{"id":181,"date":"2016-08-07T22:40:53","date_gmt":"2016-08-08T06:40:53","guid":{"rendered":"http:\/\/wonghoi.humgar.com\/blog\/?page_id=181"},"modified":"2025-09-14T14:53:55","modified_gmt":"2025-09-14T22:53:55","slug":"ideas-oversimplified-signals-and-systems-3-prerequisities-linear-algebra","status":"publish","type":"page","link":"https:\/\/wonghoi.humgar.com\/blog\/ideas-oversimplified-signals-and-systems-3-prerequisities-linear-algebra\/","title":{"rendered":"Oversimplified: Signals and Systems (3) &#8211; Prereq: Linear Algebra"},"content":{"rendered":"<p>&#8216;Signals and Systems&#8217; (the title of Oppenheim and Wilsky&#8217;s book) class is also\u00a0called &#8216;Linear Signals and Systems&#8217; (by B.P. Lathi). The keyword of the day is &#8216;<span style=\"text-decoration: underline;\">linear<\/span>&#8216;. It&#8217;s the\u00a0same &#8216;linear&#8217; as in linear algebra!<\/p>\n<p>Most ideas in this class are very basic linear algebra ideas rolled\u00a0out\u00a0in its\u00a0most tedious, explicit long form because\u00a0they assumed you haven&#8217;t taken linear algebra yet*. I think it&#8217;s a horrible compromise because:<\/p>\n<ul>\n<li>there&#8217;s too\u00a0much information to organize in your head and remember. Not compact!<\/li>\n<li>it&#8217;s a huge intellectual jump when you get to\u00a0modern (as in state-space) approach later.<\/li>\n<li>with\u00a0computer data, you end up using linear algebra implicitly in this class anyway.<\/li>\n<\/ul>\n<p>By emphasizing\u00a0the concept of linearity at the beginning, you can take advantage of the linear algebra tools (especially inner products) to simplify the ideas in this class\u00a0to a few things that you can remember and interpret in seconds.<\/p>\n<hr \/>\n<p>Before I start,\u00a0I&#8217;d like to emphasize that &#8216;signals&#8217; and &#8216;systems&#8217; are the same mathematical object (vector or function) so the math is interchangeable. Nonetheless it&#8217;s often useful to call the inputs\/outputs &#8216;signals&#8217; and the function map &#8216;systems&#8217; to give so insights to the context of the problem at hand. In this class, &#8216;systems&#8217; are almost always linear, so it&#8217;s a linear function map.<\/p>\n<p>Most definitions and usages IN THIS CLASS\u00a0allows you to interchange the roles &#8216;signals&#8217; and &#8216;systems&#8217; as the math is symmetric (swapping them will give you the same results\u00a0=\u00a0<span style=\"text-decoration: underline;\">commutative<\/span>).<\/p>\n<p>Most frequently, a signal\/system is most often described as a function of time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-fd9cb27edab3f0a8a249bc80cc9c6ee2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"\/>, but it doesn&#8217;t have to be, as any arbitrary (scalar or vector) inputs with any meaning (such as\u00a0distance) will do. If\u00a0only spotty\u00a0time instances are recorded, we call it a discrete-time signal, or <span style=\"text-decoration: underline;\">time-series<\/span>.<\/p>\n<hr \/>\n<p>The word &#8216;linear&#8217; actually has a strict mathematical definition. Out of infinite possible functions (systems or <span style=\"text-decoration: underline;\">map<\/span>) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-f5844370b6482674a233a3063f762555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\"\/>, if we declare it to be <span style=\"text-decoration: underline;\">linear<\/span>, we are limiting it to the ones\u00a0satisfying\u00a0ALL of\u00a0the following:<\/p>\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 45px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-5889de0f4d7c6a34cdb1dd86e5d03d72_l3.png\" height=\"45\" width=\"673\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#102;&#40;&#120;&#43;&#121;&#41;&#32;&#38;&#32;&#61;&#32;&#102;&#40;&#120;&#41;&#43;&#102;&#40;&#121;&#41;&#32;&#32;&#38;&#32;&#92;&#44;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#110;&#111;&#114;&#109;&#97;&#108;&#123;&#65;&#100;&#100;&#105;&#116;&#105;&#118;&#105;&#116;&#121;&#58;&#32;&#67;&#97;&#110;&#32;&#111;&#112;&#101;&#114;&#97;&#116;&#101;&#32;&#115;&#101;&#112;&#97;&#114;&#97;&#116;&#101;&#108;&#121;&#32;&#97;&#110;&#100;&#32;&#112;&#111;&#111;&#108;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#115;&#117;&#108;&#116;&#115;&#32;&#98;&#97;&#99;&#107;&#125;&#92;&#92; &#102;&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#120;&#41;&#32;&#38;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#102;&#40;&#120;&#41;&#32;&#32;&#38;&#32;&#92;&#44;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#110;&#111;&#114;&#109;&#97;&#108;&#123;&#72;&#111;&#109;&#111;&#103;&#101;&#110;&#105;&#116;&#121;&#58;&#32;&#67;&#97;&#110;&#32;&#102;&#114;&#101;&#101;&#108;&#121;&#32;&#112;&#117;&#108;&#108;&#32;&#99;&#111;&#110;&#115;&#116;&#97;&#110;&#116;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#111;&#117;&#116;&#125;&#92;&#92; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>Verify by yourself\u00a0the following <strong>commonly used<\/strong> linear operators are indeed linear by the\u00a0definitions above. Try making up examples on your own:<\/p>\n<ul>\n<li>Derivative (e.g. