From 371ffa957e08bee1a4df919a97674696c1908d43 Mon Sep 17 00:00:00 2001 From: cheese-cracker Date: Sun, 15 May 2022 17:29:28 +0000 Subject: [PATCH] deploy: 662c54e457a081c9036c14febc299fe908079895 --- categories/ideas/index.xml | 2 +- categories/index.xml | 2 +- categories/notes/index.xml | 2 +- index.xml | 2 +- posts/discrete-vs-continuous/index.html | 2 +- posts/index.xml | 2 +- search/index.xml | 2 +- tags/c++/index.xml | 2 +- tags/cs/index.xml | 2 +- tags/economics/index.xml | 2 +- tags/graphs/index.xml | 2 +- tags/index.xml | 2 +- tags/index/index.xml | 2 +- tags/math/index.xml | 2 +- tags/mentalmodels/index.xml | 2 +- tags/numericalmethods/index.xml | 2 +- tags/signalprocessing/index.xml | 2 +- 17 files changed, 17 insertions(+), 17 deletions(-) diff --git a/categories/ideas/index.xml b/categories/ideas/index.xml index e2a6e61..da039fa 100644 --- a/categories/ideas/index.xml +++ b/categories/ideas/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/categories/index.xml b/categories/index.xml index 40d674c..c711070 100644 --- a/categories/index.xml +++ b/categories/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/categories/notes/index.xml b/categories/notes/index.xml index 4cd9308..85e0fbb 100644 --- a/categories/notes/index.xml +++ b/categories/notes/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/index.xml b/index.xml index 9546e8b..0aa287a 100644 --- a/index.xml +++ b/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/posts/discrete-vs-continuous/index.html b/posts/discrete-vs-continuous/index.html index 2a5200d..25f5ccb 100644 --- a/posts/discrete-vs-continuous/index.html +++ b/posts/discrete-vs-continuous/index.html @@ -1,7 +1,7 @@ Discrete Vs Continuous – A House of Ideas

Discrete Vs Continuous

2022-01-26

#signalprocessing#math#mentalmodels#numericalmethods

Introduction

In this post, we delve into the mental model (or abstract feeling) of things being continuous Vs discrete. -In the direction of continuous vs discrete ideas, we look into signal processing, numerical methods and random ideas.

Poster originally by mathbytori.blogspot.com

Throughout this post, the terms ‘discrete’ and ‘continuous’ are somewhat loosely defined.

A simple intuition on continuous and discrete would give us,

DiscreteContinuous
Distinct, seperate and countableFluid, mostly singular
Dotted LineNormal straight Line
ScatteredUnionized
Set of Integers $Z$: Distinct points on the number lineSet of Real Numbers $\Re$: Continuous line on the number line
Sets not being dense (in any metric space)Set is dense (in any metric space)
Non-seperable Metric Space(like discrete metric space)Seperable Metric Space
Discrete Probability DistributionContinuous Probability Distributions
HistogramsKernel Density Estimates
Digital Image ProcessingAnalog Image Processing
Discrete Delaunay TriangulationsContinuous Voronoi Diagrams
Discrete Time SignalsContinuous Time Signals
Sum of many discrete rectanglesArea under a continuous curve
Discrete FactorialsContinuous Gamma Function
Discrete TransformsContinuous Transforms
Discrete Recurrence RelationsContinuous Differential Equations

A common trend to notice is that discrete things tend to take a finite or countable set of values.

Discrete Signals Vs Continuous Signals

Moving to more concrete definitions, +In the direction of continuous vs discrete ideas, we look into signal processing, numerical methods and random ideas.

Poster originally by mathbytori.blogspot.com

Throughout this post, the terms ‘discrete’ and ‘continuous’ are somewhat loosely defined.

