[amp_mcq option1=”$$ – \\frac{1}{2}X\\left( {\\frac{f}{2}} \\right){e^{ – j2\\pi f}}$$” option2=”$$ – \\frac{1}{2}X\\left( {\\frac{f}{2}} \\right){e^{j2\\pi f}}$$” option3=”$$ – X\\left( {\\frac{f}{2}} \\right){e^{j2\\pi f}}$$” option4=”$$ – X\\left( {\\frac{f}{2}} \\right){e^{ – j2\\pi f}}$$” correct=”option4″]<\/p>\n
The correct answer is $\\boxed{-\\frac{1}{2}X\\left(\\frac{f}{2}\\right)e^{-j2\\pi f}}$.<\/p>\n
The Fourier transform of a function $x(t)$ is defined as<\/p>\n
$$X(f) = \\int_{-\\infty}^{\\infty} x(t) e^{-j2\\pi ft} dt$$<\/p>\n
The inverse Fourier transform is defined as<\/p>\n
$$x(t) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} X(f) e^{j2\\pi ft} df$$<\/p>\n
Given that $x(t)$ is an even function, its Fourier transform is given by<\/p>\n
$$X(f) = \\frac{1}{2} \\int_{-\\infty}^{\\infty} x(t) e^{-j2\\pi ft} dt = \\frac{1}{2} \\int_{-\\infty}^{\\infty} x(-t) e^{j2\\pi ft} dt = \\frac{1}{2} X(-f)$$<\/p>\n
The Fourier transform of $y(t)$ is given by<\/p>\n
$$Y(f) = \\int_{-\\infty}^{\\infty} y(t) e^{-j2\\pi ft} dt = \\int_{-\\infty}^{\\infty} -x(t-\\tau) e^{-j2\\pi ft} dt = -X(f) e^{-j2\\pi \\tau f}$$<\/p>\n
where $\\tau$ is the delay parameter.<\/p>\n
In this case, $\\tau = \\frac{1}{2}$, so<\/p>\n
$$Y(f) = -X(f) e^{-j2\\pi \\frac{1}{2} f} = -\\frac{1}{2}X\\left(\\frac{f}{2}\\right)e^{-j2\\pi f}$$<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”$$ – \\frac{1}{2}X\\left( {\\frac{f}{2}} \\right){e^{ – j2\\pi f}}$$” option2=”$$ – \\frac{1}{2}X\\left( {\\frac{f}{2}} \\right){e^{j2\\pi f}}$$” option3=”$$ – X\\left( {\\frac{f}{2}} \\right){e^{j2\\pi f}}$$” option4=”$$ – X\\left( {\\frac{f}{2}} \\right){e^{ – j2\\pi f}}$$” correct=”option4″]<\/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-54340","post","type-post","status-publish","format-standard","hentry","category-signal-processing","no-featured-image-padding"],"yoast_head":"\n