{"id":1369,"date":"2019-12-04T22:07:17","date_gmt":"2019-12-05T06:07:17","guid":{"rendered":"https:\/\/linearcontrol.info\/fundamentals\/?p=1369"},"modified":"2019-12-06T20:13:13","modified_gmt":"2019-12-07T04:13:13","slug":"sampling-part-ii","status":"publish","type":"post","link":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/2019\/12\/04\/sampling-part-ii\/","title":{"rendered":"Sampling. Part II"},"content":{"rendered":"\n

In an earlier post<\/a> we discussed how to obtain the continuous-time Laplace and Fourier transforms of a sampled signal based on the Laplace and Fourier transforms of the original signal. We shall now explore other relationships between those transforms and the Z-transform of the sampled signal.<\/p>\n\n\n\n\n\n\n\n

The Z-transform<\/h2>\n\n\n\n

The Z-transform of the discrete-time signal $u[n]$, $n \\in \\mathbb{N}$, is the function of the complex variable $z$ defined by<\/p>\n\n\n\n

$$U[z]=\\mathcal{Z} \\{ u[n] \\} = \\sum_{n=0}^{\\infty} u[n] \\, z^{-n}.$$ <\/p>\n\n\n\n

The Z-transform does to a discrete-time signal what the Laplace transform does to a continuous-time signal. It has many properties that are analogous to the properties of the Laplace transform. We shall discuss other properties of the Z-transform in more detail later in this and on future posts.<\/p>\n\n\n\n

In the present post, our goal is to reveal a connection between the Laplace transform of the sampled signals $u_s$ and $u_t$ and the Z-transform of its associated sampled-signal $u[n]$.<\/p>\n\n\n\n

The Z-transform of sampled signals<\/h2>\n\n\n\n

Recall the discrete-time signal <\/p>\n\n\n\n

$$u[n] = u(n T_s), \\quad n = 0, 1, \\ldots$$<\/p>\n\n\n\n

consisting of samples obtained periodically from the continuous-time signal $u(t)$, $t \\geq 0$. As discussed here<\/a>, consider also two related continuous-time signals: the sample-and-hold signal<\/p>\n\n\n\n

$$
u_s(t) = u(n T_s) = u[k], \\quad n T_s \\leq t \\leq (n+1) T_s
$$<\/p>\n\n\n\n

and the modulated train of impulses<\/p>\n\n\n\n

$$
u_t(t)=\\sum_{n = 0}^{\\infty} u[n] \\, \\delta(t – n T_s) = u(t) \\sum_{n = 0}^{\\infty} \\delta(t – n T_s).
$$<\/p>\n\n\n\n

As seen before, $u_s$ and $u_t$ are related by<\/p>\n\n\n\n

$$u_s(t) = \\int_{0}^{t} g_h(t – \\tau) \\, u_t(\\tau) \\, d\\tau, \\quad g_h(t) = 1(t) – 1(t – T_s).$$<\/p>\n\n\n\n

We also already know how to calculate the transforms<\/p>\n\n\n\n

$$
U_t(s)=\\frac{1}{T_s}\\sum_{k=-\\infty}^{\\infty} U(s – j k \\omega_s), \\quad
U_s(s)=\\frac{1-e^{-s T_s}}{s} U_t(s).
$$<\/p>\n\n\n\n

Let us now attempt to calculate the Laplace transform of $u_t$ directly using the defintion<\/p>\n\n\n\n

$$U_t(s)=\\int_{0}^{t} u_t(t) \\, e^{- s t} \\, dt.$$<\/p>\n\n\n\n

Upon substitution of $u_t$ one obtains<\/p>\n\n\n\n

$$
\\begin{aligned}
U_t(s)&=\\int_{0}^{\\infty} \\sum_{n = 0}^{\\infty} u[n] \\delta(t – n T_s) \\, e^{- s t} \\, dt \\\\
&= \\sum_{n = 0}^{\\infty} u[n] \\int_{0}^{\\infty} \\delta(t – n T_s) \\, e^{- s t} \\, dt \\\\
&= \\sum_{n = 0}^{\\infty} u[n] \\, e^{- s n T_s}
\\end{aligned}
$$<\/p>\n\n\n\n

from which<\/p>\n\n\n\n

$$U_t(s)= \\left . U[z] \\right |_{z=e^{s T_s}} = U[e^{s T_s}].$$<\/p>\n\n\n\n

The ability to calculate $U_t(s)$ directly from $U[z]$ will be key in providing a formula for the inverse Z-transform. We leave that for a future post.<\/p>\n\n\n\n

The above relationship suggests a strong connection between the variable $z$ and the exponential $e^{s T_s}$. Indeed, $U_t(s)$ converges in the same region as $U(s)$, that is $\\operatorname{Re}\\{s\\} > \\sigma_u$, consequently, because $U[z]$ must converge for $z=e^{s T_s}$, it follows that $U[z]$ converges in the region $|z| > \\tilde{\\sigma}_u = e^{\\sigma_u T_s}$. Conversely, if $U[z]$ converges in $|z| > \\tilde{\\sigma}_u$ then $U_t(s)$ converges in $\\operatorname{Re}\\{s\\} > \\sigma_u = \\log(\\tilde{\\sigma}_u)\/T_s$. <\/p>\n\n\n\n

Similar relationships hold for the sample-and-hold signal<\/p>\n\n\n\n

$$U_s(s) = G_h(s) \\, U_t(s) = \\frac{1 – e^{-s T_s}}{s} U[e^{s T_s}],$$<\/p>\n\n\n\n

and, if $\\sigma_u < 0$, for the corresponding continuous- and discrete-time Fourier transforms (we will get to those eventually) <\/p>\n\n\n\n

$$U_t(j \\omega) = U[e^{j \\omega T_s}].$$ <\/p>\n\n\n\n

By now, it should be apparent that the Z-transform can handle sampled signals in a much more straightforward way than the Laplace transform. We will show in future posts how this capability extends naturally to discrete-time linear systems.<\/p>\n","protected":false},"excerpt":{"rendered":"

In an earlier post we discussed how to obtain the continuous-time Laplace and Fourier transforms of a sampled signal based on the Laplace and Fourier transforms of the original signal. We shall now explore other relationships between those transforms and the Z-transform of the sampled signal.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[7],"tags":[15,38,59],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/1369"}],"collection":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/comments?post=1369"}],"version-history":[{"count":78,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/1369\/revisions"}],"predecessor-version":[{"id":1633,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/posts\/1369\/revisions\/1633"}],"wp:attachment":[{"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/media?parent=1369"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/categories?post=1369"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/linearcontrol.info\/fundamentals\/index.php\/wp-json\/wp\/v2\/tags?post=1369"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}