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Download The Nonuniform Discrete Fourier Transform and Its by Sonali Bagchi, Sanjit K. Mitra PDF

By Sonali Bagchi, Sanjit K. Mitra

The progress within the box of electronic sign processing all started with the simulation of continuous-time platforms within the Nineteen Fifties, even supposing the starting place of the sector could be traced again to four hundred years whilst tools have been constructed to resolve numerically difficulties akin to interpolation and integration. over the last forty years, there were exceptional advances within the idea and alertness of electronic sign processing. in lots of functions, the illustration of a discrete-time sign or a sys­ tem within the frequency area is of curiosity. To this finish, the discrete-time Fourier rework (DTFT) and the z-transform are frequently used. on the subject of a discrete-time sign of finite size, the main general frequency-domain illustration is the discrete Fourier rework (DFT) which leads to a finite­ size series within the frequency area. The DFT is just composed of the samples of the DTFT of the series at both spaced frequency issues, or equivalently, the samples of its z-transform at both spaced issues at the unit circle. The DFT presents information regarding the spectral contents of the sign at both spaced discrete frequency issues, and hence, can be utilized for spectral research of signs. a variety of recommendations, generally called the quick Fourier remodel (FFT) algorithms, were complex for the effective com­ putation of the DFT. a tremendous instrument in electronic sign processing is the linear convolution of 2 finite-length indications, which regularly could be carried out very successfully utilizing the DFT.

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Property 1 21 Basic properties of the NDFT. :{ z;;n oX(Zk) (m = no, no + 1, ... { X*(Zk) (m = -N + 1, -N + 2, ... :{ ~ {X(Zk) - X*(Zk)} Symmetry properties For real x[n), X(Zk) = X*(zk) Re {X(Zk)} = Re {X(Zk)} 1m {X(Zk)} = - 1m {X(Zk)} IX(Zk)\ = IX(Zk)1 arg {X(Zk)} = - arg {X(Zk)} Consider the sequence, y[n] = z~x[n], n = 0,1, ... , N - 1. 20) 22 THE NDFT To interpret the meaning of the above scaling, let and k=0,1, ... ,N-1, where a and rk are non-negative. ) Zk = - = -e '+'. a Zs Therefore, as a result of multiplying the original sequence by an exponential sequence z~, the sample locations of the NDFT in the z-plane undergo a scaling in radius and a shift in angle.

The total computation is (2N + 4) real multiplications and (4N - 2) real additions. Besides, we need only the coefficients, COSWk, sinwk, and e-j(N-l)wk. If the input signal is real, Ck and 5k are also real. Therefore, the total computation reduces to (N + 2) real multiplications and (2N - 1) real additions. 4 SUBBAND NDFT In this section, we outline the subband NDFT (SB-NDFT), which is a method for computing the NDFT based on a subband decomposition of the input sequence. , 1995]. The method is useful for developing fast algorithms to approximately compute NDFT samples for signals which have their energy concentrated in only a few bands of the spectrum.

These points can be chosen arbitrarily but in such a way that the inverse transform exists. We illustrate this by a simple example. Consider the case, N l = N 2 = 2. We can express Eq. 118) In general, the 2-D NDFT matrix D is of size N l N 2 x N l N 2 . It is fully specified by the choice of the N l N 2 sampling points. For the 2-D NDFT matrix to exist uniquely, these points should be chosen so that D is nonsingular. In 41 THE NONUNIFORM DISCRETE FOURIER TRANSFORM the case of the 1-D NDFT, if the points Zk are distinct, the inverse NDFT is guaranteed to exist uniquely.

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