{"id":736,"date":"2024-05-30T17:07:29","date_gmt":"2024-05-30T17:07:29","guid":{"rendered":"https:\/\/neilfoxman.com\/?page_id=736"},"modified":"2024-06-24T03:03:53","modified_gmt":"2024-06-24T03:03:53","slug":"continuous-signals","status":"publish","type":"page","link":"https:\/\/neilfoxman.com\/?page_id=736","title":{"rendered":"Continuous Signals"},"content":{"rendered":"\n

<\/span>Complex Signals<\/span><\/h2>\n\n\n\n

<\/span>Euler Relations<\/span><\/h3>\n\n\n\n

General form of the Euler Relation is<\/p>\n\n\n\n

$$
\\begin{align}
e^{j \\theta} &= \\cos(\\theta) + j \\sin(\\theta) \\\\
\\end{align}
$$<\/p>\n\n\n\n

This relation can be inverted to yield<\/p>\n\n\n\n

$$
\\begin{align}
e^{j(-\\theta)} &= \\cos(-\\theta) + j \\sin(-\\theta) \\\\
e^{-j \\theta} &= \\cos(\\theta) – j \\sin(\\theta) \\\\
\\end{align}
$$<\/p>\n\n\n\n

Isolating $\\cos(\\theta)$ we have<\/p>\n\n\n\n

$$
\\begin{align}
e^{j \\theta} &= \\cos(\\theta) + j \\sin(\\theta) \\\\
e^{-j \\theta} &= \\cos(\\theta) – j \\sin(\\theta) \\\\
\\\\
e^{j \\theta} + e^{-j \\theta} &= 2 \\cos(\\theta) \\\\
\\cos(\\theta) &= \\frac{1}{2}\\left[ e^{j\\theta} + e^{-j\\theta}\\right] \\\\
\\end{align}
$$<\/p>\n\n\n\n

Isolating $\\sin(\\theta)$ we have<\/p>\n\n\n\n

$$
\\begin{align}
e^{j \\theta} – e^{- j \\theta} &= 2 j \\sin(\\theta) \\\\
\\sin(\\theta) &= \\frac{1}{2j}\\left[ e^{j\\theta} – e^{-j\\theta}\\right] \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Complex Expressions<\/span><\/h3>\n\n\n\n

$$
\\begin{align}
|C|e^{rt}e^{j\\omega_0t}e^{j\\phi} = \\text{ampl} \\cdot \\text{exp growth\/decay} \\cdot \\text{periodic} \\cdot \\text{phase} \\\\
\\\\
Acos(\\omega_0 t + \\phi) = \\frac{A}{2}e^{j\\phi}e^{j \\omega_0 t} + \\frac{A}{2}e^{-j\\phi}e^{j\\omega_0 t} \\\\
\\end{align}
$$<\/p>\n\n\n\n

For complex periodic functions, we have the relations<\/p>\n\n\n\n

$$
\\begin{align}
e^{j \\omega t} &= e^{j 2 \\pi f t} \\\\
\\\\
\\omega_0 &= 2 \\pi f_0 \\\\
f_0 &= \\frac{\\omega_0}{2 \\pi} \\\\
\\\\
T_0 &= \\frac{1}{f_0} \\\\
T_0 &= \\frac{2 \\pi}{|\\omega_0|} \\\\
\\omega_0 &= \\frac{2\\pi}{T_0} \\\\
\\omega_0 T_0 &= 2 \\pi \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Even\/Odd<\/span><\/h2>\n\n\n\n

<\/span>Even<\/span><\/h3>\n\n\n\n

$$
\\begin{align}
\\text{Even Signal} \\implies x(-t) &= x(t) \\\\
\\\\
Ev \\left\\{ x(t) \\right\\} &= \\frac{1}{2} \\left[ x(t) + x(-t) \\right] \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Odd<\/span><\/h3>\n\n\n\n

$$
\\begin{align}
\\text{Odd Signal} \\implies x(-t) &= -x(t) \\\\
\\\\
Od \\left\\{ x(t) \\right\\} &= \\frac{1}{2} \\left[ x(t) – x(-t) \\right] \\\\
\\end{align}
$$<\/p>\n\n\n\n

Note<\/p>\n\n\n\n

$$
\\begin{align}
x(t) &= Ev \\left\\{ x(t) \\right\\} + Od \\left\\{ x(t) \\right\\} \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Periodic Signals<\/span><\/h2>\n\n\n\n

Periodic iff<\/p>\n\n\n\n

$$
\\begin{align}
\\forall t \\exists T :x(t) &= x(t+T) \\\\
\\end{align}
$$<\/p>\n\n\n\n

