{"id":83531,"date":"2021-04-13T13:20:29","date_gmt":"2021-04-13T17:20:29","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=83531"},"modified":"2022-11-21T09:47:20","modified_gmt":"2022-11-21T14:47:20","slug":"trend-following-filters-part-3","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/","title":{"rendered":"Trend-Following Filters \u2013 Part 3"},"content":{"rendered":"\n<p><em>The article &#8220;Trend-Following Filters \u2013 Part 3&#8221; first appeared on <a href=\"https:\/\/alphaarchitect.com\/2021\/04\/08\/trend-following-filters-part-3\/\">Alpha Architect Blog<\/a>.<\/em><\/p>\n\n\n\n<p>This is the third article in a series of three, the first two are available&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2020\/12\/29\/trend-following-filters-part-1-2\/\" target=\"_blank\" rel=\"noreferrer noopener\">here&nbsp;<\/a>and&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2021\/01\/21\/trend-following-filters-part-2-2\/\" target=\"_blank\" rel=\"noreferrer noopener\">here<\/a>. Those articles focus on examining from a digital signal processing (DSP) perspective&nbsp;<sup>1<\/sup>&nbsp;various types of digital filters that are designed to model trends in time series, in order to illustrate their properties and limitations. This article discusses a different signal processing tool called a \u201cfilter bank\u201d.<\/p>\n\n\n\n<p>Filter banks are used in many areas of electrical engineering, telecommunications, and other fields in applications such as audio, video, and image noise reduction, compression, storage, and transmission. A familiar example of a filter bank is the audio graphic equalizer, which can be used to alter music sound quality, e.g., to boost bass frequencies, which are generally between 20 and 300 Hertz (cycles per second), relative to treble frequencies, which are generally between 2,000 and 16,000 Hertz.<\/p>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" width=\"400\" height=\"109\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-1.png\" alt=\"\" class=\"wp-image-83539 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-1.png 400w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-1-300x82.png 300w\" data-sizes=\"(max-width: 400px) 100vw, 400px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 400px; aspect-ratio: 400\/109;\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-1-digital-analysis-synthesis-filter-bank\">1.&nbsp;&nbsp;&nbsp; Digital Analysis-Synthesis Filter Bank<\/h2>\n\n\n\n<p>Filter banks&nbsp;<sup>2<\/sup>&nbsp;can be designed to work with either continuous-time analog signals, such as music and speech, or discrete-time digital signals, such as high-definition television and wireless communication. The general form of a digital filter bank that can be used to implement a range of different operations is called an \u201cM-band digital analysis-synthesis filter bank\u201d and is comprised of three parts:<\/p>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" width=\"1100\" height=\"460\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-2-1-1100x460.png\" alt=\"\" class=\"wp-image-83552 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-2-1-1100x460.png 1100w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-2-1-700x293.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-2-1-300x126.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-2-1-768x321.png 768w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-2-1.png 1200w\" data-sizes=\"(max-width: 1100px) 100vw, 1100px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/460;\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-center\">M-Band Digital Analysis-Synthesis Filter Bank<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Analysis filter bank \u2013 Splits the input signal x(t) at each integer time step t into an M number of frequency components (M &gt; 1), called \u201csub-bands\u201d, using a set of digital filters arranged in parallel. Each filter in the analysis filter bank is designed to pass a specific frequency sub-band \u03bd<sub>i<\/sub>(t), with each sub-band occupying a portion of the original frequency band of x(t). These filters are represented by the H<sub>i<\/sub>(z) blocks in the diagram above.<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" width=\"800\" height=\"279\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-3.png\" alt=\"\" class=\"wp-image-83542 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-3.png 800w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-3-700x244.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-3-300x105.