{"id":1130,"date":"2024-06-22T04:27:09","date_gmt":"2024-06-22T04:27:09","guid":{"rendered":"https:\/\/neilfoxman.com\/?page_id=1130"},"modified":"2024-06-24T03:02:02","modified_gmt":"2024-06-24T03:02:02","slug":"continuous-systems","status":"publish","type":"page","link":"https:\/\/neilfoxman.com\/?page_id=1130","title":{"rendered":"Continuous Systems"},"content":{"rendered":"\n

Some more rigor and time is spent developing LTI concepts, convolution, and system properties in the Discrete Systems<\/a> page. This page will consist of a “fast-motion” adaptation of this information for continuous time systems.<\/p>\n\n\n\n

Convolution and LTI Systems<\/h2>\n\n\n\n

Imagine an impulse centered at $t=t_0$ represented by $\\delta(t – t_0)$. Assume the system’s response as this impulse is fed through the system is then represented by<\/p>\n\n\n\n

$$
\\begin{align}
\\delta(t – t_0) \\rightarrow h_{t_0}(t) \\\\
\\end{align}
$$<\/p>\n\n\n\n

If the system is linear, then scaling the impulse at the input yields a correspondingly scaled version of the output.<\/p>\n\n\n\n

$$
\\begin{align}
a \\delta(t – t_0) \\rightarrow a h_{t_0}(t) && \\text{Linear System}\\\\
\\end{align}
$$<\/p>\n\n\n\n

Additionally, if the system is time invariant, the impulse response remains the same regardless of the time, that is<\/p>\n\n\n\n

$$
\\begin{align}
\\delta(t – t_0) \\rightarrow h(t – t_0) &&\\text{Time Invariant System} \\\\
\\end{align}
$$<\/p>\n\n\n\n

From the Sifting Property<\/a> of continuous time signals, we can represent any signal $x(t)$ as a sum of scaled impulses.<\/p>\n\n\n\n

\\begin{align}
x(t) &= \\int_{-\\infty}^{\\infty} x(\\tau) \\delta(t – \\tau) d\\tau \\\\
\\end{align}<\/p>\n\n\n\n

Therefore, for Linear Time Invariant systems, a sum of impulses that comprise an input signal $x(t)$ would be transformed to a sum of scaled impulse responses as an output signal $y(t)$.<\/p>\n\n\n\n

$$
\\begin{align}
\\delta(t) &\\rightarrow h(t) \\\\
x(0) \\delta(t) &\\rightarrow x(0) h(t) \\\\
x(t_0) \\delta(t-t_0) &\\rightarrow x(t_0) h(t – t_0) \\\\
x(t) = \\int_{-\\infty}^{\\infty} x(\\tau) \\delta(t – \\tau) d\\tau &\\rightarrow y(t) = \\int_{-\\infty}^{\\infty} x(\\tau) h(t – \\tau) d\\tau \\\\
\\end{align}
$$<\/p>\n\n\n\n

This is called the convolution sum which is also represented as<\/p>\n\n\n\n

$$
y(t) = x(t) * h(t)
$$<\/p>\n","protected":false},"excerpt":{"rendered":"

Some more rigor and time is spent developing LTI concepts, convolution, and system properties in the Discrete Systems page. This page will consist of a “fast-motion” adaptation of this information for continuous time systems. Convolution and LTI Systems Imagine an impulse centered at $t=t_0$ represented by $\\delta(t – t_0)$. Assume the system’s response as this […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":736,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1130","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/1130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1130"}],"version-history":[{"count":6,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/1130\/revisions"}],"predecessor-version":[{"id":1163,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/1130\/revisions\/1163"}],"up":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/736"}],"wp:attachment":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}