Methods of approximating sine waves (given other sine waves)
I'm trying to find a way to approximate sine waves using sine waves that I already have. I have sine waves that are define by:
$$f(n) = 2^{frac{n-49}{12}}times440$$
$$sin(2pi f(n)x), sin(2pi f(n+4)x)$$
What I have already tried is to just use linear interpolation on all the points like so:
$$sin(2pi f(n+c)x) approx frac{csin(2pi f(n)x)+(4-c)sin(2pi f(n+4)x)}{4}$$
I only need to approximate it for one period, but this is still only a mediocre approximation at best. I can only use basic arithmetic, and there is no way for me to calculate the actual values for $f(n)$ and $sin(n)$ on the fly, because I am working with a machine that does not implement floating point arithmetic. (To represent the output values of the sine wave, I'm just mapping it from -1 to 1 to whole numbers in the range 0 to 255). I want to precompute the values of the sine waves for certain n-values to conserve space, but I don't want to do it for all possible n-values (there would be 88 in total). Is there a better approximation, given these limitations?
trigonometry approximation-theory
asked Nov 26 '18 at 19:17
Kristbp2
11
add a comment |
I'm trying to find a way to approximate sine waves using sine waves that I already have. I have sine waves that are define by:
$$f(n) = 2^{frac{n-49}{12}}times440$$
$$sin(2pi f(n)x), sin(2pi f(n+4)x)$$
What I have already tried is to just use linear interpolation on all the points like so:
$$sin(2pi f(n+c)x) approx frac{csin(2pi f(n)x)+(4-c)sin(2pi f(n+4)x)}{4}$$
I only need to approximate it for one period, but this is still only a mediocre approximation at best. I can only use basic arithmetic, and there is no way for me to calculate the actual values for $f(n)$ and $sin(n)$ on the fly, because I am working with a machine that does not implement floating point arithmetic. (To represent the output values of the sine wave, I'm just mapping it from -1 to 1 to whole numbers in the range 0 to 255). I want to precompute the values of the sine waves for certain n-values to conserve space, but I don't want to do it for all possible n-values (there would be 88 in total). Is there a better approximation, given these limitations?
trigonometry approximation-theory
asked Nov 26 '18 at 19:17
Kristbp2
11
Why not just scale $x$ by $f(n+c)/f(n)$ and be done with it? You never said what operations are allowed.
– Andy Walls
Nov 26 '18 at 20:29
I tried explaining better the limitations, I hope its more clear now. Sorry.
– Kristbp2
Nov 26 '18 at 20:41
add a comment |
I'm trying to find a way to approximate sine waves using sine waves that I already have. I have sine waves that are define by:
$$f(n) = 2^{frac{n-49}{12}}times440$$
$$sin(2pi f(n)x), sin(2pi f(n+4)x)$$
What I have already tried is to just use linear interpolation on all the points like so:
$$sin(2pi f(n+c)x) approx frac{csin(2pi f(n)x)+(4-c)sin(2pi f(n+4)x)}{4}$$
I only need to approximate it for one period, but this is still only a mediocre approximation at best. I can only use basic arithmetic, and there is no way for me to calculate the actual values for $f(n)$ and $sin(n)$ on the fly, because I am working with a machine that does not implement floating point arithmetic. (To represent the output values of the sine wave, I'm just mapping it from -1 to 1 to whole numbers in the range 0 to 255). I want to precompute the values of the sine waves for certain n-values to conserve space, but I don't want to do it for all possible n-values (there would be 88 in total). Is there a better approximation, given these limitations?
trigonometry approximation-theory
asked Nov 26 '18 at 19:17
Kristbp2
11
I'm trying to find a way to approximate sine waves using sine waves that I already have. I have sine waves that are define by:
$$f(n) = 2^{frac{n-49}{12}}times440$$
$$sin(2pi f(n)x), sin(2pi f(n+4)x)$$
What I have already tried is to just use linear interpolation on all the points like so:
$$sin(2pi f(n+c)x) approx frac{csin(2pi f(n)x)+(4-c)sin(2pi f(n+4)x)}{4}$$
I only need to approximate it for one period, but this is still only a mediocre approximation at best. I can only use basic arithmetic, and there is no way for me to calculate the actual values for $f(n)$ and $sin(n)$ on the fly, because I am working with a machine that does not implement floating point arithmetic. (To represent the output values of the sine wave, I'm just mapping it from -1 to 1 to whole numbers in the range 0 to 255). I want to precompute the values of the sine waves for certain n-values to conserve space, but I don't want to do it for all possible n-values (there would be 88 in total). Is there a better approximation, given these limitations?
trigonometry approximation-theory
trigonometry approximation-theory
asked Nov 26 '18 at 19:17
Kristbp2
11
asked Nov 26 '18 at 19:17
Kristbp2
11
edited Nov 26 '18 at 20:41
asked Nov 26 '18 at 19:17
Kristbp2
11
asked Nov 26 '18 at 19:17
Kristbp2
11
asked Nov 26 '18 at 19:17
Kristbp2
11
11
Why not just scale $x$ by $f(n+c)/f(n)$ and be done with it? You never said what operations are allowed.
– Andy Walls
Nov 26 '18 at 20:29
I tried explaining better the limitations, I hope its more clear now. Sorry.
– Kristbp2
Nov 26 '18 at 20:41
add a comment |
Why not just scale $x$ by $f(n+c)/f(n)$ and be done with it? You never said what operations are allowed.
– Andy Walls
Nov 26 '18 at 20:29
I tried explaining better the limitations, I hope its more clear now. Sorry.
– Kristbp2
Nov 26 '18 at 20:41
Why not just scale $x$ by $f(n+c)/f(n)$ and be done with it? You never said what operations are allowed.
– Andy Walls
Nov 26 '18 at 20:29
Why not just scale $x$ by $f(n+c)/f(n)$ and be done with it? You never said what operations are allowed.
– Andy Walls
Nov 26 '18 at 20:29
I tried explaining better the limitations, I hope its more clear now. Sorry.
– Kristbp2
Nov 26 '18 at 20:41
I tried explaining better the limitations, I hope its more clear now. Sorry.
– Kristbp2
Nov 26 '18 at 20:41
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014783%2fmethods-of-approximating-sine-waves-given-other-sine-waves%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014783%2fmethods-of-approximating-sine-waves-given-other-sine-waves%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Why not just scale $x$ by $f(n+c)/f(n)$ and be done with it? You never said what operations are allowed.
– Andy Walls
Nov 26 '18 at 20:29
I tried explaining better the limitations, I hope its more clear now. Sorry.
– Kristbp2
Nov 26 '18 at 20:41