DE2436013C2 - Non-recursive digital filter - Google Patents

Description

The invention relates to a non-recursive digital filter according to the preamble of claim 1. One such Filter is known from IEEE Trans. AU-15 (1967), pp. 85-90.

The cited publication describes the use of the fast Fourier transform (FFT) in a digital filter for a channel. Dr.'s input signal is available here as a sequence of numbers {x _k \ , which is obtained by sampling and digitizing the originally continuous input signal χ (/). The sequence of numbers {y _s ) corresponding to the output signal of the filter results from discrete convolution

ΛΤ-1

Λ - Σ ^ -Μ'-Ο · —ΛΜ)

of the input signal with the impulse response \ h ") of the filter. The advantage of performing the discrete convolution indirectly via the fast Fourier transform (FFT) ("fast convolution") becomes apparent when one compares the number of multiplications required and neglects the additions that are less important for the complexity of a digital filter. In order to obtain a block with N output values y ^1, N 2 multiplications are required when performing the discrete convolution directly. In principle, there are 2N- log ₂ N multiplications in the FFT and 2JV multiplications in the spectral range, ie a total of 2N (1 + Iog ₂ / V) multiplications for fast convolution. So the advantage of fast convolution with large N is obvious. This advantage is also mentioned in IEEE Trans. AU-16, No. 3 (Sept. 1968), pp. 336 to 342 and in Gold / Rader, Digital Processing of Signals, Mc Graw-Hill Book Company 1969, pp. 203 to 232.

In detail, the procedure for fast folding is as follows:

First, the input signal (x _k ) is subjected to a discrete Fourier transformation (DFT) with the aid of an FFT computer, whereby the spectra lines

.V- I

Χι = Σ Xk ^w '"U = O / VI and W = e -" * with j = V ^

k = 0

receives. Each spectral line X _n is then multiplied by the associated filter coefficient H _{m of the} same frequency. The filter coefficients

iV-l

H, = Σ ^Λ » W- * U = O, ..., NI)

»-0

are the Fourier transforms of the filter parameters H _n . Finally, by inverse discrete Fourier transform (IDFT) of the products X _n , ■ H _m, the output signal is obtained

V = J-Yx H W "" (J = O NU

" m = 0

Because of the periodicity of the input signal and the impulse response required for DFT, the convolution carried out by means of DFT is cyclical. In order to obtain the desired discrete convolution instead of the cyclic convolution, special precautions must be taken. The continuous sequence of values to be filtered is grouped together -Ln blocks [x _k ) of length Λ ', in such a way that aufcinanderhaben.de blocks overlap each other in LI value «(JV> L). The number L corresponds to the number of filter parameters h " in the original impulse response. The impulse response is then brought to the block length / V by adding NL zeros. The segment of the impulse response consisting of zeros leads to the fact that the output block [x _s ) of the cyclical fast convolution in the last NL + 1 values is identical to the block that is obtained with the discrete convolution. Therefore only the last NL + 1 values of the JV of the fast convolution occurring are given to the output of the filter.

The disadvantage of the known filter mentioned at the beginning is the fact that it is only for one channel is designed. Such filters are less expensive only in multiplex operation with a sufficiently high number of channels as a corresponding analog filter. Including the effort for the analog-to-digital conversion, the benefit limit lies currently on the order of 100 channels.

From DE-OS 22 62 652 a digital filter for Filiermultiplexbetrieb is known, its transmission characteristic from equidistant and non-overlapping bandpass curves of the same shape and the same bandwidth consists. Multiplex bandpass filters, the channels of which can be spaced from one another and are of different widths cannot be implemented with this filter. However, it is precisely such possible applications that are available at a number of applications, for example message receivers in radio control centers with high traffic volumes, desirable.

The invention is based on the object of an economically working non-recursive digital filter for To create filter multiplex operation, the channels of which are of different widths and selectable distance from one another to have.

The solution to this problem comes from the characteristic. Part of claim 1 emerges. An inventive one Further development is characterized in claim 2, accordingly the spectral values of interest of a channel, which are combined into a group, are defined by the lower channel edge and the Total number of lines in the group.

The further claims contain advantageous embodiments of the invention.

In the case of the filter according to the invention, the economic outlay is lower than in the case of corresponding analog filters. The effort is essentially the same as for a conventional application of fast convolution on only one channel.

