{"id":196982,"date":"2025-03-06T11:28:25","date_gmt":"2025-03-06T11:28:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196982"},"modified":"2025-03-06T11:28:27","modified_gmt":"2025-03-06T11:28:27","slug":"design-a-digital-system-whose-output-is-defined-as-logically-low-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/06\/design-a-digital-system-whose-output-is-defined-as-logically-low-2\/","title":{"rendered":"Design a digital system whose output is defined as logically low"},"content":{"rendered":"\n

Design a digital system whose output is defined as logically low. If the 4 bit inpur binary number is a multiple of 3. Otherwise, the output will be logically high. The output is defined if and only if the input binary number is greater than 2<\/p>\n\n\n\n

Design of a Digital System for Multiples of 3 Detection<\/strong><\/h3>\n\n\n\n

Problem Statement:<\/strong><\/h4>\n\n\n\n

We need to design a digital system that outputs a logically low (0) when a 4-bit input binary number is a multiple of 3. Otherwise, the output should be logically high (1). Additionally, the output should only be defined if the input number is greater than 2.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Identifying Multiples of 3 in 4-bit Binary<\/strong><\/h3>\n\n\n\n

A 4-bit binary input ranges from 0000 (0) to 1111 (15). However, we only consider numbers greater than 2 (i.e., from 3 to 15). The multiples of 3 in this range are:<\/p>\n\n\n\n

    \n
  • 3 (0011)<\/strong><\/li>\n\n\n\n
  • 6 (0110)<\/strong><\/li>\n\n\n\n
  • 9 (1001)<\/strong><\/li>\n\n\n\n
  • 12 (1100)<\/strong><\/li>\n\n\n\n
  • 15 (1111)<\/strong><\/li>\n<\/ul>\n\n\n\n

    For these inputs, the output should be low (0)<\/strong>. For all other numbers, the output should be high (1)<\/strong>.<\/p>\n\n\n\n

    \n\n\n\n

    Step 2: Karnaugh Map (K-Map) for Optimization<\/strong><\/h3>\n\n\n\n

    Using a 4-variable Karnaugh map (K-map), we map the outputs for inputs 3 to 15.<\/p>\n\n\n\n

    AB\/CD<\/th>00<\/th>01<\/th>10<\/th>11<\/th><\/tr><\/thead>
    00<\/strong><\/td>X<\/td>X<\/td>1<\/td>0<\/td><\/tr>
    01<\/strong><\/td>1<\/td>0<\/td>1<\/td>1<\/td><\/tr>
    10<\/strong><\/td>0<\/td>1<\/td>1<\/td>X<\/td><\/tr>
    11<\/strong><\/td>0<\/td>X<\/td>X<\/td>X<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    Where X<\/strong> indicates an undefined condition (input < 3).<\/p>\n\n\n\n

    The minimized Boolean equation for the output (Y) is:
    [
    Y = \\overline{(A \\oplus B)C + \\overline{B}D}
    ]<\/p>\n\n\n\n

    \n\n\n\n

    Step 3: Logic Circuit Design<\/strong><\/h3>\n\n\n\n

    Using the equation derived, the digital system can be designed using XOR gates, AND gates, and NOT gates<\/strong>.<\/p>\n\n\n\n

    \n\n\n\n

    Final Design<\/strong><\/h3>\n\n\n\n
      \n
    • Input:<\/strong> 4-bit binary (A, B, C, D)<\/li>\n\n\n\n
    • Output:<\/strong> Y (0 if multiple of 3, 1 otherwise)<\/li>\n\n\n\n
    • Components Used:<\/strong> XOR gates, AND gates, NOT gates<\/li>\n<\/ul>\n\n\n\n
      \"\"<\/figure>\n\n\n\n

      Here is the digital logic circuit diagram that detects multiples of 3 for a 4-bit binary input. The circuit follows the equation:<\/p>\n\n\n\n

      [
      Y = \\overline{(A \\oplus B)C + \\overline{B}D}
      ]<\/p>\n\n\n\n

      The output is logically low (0) when the input is a multiple of 3 and logically high (1) otherwise. <\/p>\n","protected":false},"excerpt":{"rendered":"

      Design a digital system whose output is defined as logically low. If the 4 bit inpur binary number is a multiple of 3. Otherwise, the output will be logically high. The output is defined if and only if the input binary number is greater than 2 Design of a Digital System for Multiples of 3 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-196982","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196982","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=196982"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196982\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196982"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196982"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}