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-be778e482f76370006c410570898b856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#32;&#61;&#32;&#92;&#115;&#105;&#110;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-17d82083272a99c1141f6447ae47fa67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#108;&#110;&#40;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"67\" style=\"vertical-align: -5px;\"\/>)<\/li>\n<li>Integral<\/li>\n<li>Expectation (aka the mean)<\/li>\n<\/ul>\n<p>These imposed\u00a0restrictions makes the math much much easier, something an engineer so eagerly\u00a0desires that they will try to\u00a0bend the problem to make the\u00a0system almost linear. It goes under the name of <span style=\"text-decoration: underline;\">linearization<\/span>. Transistors amplifies with approximate linearity.<\/p>\n<p>Also note that an\u00a0<span style=\"text-decoration: underline;\">affine<\/span> function, which is a linear function plus a constant <span style=\"text-decoration: underline;\">offset<\/span>, is\u00a0NOT\u00a0linear. Linear functions has to cross the <span style=\"text-decoration: underline;\">origin<\/span>. An example would be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-56bdc8c904d19ad365a6a1104910afcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#40;&#120;&#41;&#61;&#50;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"\/>. Try the definition above\u00a0with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-389c665cd6bee33adc8e40e1005726e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#120;&#95;&#49;&#43;&#120;&#95;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -3px;\"\/>, you will get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-2295be99080f6a422a3450f44715d890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#50;&#120;&#95;&#49;&#43;&#49;&#41;&#32;&#43;&#32;&#40;&#50;&#120;&#95;&#50;&#32;&#43;&#32;&#49;&#41;&#32;&#61;&#32;&#50;&#40;&#120;&#95;&#49;&#43;&#120;&#95;&#50;&#41;&#32;&#43;&#32;&#50;&#32;&#61;&#32;&#102;&#40;&#120;&#41;&#32;&#43;&#49;&#32;&#92;&#110;&#101;&#113;&#32;&#102;&#40;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"443\" style=\"vertical-align: -5px;\"\/>. The constant offset is\u00a0added twice instead of once!<\/p>\n<hr \/>\n<p>Students often think the definition of linearity is\u00a0stating the obvious, and didn&#8217;t\u00a0pay too much attention to it. Turns out it&#8217;s one of the golden tricks for the class called <strong><span style=\"text-decoration: underline;\">superposition<\/span><\/strong>.\u00a0You can use these properties freely in this class because almost everything (except a few trick problems) is declared\u00a0<span style=\"text-decoration: underline;\">linear<\/span>.<\/p>\n<p>As with\u00a0superposition in LINEAR\u00a0circuits, you can shut-off\u00a0any combinations of individual ADDITIVE input components (sources), process them separately with the same system, then\u00a0pool (add) the individual outputs together as the overall output.<\/p>\n<p>ONLY with\u00a0linear systems, the outputs <span style=\"text-decoration: underline;\">caused<\/span> by different\u00a0input components (however way you want to break it) will have nothing to do with each other (<span style=\"text-decoration: underline;\">no coupling<\/span>) as far as the output is concerned. If they couple, you are looking at a non-linear system, which is nasty to work with as you have to watch out for all possible\u00a0interactions.<\/p>\n<p>A useful\u00a0alternative way of saying the same thing is that any signal can be seen as a <span style=\"text-decoration: underline;\">linear combination<\/span> of multiple sources (feel free to define the sources\/basis that are convenient for you). I kept it generic here because you will see the same concept in convolution as well as all the transforms in this class.<\/p>\n<hr \/>\n<p>Now go and grab a basic linear algebra <a href=\"http:\/\/www.umassd.edu\/cas\/math\/people\/facultyandstaff\/steveleon\/\">book <\/a>and learn these basics, really understanding them:<\/p>\n<ul>\n<li>Elementary matrices (identity matrices, elementary vectors)<\/li>\n<li>Vector spaces (what I just covered above: <span style=\"text-decoration: underline;\">superposition<\/span>! Understand\u00a0<span style=\"text-decoration: underline;\">basis<\/span>\u00a0well)<\/li>\n<li>Linear transformations (they&#8217;re 100% of what linear systems are. Know\u00a0inverses)<\/li>\n<li>Orthorgonality and\u00a0INNER PRODUCTS (it&#8217;s the compact form the transforms in this class)<\/li>\n<\/ul>\n<p>These can\u00a0wait until you take the linear algebra class**:<\/p>\n<ul>\n<li>Gaussian elimination (goes under row echelon form), determinants<\/li>\n<li>Similarity transformations, eigen-decomposition \/ SVD<\/li>\n<li>Quadratic forms and least squares<\/li>\n<li>Gram-Schmidt (QR) factorization and numerical linear algebra.<\/li>\n<\/ul>\n<p>I recommend the 4th or\u00a05th edition of Steve Leon&#8217;s &#8220;Linear Algebra with Applications&#8221; because it&#8217;s short and gets to the point very quickly. Anything more is just clutter.