The required sampling frequency is determined by the bandwidth of the signal. The output of a 5C Bandpass is limited to a lower bandwidth than the input signal, so the sampling frequency is allowed be lower at the output, and only every Me sample is taken (bandpass sampling). There a If the non-recussive digital filter does not contain any feedback, the unsampled output values need to be used not to be charged. This reduces the technical complexity of the filter - apart from the memories - the ratio of input to output bandwidth, which is particularly important with highly selective filters Weight falls.

The invention is explained in more detail below.

In a first embodiment of the filter according to the invention for performing the fast convolution with M channels eingangssei tig then to a scanning device, and an analog-to-digital converter in addition to a first memory in which the digital numbers x _k resulting input ": rte into blocks \ x _k) the length JV are combined, a first Fourier computer is provided, with the aid of which the discrete Fourier transform (DFT) of the input values is carried out, also a second memory in which the blocks [X ₁ ) of the Fourier transform are stored, and a third memory in which the characteristic data of each channel are recorded: the filter coefficients H}, ⁿ , a parameter //, which characterizes the position of the lower channel edge, and a number JV ,, which indicates how many Fourier coefficients (spectral lines) are allocated to the channel. In addition, the filter contains a control section which compiles the Fcarier coefficients into groups in accordance with the channel parameters n, and Jv ", supplies the groups to a multiplier in which each Fourier coefficient is weighted with a filter coefficient, and the weighted groups in

stores in a fourth memory. Finally, a second Fourier computer is provided, which carries out the inverse discrete Fourier transformation (IDFT) of the groups, as well as a fifth memory, at the outputs of which the output values of the individual filter channels can be taken. The channel parameters Λ?, '\ Η, and N ₁ are selected so that the filter operates in bandpass scanning on the output side.

In a second embodiment of the filter according to the invention, the first and the second Fourier computer work according to the FFT algorithm. ^; In the event that (Λβ ,,, _αλ < / Vist, it is more advantageous if only the first Fourier computer is an FFT computer. In an expedient embodiment of the filter according to the invention, the third memory is a read-only memory, while the first, second, fourth and fifth memories are buffer memories.

ίο In a white embodiment, to avoid cyclical convolution, the number N, which corresponds to the length of the input blocks, is selected to be greater than the number /. of the filter parameters in the impulse response. The impulse response is then extended to the length / V of the input blocks by adding NL zeros, and the block formation at the filter input takes place in such a way that successive blocks each overlap in N-L + X values, and finally the / V output values per F. Entrance block only the .V ₁

υ last /V-Z.+1 values allowed as output values of the filter. ..- ^

N ■■ '*' ■■

It has proven to be advantageous to choose the length L of the impulse response to be the same. The overlap of the «j

Input blockc is then 50%. ; ·;

The scheme of discrete folding with several cannulas was the same. The continuous real-valued input Li

signal v _t is combined into blocks of length / V and transformed (DFT). The $

/ V Spectral lines are combined in groups W according to the desired bandpass channels. Each ψί

Group (») is weighted with a set of filter coefficients ( H _n * I. This gives the desired throughput λ *

adjusted and also by appropriate selection of the values in the transition area of the filter '

curve prevents crosstalk from neighboring channels. Then the groups are individually transformed back into the time domain (IDFT). As many samples are generated per data block and channel as there are lines in the group. The uniqueness of this bandpass scanning is preserved, because the combination of N ₁ adjacent lines to form a group increases the bandwidth and thus the required scanning frequency by N,.

To avoid cyclicality, a transformation block JO size N = r / J. 'is required for the rapid convolution of continuous signals, which is greater than the length of the impulse response, where Atazm is a time interval between two

successive scans and Tdei time expansion of a block corresponds. It is here "

It is expedient to choose the maximum length of the impulse response equal to 772. The overlap of the transformation blocks is then 50%, i.e. H. the data throughput doubles. The discreet fold for the Men Canal reads:

N-I V-I

y, lk ⁼ } \ ll ♦ t - - 2-1 To Al ·! » M ^e " it & >

ⁿ «-0 <n-0

k = N / 2, N / 2 + 1 / VI,

where the following definitions are used:

VI

DFT: X _k = Y ₁ x, e " ^{2 T} ' ^yuv with k : = 0, 1 / VI

. V-I

IDFT: χ, = - £ Χι. e ^2r '^{</ v} with / = 0, 1 / VI and j = / ¹ T.

