Find the output of the filter with the given input
$begingroup$
Suppose we have a filter $L$ defined by $(Lcirc x)_n=(xast h)_n,$ where $$h_n=begin{cases}frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n},&nneq 0,\ 2/3,&n=0.end{cases}$$
Here $n$ is an integer. Determine the output of this filter with the input $x_n=sin(frac{pi}{8}n)-3cos(frac{pi}{4}n).$
My attempt: First I tried to calculate the discrete-time Fourier transform of both $x$ and $h$, namely, $$X(e^{ilambda})=sum_{ninmathbb{Z}}left(sin(frac{pi}{8}n)-3cos(frac{pi}{4}n)right)e^{-inlambda}$$ and $$H(e^{ilambda})=sum_{ninmathbb{Z}}left(frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n}right)e^{-inlambda}.$$
Since the Fourier transform of the convolution is equal to the product of Fourier transforms, the DTFT of $xast h$ is $X(e^{ilambda})H(e^{ilambda}).$
However, how should I calculate this product and take the inverse transform?
Thanks in advance!
fourier-analysis convolution signal-processing
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
$endgroup$
add a comment |
$begingroup$
Suppose we have a filter $L$ defined by $(Lcirc x)_n=(xast h)_n,$ where $$h_n=begin{cases}frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n},&nneq 0,\ 2/3,&n=0.end{cases}$$
Here $n$ is an integer. Determine the output of this filter with the input $x_n=sin(frac{pi}{8}n)-3cos(frac{pi}{4}n).$
My attempt: First I tried to calculate the discrete-time Fourier transform of both $x$ and $h$, namely, $$X(e^{ilambda})=sum_{ninmathbb{Z}}left(sin(frac{pi}{8}n)-3cos(frac{pi}{4}n)right)e^{-inlambda}$$ and $$H(e^{ilambda})=sum_{ninmathbb{Z}}left(frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n}right)e^{-inlambda}.$$
Since the Fourier transform of the convolution is equal to the product of Fourier transforms, the DTFT of $xast h$ is $X(e^{ilambda})H(e^{ilambda}).$
However, how should I calculate this product and take the inverse transform?
Thanks in advance!
fourier-analysis convolution signal-processing
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
$endgroup$
add a comment |
$begingroup$
Suppose we have a filter $L$ defined by $(Lcirc x)_n=(xast h)_n,$ where $$h_n=begin{cases}frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n},&nneq 0,\ 2/3,&n=0.end{cases}$$
Here $n$ is an integer. Determine the output of this filter with the input $x_n=sin(frac{pi}{8}n)-3cos(frac{pi}{4}n).$
My attempt: First I tried to calculate the discrete-time Fourier transform of both $x$ and $h$, namely, $$X(e^{ilambda})=sum_{ninmathbb{Z}}left(sin(frac{pi}{8}n)-3cos(frac{pi}{4}n)right)e^{-inlambda}$$ and $$H(e^{ilambda})=sum_{ninmathbb{Z}}left(frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n}right)e^{-inlambda}.$$
Since the Fourier transform of the convolution is equal to the product of Fourier transforms, the DTFT of $xast h$ is $X(e^{ilambda})H(e^{ilambda}).$
However, how should I calculate this product and take the inverse transform?
Thanks in advance!
fourier-analysis convolution signal-processing
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
$endgroup$
Suppose we have a filter $L$ defined by $(Lcirc x)_n=(xast h)_n,$ where $$h_n=begin{cases}frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n},&nneq 0,\ 2/3,&n=0.end{cases}$$
Here $n$ is an integer. Determine the output of this filter with the input $x_n=sin(frac{pi}{8}n)-3cos(frac{pi}{4}n).$
My attempt: First I tried to calculate the discrete-time Fourier transform of both $x$ and $h$, namely, $$X(e^{ilambda})=sum_{ninmathbb{Z}}left(sin(frac{pi}{8}n)-3cos(frac{pi}{4}n)right)e^{-inlambda}$$ and $$H(e^{ilambda})=sum_{ninmathbb{Z}}left(frac{sin(frac{pi}{6}n)+sin(frac{pi}{2}n)}{pi n}right)e^{-inlambda}.$$
Since the Fourier transform of the convolution is equal to the product of Fourier transforms, the DTFT of $xast h$ is $X(e^{ilambda})H(e^{ilambda}).$
However, how should I calculate this product and take the inverse transform?
Thanks in advance!
fourier-analysis convolution signal-processing
fourier-analysis convolution signal-processing
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
asked Dec 3 '18 at 13:37
bellcirclebellcircle
1,331411
1,331411
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The DTFT and the Convolution Theorem for the DTFT are the correct tools to use.
This problem can be easily solved with table lookups, instead of direct computation. (I find the Inverse DTFT a lot easier to directly compute than the forward DTFT anyway.)
Here's a convenient table
Table of Discrete Time Fourier Transform Pairs . Since the DTFT is cyclic in the frequency domain, you need only concern yourself with the interval $[-pi, pi]$ in the frequency domain.
