Draw the autocorrelation function for the following codes: 1- 0010111 2- 0000000 Find which code has a better correlation function and why.<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n To find the autocorrelation function<\/strong> for the two binary codes and determine which has better autocorrelation properties, we follow these steps:<\/p>\n\n\n\n We replace:<\/p>\n\n\n\n The autocorrelation function R(k)R(k) of a sequence x(n)x(n) is calculated as: R(k)=\u2211n=0N\u2212k\u22121x(n)\u22c5x(n+k)R(k) = \\sum_{n=0}^{N-k-1} x(n) \\cdot x(n+k)<\/p>\n\n\n\n Where kk is the lag (0 to N-1).<\/p>\n\n\n\n Let\u2019s compute the autocorrelation functions manually for both codes.<\/p>\n\n\n\n Let x=[\u22121,\u22121,+1,\u22121,+1,+1,+1]x = [-1, -1, +1, -1, +1, +1, +1]<\/p>\n\n\n\n So, ACF = [7, 1, 1, 2, -1, -2, -1<\/strong>]<\/p>\n\n\n\n Let x=[\u22121,\u22121,\u22121,\u22121,\u22121,\u22121,\u22121]x = [-1, -1, -1, -1, -1, -1, -1]<\/p>\n\n\n\n So, ACF = [7, 6, 5, 4, 3, 2, 1<\/strong>]<\/p>\n\n\n\n Autocorrelation is a measure of how well a signal matches a shifted version of itself. A good autocorrelation function for communication systems has a sharp peak at lag 0<\/strong> and low values (ideally zero)<\/strong> elsewhere. This property helps in distinguishing the signal from noise and in detecting signal timing accurately.<\/p>\n\n\n\n From the results above:<\/p>\n\n\n\n While both codes have the same peak value at lag 0 (which is expected, as this represents total signal energy), Code 1 shows much lower side-lobes<\/strong>, and some values are even negative<\/strong> or close to zero<\/strong>. These lower values away from the peak (lag 0) mean that Code 1 exhibits better autocorrelation properties<\/strong>.<\/p>\n\n\n\n In contrast, Code 2 is a constant sequence (all zeros), leading to high correlation at every lag. This uniformity makes it poor for signal detection because it lacks uniqueness \u2014 the receiver cannot easily distinguish the start of the code, increasing the risk of synchronization errors.<\/p>\n\n\n\n Thus, Code 1 is better<\/strong>, as it has a distinct, narrow peak and relatively low off-peak values \u2014 ideal for minimizing cross-talk and ensuring accurate detection in digital communication systems like CDMA and spread-spectrum.<\/p>\n\n\n\n \u2705 Correct Answer<\/strong>: Code 1 (0010111)<\/strong> has a better correlation function because it has lower off-peak autocorrelation values, which improves signal detection and synchronization.<\/p>\n","protected":false},"excerpt":{"rendered":" Draw the autocorrelation function for the following codes: 1- 0010111 2- 0000000 Find which code has a better correlation function and why. The correct answer and explanation is: To find the autocorrelation function for the two binary codes and determine which has better autocorrelation properties, we follow these steps: Step 1: Convert binary to bipolar […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-16622","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/16622","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=16622"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/16622\/revisions"}],"predecessor-version":[{"id":16623,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/16622\/revisions\/16623"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=16622"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=16622"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=16622"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nStep 1: Convert binary to bipolar format<\/strong><\/h3>\n\n\n\n
\n
0<\/code> \u2192 -1<\/code><\/li>\n\n\n\n1<\/code> \u2192 +1<\/code><\/li>\n<\/ul>\n\n\n\nCode 1: 0010111 \u2192 Bipolar: -1 -1 +1 -1 +1 +1 +1<\/h4>\n\n\n\n
Code 2: 0000000 \u2192 Bipolar: -1 -1 -1 -1 -1 -1 -1<\/h4>\n\n\n\n
\n\n\n\nStep 2: Compute autocorrelation function (ACF)<\/strong><\/h3>\n\n\n\n
\n\n\n\nCode 1: ACF of 0010111 (bipolar)<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\nCode 2: ACF of 0000000 (bipolar)<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\nAnalysis & Conclusion (300 Words)<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\n