Fitting a sinusoid vs. DTFT
$begingroup$
I want to fit a sinusoid to a given discrete finite real signal $X(n)$ with length $N$. Of particular interest is the frequency. For the estimation for example least squares can be utilized:
$$text{min}_{r,phi,k}(sum_{n=0}^{N-1}(X(n)-rtext{cos}(frac{kncdot2pi}{T}-phi))^2)$$
$$=text{min}_{a,b,k}(sum_{n=0}^{N-1}(X(n)-atext{cos}(frac{kncdot2pi}{T})-btext{sin}((frac{kncdot2pi}{T}))^2)$$
The optimization problem looks a lot like solving for the argmax of a Fourier transform, or more precisely $DTFT$. Quite naturally one may start to think whether maximizing the absolute value of the $DTFT$ gives the same result.
$$DTFT_{frac{1}{T}}(k)=sum_{n=0}^{N-1}X(t)e^{frac{-1jcdot 2picdot kn}{T}}$$
So that we get a maximization problem with only the frequency as an argument:
$$text{max}_k(abs(DTFT_frac{1}{T}(k)))$$
Basically it boils down to checking the correctness of the following two steps.
1) DTFT solves the coefficients $a,b$ correctly. IE. as output we get $(a,-ib)$.
2) At the minimum $r=abs(a,-ib)$ is maximized.
The result seems that at least 1) does not hold, which is quite interesting since ome might think that $DTFT$ solves those coefficients. It only holds for integer valued $k$. Why is this?
It's doubtful that 2) holds either. Any thoughts?
Overall, does this seem like a sound method of estimation? (At least for a particular problem I had it worked).
fourier-analysis estimation
asked Jan 8 at 0:05
DoleDole
899514
$endgroup$
add a comment |
$begingroup$
I want to fit a sinusoid to a given discrete finite real signal $X(n)$ with length $N$. Of particular interest is the frequency. For the estimation for example least squares can be utilized:
$$text{min}_{r,phi,k}(sum_{n=0}^{N-1}(X(n)-rtext{cos}(frac{kncdot2pi}{T}-phi))^2)$$
$$=text{min}_{a,b,k}(sum_{n=0}^{N-1}(X(n)-atext{cos}(frac{kncdot2pi}{T})-btext{sin}((frac{kncdot2pi}{T}))^2)$$
The optimization problem looks a lot like solving for the argmax of a Fourier transform, or more precisely $DTFT$. Quite naturally one may start to think whether maximizing the absolute value of the $DTFT$ gives the same result.
$$DTFT_{frac{1}{T}}(k)=sum_{n=0}^{N-1}X(t)e^{frac{-1jcdot 2picdot kn}{T}}$$
So that we get a maximization problem with only the frequency as an argument:
$$text{max}_k(abs(DTFT_frac{1}{T}(k)))$$
Basically it boils down to checking the correctness of the following two steps.
1) DTFT solves the coefficients $a,b$ correctly. IE. as output we get $(a,-ib)$.
2) At the minimum $r=abs(a,-ib)$ is maximized.
The result seems that at least 1) does not hold, which is quite interesting since ome might think that $DTFT$ solves those coefficients. It only holds for integer valued $k$. Why is this?
It's doubtful that 2) holds either. Any thoughts?
Overall, does this seem like a sound method of estimation? (At least for a particular problem I had it worked).
fourier-analysis estimation
asked Jan 8 at 0:05
DoleDole
899514
$endgroup$
add a comment |
$begingroup$
I want to fit a sinusoid to a given discrete finite real signal $X(n)$ with length $N$. Of particular interest is the frequency. For the estimation for example least squares can be utilized:
$$text{min}_{r,phi,k}(sum_{n=0}^{N-1}(X(n)-rtext{cos}(frac{kncdot2pi}{T}-phi))^2)$$
$$=text{min}_{a,b,k}(sum_{n=0}^{N-1}(X(n)-atext{cos}(frac{kncdot2pi}{T})-btext{sin}((frac{kncdot2pi}{T}))^2)$$
The optimization problem looks a lot like solving for the argmax of a Fourier transform, or more precisely $DTFT$. Quite naturally one may start to think whether maximizing the absolute value of the $DTFT$ gives the same result.
$$DTFT_{frac{1}{T}}(k)=sum_{n=0}^{N-1}X(t)e^{frac{-1jcdot 2picdot kn}{T}}$$
So that we get a maximization problem with only the frequency as an argument:
$$text{max}_k(abs(DTFT_frac{1}{T}(k)))$$
Basically it boils down to checking the correctness of the following two steps.
1) DTFT solves the coefficients $a,b$ correctly. IE. as output we get $(a,-ib)$.
2) At the minimum $r=abs(a,-ib)$ is maximized.
The result seems that at least 1) does not hold, which is quite interesting since ome might think that $DTFT$ solves those coefficients. It only holds for integer valued $k$. Why is this?
It's doubtful that 2) holds either. Any thoughts?
Overall, does this seem like a sound method of estimation? (At least for a particular problem I had it worked).
fourier-analysis estimation
asked Jan 8 at 0:05
DoleDole
899514
$endgroup$
I want to fit a sinusoid to a given discrete finite real signal $X(n)$ with length $N$. Of particular interest is the frequency. For the estimation for example least squares can be utilized:
$$text{min}_{r,phi,k}(sum_{n=0}^{N-1}(X(n)-rtext{cos}(frac{kncdot2pi}{T}-phi))^2)$$
$$=text{min}_{a,b,k}(sum_{n=0}^{N-1}(X(n)-atext{cos}(frac{kncdot2pi}{T})-btext{sin}((frac{kncdot2pi}{T}))^2)$$
The optimization problem looks a lot like solving for the argmax of a Fourier transform, or more precisely $DTFT$. Quite naturally one may start to think whether maximizing the absolute value of the $DTFT$ gives the same result.
$$DTFT_{frac{1}{T}}(k)=sum_{n=0}^{N-1}X(t)e^{frac{-1jcdot 2picdot kn}{T}}$$
So that we get a maximization problem with only the frequency as an argument:
$$text{max}_k(abs(DTFT_frac{1}{T}(k)))$$
Basically it boils down to checking the correctness of the following two steps.
1) DTFT solves the coefficients $a,b$ correctly. IE. as output we get $(a,-ib)$.
2) At the minimum $r=abs(a,-ib)$ is maximized.
The result seems that at least 1) does not hold, which is quite interesting since ome might think that $DTFT$ solves those coefficients. It only holds for integer valued $k$. Why is this?
It's doubtful that 2) holds either. Any thoughts?
Overall, does this seem like a sound method of estimation? (At least for a particular problem I had it worked).
fourier-analysis estimation
fourier-analysis estimation
asked Jan 8 at 0:05
DoleDole
899514
asked Jan 8 at 0:05
DoleDole
899514
edited Jan 8 at 0:16
Dole
asked Jan 8 at 0:05
DoleDole
899514
asked Jan 8 at 0:05
DoleDole
899514
asked Jan 8 at 0:05
DoleDole
899514
899514
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