{"id":786,"date":"2024-06-03T16:51:03","date_gmt":"2024-06-03T16:51:03","guid":{"rendered":"https:\/\/neilfoxman.com\/?page_id=786"},"modified":"2024-06-03T16:51:58","modified_gmt":"2024-06-03T16:51:58","slug":"sampling-rate-variations","status":"publish","type":"page","link":"https:\/\/neilfoxman.com\/?page_id=786","title":{"rendered":"Sampling Rate Variations"},"content":{"rendered":"\n

<\/span>Sampling of Discrete Signals<\/span><\/h1>\n\n\n\n

Taking a sample every $N$ steps of the input signal and setting $N-1$ samples in between to zero<\/p>\n\n\n\n

$$
x_p[n] = \\cases{
\\begin{align}
x[n] && n\/N \\in \\mathbb{Z} \\\\
0 &&\\text{Otherwise}
\\end{align}}
$$<\/p>\n\n\n\n

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

We can also describe $x_p[n]$ in another form<\/p>\n\n\n\n

$$
\\begin{align}
x_p[n] &= p[n] x[n] \\\\
\\end{align}
$$<\/p>\n\n\n\n

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

$$
\\begin{align}

p[n] = \\sum_{k=-\\infty}^{\\infty} \\delta[n – kN] &\\stackrel{\\mathcal{F}}{\\leftrightarrow} P(e^{j \\omega}) = \\sum_{k=-\\infty}^{\\infty} \\frac{2 \\pi}{N} \\delta(\\omega – k \\frac{2 \\pi}{N}) \\\\

p[n] = \\sum_{k=-\\infty}^{\\infty} \\delta[n – kN] &\\stackrel{\\mathcal{F}}{\\leftrightarrow} P(e^{j \\omega}) = \\omega_s \\sum_{k=-\\infty}^{\\infty} \\delta(\\omega – k \\omega_s) \\\\
\\end{align}
$$<\/p>\n\n\n\n

Then, using the multiplication property<\/p>\n\n\n\n

$$
\\begin{align}
X_p(e^{j \\omega}) &= \\frac{1}{2 \\pi} \\int_{2 \\pi} P(e^{j \\theta}) X(e^{j (\\omega – \\theta)}) d \\theta \\\\

X_p(e^{j \\omega}) &= \\frac{1}{2 \\pi} \\int_{2 \\pi} \\left[ \\frac{2 \\pi}{N} \\sum_{k=-\\infty}^{\\infty} \\delta(\\theta – k \\frac{2 \\pi}{N}) \\right] X(e^{j (\\omega – \\theta)}) d \\theta \\\\

X_p(e^{j \\omega}) &= \\frac{1}{N} \\int_{2 \\pi} \\sum_{k=-\\infty}^{\\infty} \\delta(\\theta – k \\frac{2 \\pi}{N}) X(e^{j (\\omega – \\theta)}) d \\theta \\\\
\\end{align}
$$<\/p>\n\n\n\n

Note that the inner term is only nonzero when<\/p>\n\n\n\n

$$
\\theta = 0, \\pm \\frac{2 \\pi}{N}, \\pm \\frac{4 \\pi}{N}, \\dots
$$<\/p>\n\n\n\n

Note also that there are only $N$ distinct elements in the series $X$ is periodic every $2 \\pi$. Therefore we could also write<\/p>\n\n\n\n

$$
\\begin{align}
X_p(e^{j \\omega}) &= \\frac{1}{N} \\int_{2 \\pi} \\sum_{k=\\langle N \\rangle} \\delta(\\theta – k \\frac{2 \\pi}{N}) X(e^{j (\\omega – \\theta)}) d \\theta \\\\

X_p(e^{j \\omega}) &= \\frac{1}{N} \\left[ \\int_{2 \\pi} \\delta(\\theta) X(e^{j (\\omega – \\theta)}) d \\theta + \\int_{2 \\pi} \\delta(\\theta – \\frac{2 \\pi}{N}) X(e^{j (\\omega – \\theta)}) d \\theta + \\int_{2 \\pi} \\delta(\\theta – \\frac{4 \\pi}{N}) X(e^{j (\\omega – \\theta)}) d \\theta + \\dots + \\int_{2 \\pi} \\delta(\\theta – (N-1)\\frac{2 \\pi}{N}) X(e^{j (\\omega – \\theta)}) d \\theta \\right] \\\\

