{"id":52530,"date":"2024-04-15T23:40:43","date_gmt":"2024-04-15T23:40:43","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=52530"},"modified":"2024-04-15T23:40:43","modified_gmt":"2024-04-15T23:40:43","slug":"the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/","title":{"rendered":"The signal $$\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t &#8211; \\frac{\\pi }{2}} \\right).$$ The filter output is"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a975bf52aa17\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    $$\frac{{15}}{2}cos left( {40pi t - \frac{pi }{4}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    $$\frac{{15}}{2}left( {\frac{{sin left( {pi t} \right)}}{{pi t}}} \right)cos left( {10pi t + \frac{pi }{4}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    $$\frac{{15}}{2}cos left( {10pi t - \frac{pi }{4}} \right)$$                <\/div>\r\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{{15}}{2}\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi t}}} \\right)\\cos \\left( {10\\pi t &#8211; \\frac{\\pi }{4}} \\right)}$.<\/p>\n<p>The signal $cos(10\\pi t + \\frac{\\pi}{4})$ is ideally sampled at a sampling frequency of 15 Hz. This means that the signal is sampled at regular intervals of $T = \\frac{1}{f_s} = \\frac{1}{15} = 0.067$ seconds. The sampled signal is then passed through a filter with impulse response $\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t &#8211; \\frac{\\pi }{2}} \\right)$. The impulse response of a filter is a function that describes how the filter will respond to a single input pulse. In this case, the impulse response is a decaying sinusoid with a period of $T_p = \\frac{2\\pi}{\\omega} = 0.052$ seconds. The filter output is the convolution of the sampled signal and the impulse response.<\/p>\n<p>The convolution of two signals $x(t)$ and $h(t)$ is defined as<\/p>\n<p>$$y(t) = \\int_{-\\infty}^{\\infty} x(\\tau) h(t-\\tau) d\\tau$$<\/p>\n<p>In this case, the sampled signal is $x(t) = cos(10\\pi t + \\frac{\\pi}{4})$ and the impulse response is $h(t) = \\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t &#8211; \\frac{\\pi }{2}} \\right)$. The convolution of these two signals can be evaluated using the following steps:<\/p>\n<ol>\n<li>Multiply the two signals pointwise.<\/li>\n<li>Integrate the product over all time.<\/li>\n<\/ol>\n<p>The result of the convolution is a signal that is a sum of two sinusoids, one at a frequency of 10 Hz and one at a frequency of 40 Hz. The amplitude of the 10 Hz sinusoid is $\\frac{{15}}{2}\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi t}}} \\right)$ and the amplitude of the 40 Hz sinusoid is $\\frac{{15}}{2}\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi t}}} \\right)\\cos \\left( {40\\pi t &#8211; \\frac{\\pi }{2}} \\right)$. The phase of the 10 Hz sinusoid is $\\frac{\\pi}{4}$ and the phase of the 40 Hz sinusoid is $-\\frac{\\pi}{2}$.<\/p>\n<p>The filter output is therefore<\/p>\n<p>$$y(t) = \\frac{{15}}{2}\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi t}}} \\right)\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right) + \\frac{{15}}{2}\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi t}}} \\right)\\cos \\left( {40\\pi t &#8211; \\frac{\\pi }{2}} \\right)$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/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":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The signal $$\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t - \\frac{\\pi }{2}} \\right).$$ The filter output is<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The signal $$\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t - \\frac{\\pi }{2}} \\right).$$ The filter output is\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T23:40:43+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"The signal $$\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t - \\frac{\\pi }{2}} \\right).$$ The filter output is","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/","og_locale":"en_US","og_type":"article","og_title":"The signal $$\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\\left( {\\frac{{\\sin \\left( {\\pi t} \\right)}}{{\\pi \\tau }}} \\right)\\cos \\left( {40\\pi t - \\frac{\\pi }{2}} \\right).$$ The filter output is","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T23:40:43+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-signal-cos-left-10pi-t-fracpi-4-right-is-ideally-sampled-at-a-sampling-frequency-of-15-hz-the-sampled-signal-is-passed-through-a-filter-with-impulse-response-left\/","name":"The signal $$\\cos \\left( {10\\pi t + \\frac{\\pi }{4}} \\right)$$ is ideally sampled at a sampling frequency of 15 Hz. 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