{"id":251,"date":"2023-12-19T04:19:49","date_gmt":"2023-12-19T04:19:49","guid":{"rendered":"https:\/\/neilfoxman.com\/?page_id=251"},"modified":"2024-05-30T17:36:51","modified_gmt":"2024-05-30T17:36:51","slug":"discrete-signals-and-systems","status":"publish","type":"page","link":"https:\/\/neilfoxman.com\/?page_id=251","title":{"rendered":"Discrete Signals"},"content":{"rendered":"\n

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

$$
\\delta[n] = \\cases{
\\begin{align}
0 &&& n \\neq 0 \\\\
1 &&& n = 0 \\\\
\\end{align}
}
$$<\/p>\n\n\n\n

$$
u[n] = \\cases{
\\begin{align}
0 &&& n < 0 \\\\
1 &&& n \\ge 0 \\\\
\\end{align}
}
$$<\/p>\n\n\n\n

Relationships of note<\/p>\n\n\n\n

$$
\\begin{align}
\\delta[n] &= u[n] – u[n-1] \\\\
u[n] &= \\sum_{k = -\\infty}^n \\delta[k] \\\\
\\end{align}
$$<\/p>\n\n\n\n

<\/span>Discrete Time Frequency<\/span><\/h2>\n\n\n\n

\\(\\Omega\\) represents the discrete frequency<\/p>\n\n\n\n

\\[
\\begin{align}
\\Omega &= \\frac{2 \\pi M}{N}
\\end{align}
\\]<\/p>\n\n\n\n

So, the complex exponential signal<\/p>\n\n\n\n

\\[
\\begin{align}
x[n] &= e^{j \\Omega n} \\\\
x[n] &= e^{j (2 \\pi M\/N) n} \\
\\end{align}
\\]<\/p>\n\n\n\n

will oscillate \\(M\\) times over \\(N\\) discrete time steps.<\/p>\n\n\n\n

<\/span>Discrete Time Periodicity<\/span><\/h2>\n\n\n\n

For a discrete time signal \\(x[n]\\) to be periodic, it must satisfy<\/p>\n\n\n\n

\\[
\\begin{align}
\\forall n \\in \\mathbb{Z}, \\exists N \\in \\mathbb{Z} : x[n] = x[n+N]
\\end{align}
\\]<\/p>\n\n\n\n

Therefore, in the discrete case, we have the phenomenon where high frequency signals may appear like low frequency signals due to aliasing. This is shown by analysis of the periodic signal<\/p>\n\n\n\n

\\[
\\begin{align}
x[n] &= x[n+N] \\\\
e^{j \\Omega n} &= e^{j \\Omega (n+N)} \\\\
e^{j \\Omega n} &= e^{j \\Omega n} e^{j \\Omega N} \\\\
1 &= e^{j \\Omega N} \\\\
\\end{align}
\\]<\/p>\n\n\n\n

In order to satisfy this result, we must have<\/p>\n\n\n\n

\\[
\\begin{align}
\\Omega N = 2 \\pi k &&k \\in \\mathbb{Z} \\\\
\\end{align}
\\]<\/p>\n\n\n\n

Another way to look at this is that for a signal to be periodic in discrete time, the quantity \\(\\Omega \/ 2 \\pi\\) must be rational.<\/p>\n\n\n\n

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

<\/span>Discrete Frequency Periodicity<\/span><\/h2>\n\n\n\n

Consider complex exponential with frequency \\(\\Omega_1 = \\Omega_0 + 2\\pi\\). <\/p>\n\n\n\n

\\[
\\begin{align}
e^{j \\Omega_1 n} = e^{j (\\Omega_0 + 2\\pi) n} \\\\
e^{j \\Omega_1 n} = e^{j \\Omega_0 n} e^{j 2 \\pi n} \\\\
\\end{align}
\\]<\/p>\n\n\n\n

But because \\(\\forall n \\in \\mathbb{Z}, e^{j (2\\pi n)} = e^{j 2\\pi} = 1\\), this is simplified to<\/p>\n\n\n\n

\\[
e^{j \\Omega_1 n} = e^{j \\Omega_0 n}
\\]<\/p>\n\n\n\n

so effectively in discrete time signals, the discrete frequency will repeat every \\(2 \\pi\\)<\/p>\n\n\n\n

\\[
\\begin{align}
\\Omega = \\Omega_0 \\pm 2\\pi k &&k \\in \\mathbb{Z}
\\end{align}
\\]<\/p>\n\n\n\n

Lower frequencies are near \\(0\\) or \\(2\\pi\\) while higher frequencies are near \\(\\pm\\pi\\)<\/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

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

An impulse at time $n_0$ scaled by an entire signal is equivalent to an impulse at $n_0$ scaled just by the value $x[n_0]$. That is<\/p>\n\n\n\n

$$
x[n] \\delta[n – n_0] = x[n_0] \\delta[n – n_0]
$$<\/p>\n\n\n\n

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

The Sifting property is essentially an extension of the sampling property described above, where an alternative representation for $x[n]$ can be produced as a sum of all sampled values. In other words, a signal can be represented as a sum of unit impulses scaled by the signal at that time, $n$.<\/p>\n\n\n\n

$$
\\begin{align}

x[n] &= \\dots + x[-2] \\delta[n + 2] + x[-1] \\delta[n + 1] + x[0] \\delta[n] + x[1] \\delta[n – 1] + x[2] \\delta[n – 2] + \\dots \\\\

x[n] &= \\sum_{k = -\\infty}^{\\infty} x[k] \\delta[n-k] \\\\

\\end{align}
$$<\/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

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

Power\/Energy quantities tend to use squared terms

$$
\\begin{align}
E_{\\infty} &= \\sum_{n = n_1}^{n_2} |x[n]|^2 \\\\
P_\\infty &= \\lim_{N \\to \\infty} \\left[ \\frac{1}{2N+1} \\sum_{n=-N}^{N} |x[n]|^2 \\right] \\\\
\\end{align}
$$<\/p>\n","protected":false},"excerpt":{"rendered":"

Impulse and Step Functions $$\\delta[n] = \\cases{\\begin{align}0 &&& n \\neq 0 \\\\1 &&& n = 0 \\\\\\end{align}}$$ $$u[n] = \\cases{\\begin{align}0 &&& n < 0 \\\\1 &&& n \\ge 0 \\\\\\end{align}}$$ Relationships of note $$\\begin{align}\\delta[n] &= u[n] – u[n-1] \\\\u[n] &= \\sum_{k = -\\infty}^n \\delta[k] \\\\\\end{align}$$ Discrete Time Frequency \\(\\Omega\\) represents the discrete frequency \\[\\begin{align}\\Omega &= […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":106,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-251","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/251","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=251"}],"version-history":[{"count":21,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/251\/revisions"}],"predecessor-version":[{"id":751,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/251\/revisions\/751"}],"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=251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}