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-3-768x268.png 768w\" data-sizes=\"(max-width: 800px) 100vw, 800px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 800px; aspect-ratio: 800\/279;\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\"><li>Processing \u2013 Performs transformations, such as compression, on the individual sub-bands, based on the filter bank design objective. The processed sub-band outputs are represented by \u03bd^<sub>i<\/sub>(t).<\/li><li>Synthesis filter bank \u2013 Combines the processed sub-band outputs into a single output signal y(t) using addition. The synthesis operation can include ancillary filtering operations, represented by the G<sub>i<\/sub>(z) blocks.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-2-example-digital-analysis-synthesis-filter-bank\">2. Example Digital Analysis-Synthesis Filter Bank<\/h2>\n\n\n\n<p>The following example illustrates the digital analysis-synthesis filter bank concept. The design objective is to produce a \u201cde-noised\u201d output y(t) of the input time series x(t).<\/p>\n\n\n\n<p>The three parts of the example filter bank are as follows:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Analysis filter bank \u2013 The analysis filter bank is comprised of a set of six second-order IIR bandpass digital filters (M = 6). The details of the bandpass filter design are described in the Appendix. The center periods P<sub>0<\/sub>&nbsp;of the six filters are set to 15, 88, 513, 2990, 17427, and 101572 time samples, respectively, in order to have nonoverlapping passbands at the -3 dB (half-power) period cutoff points of the filters. This arrangement effectively attenuates short periods below around 6 time samples, which are considered to be \u201cnoise\u201d with respect to daily data (i.e., an input x(t) sampling interval of once per trading day at the market close) for the purposes of the example, while passing the longer periods. The quality factor Q of each filter is set to 0.5, i.e., critically damped (see the Appendix), in order to balance filter output responsiveness versus lag.<\/li><li>Processing \u2013 No additional processing is performed. The analysis filter bank outputs \u03bd<sub>i<\/sub>(t) are passed directly to the synthesis filter bank inputs \u03bd^<sub>i<\/sub>(t).<\/li><li>Synthesis filter bank \u2013 The G<sub>i<\/sub>(z) filters adjust the gain of each \u03bd^<sub>i<\/sub>(t) using multiplication so that the average maximum passband magnitude of the output y(t) is approximately 1.0 (0 dB):<\/li><\/ul>\n\n\n\n<p class=\"has-text-align-center has-cyan-bluish-gray-background-color has-background\">G<sub>i<\/sub>(z)&nbsp;= 0.886<\/p>\n\n\n\n<p>The graphs below show the magnitude and phase spectrums of the individual analysis filter bank outputs \u03bd<sub>i<\/sub>(t) and the synthesis filter bank output y(t). Note that the horizontal x-axis of both graphs uses a base-10 logarithmic scale, which is nonlinear.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"800\" height=\"524\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-4.png\" alt=\"Trend-Following Filter\" class=\"wp-image-83563 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-4.png 800w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-4-700x459.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-4-300x197.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-4-768x503.png 768w\" data-sizes=\"(max-width: 800px) 100vw, 800px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 800px; aspect-ratio: 800\/524;\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"800\" height=\"519\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-5.png\" alt=\"\" class=\"wp-image-83592 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-5.png 800w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-5-700x454.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-5-300x195.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/alpha-architect-filter-5-768x498.png 768w\" data-sizes=\"(max-width: 800px) 100vw, 800px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 800px; aspect-ratio: 800\/519;\" \/><\/figure>\n\n\n\n<p><em>Visit <a href=\"https:\/\/alphaarchitect.com\/2021\/04\/08\/trend-following-filters-part-3\/\">Alpha Architect Blog<\/a> to read the rest of the article.<\/em><\/p>\n\n\n\n<p>Notes:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>For background information on DSP, read my post&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2020\/08\/13\/an-introduction-to-digital-signal-processing-for-trend-following\/\" target=\"_blank\" rel=\"noreferrer noopener\">An Introduction to Digital Signal Processing for Trend Following<\/a>&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2021\/04\/08\/trend-following-filters-part-3\/#return-note-60930-1\">\u21a9<\/a><\/li><li>A detailed discussion of digital filter banks can be found in: Diniz, P. S. R., da Silva, E. A., B., and Netto, S. L.,&nbsp;<em><a href=\"https:\/\/www.amazon.com\/Digital-Signal-Processing-System-Analysis\/dp\/0521887755\" target=\"_blank\" rel=\"noreferrer noopener\">Digital Signal Processing \u2013 System Analysis and Design<\/a><\/em>, Cambridge University Press, 2010.