A simple intuition on continuous and discrete would give us,

DiscreteContinuous
Distinct, seperate and countableFluid, mostly singular
Dotted LineNormal straight Line
ScatteredUnionized
Set of Integers $Z$: Distinct points on the number lineSet of Real Numbers $\Re$: Continuous line on the number line
Sets not dense (in any metric space)Set is dense (in any metric space)
Non-seperable Metric Space(like discrete metric space)Seperable Metric Space
Discrete Probability DistributionContinuous Probability Distributions
HistogramsKernel Density Estimates
Digital Image ProcessingAnalog Image Processing
Discrete Delaunay TriangulationsContinuous Voronoi Diagrams
Discrete Time SignalsContinuous Time Signals
Sum of many discrete rectanglesArea under a continuous curve
Discrete FactorialsContinuous Gamma Function
Discrete TransformsContinuous Transforms
Discrete Recurrence RelationsContinuous Differential Equations

A common trend to notice is that discrete things tend to take a finite or countable set of values.

Discrete Signals Vs Continuous Signals

Moving to more concrete definitions, Let us take the example of Continuous vs Discrete in the context of Signals. This divides us into four parts shown below,

Four Divisions of Signals

Clockwise: CT Analog, CT Digital, DT Digital, DT Analog

The differentiating factors are,

  • Discrete time (DT) Vs Continuous time (CT)
  • Analog signal Vs Digital signal

Here, digital signal and analog signals are both continuous in time but the CT digital signal has only a discrete set of values for the ouput $y(t)$.

In this post, the ideas of ‘digital’ and ‘discrete’ may be mixed because ‘digital’ signals are still discrete in the y-axis.

Area under a Continuous Curve Vs Sum of many Discrete Rectangles

Considering the area under the curve for any continuous function $y(t)$, diff --git a/posts/index.xml b/posts/index.xml index 5859c22..5a9696b 100644 --- a/posts/index.xml +++ b/posts/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/search/index.xml b/search/index.xml index b3f49a1..52fb9f5 100644 --- a/search/index.xml +++ b/search/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/c++/index.xml b/tags/c++/index.xml index f7a4ad6..3ae9f55 100644 --- a/tags/c++/index.xml +++ b/tags/c++/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/cs/index.xml b/tags/cs/index.xml index b29cabc..a360b15 100644 --- a/tags/cs/index.xml +++ b/tags/cs/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/economics/index.xml b/tags/economics/index.xml index 7261fb1..522b742 100644 --- a/tags/economics/index.xml +++ b/tags/economics/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/graphs/index.xml b/tags/graphs/index.xml index 09cc27d..d72b451 100644 --- a/tags/graphs/index.xml +++ b/tags/graphs/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/index.xml b/tags/index.xml index 9847e53..f5e08d1 100644 --- a/tags/index.xml +++ b/tags/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/index/index.xml b/tags/index/index.xml index 65dac6d..4a67c3a 100644 --- a/tags/index/index.xml +++ b/tags/index/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/math/index.xml b/tags/math/index.xml index 4ed8fd9..e8151bf 100644 --- a/tags/math/index.xml +++ b/tags/math/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/mentalmodels/index.xml b/tags/mentalmodels/index.xml index d36c631..550fba6 100644 --- a/tags/mentalmodels/index.xml +++ b/tags/mentalmodels/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/numericalmethods/index.xml b/tags/numericalmethods/index.xml index 191c54b..ccf7c12 100644 --- a/tags/numericalmethods/index.xml +++ b/tags/numericalmethods/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr> diff --git a/tags/signalprocessing/index.xml b/tags/signalprocessing/index.xml index d2d2d42..41bc12a 100644 --- a/tags/signalprocessing/index.xml +++ b/tags/signalprocessing/index.xml @@ -34,7 +34,7 @@ In the direction of continuous vs discrete ideas, we look into signal processing <td style="text-align:center">Set of Real Numbers $\Re$: Continuous line on the number line</td> </tr> <tr> -<td style="text-align:center">Sets not being dense <small>(in any metric space)</small></td> +<td style="text-align:center">Sets not dense <small>(in any metric space)</small></td> <td style="text-align:center">Set is dense <small>(in any metric space)</small></td> </tr> <tr>