![\"Sampling](\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/sampling-theorem.jpg\")
<\/h2>\nThe process of sampling is a bridge between continuous-time and discrete-time domains. Sampling is a process of converting a continuous-time signal to discrete-time signal and under certain condition the continuous time signal can be completely recovered from its sampled sequence.<\/p>\n
<\/h2>\nAccording to sampling theorem \u201cA band limited signal of finite energy can be completely reconstructed from its samples taken uniformly at a rate \u03c9s<\/sub> \u2265 2\u03c9m<\/sub> samples\/rad\/sec (or, fs<\/sub> \u2265 2fm<\/sub> sample\/sec)\u201d.<\/p>\n In other words, the minimum sampling rate is To prove sampling theorem, let x(t) be continuous-time finite energy signal whose spectrum is band limited to \u03c9m<\/sub> rad\/sec (i.e. X(\u03c9) = 0 for | \u03c9 | > \u03c9m<\/sub>).<\/p>\n The input x(t) is sampled at the rate of fs<\/sub> Hz by multiplying x(t) by a periodic impulse train \u03b4T<\/sub> (t) with period Ts<\/sub> = 1\/fs<\/sub>.<\/p>\n The sampled signal \u2018xs<\/sub>(t)\u2019 consists of impulses spaced every Ts<\/sub> second whose strength (area) is equal to instantaneous value of input signal x(t)<\/p>\n Taking Fourier transform of equation (ii) and using multiplication property of Fourier transform<\/p>\n The sampling frequency can take three possible values with respect to spectrum width (2 \u03c9m<\/sub>)<\/p>\n (i) \u03c9s<\/sub> > 2 \u03c9m<\/sub> Let, \u03c9s<\/sub> = 3 \u03c9m We see for \u03c9s<\/sub> > 2\u03c9m<\/sub>, there is no overlap between the shifted spectrum of X(\u03c9). Thus, as long as the sampling\u00a0 frequency \u03c9s is greater than the twice the signal bandwidth (2 \u03c9m<\/sub>), x(t) can be recovered by passing the sampled signal xs<\/sub> (t) through ideal or practical low pass filter having bandwidth between \u03c9m and (\u03c9s<\/sub> \u2013 \u03c9m<\/sub>) rad\/sec.<\/p>\n We see for \u03c9s<\/sub> = 2 \u03c9m<\/sub> , there is no overlap between shifted spectrum of X(\u03c9). Consequently, x(t) can be recovered from its sampled signal by means of an ideal low pass filter. The minimum sampling rate \u03c9s<\/sub> = 2 \u03c9m<\/sub> (or, fs<\/sub> = 2fm<\/sub>) required to recover x(t) is called Nyquist rate.<\/p>\n We see for \u03c9s<\/sub> < 2\u03c9m<\/sub>, there is overlap between shifted spectrum of X(\u03c9). Consequently, the signal x(t) cannot be exactly recovered from its sampled signal.<\/p>\n The overlap in shifted spectrum of original signal is termed as \u201caliasing\u201d. <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" The process of sampling is a bridge between continuous-time and discrete-time domains. Sampling is a process of converting a continuous-time<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[729],"tags":[740,742,741],"class_list":["post-2730","post","type-post","status-publish","format-standard","hentry","category-signals-systems","tag-aliasing","tag-nyquist-rate","tag-spectrum-folding"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2730","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=2730"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2730\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=2730"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=2730"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=2730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nfs<\/sub> = 2fm<\/sub> Hz.<\/p>\n![\"Periodic](\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/periodic-impulse.jpg\")
<\/p>\n![\"Spectrum\"](\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/spectrum.jpg\")
<\/p>\n
\n(ii) \u03c9s<\/sub> = 2 \u03c9m<\/sub>
\n(iii) \u03c9s<\/sub> < 2 \u03c9m<\/sub>![\"Spectrum\"](\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/spectrum-2.jpg\")
<\/p>\nCase-1 \u03c9> 2 \u03c9m<\/sub><\/strong><\/h3>\n
\n<\/sub><\/p>\nCase-2 \u03c9s<\/sub> = 2 \u03c9m
\n![\"](\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/nyquist-rate.jpg\")
\n<\/sub><\/strong><\/h3>\nCase-3 \u03c9s<\/sub> < 2 \u03c9m<\/sub><\/strong>
![\"Spectrum\"](\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/spectrum-3.jpg\")
<\/h3>\nAliasing or Spectrum Folding<\/h3>\n
\nIn other words, \u201cThe phenomenon of high frequency component taken on the identity of low frequency component is known as aliasing\u201d. To avoid aliasing, the condition is \u03c9s<\/sub> \u2265 2\u03c9m<\/sub>.<\/p>\n