{"id":37736,"date":"2025-06-26T05:10:23","date_gmt":"2025-06-26T05:10:23","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=37736"},"modified":"2025-06-26T05:10:25","modified_gmt":"2025-06-26T05:10:25","slug":"test-if-the-following-is-time-invariant-or-time-variant","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/test-if-the-following-is-time-invariant-or-time-variant\/","title":{"rendered":"Test if the following is time-invariant or time variant"},"content":{"rendered":"\n

Test if the following is time-invariant or time variant: a. y(n) = x(-n + 2) b. y(n) = x(n) – x(n – 1) 2. Test if the following is linear or non-linear: a. y(n) = n^2x(n – 1) b. y(n) = x(-2n)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Here’s a complete analysis of the systems based on time-invariance and linearity:<\/p>\n\n\n\n

1. Time-Invariance<\/strong><\/h3>\n\n\n\n

a. y(n)=x(\u2212n+2)y(n) = x(-n + 2)<\/h4>\n\n\n\n

Apply a shift n\u2192n\u2212n0n \\rightarrow n – n_0:<\/p>\n\n\n\n

y(n\u2212n0)=x(\u2212(n\u2212n0)+2)=x(\u2212n+n0+2)y(n – n_0) = x(-(n – n_0) + 2) = x(-n + n_0 + 2)<\/p>\n\n\n\n

Now shift the input instead:<\/p>\n\n\n\n

x\u2032(n)=x(n\u2212n0)\u21d2y\u2032(n)=x\u2032(\u2212n+2)=x(\u2212n+2\u2212n0)x'(n) = x(n – n_0) \\Rightarrow y'(n) = x'(-n + 2) = x(-n + 2 – n_0)<\/p>\n\n\n\n

Since y(n\u2212n0)\u2260y\u2032(n)y(n – n_0) \\ne y'(n), the system output depends on the absolute value of nn, not just the relative shift. Conclusion:<\/strong> Time-variant.<\/p>\n\n\n\n

b. y(n)=x(n)\u2212x(n\u22121)y(n) = x(n) – x(n – 1)<\/h4>\n\n\n\n

Shift input: x(n\u2212n0)\u21d2y\u2032(n)=x(n\u2212n0)\u2212x(n\u2212n0\u22121)x(n – n_0) \\Rightarrow y'(n) = x(n – n_0) – x(n – n_0 – 1) Now compute y(n\u2212n0)y(n – n_0):<\/p>\n\n\n\n

y(n\u2212n0)=x(n\u2212n0)\u2212x(n\u2212n0\u22121)y(n – n_0) = x(n – n_0) – x(n – n_0 – 1)<\/p>\n\n\n\n

Both expressions match. Conclusion:<\/strong> Time-invariant.<\/p>\n\n\n\n

2. Linearity<\/strong><\/h3>\n\n\n\n

a. y(n)=n2x(n\u22121)y(n) = n^2 x(n – 1)<\/h4>\n\n\n\n

Let y1(n)=n2x1(n\u22121)y_1(n) = n^2 x_1(n – 1) and y2(n)=n2x2(n\u22121)y_2(n) = n^2 x_2(n – 1). Check additivity:<\/p>\n\n\n\n

y(n)=n2[x1(n\u22121)+x2(n\u22121)]=y1(n)+y2(n)y(n) = n^2 [x_1(n – 1) + x_2(n – 1)] = y_1(n) + y_2(n)<\/p>\n\n\n\n

Check homogeneity:<\/p>\n\n\n\n

ay(n)=n2ax(n\u22121)ay(n) = n^2 a x(n – 1)<\/p>\n\n\n\n

Both properties hold. Conclusion:<\/strong> Linear.<\/p>\n\n\n\n

b. y(n)=x(\u22122n)y(n) = x(-2n)<\/h4>\n\n\n\n

Check additivity:<\/p>\n\n\n\n

y(n)=x1(\u22122n)+x2(\u22122n)=x(\u22122n) for x=x1+x2y(n) = x_1(-2n) + x_2(-2n) = x(-2n) \\text{ for } x = x_1 + x_2<\/p>\n\n\n\n

Check homogeneity:<\/p>\n\n\n\n

ay(n)=ax(\u22122n)a y(n) = a x(-2n)<\/p>\n\n\n\n

Both properties hold. Conclusion:<\/strong> Linear.<\/p>\n\n\n\n

Final Answers:<\/h3>\n\n\n\n
    \n
  • 1a: Time-variant<\/li>\n\n\n\n
  • 1b: Time-invariant<\/li>\n\n\n\n
  • 2a: Linear<\/li>\n\n\n\n
  • 2b: Linear<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Test if the following is time-invariant or time variant: a. y(n) = x(-n + 2) b. y(n) = x(n) – x(n – 1) 2. Test if the following is linear or non-linear: a. y(n) = n^2x(n – 1) b. y(n) = x(-2n) The Correct Answer and Explanation is: Here’s a complete analysis of the systems […]<\/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-37736","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37736","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=37736"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37736\/revisions"}],"predecessor-version":[{"id":37739,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37736\/revisions\/37739"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=37736"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=37736"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=37736"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}