Fundamental Period<\/span> $T_0$ is minimum positive nonzero value of $T$ for which above equation is satisfied.<\/p>\n\n\n\n

Fundamental Frequency<\/span> $\\omega_0$ is then defined as<\/p>\n\n\n\n

$$
\\begin{align}
\\omega_0 = \\frac{2 \\pi}{T_0} \\\\
\\end{align}
$$<\/p>\n\n\n\n

Periodic complex exponentials representing the fundamental can then be written as<\/p>\n\n\n\n

$$
\\begin{align}
e^{j \\frac{2 \\pi}{T_0} t} \\\\
e^{j \\omega_0 t} \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Harmonics<\/span><\/h3>\n\n\n\n

Harmonics are the set of harmonically related complex exponentials with fundamental frequencies that are all multiples of a single positive frequency $\\omega_0$.<\/p>\n\n\n\n

$$
\\begin{align}
&e^{j \\omega t} \\text{ is periodic with Fundamental Period } T_0 \\text{ and Fundamental Frequency } \\omega_0 \\implies \\\\
&e^{j \\omega_0 T_0} = 1 \\implies \\\\
&\\omega_0 T_0 = 2 \\pi k, k=0, \\pm1, \\pm2 \\dots \\\\
\\end{align}
$$<\/p>\n\n\n\n

Note that a continuous complex exponential may be periodic across infinite frequencies\/periods, but there is only one fundamental that defines the set of harmonics. We then define the kth harmonic as<\/p>\n\n\n\n

$$
\\begin{align}
\\phi_k(t) = e^{j k \\frac{2 \\pi}{T_0} t} \\\\
\\phi_k(t) = e^{j k \\omega_0 t} \\\\
\\end{align}
$$<\/p>\n\n\n\n

Note that when $k = 0$, the harmonic is a constant.<\/p>\n\n\n\n

<\/span>Power<\/span><\/h2>\n\n\n\n

Power\/Energy quantities tend to use squared terms

$$
\\begin{align}
E_{\\infty} &= \\int_{t_1}^{t_2} |x(t)|^2 dt \\\\
P_{\\infty} &= \\lim_{T \\to \\infty} \\left[ \\frac{1}{2T} \\int_{-T}^{T} |x(t)|^2 dt \\right] \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Periodic Power<\/span><\/h3>\n\n\n\n

$$
\\begin{align}
E_{\\text{period}} = \\int_{0}^{T_0} |e^{j \\omega_0 t}|^2 dt \\\\

E_{\\text{period}} = \\int_{0}^{T_0} 1^2 dt \\\\

E_{\\text{period}} = \\left[ t \\right]_{0}^{T_0} \\\\

E_{\\text{period}} = T_0 \\\\

P_{\\text{period}} = \\frac{1}{T_0}E_{\\text{period}} = 1 \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Impulse and Step Functions<\/span><\/h2>\n\n\n\n

$$
u(t) = \\begin{cases}
\\begin{align}
0 &&t < 0 \\\\
1 &&t > 0 \\\\
\\end{align}
\\end{cases}
$$<\/p>\n\n\n\n

Note that $u(t)$ is undefined for $t=0$<\/p>\n\n\n\n

Desired relationships that mirror the discrete time case<\/p>\n\n\n\n

$$
\\begin{align}
u(t) &= \\int_{- \\infty}^t \\delta(\\tau) d \\tau \\\\
\\delta(t) &= \\frac{d u(t)}{dt} \\\\
\\end{align}
$$<\/p>\n\n\n\n

These relationships can be used to determine how to define $\\delta(t)$. Start with a continuous function with real life step over time $\\Delta$.<\/p>\n\n\n\n

$$
\\begin{align}
\\delta_{\\Delta}(t) = \\frac{du_{\\Delta}(t)}{dt}
\\end{align}
$$<\/p>\n\n\n\n

Note that the $\\delta_{\\Delta}(t)$ function has height $1\/\\Delta$ and width $\\Delta$, so total area is 1.<\/p>\n\n\n\n

$$
\\delta_{\\Delta}(t) =
\\begin{cases}
1\/\\Delta &&0 \\leq t < \\Delta \\\\
0 &&\\text{otherwise} \\\\
\\end{cases}
$$<\/p>\n\n\n\n

If you now take $\\lim_{\\Delta \\to 0} \\delta_{\\Delta}(t)$ we get a function that is infinitely high and infinitely thin at $t=0$ with integrated area 1. This is usually represented as a vertical arrow with height 1. When the function is scaled, the arrow changes height.<\/p>\n\n\n\n

