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| In applied mathematics, the '''non-uniform discrete Fourier transform (NDFT)''' of a signal is a type of [[Fourier transform]], related to a [[discrete Fourier transform]] or [[discrete-time Fourier transform]], but in which the input signal is not sampled at equally-spaced intervals. As a result of this, the computed Discrete Fourier Transform can also consist of unevenly sampled frequency values. It is however also possible to compute uniformly sampled frequency values from an unevenly sampled input signal.
| | By investing in a premium Word - Press theme, you're investing in the future of your website. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Should you go with simple HTML or use a platform like Wordpress. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>If you have any questions concerning where and ways to use [http://twsi.in/wordpress_backup_428575 wordpress backup plugin], you can call us at the web site. The least difficult and very best way to do this is by acquiring a Word - Press site. To sum up, ensure that the tactics are aiming to increase the ranking and attracting the maximum intended traffic in the major search engines. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Provide the best and updated information to the web searchers and make use of these wonderful free themes and create beautiful websites. Customization of web layout is easy due to the availability of huge selection of templates. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. An ease of use which pertains to both internet site back-end and front-end users alike. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Thus, Word - Press is a good alternative if you are looking for free blogging software. Get started today so that people searching for your type of business will be directed to you. |
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| As a generalized approach for [[nonuniform sampling]], NDFT can help us to get the information of a finite length signal in frequency domain at any frequency. It can also be used to design the FIR filters as the role of DFT, no matter if it's 1-D or 2-D.
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| One of the reason to adopt NDFT is that most signals have their energy distributed nonuniformly in the frequency domain. Therefore, a nonuniform sampling scheme could be more convenient and useful in lots of applications of '''[[Digital signal processing|Digital Signal Processing (DSP)]]'''. For example, NDFT provides a variable spectral resolution controlled by the users.
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| Note that NDFT reduces to DFT when the sampling points are located on the unit circle at equally spaced angles.
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| ==1-D NDFT==
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| ===Definition===
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| 1-D NDFT of a sequence x[n] of length N is<ref>Marvasti 2001, p. 326.</ref>
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| :<math>X(z_k)=X(z)|_{z=z_k}=\sum_{n=0}^{N-1}x[n]z_k^{-n},\quad k=0, 1, ..., N-1,</math>
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| where <math>X(z)</math> is the z-transform of <math>x[n]</math>, and <math>\{z_i\}_{i=0, 1, ..., N-1}</math> are arbitrarily distinct points in the z-plane.
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| Expressing the above as matrix, we get
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| :<math>\mathbf{X}=\mathbf{D}\mathbf{x}</math>
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| where
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| :<math>
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| \mathbf{X}=\begin{bmatrix}
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| X(z_0)\\
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| X(z_1)\\
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| \vdots\\
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| X(z_{N-1})
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| \end{bmatrix},\quad
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| \mathbf{x}=\begin{bmatrix}
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| x[0]\\
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| x[1]\\
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| \vdots\\
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| x[N-1]
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| \end{bmatrix},\text{ and}\quad
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| \mathbf{D}=\begin{bmatrix}
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| 1 & z_0^{-1} & z_0^{-2} & \cdots & z_0^{-(N-1)}\\
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| 1 & z_1^{-1} & z_1^{-2} & \cdots & z_1^{-(N-1)}\\
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| \vdots & \vdots & \vdots & \ddots & \vdots\\
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| 1 & z_{N-1}^{-1} & z_{N-1}^{-2} & \cdots & \cdots & z_{N-1}^{-(N-1)}
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| \end{bmatrix}.
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| </math>
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| As we can see, the NDFT is characterized by <math>\mathbf{D}</math> and hence the N <math>{z_k}</math> points. If we further factorize <math>det(\mathbf{D})</math>, we can see that <math>\mathbf{D}</math> is nonsingular provided the N <math>{z_k}</math> points are distinct. If <math>\mathbf{D}</math> is nonsingular, we can get a unique inverse NDFT as following:
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| :<math>\mathbf{x}=\mathbf{D^{-1}}\mathbf{X}</math>
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| Given <math>\mathbf{X}\text{ and }\mathbf{D}</math>, we can use [[Gaussian elimination]] to solve <math>\mathbf{x}</math>. However, the complexity of this method is <math>O(N^3)</math>. To solve this problem more efficiently, we first determine <math>X(z)</math> directly by polynomial interpolation
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| :<math>\hat X [k]=X(z_k),\quad k=0, 1, ..., N-1,</math>
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| then x[n] is the coefficients of the above interpolating polynomial which can be solved more efficiently. This is illustrated in the next subsection.