Note that your $h[n]$ can be written as
$$h[n] = dfrac{1}{6}mathrm{sinc}left(dfrac{n}{6}right) + dfrac{1}{2}mathrm{sinc}left(dfrac{n}{2}right)$$
which transforms to a sum of 2 rectangle functions of differing widths and magnitudes.
Your $x[n]$ will transform to a sum of pairs of delta functions at the 4 frequencies.
The multiplication of $Hleft(e^{ilambda}right)Xleft(e^{ilambda}right)$ should be simple. That product's inverse DTFT should be simple to compute directly, but may also be amenable to table lookup..
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
$endgroup$
add a comment |
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%2f3024070%2ffind-the-output-of-the-filter-with-the-given-input%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The DTFT and the Convolution Theorem for the DTFT are the correct tools to use.
This problem can be easily solved with table lookups, instead of direct computation. (I find the Inverse DTFT a lot easier to directly compute than the forward DTFT anyway.)
Here's a convenient table
Table of Discrete Time Fourier Transform Pairs . Since the DTFT is cyclic in the frequency domain, you need only concern yourself with the interval $[-pi, pi]$ in the frequency domain.
Note that your $h[n]$ can be written as
$$h[n] = dfrac{1}{6}mathrm{sinc}left(dfrac{n}{6}right) + dfrac{1}{2}mathrm{sinc}left(dfrac{n}{2}right)$$
which transforms to a sum of 2 rectangle functions of differing widths and magnitudes.
Your $x[n]$ will transform to a sum of pairs of delta functions at the 4 frequencies.
The multiplication of $Hleft(e^{ilambda}right)Xleft(e^{ilambda}right)$ should be simple. That product's inverse DTFT should be simple to compute directly, but may also be amenable to table lookup..
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
$endgroup$
add a comment |
$begingroup$
The DTFT and the Convolution Theorem for the DTFT are the correct tools to use.
This problem can be easily solved with table lookups, instead of direct computation. (I find the Inverse DTFT a lot easier to directly compute than the forward DTFT anyway.)
Here's a convenient table
Table of Discrete Time Fourier Transform Pairs . Since the DTFT is cyclic in the frequency domain, you need only concern yourself with the interval $[-pi, pi]$ in the frequency domain.
Note that your $h[n]$ can be written as
$$h[n] = dfrac{1}{6}mathrm{sinc}left(dfrac{n}{6}right) + dfrac{1}{2}mathrm{sinc}left(dfrac{n}{2}right)$$
which transforms to a sum of 2 rectangle functions of differing widths and magnitudes.
Your $x[n]$ will transform to a sum of pairs of delta functions at the 4 frequencies.
The multiplication of $Hleft(e^{ilambda}right)Xleft(e^{ilambda}right)$ should be simple. That product's inverse DTFT should be simple to compute directly, but may also be amenable to table lookup..
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
$endgroup$
add a comment |
$begingroup$
The DTFT and the Convolution Theorem for the DTFT are the correct tools to use.
This problem can be easily solved with table lookups, instead of direct computation. (I find the Inverse DTFT a lot easier to directly compute than the forward DTFT anyway.)
Here's a convenient table
Table of Discrete Time Fourier Transform Pairs . Since the DTFT is cyclic in the frequency domain, you need only concern yourself with the interval $[-pi, pi]$ in the frequency domain.
Note that your $h[n]$ can be written as
$$h[n] = dfrac{1}{6}mathrm{sinc}left(dfrac{n}{6}right) + dfrac{1}{2}mathrm{sinc}left(dfrac{n}{2}right)$$
which transforms to a sum of 2 rectangle functions of differing widths and magnitudes.
Your $x[n]$ will transform to a sum of pairs of delta functions at the 4 frequencies.
The multiplication of $Hleft(e^{ilambda}right)Xleft(e^{ilambda}right)$ should be simple. That product's inverse DTFT should be simple to compute directly, but may also be amenable to table lookup..
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
$endgroup$
The DTFT and the Convolution Theorem for the DTFT are the correct tools to use.
This problem can be easily solved with table lookups, instead of direct computation. (I find the Inverse DTFT a lot easier to directly compute than the forward DTFT anyway.)
Here's a convenient table
Table of Discrete Time Fourier Transform Pairs . Since the DTFT is cyclic in the frequency domain, you need only concern yourself with the interval $[-pi, pi]$ in the frequency domain.
Note that your $h[n]$ can be written as
$$h[n] = dfrac{1}{6}mathrm{sinc}left(dfrac{n}{6}right) + dfrac{1}{2}mathrm{sinc}left(dfrac{n}{2}right)$$
which transforms to a sum of 2 rectangle functions of differing widths and magnitudes.
Your $x[n]$ will transform to a sum of pairs of delta functions at the 4 frequencies.
The multiplication of $Hleft(e^{ilambda}right)Xleft(e^{ilambda}right)$ should be simple. That product's inverse DTFT should be simple to compute directly, but may also be amenable to table lookup..
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
answered Dec 3 '18 at 14:34
Andy WallsAndy Walls
1,594128
1,594128
add a comment |
add a comment |
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.
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%2f3024070%2ffind-the-output-of-the-filter-with-the-given-input%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
