[amp_mcq option1=”$${{{t^2}} \\over 2}u\\left( t \\right)$$” option2=”$${{t\\left( {t – 1} \\right)} \\over 2}u\\left( {t – 1} \\right)$$” option3=”$${{{{\\left( {t – 1} \\right)}^2}} \\over 2}u\\left( {t – 1} \\right)$$” option4=”$${{{t^2} – 1} \\over 2}u\\left( {t – 1} \\right)$$” correct=”option1″]<\/p>\n
The correct answer is $\\boxed{{t\\left( {t – 1} \\right)} \\over 2}u\\left( {t – 1} \\right)$.<\/p>\n
The output of a system is the convolution of the input and the impulse response. The impulse response is a function that describes how the system responds to a single impulse. The input is the function that is applied to the system. The convolution is a mathematical operation that combines two functions.<\/p>\n
In this case, the impulse response is $h(t) = tu(t)$. The input is $u(t – 1)$. The convolution of these two functions is given by<\/p>\n
$$y(t) = \\int_{-\\infty}^{\\infty} h(t – \\tau) u(\\tau) d\\tau = \\int_{-\\infty}^{t-1} tu(\\tau) d\\tau = {t\\left( {t – 1} \\right)} \\over 2}u\\left( {t – 1} \\right).$$<\/p>\n
The output is a function that is zero for $t < -1$, and is equal to ${t\\left( {t – 1} \\right)} \\over 2$ for $t \\ge -1$. The output is also a unit step function, delayed by one unit.<\/p>\n
The other options are incorrect because they do not match the convolution of the impulse response and the input.<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”$${{{t^2}} \\over 2}u\\left( t \\right)$$” option2=”$${{t\\left( {t – 1} \\right)} \\over 2}u\\left( {t – 1} \\right)$$” option3=”$${{{{\\left( {t – 1} \\right)}^2}} \\over 2}u\\left( {t – 1} \\right)$$” option4=”$${{{t^2} – 1} \\over 2}u\\left( {t – 1} \\right)$$” correct=”option1″]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[959],"tags":[],"class_list":["post-51186","post","type-post","status-publish","format-standard","hentry","category-signal-processing","no-featured-image-padding"],"yoast_head":"\n