By Sonali Bagchi, Sanjit K. Mitra
The development within the box of electronic sign processing all started with the simulation of continuous-time platforms within the Nineteen Fifties, although the foundation of the sphere might be traced again to four hundred years while tools have been constructed to resolve numerically difficulties resembling interpolation and integration. over the last forty years, there were extra special advances within the thought 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 remodel (DTFT) and the z-transform are usually used. in terms of a discrete-time sign of finite size, the main commonly used frequency-domain illustration is the discrete Fourier rework (DFT) which ends up in 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 therefore, can be utilized for spectral research of signs. numerous strategies, generally known as the quick Fourier rework (FFT) algorithms, were complex for the effective com putation of the DFT. a huge software in electronic sign processing is the linear convolution of 2 finite-length indications, which regularly should be applied very successfully utilizing the DFT.
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Additional resources for The Nonuniform Discrete Fourier Transform and Its Applications in Signal Processing
Example text
36) h[k] = 1m {X(Zk)}' Conjugating both sides of Eq. 37) But, from Eq. 36), X*(Zk) = g[k] - jh[k]. 38) Comparing Eqs. 38), we obtain g[k] = Re {X(Zk)} = Re {X(zZ)}. 40) Comparing Eqs. 38), we obtain h[k] = 1m {X(Zk)} = - 1m {X(zZ)}. 42) This follows directly from Eq. 37). (v) arg {X(Zk)} This also follows from Eq. 37). 2 Linear Convolution Using the NDFT We now show that the linear convolution of two sequences can be computed using the NDFT. Let yL[n) be the linear convolution of two N-point sequences, x[nJ and h[nJ.
21) 54 THE NDFT The expression for Hs(w) in Eq. 9) implies that Using the definition of TsC) given in Eq. 10), we have Xs(w s ) = 1. If we substitute for Xs(w) from Eq. 14), we obtain c cos(cw s + d) + D = 1. Substituting for d from Eq. 18), we have c 1 = (w s _ 71') cos- 1 (D--C - 1) . 22) Step 5 We constrain Hp(w) so that an extremum occurs at w H (0) p = { 1 + c5 p , 1 - c5p , = O. Therefore, P = odd, P = even. 23) Using the definition of Hp(w) in Eq. 8), we obtain the condition, Tp(Xp(O» ={ -1, 1, Xp(O) = -1.
The filter coefficients. The frequency response H(e jW ) of the resulting filter is, therefore, an interpolation between the frequency samples. Although exact frequency response values are obtained at the specified frequencies, the approximation error between the interpolated response and the desired response can be unacceptably large at intermediate frequencies. , near the band edges of band-selective filters. This occurs because the desired frequency response is usually piecewise constant, with step-like transitions between successive bands.