离散时间和连续时间

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In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.

In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time.

在数学动力学中,离散时间和连续时间是两种可供选择的框架,在这两种框架内可以对随时间演化的变量建模。


Discrete time

文件:Sampled.signal.svg
Discrete sampled signal

Discrete sampled signal

离散采样信号


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(”时间周期”)中没有变化——也就是说,时间被视为一个离散变量。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。这种时间观相当于一个数字时钟,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。测量通常按照变量“时间”的连续整数值进行。


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系列量组成的时间序列。


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 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 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,它可能是从连续时间信号中采样得到的。当一个离散时间信号是通过采样序列在均匀间隔时间,它有一个相关的采样率。


Discrete-time signals may have several origins, but can usually be classified into one of two groups:[1]

Discrete-time signals may have several origins, but can usually be classified into one of two groups:

离散时间信号可能有几种来源,但通常可分为两类:


  • By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.


Continuous time

In contrast, continuous time views variables as having a particular value for potentially only an infinitesimally short amount of time. Between any two points in time there are an 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 infinitesimally short amount of time. Between any two points in time there are an 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.

相比之下,连续时间将变量视为具有特定值的变量,其时间可能只有无限短的时间。在任意两个时间点之间,还有无数其他时间点。变量“时间”的范围是整个实数行,或者取决于上下文,取决于它的某个子集,如非负雷亚尔。因此,时间被看作是一个连续变量。


A continuous signal or a continuous-time signal is a varying quantity (a signal)

A continuous signal or a continuous-time signal is a varying quantity (a signal)

连续信号或连续时间信号是可变量(信号)

whose domain, which is often time, is a continuum (e.g., a connected interval of the reals). That is, the function's domain is an uncountable set. The function itself need not be continuous. To contrast, a discrete time signal has a countable domain, like the natural numbers.

whose domain, which is often time, is a continuum (e.g., a connected interval of the reals). That is, the function's domain is an uncountable set. The function itself need not be continuous. To contrast, a discrete time signal has a countable domain, like the natural numbers.

其域(通常是时间)是一个连续统一体(例如,雷亚尔的连通区间)。也就是说,函数的域是一个不可数的集合。函数本身不需要是连续的。相比之下,离散时间信号像自然数一样有一个可数域。


A signal of continuous amplitude and time is known as a continuous-time signal or an analog signal. This (a 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) 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。这(一个信号)在任何时刻都有一定的价值。电信号与温度、压力、声音等物理量成正比。通常是连续信号。连续信号的其他例子有正弦波、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:


[math]\displaystyle{ f(t) = \sin(t), \quad t \in \mathbb{R} }[/math]

[math]\displaystyle{ f(t) = \sin(t), \quad t \in \mathbb{R} }[/math]

数学中的四个 t = sin (t)


A finite duration counterpart of the above signal could be:

A finite duration counterpart of the above signal could be:

上述信号的一个有限持续时间的副本可以是:


[math]\displaystyle{ f(t) = \sin(t), \quad t \in [-\pi,\pi] }[/math] and [math]\displaystyle{ f(t) = 0 }[/math] otherwise.

[math]\displaystyle{ f(t) = \sin(t), \quad t \in [-\pi,\pi] }[/math] and [math]\displaystyle{ f(t) = 0 }[/math] otherwise.

[-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,


[math]\displaystyle{ f(t) = \frac{1}{t}, \quad t \in [0,1] }[/math] and [math]\displaystyle{ f(t) = 0 }[/math] otherwise,

[math]\displaystyle{ f(t) = \frac{1}{t}, \quad t \in [0,1] }[/math] and [math]\displaystyle{ f(t) = 0 }[/math] otherwise,

[0,1] </math > 和 < math > f (t) = 0 </math > 否则,


is a finite duration signal but it takes an infinite value for [math]\displaystyle{ t = 0\, }[/math].

is a finite duration signal but it takes an infinite value for [math]\displaystyle{ t = 0\, }[/math].

是一个有限持续时间的信号,但是对于 < math > t = 0,</math > 它取一个无限值。


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。


For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the [math]\displaystyle{ t^{-1} }[/math] signal is not integrable at infinity, but [math]\displaystyle{ t^{-2} }[/math] is).

For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the [math]\displaystyle{ t^{-1} }[/math] signal is not integrable at infinity, but [math]\displaystyle{ 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 > 信号在无穷远处不可积,但 < math > t ^ {-2} </math > 可积)。


Any analog signal is continuous by nature. Discrete-time signals, used in digital signal processing, can be obtained by sampling and quantization of continuous signals.

Any analog signal is continuous by nature. Discrete-time signals, used in digital signal processing, can be obtained by sampling and quantization of continuous signals.

