{"id":1942,"date":"2026-08-17T05:00:48","date_gmt":"2026-08-17T05:00:48","guid":{"rendered":"https:\/\/smartdata.ece.ufl.edu\/?p=1942"},"modified":"2026-08-17T05:12:59","modified_gmt":"2026-08-17T05:12:59","slug":"sampling","status":"publish","type":"post","link":"https:\/\/smartdata.ece.ufl.edu\/index.php\/2026\/08\/17\/sampling\/","title":{"rendered":"Sampling"},"content":{"rendered":"\n

Sampling: Continuous-to-discrete<\/h2>\n\n\n\n

What is Sampling?<\/h3>\n\n\n\n

Sampling<\/strong> is the process of converting a continuous-time signal into a discrete time signal.<\/p>\n\n\n\n

Sampling is accomplished by multiplying a continuous-time signal \"x(t)\" with an impulse train<\/strong>, a periodic sequence of impulses with a fundamental period \"T_s\" such that

  <\/span>   <\/span>\"\[\delta_{T_s}(t) = \sum_{k=-\infty}^{\infty} \delta(t - k T_s) \; .\]\"<\/p> We will use the notation \"\delta_{T_s}(t)\" to represent a pulse train with sampling period<\/strong> \"T_s\". When we multiply \"x(t)\" and \"\delta_{T_s}(t)\", we get an new train of impulses with amplitudes taken from \"x(t)\". Mathematically, we get

  <\/span>   <\/span>\"\[x_s(t) = x(t) \delta_{T_s}(t) = \sum_{k=-\infty}^{\infty} x(k T_s) \delta(t - k T_s) \; .\]\"<\/p> The question address in the rest of this class is: does the sampling process uniquely preserve all information about our signal?<\/strong><\/p>\n\n\n\n

\n
\"Pulse
Pulse train in time with T<\/em>s<\/em>=1\/2.<\/figcaption><\/figure>\n\n\n\n
\"Sinusoid
Sinusoid with a period of 10.<\/figcaption><\/figure>\n\n\n\n
\"Sampled
Sampled sinusoid with sampling period of T<\/em>s<\/em>=1.<\/figcaption><\/figure>\n<\/div>\n\n\n\n

Fourier Series\/Transform of an impulse train<\/h3>\n\n\n\n

To answer this question, we look at the Fourier series\/transform of the impulse train. Since \"\delta_{T_s}(t)\" is periodic, we use the Fourier series. When we plug \"\delta_{T_s}(t)\" into the Fourier analysis equation, we get

  <\/span>   <\/span>\"\[c_k = \frac{1}{T_s} \int_{T_s} \delta_{T_s}(t) e^{-j k \Omega_s t} dt \; .\]\"<\/p> Over one period, \"\delta_{T_s}(t)\" is just an impulse. Therefore, the Fourier analysis equation simplifies to <\/p>\n\n\n\n

  <\/span>   <\/span>\"\begin{align*} c_k &=& \frac{1}{T_s} \int_{-T_s/2}^{T_s/2} \delta(t) e^{-j k \Omega_s t} dt \\ &=& \frac{1}{T_s} (1) \end{align*}\"<\/p><\/p>\n\n\n\n

We refer to \"\Omega_s = 2 \pi / T_s\" as the [angular]\u00a0sampling frequency<\/strong>.<\/p>\n\n\n\n

Since the Fourier series coefficients are defined by \"c_k = \frac{1}{T_s}\", the Fourier transform has a value of \"2 \pi / T_{s}\" (the \"2 \pi\" is introduced because we are in the angular frequency domain) at \"k\" times the fundamental angular frequency, i.e. \"k \Omega_s = k 2 \pi / T_s\", where \"k\" is an any integer. Therefore, in the continuous-frequency, we get a pulse train with spacing of \"\Omega_s\", <\/p>\n\n\n\n

  <\/span>   <\/span>\"\begin{align*} \delta_{T_s}(t) &\stackrel{\mathcal{F}}{\longleftrightarrow}& \frac{2 \pi}{T_s} \delta_{\Omega_s}(\Omega) \\ \end{align*}\"<\/p><\/p>\n\n\n\n

Therefore, the Fourier transform of an infinite pulse train with period \"T_s\" (i.e., \"\delta_{T_s}(t))\" is a infinite pulse train with spacing \"\Omega_s = 2 \pi / T_s\" (i.e., \"({2 \pi}/{T_s}) \delta_{\Omega_s}(\Omega)\").<\/p>\n\n\n\n