&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2021\/04\/08\/trend-following-filters-part-3\/#return-note-60930-2\">\u21a9<\/a><\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Henry Stern explores a signal processing tool called a &#8220;filter bank&#8221; and demonstrates analysis filter bank and synthesis filter bank.<\/p>\n","protected":false},"author":575,"featured_media":83578,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341,352,344],"tags":[9537,9536,9116,9538,5761,6068],"contributors-categories":[13651],"class_list":{"0":"post-83531","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"category-quant-north-america","11":"category-quant-regions","12":"tag-analysis-filter-bank","13":"tag-continuous-time-analog-signals","14":"tag-digital-signal-processing-dsp","15":"tag-synthesis-filter-bank","16":"tag-trend-following","17":"tag-trend-following-strategy","18":"contributors-categories-alpha-architect"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Trend-Following Filters \u2013 Part 3 | IBKR Quant<\/title>\n<meta name=\"description\" content=\"Henry Stern explores a signal processing tool called a &quot;filter bank&quot; and demonstrates analysis filter bank and synthesis filter bank.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/83531\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trend-Following Filters \u2013 Part 3 | IBKR Quant Blog\" \/>\n<meta property=\"og:description\" content=\"Henry Stern explores a signal processing tool called a &quot;filter bank&quot; and demonstrates analysis filter bank and synthesis filter bank.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/\" \/>\n<meta property=\"og:site_name\" content=\"IBKR Campus US\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-13T17:20:29+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-11-21T14:47:20+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1000\" \/>\n\t<meta property=\"og:image:height\" content=\"563\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Henry Stern\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Henry Stern\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\n\t    \"@context\": \"https:\\\/\\\/schema.org\",\n\t    \"@graph\": [\n\t        {\n\t            \"@type\": \"NewsArticle\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/#article\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/\"\n\t            },\n\t            \"author\": {\n\t                \"name\": \"Henry Stern\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/30f94162d86f3b7bb5d5068c0d04f8d4\"\n\t            },\n\t            \"headline\": \"Trend-Following Filters \u2013 Part 3\",\n\t            \"datePublished\": \"2021-04-13T17:20:29+00:00\",\n\t            \"dateModified\": \"2022-11-21T14:47:20+00:00\",\n\t            \"mainEntityOfPage\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/\"\n\t            },\n\t            \"wordCount\": 719,\n\t            \"publisher\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/#primaryimage\"\n\t            },\n\t            \"thumbnailUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/04\\\/tech-dashboard-new.jpg\",\n\t            \"keywords\": [\n\t                \"Analysis filter bank\",\n\t                \"Continuous-time Analog Signals\",\n\t                \"digital signal processing (DSP)\",\n\t                \"Synthesis filter bank\",\n\t                \"Trend Following\",\n\t                \"Trend Following Strategy\"\n\t            ],\n\t            \"articleSection\": [\n\t                \"Data Science\",\n\t                \"Quant\",\n\t                \"Quant Development\",\n\t                \"Quant North America\",\n\t                \"Quant Regions\"\n\t            ],\n\t            \"inLanguage\": \"en-US\"\n\t        },\n\t        {\n\t            \"@type\": \"WebPage\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/\",\n\t            \"name\": \"Trend-Following Filters \u2013 Part 3 | IBKR Quant Blog\",\n\t            \"isPartOf\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#website\"\n\t            },\n\t            \"primaryImageOfPage\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/#primaryimage\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/#primaryimage\"\n\t            },\n\t            \"thumbnailUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/04\\\/tech-dashboard-new.jpg\",\n\t            \"datePublished\": \"2021-04-13T17:20:29+00:00\",\n\t            \"dateModified\": \"2022-11-21T14:47:20+00:00\",\n\t            \"description\": \"Henry Stern explores a signal processing tool called a \\\"filter bank\\\" and demonstrates analysis filter bank and synthesis filter bank.