\"\"\n\t\t\t\n\t\t\t\t\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n\n\n\n

This idealized spike is then described as<\/p>\n\n\n\n

$$
\\begin{align}
\\delta(t) = \\lim_{\\Delta \\to 0} \\delta_{\\Delta}(t) \\\\
\\end{align}
$$<\/p>\n\n\n\n

Using the sifting property<\/a>, we can also find that we also have the relationship<\/p>\n\n\n\n

$$
\\begin{align}
u(t) &= \\int_{-\\infty}^{\\infty} u(\\tau) \\delta(t-\\tau) d \\tau \\\\
u(t) &= \\int_{0}^{\\infty} \\delta(t-\\tau) d \\tau \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Sampling Property<\/span><\/h2>\n\n\n\n

Imagine the sample of a function $x_1(t)$ that is a function $x(t)$ multiplied by $\\delta_{\\Delta}(t)$, or <\/p>\n\n\n\n

$$
x_1(t) = x(t) \\delta_{\\Delta}(t)
$$<\/p>\n\n\n\n

Recalling that<\/p>\n\n\n\n

$$
\\delta(t) = \\lim_{\\Delta \\rightarrow 0} \\delta_{\\Delta}(t)
$$<\/p>\n\n\n\n

As $\\Delta$ gets smaller and smaller, we can approximate $x(t)$ as constant in the interval $[0,\\Delta]$, or <\/p>\n\n\n\n

$$
\\lim_{\\Delta \\rightarrow 0} x_1(t) = x(0) \\delta_{\\Delta}(t)
$$<\/p>\n\n\n\n

Putting these properties together we get the sampling property<\/p>\n\n\n\n

$$
\\begin{align}
x(t) \\delta(t) &= x(0) \\delta(t) \\\\
x(t) \\delta(t-t_0) &= x(t_0) \\delta(t-t_0) \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Sifting Property<\/span><\/h2>\n\n\n\n

A signal may be alternatively represented as the summation (integral) of samples. This could be conceptualized as impulses across all $t$ each scaled by the original signal’s amplitude $x(t)$ at each $t$.<\/p>\n\n\n\n

\"\"\n\t\t\t\n\t\t\t\t\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n\n\n\n

In other words, define<\/p>\n\n\n\n

$$
\\begin{align}
x_1(t) &=
\\dots
x(-2\\Delta) \\delta_\\Delta(t + 2\\Delta) +
x(-\\Delta) \\delta_\\Delta(t + \\Delta) +
x(0) \\delta_\\Delta(t) +
x(\\Delta) \\delta_\\Delta(t – \\Delta) +
x(2\\Delta) \\delta_\\Delta(t – 2\\Delta) +
\\dots \\\\
\\end{align}
$$<\/p>\n\n\n\n

Taking the limit as $\\Delta \\to 0$, the right hand side approaches an integral and the result appears closer to the original signal, so we can then say<\/p>\n\n\n\n

$$
\\begin{align}
x(t) &= \\lim_{\\Delta \\to 0} x_1(t) \\\\

x(t) &= \\lim_{\\Delta \\to 0} \\left[
\\dots
x(-2\\Delta) \\delta_\\Delta(t + 2\\Delta) +
x(-\\Delta) \\delta_\\Delta(t + \\Delta) +
x(0) \\delta_\\Delta(t) +
x(\\Delta) \\delta_\\Delta(t – \\Delta) +
x(2\\Delta) \\delta_\\Delta(t – 2\\Delta) +
\\dots
\\right] \\\\

x(t) &= \\int_{-\\infty}^{\\infty} x(\\tau) \\delta(t – \\tau) d \\tau \\\\
\\end{align}
$$<\/p>\n","protected":false},"excerpt":{"rendered":"

Complex Signals Euler Relations General form of the Euler Relation is $$\\begin{align}e^{j \\theta} &= \\cos(\\theta) + j \\sin(\\theta) \\\\\\end{align}$$ This relation can be inverted to yield $$\\begin{align}e^{j(-\\theta)} &= \\cos(-\\theta) + j \\sin(-\\theta) \\\\e^{-j \\theta} &= \\cos(\\theta) – j \\sin(\\theta) \\\\\\end{align}$$ Isolating $\\cos(\\theta)$ we have $$\\begin{align}e^{j \\theta} &= \\cos(\\theta) + j \\sin(\\theta) \\\\e^{-j \\theta} &= \\cos(\\theta) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":106,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-736","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/736","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=736"}],"version-history":[{"count":41,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/736\/revisions"}],"predecessor-version":[{"id":1165,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/736\/revisions\/1165"}],"up":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/106"}],"wp:attachment":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=736"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}