\n
\"Pulse
Pulse train in time with T<\/em>s<\/em>=1\/2.<\/figcaption><\/figure>\n\n\n\n
\"Fourier
Fourier transform of the above pulse train.<\/figcaption><\/figure>\n<\/div>\n\n\n\n

Apply multiplication property<\/h3>\n\n\n\n

To determine the Fourier transform of \"x(t) \delta_{T_s}(t)\", we apply the multiplication property. That is, multiplication in time is equivalent to convolution in frequency. We get the relationship <\/p>\n\n\n\n

  <\/span>   <\/span>\"\begin{align*} x(t) \delta_{T_s}(t) &\stackrel{\mathcal{F}}{\longleftrightarrow}& \frac{1}{2 \pi} \left[ X(\Omega) * \left( \frac{2 \pi}{T_s} \delta_{\Omega_s}(\Omega) \right) \right] \\ x(t) \delta_{T_s}(t) &\stackrel{\mathcal{F}}{\longleftrightarrow}& \frac{1}{T_s} \sum_{k=-\infty}^{\infty} X(\Omega - k \Omega_s) \end{align*}\"<\/p><\/p>\n\n\n\n

Therefore, sampling in time results in the sum of shifted replicas of the frequency domain representation \"X(\Omega)\".<\/p>\n\n\n\n

\n
\"A
A Gaussian pulse signal.<\/figcaption><\/figure>\n\n\n\n
\"The
The Fourier transform of the above Gaussian pulse.<\/figcaption><\/figure>\n\n\n\n
\"A
A sampled Gaussian pulse signal with sampling period of Ts<\/em>=1\/2 and sampling rate of \u03a9s<\/em>=4\u03c0<\/em>.<\/figcaption><\/figure>\n\n\n\n
\"The
The Fourier transform the sampled Gaussian pulse with a sampling rate of \u03a9s<\/em>=4\u03c0<\/em>.<\/figcaption><\/figure>\n<\/div>\n\n\n\n

The sampling theorem<\/h3>\n\n\n\n

We can now answer our question, does the sampling process uniquely preserve all information about our signal?<\/strong> The answer is: yes<\/strong>, but under certain conditions.<\/p>\n\n\n\n

Let the Fourier transform of \"x(t)\" be \"X(\Omega)\" with a largest angular frequency \"\Omega_{max}\". That is

  <\/span>   <\/span>\"\[X(\Omega) = 0 \quad \textrm{for} \quad |\Omega| > \Omega_{max} \; .\]\"<\/p> Also, let \"X_{s}(\Omega)\" be defined as the Fourier transform of the sampled signal \"x_s(t) = x(t) \delta_{T_s}(t)\" such that

  <\/span>   <\/span>\"\[X_{s}(\Omega) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} X(\Omega - k \Omega_s) \; .\]\"<\/p><\/p>\n\n\n\n

If the bandwidth of the \"X(\Omega)\", \"X(\Omega-\Omega_s)\", and \"X(\Omega+\Omega_s)\" never overlap, then the original frequency information is remains changed. That is, to preserve all information about the original signal, we require

  <\/span>   <\/span>\"\[\Omega_s > 2 \Omega_{max}\]\"<\/p> or, equivalently, <\/p>\n\n\n\n

  <\/span>   <\/span>\"\begin{align*} f_s &>& 2 f_{max} \\ T_s &<& \frac{1}{2 f_{max}} \\ T_s &<& \frac{\pi}{\Omega_{max}} \end{align*}\"<\/p> <\/p>\n\n\n\n

Therefore, the sampling frequency, or sampling rate, must be two times larger than the maximum frequency in the signal. This is known as the\u00a0sampling theorem<\/strong>\u00a0or\u00a0sampling criteria<\/strong>. The frequency \"2 f_{max}\" or \"2 \Omega_{max}\" is known as the\u00a0Nyquist sampling rate<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Sampling: Continuous-to-discrete What is Sampling? Sampling is the process of converting a continuous-time signal into a discrete time signal. Sampling is accomplished by […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"post-templates\/post_noheader_nofooter2.php","format":"standard","meta":{"footnotes":""},"categories":[76,108,1],"tags":[],"class_list":["post-1942","post","type-post","status-publish","format-standard","hentry","category-education","category-foundations-of-digital-signal-processing","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1942","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/comments?post=1942"}],"version-history":[{"count":5,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1942\/revisions"}],"predecessor-version":[{"id":2353,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1942\/revisions\/2353"}],"wp:attachment":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/media?parent=1942"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/categories?post=1942"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/tags?post=1942"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}