Abstract: An apparatus for computing the two-dimensional discrete Fourier transform (DFT) of an image comprised of N.times.N samples. The samples within each row are respectively multiplied by W.sup.-n.sbsp.1, n.sub.1 =0, 1, . . . , N-1 and stored in a memory 17. A device 20 derives therefrom N polynomials of N terms by means of a polynomial transform. The terms of each of these polynomials are multiplied by W.sup.n.sbsp.1 and a device 28 computes the one-dimensional DFT thereof, thereby providing the N.sup.2 terms of the transform of said image.

Abstract: When using the family of Discrete Fourier Transform (DFT) and Finite Field Transforms, the continuous convolution process corresponding to the filtering operation is converted into a series of finite aperiodic convolutions which are in turn computed as a series of circular convolutions by dividing the sequences to be convolved into blocks and lengthening these blocks either by appending zeros (overlap-add) or by repeating part of the sequences (overlap-save).This causes a serious increase in the number of operations over what would have been required if the filtering process had been a true circular convolution.This disclosure provides means for performing the filtering operation without appending any zeros and without repeating part of the sequences.The flow of input samples is split into fixed length blocks which are simultaneously fed into two circular convolutors using different DFT's. More particularly, one convolutor operates in a conventional DFT domain, i.e., a DFT providing: ##EQU1## Where x.sub.

Abstract: Digital filtering of digitally represented samples of an output analog signal may be greatly simplified when the required processing is done in a transformed domain instead of in the object domain. Such a transform is performed by converting blocks of samples ##EQU1## into blocks of signals ##EQU2## and vice versa where ##EQU3## It will be obvious that the filtering operation can be performed efficiently when the computing power required for the transforms is not too great or expensive.It has been shown that whenever the number of samples N is the square of an integer M, any Fourier transform with ##EQU4## may be performed using a bank of very simple filters. However, the technique is not as efficient in reducing the number of required computing operations as is the Fast Fourier transform decomposition.The present invention proposes a design using a bank of simple filters to adapt the above method to the computation of transform operations performed in a ring (e.g., Mersenne or Fermat transforms).

Abstract: The invention covers a precision digital filter for signal samples represented in a digital form. The flow of samples x.sub.o, x.sub.1, x.sub.2, . . . x.sub.n, of the signal is split into blocks of a fixed length which are simultaneously transmitted to two circular convolution generators operating to different prime number modulos. The outputs of the generators are each submitted to a correcting device and the corresponding corrected terms are added. The added terms are each then added to a delayed prior term to generate representations of samples of a filtered signal.

Abstract: A digital filter is disclosed which includes a circular convolution device using the Complex Mersenne transform to convert a sequence of values A.sub.n into another sequence A.sub.k in which ##EQU1## WHERE P IS PRIME NUMBER AND J IS THE SQUARE ROOT OF MINUS ONE. The convolutor is provided with an input for applying fixed length data blocks made up of input samples appended with an equal number of zeros; circuits for recirculating and accumulating said data; a register for storing said accumulated data; switches for selectively connecting the output of the storage to the inputs of an adder-subtractor; a product device for term-by-term multiplying of the output of the adder-subtractor with the Complex Mersenne transforms of the filter coefficients set appended with zeros; and an inverse transform device for performing the inverse Complex Mersenne transform on the multiplier output blocks of data.

Abstract: In a Fermat representation of numbers, the imaginary operator j is a real number which is a power of two. This is applied here to the implementation of complex digital filters, i.e., filters deriving the real and imaginary components of the desired filtered signal y by processing the components of the input signal x. Let (xm, xm) and (ym, ym) be samples of the real and imaginary components of x and y, respectively. The samples xm are delayed to provide j xm in the Fermat system, then both xm and j xm are fed to digital adders to provide xm + j xm and xm - j xm, which are separately applied to the input of transversal digital filters. The outputs of said filters are combined in simple digital adders to provide the desired y components.

Abstract: A digital filter for a plural digitally expressed pulse code modulated (PCM) signal utilizes a storage register to retain several samples of the signal and obviates the need for digital multiplication circuits by using a delta coding for the filter coefficients. Processing steps are at a multiple of the sampling rate and at each step, the value of one stored sample is combined additively or subtractively with previously stored sample values as determined by stored delta coefficient values. The summed values are periodically transmitted as output samples of a filtered PCM signal.