[Category:Time in science]]
[[Category:Time in science]]
Category:Time in science
Category:Time in science
分类: 科学时间
分类: 时间科学
[[Category:Dynamical systems]]
[[Category:Dynamical systems]]