任何模拟信号本质上都是连续的。数字信号处理中使用的离散时间信号,可以通过对连续信号的采样和量化来获得。


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,它在图像处理中特别有用,在图像处理中使用两个空间维度。


Relevant contexts

Discrete time is often employed when empirical measurements 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 quarterly values.

Discrete time is often employed when empirical measurements 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 quarterly values.

当涉及到经验测量时,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(gdp)数据将显示一系列的季度价值。


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 methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, yt might refer to the value of income observed in unspecified time period t, y3 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 methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, yt might refer to the value of income observed in unspecified time period t, y3 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,他们使用时间序列或回归方法,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 可能是指在非特定时间段 t 观察到的收入值,y < sub > 3 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,以解释什么是观察在离散时间,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 tractable to construct theoretical models 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 tractable to construct theoretical models 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,在连续时间内建立理论模型在数学上更易于处理,而在物理学等领域,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。


Types of equations

Discrete time

Discrete time makes use of difference equations, also known as recurrence relations. An example, known as the logistic map or logistic equation, is

Discrete time makes use of difference equations, 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,也称为递推关系。一个被称为逻辑斯谛图或逻辑斯谛方程的例子是


[math]\displaystyle{ x_{t+1} = rx_t(1-x_t), }[/math]

[math]\displaystyle{ 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 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 nonlinearly affects its value in the next period, t+1. For example, if [math]\displaystyle{ r=4 }[/math] and [math]\displaystyle{ x_1 = 1/3 }[/math], then for t=1 we have [math]\displaystyle{ x_2=4(1/3)(2/3)=8/9 }[/math], and for t=2 we have [math]\displaystyle{ x_3=4(8/9)(1/9)=32/81 }[/math].

in which r is a 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 nonlinearly affects its value in the next period, t+1. For example, if [math]\displaystyle{ r=4 }[/math] and [math]\displaystyle{ x_1 = 1/3 }[/math], then for t=1 we have [math]\displaystyle{ x_2=4(1/3)(2/3)=8/9 }[/math], and for t=2 we have [math]\displaystyle{ 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 p 的调整,以响应非零过剩需求的产品如下


[math]\displaystyle{ P_{t+1} = P_t + \delta \cdot f(P_t,...) }[/math]

[math]\displaystyle{ P_{t+1} = P_t + \delta \cdot f(P_t,...) }[/math]

< math > p _ { t + 1} = p _ t + delta cdot f (p _ t,...) </math >


where [math]\displaystyle{ \delta }[/math] is the positive speed-of-adjustment parameter which is less than or equal to 1, and where [math]\displaystyle{ f }[/math] is the excess demand function.

where [math]\displaystyle{ \delta }[/math] is the positive speed-of-adjustment parameter which is less than or equal to 1, and where [math]\displaystyle{ 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,< math > f </math > 是超额需求函数。


Continuous time

Continuous time makes use of differential equations. 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 equations. 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

连续时间利用微分方程。例如,对产品非零过剩需求的价格 p 的调整可以在连续时间内建模为


[math]\displaystyle{ \frac{dP}{dt}=\lambda \cdot f(P,...) }[/math]

[math]\displaystyle{ \frac{dP}{dt}=\lambda \cdot f(P,...) }[/math]

(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]\displaystyle{ \lambda }[/math] is the speed-of-adjustment parameter which can be any positive finite number, and [math]\displaystyle{ 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]\displaystyle{ \lambda }[/math] is the speed-of-adjustment parameter which can be any positive finite number, and [math]\displaystyle{ f }[/math] is again the excess demand function.

其中左边是价格对时间的一阶导数(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。


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,其中每个时间周期在水平轴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.

由于时间域被认为是整个实轴,或者至少是其中的某个连接部分,因此将连续时间中测量的变量的值绘制成连续函数。


See also


References

  1. "Digital Signal Processing" Prentice Hall - Pages 11-12
  2. "Digital Signal Processing: Instant access." Butterworth-Heinemann - Page 8


  • Gershenfeld, Neil A.

1 = Gershenfeld,Neil a. (1999

1999年). [2012年10月15日 The Nature of mathematical Modeling

数学建模的本质]. Cambridge University Press

剑桥大学出版社. pp. 2012年10月11日. ISBN 0-521-57095-6. 2012年10月22日. 2012年10月15日. 

| isbn = 0-521-57095-6}}

| isbn = 0-521-57095-6}}


  • Wagner, Thomas Charles Gordon

1 = Wagner,Thomas Charles Gordon (1959

1959年). [2012年10月15日 分析瞬态]. Wiley. pp. 2012年10月11日. 2012年10月22日. 2012年10月15日. 

| isbn = }}

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Category:Time in science

分类: 科学时间

Category:Dynamical systems

类别: 动力系统


This page was moved from wikipedia:en:Discrete time and continuous time. Its edit history can be viewed at 离散时间和连续时间/edithistory