[amp_mcq option1=”$$F\\left( s \\right) = \\frac{1}{{1 + {e^{ – \\frac{{sT}}{2}}}}}$$” option2=”$$F\\left( s \\right) = \\frac{1}{{s\\left( {1 + {e^{ – \\frac{{sT}}{2}}}} \\right)}}$$” option3=”$$F\\left( s \\right) = \\frac{1}{{s\\left( {1 – {e^{ – \\frac{{sT}}{2}}}} \\right)}}$$” option4=”$$F\\left( s \\right) = \\frac{1}{{1 – {e^{ – sT}}}}$$” correct=”option3″]<\/p>\n
The correct answer is: $\\boxed{\\text{D}. F\\left( s \\right) = \\frac{1}{{1 – {e^{ – sT}}}}}$<\/p>\n
The Laplace transform of a periodic function $f(t)$ with period $T$ is given by:<\/p>\n
$$F(s) = \\sum_{n=-\\infty}^{\\infty} \\frac{f(nT)}{s+nT}$$<\/p>\n
In this case, $f(t)$ is a square wave with period $T$, so:<\/p>\n
$$f(t) = \\begin{cases} 1 & \\text{for } 0 \\leq t \\leq T \\\\ 0 & \\text{for } t > T \\end{cases}$$<\/p>\n
Substituting this into the Laplace transform formula, we get:<\/p>\n
$$F(s) = \\sum_{n=-\\infty}^{\\infty} \\frac{1}{s+nT} = \\frac{1}{1-e^{-sT}}$$<\/p>\n
Therefore, the Laplace transform of the causal periodic square wave of period $T$ is $\\frac{1}{1-e^{-sT}}$.<\/p>\n
Option A is incorrect because it does not include the factor of $1-e^{-sT}$. Option B is incorrect because it does not include the factor of $s$. Option C is incorrect because it does not include the factor of $1-e^{-sT}$.<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”$$F\\left( s \\right) = \\frac{1}{{1 + {e^{ – \\frac{{sT}}{2}}}}}$$” option2=”$$F\\left( s \\right) = \\frac{1}{{s\\left( {1 + {e^{ – \\frac{{sT}}{2}}}} \\right)}}$$” option3=”$$F\\left( s \\right) = \\frac{1}{{s\\left( {1 – {e^{ – \\frac{{sT}}{2}}}} \\right)}}$$” option4=”$$F\\left( s \\right) = \\frac{1}{{1 – {e^{ – sT}}}}$$” correct=”option3″]<\/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-45710","post","type-post","status-publish","format-standard","hentry","category-signal-processing","no-featured-image-padding"],"yoast_head":"\n