\",\n\t            \"inLanguage\": \"en-US\",\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"ReadAction\",\n\t                    \"target\": [\n\t                        \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/\"\n\t                    ]\n\t                }\n\t            ]\n\t        },\n\t        {\n\t            \"@type\": \"ImageObject\",\n\t            \"inLanguage\": \"en-US\",\n\t            \"@id\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/ibkr-quant-news\\\/trend-following-filters-part-3\\\/#primaryimage\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/04\\\/tech-dashboard-new.jpg\",\n\t            \"contentUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2021\\\/04\\\/tech-dashboard-new.jpg\",\n\t            \"width\": 1000,\n\t            \"height\": 563,\n\t            \"caption\": \"Quant\"\n\t        },\n\t        {\n\t            \"@type\": \"WebSite\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#website\",\n\t            \"url\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/\",\n\t            \"name\": \"IBKR Campus US\",\n\t            \"description\": \"Financial Education from Interactive Brokers\",\n\t            \"publisher\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\"\n\t            },\n\t            \"potentialAction\": [\n\t                {\n\t                    \"@type\": \"SearchAction\",\n\t                    \"target\": {\n\t                        \"@type\": \"EntryPoint\",\n\t                        \"urlTemplate\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/?s={search_term_string}\"\n\t                    },\n\t                    \"query-input\": {\n\t                        \"@type\": \"PropertyValueSpecification\",\n\t                        \"valueRequired\": true,\n\t                        \"valueName\": \"search_term_string\"\n\t                    }\n\t                }\n\t            ],\n\t            \"inLanguage\": \"en-US\"\n\t        },\n\t        {\n\t            \"@type\": \"Organization\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#organization\",\n\t            \"name\": \"Interactive Brokers\",\n\t            \"alternateName\": \"IBKR\",\n\t            \"url\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/\",\n\t            \"logo\": {\n\t                \"@type\": \"ImageObject\",\n\t                \"inLanguage\": \"en-US\",\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/logo\\\/image\\\/\",\n\t                \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2024\\\/05\\\/ibkr-campus-logo.jpg\",\n\t                \"contentUrl\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/wp-content\\\/uploads\\\/sites\\\/2\\\/2024\\\/05\\\/ibkr-campus-logo.jpg\",\n\t                \"width\": 669,\n\t                \"height\": 669,\n\t                \"caption\": \"Interactive Brokers\"\n\t            },\n\t            \"image\": {\n\t                \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/logo\\\/image\\\/\"\n\t            },\n\t            \"publishingPrinciples\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/about-ibkr-campus\\\/\",\n\t            \"ethicsPolicy\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/cyber-security-notice\\\/\"\n\t        },\n\t        {\n\t            \"@type\": \"Person\",\n\t            \"@id\": \"https:\\\/\\\/ibkrcampus.com\\\/campus\\\/#\\\/schema\\\/person\\\/30f94162d86f3b7bb5d5068c0d04f8d4\",\n\t            \"name\": \"Henry Stern\",\n\t            \"url\": \"https:\\\/\\\/www.interactivebrokers.com\\\/campus\\\/author\\\/henrystern\\\/\"\n\t        }\n\t    ]\n\t}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Trend-Following Filters \u2013 Part 3 | IBKR Quant","description":"Henry Stern explores a signal processing tool called a \"filter bank\" and demonstrates analysis filter bank and synthesis filter bank.