<\/p>\n<p>The only important\u00a0idea that&#8217;s missing from Leon&#8217;s\u00a0book, which is covered in the first lecture by Gilbert Strang is this:<\/p>\n<ul>\n<li>Left multiplication by\u00a0a matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-23b3267205a9cb87b6ad7e4302f40dcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"\/> over input vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-5326c7e41f1a35d50be59758b3632140_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#102;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>, i.e <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-cb99297d4d7942e8731fedda0e939e4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#102;&#123;&#65;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\"\/> means scaling each column of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-23b3267205a9cb87b6ad7e4302f40dcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#102;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-7bc7ac9d08cd1d5b326679dba4ad5501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#98;&#102;&#123;&#40;&#97;&#95;&#105;&#41;&#125;&#125;&#32;&#92;&#102;&#111;&#114;&#97;&#108;&#108;&#32;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -5px;\"\/> by the weights (elements) of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-59099ff0ab1ea851291b419311c8bfd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> and add them all up:\n<p class=\"ql-center-displayed-equation\" style=\"line-height: 38px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/ql-cache\/quicklatex.com-00dd0ec4eb285a3071e9e689f83e0f58_l3.png\" height=\"38\" width=\"123\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#98;&#102;&#123;&#65;&#120;&#125;&#32;&#38;&#32;&#61;&#32;&#92;&#115;&#117;&#109;&#95;&#123;&#105;&#125;&#32;&#123;&#92;&#98;&#102;&#123;&#40;&#97;&#95;&#105;&#41;&#125;&#125;&#32;&#120;&#95;&#105; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>This is the foundation of &#8216;basis&#8217;, which I&#8217;ll explain later how it relates to all the transformations in this class.<\/li>\n<\/ul>\n<hr \/>\n<p>* Gilbert Strang propose that the math curriculum can simply <a href=\"http:\/\/www-math.mit.edu\/~gs\/videos\/index.html\">start with linear algebra without Calculus<\/a>. The neat thing about them is they don&#8217;t depend on each other, so you can learn it in different order.<\/p>\n<p>So far almost all schools starts with calculus because it was historically way more established than linear algebra. Integrals and derivatives are linear operators, which can be expressed in matrices as well, so I can say linear algebra is in no way less powerful. Nowadays, since most digital information is stored in vectors, they are more readily consumed by tools of linear algebra than calculus, so I expect the balance of power will change in the coming decades.<\/p>\n<p>What the\u00a0gateway &#8216;signals and systems&#8217;\u00a0class teaches is\u00a0the &#8216;classic&#8217; approach, which is\u00a0heavy on\u00a0the 4 basic transforms written in calculus forms. The &#8216;modern&#8217; approach, which is always\u00a0taught in state-space control, sees everything as matrices and vectors: each of the 4 transforms\u00a0are nothing but\u00a0left multiplying by a matrix (a linear map). As we will see later, DFT (Discrete Fourier Transform) is just left multiplying your data (vector) by\u00a0a matrix.<\/p>\n<p>** These are very important materials if you get deep into signal processing, but won&#8217;t help you in the gateway class. For example, the\u00a0connection between eigenvalues and transformations in the class is a very advanced topic way after this class. Least squares are crucial when you get to statistical signal processing.<\/p>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_181\" class=\"pvc_stats all  \" data-element-id=\"181\" style=\"\"><i class=\"pvc-stats-icon medium\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>&#8216;Signals and Systems&#8217; (the title of Oppenheim and Wilsky&#8217;s book) class is also\u00a0called &#8216;Linear Signals and Systems&#8217; (by B.P. Lathi). The keyword of the day is &#8216;linear&#8216;. It&#8217;s the\u00a0same &#8216;linear&#8217; as in linear algebra! Most ideas in this class are &hellip; <a href=\"https:\/\/wonghoi.humgar.com\/blog\/ideas-oversimplified-signals-and-systems-3-prerequisities-linear-algebra\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_181\" class=\"pvc_stats all  \" data-element-id=\"181\" style=\"\"><i class=\"pvc-stats-icon medium\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/wonghoi.humgar.com\/blog\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-181","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages\/181","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/comments?post=181"}],"version-history":[{"count":34,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages\/181\/revisions"}],"predecessor-version":[{"id":6564,"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/pages\/181\/revisions\/6564"}],"wp:attachment":[{"href":"https:\/\/wonghoi.humgar.com\/blog\/wp-json\/wp\/v2\/media?parent=181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}