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| ===Solving The Inverse NDFT===
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| Expressing <math>X(z)</math> as the Lagrange polynomial of order N-1, we get
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| :<math>X(z)=\sum_{k=0}^{N-1}\frac{L_k(z)}{L_k(z_k)}\hat X [k],</math>
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| where <math>\{L_i(z)\}_{i=0, 1, ..., N-1}</math> are the fundamental polynomials:
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| :<math>L_k(z)=\prod_{i\ne k}(1-z_iz^{-1}),\quad k=0, 1, ..., N-1</math>
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| Expressing <math>X(z)</math> by Newton interpolation method, we get
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| :<math>X(z)=c_0 + c_1(1-z_0z^{-1}) + c_2(1-z_0z^{-1})(1-z_1z^{-1}) + ... + C_{N-1}\prod_{k=0}^{N-2}(1-z_kz^{-1}),</math>
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| where <math>c_j</math> is the divided difference of the jth order of <math>\hat X [0], \hat X [1], ..., \hat X [j]</math> with respect to <math>z_0, z_1, ..., z_j</math>:
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| :<math>c_0 = \hat X [0],</math>
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| :<math>c_1 = \frac{\hat X [1]-c_0}{1-z_0z_1^{-1}},</math>
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| :<math>c_2 = \frac{\hat X [2]-c_0-c_1(1-z_0z^{-1})}{(1-z_0z_2^{-1})(1-z_1z_2^{-1})},</math>
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| ::<math>\vdots</math>
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| The disadvantage of Lagrange representation is that any additional point included will increase the order of the interpolating polynomial, leading to the need of recomputing all the fundamental polynomials. However, any additional point included in Newton representation only require one more term.
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| We can use a lower triangular system to solve <math>\{c_j\}</math>:
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| :<math>\mathbf{L}\mathbf{c}=\mathbf{X}</math>
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| where
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| :<math>
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| \mathbf{X}=\begin{bmatrix}
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| \hat X [0]\\
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| \hat X [1]\\
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| \vdots\\
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| \hat X [N-1]
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| \end{bmatrix},\quad
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| \mathbf{c}=\begin{bmatrix}
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| c_0\\
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| c_1\\
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| \vdots\\
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| c_{N-1}
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| \end{bmatrix},\text{ and}\quad
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| \mathbf{L}=\begin{bmatrix}
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| 1 & 0 & 0 & 0 & \cdots & 0\\
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| 1 & (1-z_0z_1^{-1}) & 0 & 0 & \cdots & 0\\
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| 1 & (1-z_0z_2^{-1}) & (1-z_0z_2^{-1})(1-z_1z_2^{-1}) & 0 & \cdots & 0\\
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| \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\
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| 1 & (1-z_0z_{N-1}^{-1}) & (1-z_0z_{N-1}^{-1})(1-z_1z_{N-1}^{-1}) & \cdots & \prod_{k=0}^{N-2}(1-z_kz_{N-1}^{-1})
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| \end{bmatrix}.
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| </math>
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| By the above equation, <math>\{c_j\}</math> can be computed within <math>O(N^3)</math> operations. In this way Newton interpolation is more efficient than Lagrange Interpolation unless the latter is modified by
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| :<math>L_{k+1}(z) = \frac{(1-z_{k+1}z^{-1})}{(1-z_kz^{-1})}L_k(z),\quad k=0, 1, ..., N-1</math>
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| ==2-D NDFT==
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| 2-D NDFT of a sequence <math>x[n_1,n_2]</math> of size <math>N_1\times N_2</math> is<ref>Marvasti 2001, p. 334.</ref>
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| :<math>\hat X(z_{1k},z_{2k})=\sum_{n_1=0}^{N_1-1}\sum_{n_2=0}^{N_2-1}x[n_1,n_2]z_{1k}^{-n_1}z_{2k}^{-n_2},\quad k=0, 1, ..., N_1N_2-1,</math>
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| where <math>\hat X(z_{1k},z_{2k})</math> is the 2-D z-transform of <math>x[n_1,n_2]</math>, and <math>(z_{1k},z_{2k})</math> are arbitrarily distinct <math>N_1N_2</math> points in the 4-D <math>(z_1,z_2)</math> space.
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| As in the 1-D case, we can express the above equation as
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| :<math>\mathbf{\hat X} = \mathbf{D}\mathbf{X},</math>
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| and the matrix <math>\mathbf{D}</math> is also depends only on the choice of those sampling points. However, even if those sampling points are distinct, <math>\mathbf{D}</math> could still be singular. No rules for determining whether the matrix is nonsingular or not have been found. Therefore, for all implementation of 2-D NDFT, we just check <math>det(\mathbf{D})</math> for a specific set of sampling points. In general, the complexity of 2-D NDFT <math>O(N_1^3N_2^3)</math>.
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| ==Applications==
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| The applications of NDFT are:
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| *Digital filter design
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| *Spectral analysis
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| *Antenna array design
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| *Antenna pattern synthesis with prescribed nulls
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| *Decoding of dual-tone multi-frequency(DTMF) signals
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| *Dual-tone multi-frequency tone detection
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| ==See also==
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| * [[Spectral estimation]]
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| == External links==
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| * [http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/PIRODDI1/NUFT/NUFT.html Non-Uniform Fourier Transform: A Tutorial].
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| * {{cite paper | id = {{citeseerx|10.1.1.15.3781}} | title = Nonuniform fast Fourier transforms using min-max interpolation }}
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| * [http://www-user.tu-chemnitz.de/~potts/nfft/guide/html/node2.html Notation, the NDFT and the NFFT]
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| * [http://www-user.tu-chemnitz.de/~potts/nfft/guide3/html/index.html NFFT 3.0 – Tutorial]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * F. Marvasti, Nonuniform sampling: Theory and Practice. Plenum Publishers Co., 2001, pp. 325–360.
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| [[Category:Fourier analysis]]
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| [[Category:Transforms]]
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| [[Category:Digital signal processing]]
| |
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Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Many times the camera is following Mia, taking in her point of view in almost every frame.
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