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.interactivebrokers.com\/campus\/wp-json\/wp\/v2\/posts\/83531\/","og_locale":"en_US","og_type":"article","og_title":"Trend-Following Filters \u2013 Part 3 | IBKR Quant Blog","og_description":"Henry Stern explores a signal processing tool called a \"filter bank\" and demonstrates analysis filter bank and synthesis filter bank.","og_url":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/","og_site_name":"IBKR Campus US","article_published_time":"2021-04-13T17:20:29+00:00","article_modified_time":"2022-11-21T14:47:20+00:00","og_image":[{"width":1000,"height":563,"url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg","type":"image\/jpeg"}],"author":"Henry Stern","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Henry Stern","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"NewsArticle","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/#article","isPartOf":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/"},"author":{"name":"Henry Stern","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/person\/30f94162d86f3b7bb5d5068c0d04f8d4"},"headline":"Trend-Following Filters \u2013 Part 3","datePublished":"2021-04-13T17:20:29+00:00","dateModified":"2022-11-21T14:47:20+00:00","mainEntityOfPage":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/"},"wordCount":719,"publisher":{"@id":"https:\/\/ibkrcampus.com\/campus\/#organization"},"image":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/#primaryimage"},"thumbnailUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg","keywords":["Analysis filter bank","Continuous-time Analog Signals","digital signal processing (DSP)","Synthesis filter bank","Trend Following","Trend Following Strategy"],"articleSection":["Data Science","Quant","Quant Development","Quant North America","Quant Regions"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/","url":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/","name":"Trend-Following Filters \u2013 Part 3 | IBKR Quant Blog","isPartOf":{"@id":"https:\/\/ibkrcampus.com\/campus\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/#primaryimage"},"image":{"@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/#primaryimage"},"thumbnailUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg","datePublished":"2021-04-13T17:20:29+00:00","dateModified":"2022-11-21T14:47:20+00:00","description":"Henry Stern explores a signal processing tool called a \"filter bank\" and demonstrates analysis filter bank and synthesis filter bank.","inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-3\/#primaryimage","url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg","contentUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg","width":1000,"height":563,"caption":"Quant"},{"@type":"WebSite","@id":"https:\/\/ibkrcampus.com\/campus\/#website","url":"https:\/\/ibkrcampus.com\/campus\/","name":"IBKR Campus US","description":"Financial Education from Interactive Brokers","publisher":{"@id":"https:\/\/ibkrcampus.com\/campus\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/ibkrcampus.com\/campus\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/ibkrcampus.com\/campus\/#organization","name":"Interactive Brokers","alternateName":"IBKR","url":"https:\/\/ibkrcampus.com\/campus\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/logo\/image\/","url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/05\/ibkr-campus-logo.jpg","contentUrl":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/05\/ibkr-campus-logo.jpg","width":669,"height":669,"caption":"Interactive Brokers"},"image":{"@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/logo\/image\/"},"publishingPrinciples":"https:\/\/www.interactivebrokers.com\/campus\/about-ibkr-campus\/","ethicsPolicy":"https:\/\/www.interactivebrokers.com\/campus\/cyber-security-notice\/"},{"@type":"Person","@id":"https:\/\/ibkrcampus.com\/campus\/#\/schema\/person\/30f94162d86f3b7bb5d5068c0d04f8d4","name":"Henry Stern","url":"https:\/\/www.interactivebrokers.com\/campus\/author\/henrystern\/"}]}},"jetpack_featured_media_url":"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/04\/tech-dashboard-new.jpg","_links":{"self":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts\/83531","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/users\/575"}],"replies":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/comments?post=83531"}],"version-history":[{"count":0,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/posts\/83531\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/media\/83578"}],"wp:attachment":[{"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/media?parent=83531"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/categories?post=83531"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/tags?post=83531"},{"taxonomy":"contributors-categories","embeddable":true,"href":"https:\/\/ibkrcampus.com\/campus\/wp-json\/wp\/v2\/contributors-categories?post=83531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}