X_p(e^{j \\omega}) &= \\frac{1}{N} \\left[ X(e^{j \\omega}) + X(e^{j (\\omega – \\frac{2 \\pi}{N})}) + X(e^{j (\\omega – \\frac{4 \\pi}{N})}) + \\dots + X(e^{j (\\omega – (N-1)\\frac{2 \\pi}{N})}) \\right] \\\\

X_p(e^{j \\omega}) &= \\frac{1}{N} \\sum_{k=\\langle N \\rangle} X(e^{j(\\omega – k\\frac{2 \\pi}{N})}) \\\\

X_p(e^{j \\omega}) &= \\frac{\\omega_s}{2 \\pi} \\sum_{k=\\langle N \\rangle} X(e^{j (\\omega – k \\omega_s)}) \\\\

\\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

Note in the above graphs you can see that $X_p$ is replicated $N$ times on the interval $\\omega \\in [0, 2 \\pi)$<\/p>\n\n\n\n

Aliasing is avoided when<\/p>\n\n\n\n

$$
\\begin{align}
\\omega_M &< \\omega_s\/2 \\\\
\\omega_M &< \\pi \/ N \\\\
N &< \\pi \/ \\omega_M \\\\
\\end{align}
$$<\/p>\n\n\n\n

Note also that recovery can occur with an ideal lowpass filter<\/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>Downsampling (Decimation) and Upsampling<\/span><\/h2>\n\n\n\n

Called decimation to represent reducing samples by factor of 10.
$$
\\begin{align}
x_b[n] &= x_p[nN] \\\\
x_b[n] &= x[nN] \\\\
\\\\
X_b(e^{j \\omega}) &= \\sum_{k=-\\infty}^{\\infty} x_b[k] e^{-j \\omega k} \\\\
X_b(e^{j \\omega}) &= \\sum_{k=-\\infty}^{\\infty} x_p[kN] e^{-j \\omega k} \\\\
\\end{align}
$$<\/p>\n\n\n\n

Using a new variable $n$ that is multiple integers of $N$, $n = kN$ and $k = n\/N$ where<\/p>\n\n\n\n

$$
\\begin{align}

X_b(e^{j \\omega}) &= \\sum_{n \\in { kN | k, N \\in \\mathbb{Z} }} x_p[n] e^{-j \\omega n \/ N} \\\\

\\end{align}
$$<\/p>\n\n\n\n

and because all values of $x_p[k]$ are 0 outside of $n$ we can also say<\/p>\n\n\n\n

$$
\\begin{align}
X_b(e^{j \\omega}) &= \\sum_{n=-\\infty}^{\\infty} x_p[n] e^{-j \\omega n \/ N} \\\\
X_b(e^{j \\omega}) &= X_p \\left( e^{j \\omega \/ N} \\right) \\\\
\\end{align}
$$<\/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
\"\"\n\t\t\t\n\t\t\t\t\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n\n\n\n

Note that $X_b$ is effectively a stretched out version of $X_p$. This makes sense intuitively as decimation\/downsampling is effectively sampling the signal at a lower sampling rate.<\/p>\n\n\n\n

Upsampling may also be employed using a lowpass filter to convert to a higher equivalent sampling rate. This process is effctively the reverse of downsampling and is shown below.<\/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","protected":false},"excerpt":{"rendered":"

Sampling of Discrete Signals Taking a sample every $N$ steps of the input signal and setting $N-1$ samples in between to zero $$x_p[n] = \\cases{\\begin{align}x[n] && n\/N \\in \\mathbb{Z} \\\\0 &&\\text{Otherwise}\\end{align}}$$ $$\\begin{align}x_p[n] &= x[n] \\sum_{k=-\\infty}^{\\infty} \\delta[n – kN] \\\\x_p[n] &= \\sum_{k=-\\infty}^{\\infty} x[n] \\delta[n – kN] \\\\x_p[n] &= \\sum_{k=-\\infty}^{\\infty} x[kN] \\delta[n – kN] \\\\\\end{align}$$ We can […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":169,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-786","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/786","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=786"}],"version-history":[{"count":2,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/786\/revisions"}],"predecessor-version":[{"id":788,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/786\/revisions\/788"}],"up":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/169"}],"wp:attachment":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=786"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}