62:. 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
47:
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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
205:) 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.
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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.
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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
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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
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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,
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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.
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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
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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.
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from a continuous-time signal. When a discrete-time signal is obtained by sampling a sequence at uniformly spaced times, it has an associated
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Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by
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For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the
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methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example,
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When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses
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By observing an inherently discrete-time process, such as the weekly peak value of a particular economic indicator.
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an exact description requires the use of continuous time. In a continuous time context, the value of a variable
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are involved, because normally it is only possible to measure variables sequentially. For example, while
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Discrete-time signals may have several origins, but can usually be classified into one of two groups:
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The value of a finite (or infinite) duration signal may or may not be finite. For example,
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is the positive speed-of-adjustment parameter which is less than or equal to 1, and where
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in response to non-zero excess demand for a product can be modeled in continuous time as
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A signal of continuous amplitude and time is known as a continuous-time signal or an
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is the speed-of-adjustment parameter which can be any positive finite number, and
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of the price with respect to time (that is, the rate of change of the price),
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number of other points in time. The variable "time" ranges over the entire
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is a variable in the range from 0 to 1 inclusive whose value in period
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The values of a variable measured in continuous time are plotted as a
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short amount of time. Between any two points in time there are an
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to the value of income observed in the third time period, etc.
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is a finite duration signal but it takes an infinite value for
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A finite duration counterpart of the above signal could be:
19:"Discrete signal" redirects here. Not to be confused with
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A variable measured in discrete time can be plotted as a
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views variables as having a particular value only for an
1470:"Digital Signal Processing", Prentice Hall - pages 11–12
1322:{\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...)}
1186:{\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...)}
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Frameworks for modeling variables that evolve over time
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A typical example of an infinite duration signal is:
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at constant or variable rate. This process is called
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274:{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }
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34:are two alternative frameworks within which
1093:Another example models the adjustment of a
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606:Any analog signal is continuous by nature.
162:) whose domain, which is often time, is a
573:signal is not integrable at infinity, but
1252:. For example, the adjustment of a price
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838:in the range from 2 to 4 inclusive, and
820:{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}
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355:{\displaystyle f(t)=\sin(t),\quad t\in }
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1083:{\displaystyle x_{3}=4(8/9)(1/9)=32/81}
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1509:Wagner, Thomas Charles Gordon (1959).
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849:affects its value in the next period,
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998:{\displaystyle x_{2}=4(1/3)(2/3)=8/9}
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637:Discrete time is often employed when
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1490:The Nature of mathematical Modeling
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913:{\displaystyle x_{1}=1/3}
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1009:=2 we have
924:=1 we have
920:, then for
847:nonlinearly
662:time series
79:time series
1522:Categories
1458:References
1005:, and for
666:regression
610:, used in
506:otherwise,
397:otherwise.
201:. This (a
180:continuous
1344:λ
1293:⋅
1290:λ
1204:δ
1150:⋅
1147:δ
836:parameter
830:in which
799:−
703:tractable
655:quarterly
639:empirical
586:−
556:−
444:∈
347:π
341:π
338:−
332:∈
313:
264:∈
245:
188:countable
168:connected
166:(e.g., a
164:continuum
36:variables
1513:. Wiley.
1411:Aliasing
1404:See also
657:values.
616:sampling
156:quantity
133:infinite
109:sampling
90:sampling
83:sequence
1236:is the
711:physics
64:integer
1496:
1196:where
679:income
203:signal
160:signal
1095:price
834:is a
603:is).
172:reals
150:or a
77:is a
1494:ISBN
879:and
618:and
471:and
362:and
30:and
664:or
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310:sin
242:sin
158:(a
73:or
1524::
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1139:t
1135:P
1131:=
1126:1
1123:+
1120:t
1116:P
1098:P
1074:/
1067:=
1064:)
1061:9
1057:/
1053:1
1050:(
1047:)
1044:9
1040:/
1036:8
1033:(
1030:4
1027:=
1022:3
1018:x
1007:t
993:9
989:/
985:8
982:=
979:)
976:3
972:/
968:2
965:(
962:)
959:3
955:/
951:1
948:(
945:4
942:=
937:2
933:x
922:t
908:3
904:/
900:1
897:=
892:1
888:x
867:4
864:=
861:r
851:t
844:t
840:x
832:r
815:,
812:)
807:t
803:x
796:1
793:(
788:t
784:x
780:r
777:=
772:1
769:+
766:t
762:x
727:y
723:t
721:(
719:y
715:y
691:3
687:y
683:t
674:t
670:y
589:2
582:t
559:1
552:t
524:0
521:=
518:t
494:0
491:=
488:)
485:t
482:(
479:f
459:]
456:1
453:,
450:0
447:[
441:t
437:,
432:t
429:1
424:=
421:)
418:t
415:(
412:f
385:0
382:=
379:)
376:t
373:(
370:f
350:]
344:,
335:[
329:t
325:,
322:)
319:t
316:(
307:=
304:)
301:t
298:(
295:f
268:R
261:t
257:,
254:)
251:t
248:(
239:=
236:)
233:t
230:(
227:f
111:.
23:.
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