> · And χ are real samples in low-pass sampling, / is the serial number of the data block and {Aj ' ¹ ] with η = 0, 1, 2, ..., N ~ \ is the point-wise transmission factor of the channel filter menu, which is discrete Fourier transform of the impulse response {/ &' ^} | results.
In order to avoid a cyclic convolution, {tf „ ⁿ } must apply to the channel characteristic

-i- Σ # "e ² - ' ^am - ^v = h _m = 0 for m = N / 2, / V / 2 + 1 NI.

»■«

The transition to bandpass sampling of the output values (y - » y) reduces the number of output channel samples per block from NIl to N ₁ Il:

** = \ Σ Σ ***** e- ² ^ - "Λί ° e ^J - ^

"» = G m = Q

Zc = N ₁ U, ΛΪ- / 2 + Ι, ..., Ni-I.
If {«5 ^/) } is chosen so that all points in the stop band and in the negative frequency band are zero,

/ β ' ¹ - »0 for η <η, and ri> ti, + N ₁ ; η, = left channel edge,
in this way the range of «can be restricted to, V, values. Then applies

M = 4; £ Σ * wa * ». e " ^{2 rX} ""'■""" Wi, e ^J -' * »\ / V _n ^ _n ^ J

The reverse transformation of the now analytical signal (/ # "= 0 for negative frequencies) into the time domain is thus represented as IDFT with N, points. The quotient η, / Ν, was assumed to be an integer, which can be achieved by changing N ₁ and // , can be reached at any time through zero-value support points of the channel characteristics.
The desired curve shape of the channel filter lays together with the conditions

^Λ «= 4; Σ W e ^2T """' ^v = 0 for m = - ^, - + 1 / VI and

"» -Ο 2 2

Λ * ⁰ - »0 for ri <n, and n ^ n, + N ₁

the parameter set {/ £ °). The starting point of the filter sizing is a target characteristic // "'(/), / i> 0, which is to throw approximated. The number of degrees of freedom for the approximation is bisected by the above conditions. Of the possible N parameters so you can only N , / 2 freely choose, every second one, e.g. so

_ // (2 / i / T) for 2n = n " //, + 2, .... /), + N ₁ (« ,, N ₁ even).
0 otherwise.

A filter is obtained which exactly assumes the target characteristic Ä at the (positive) frequencies 2λ = 0,2,4, ..., N / 2-2 , but can deviate considerably between these points. This occurs with rectangular selectivity) o-tions urven ft {f) are particularly hard on (Gibbs phenomenon). It is therefore advisable not to define some points at both edges of the selection curve, i.e. the transition areas, from the start, but to vary them until the fluctuation range of the blocking area is minimal for a given tolerance band for the pass band, or vice versa. The transition areas must be wider, the narrower the tolerance bands are to be.

The odd-numbered support points Hi'L .., which are not freely available because Λ- = Q for m> N / 2 , result from the filter curve

. V2-1 <(!, + IV) / 2-1

IAO ff ~ \ = - V> Hl.n ~ 2-jm (2n- / T) / \

"Ul *! ■ £ -> 4-1 ¹⁷ J " ^e

'"/ n = O HH ₁ II

They are very small in the stop band corresponding to the stop attenuation, but not zero.

With Ny- 3 optimized points of H {'] in each of the two transition areas of the bandpass filters, a minimum blocking attenuation of approx. 10OdB can be achieved with a ripple in the pass band of 0.5 dB.

The form factor of the filters results in 1 + 4 N _S IN ,.

In Fig. 2 an embodiment of the multiplex bandpass system according to the invention is shown. The channels (/ ') can be defined as desired, also overlapping. The input signal χ (/) is stored after sampling in a sampling device (1) and digitization in the analog-digital converter (2) as a sequence of digital numbers x _k in a first buffer memory (3) and combined into blocks of length ^, with successive blocks overlap by 50%. Two Fourier computers (4 and 10) are provided.

The first Fourier computer (4), whose transformation block size / V defines the achievable selectivity of the bandpass system, works according to the FFT algorithm. It transforms the input blocks into the spectral range. The spectral data are stored in a second buffer memory (5) and combined according to the channel parameters «, and Ν, ίμ groups. The parameters n, and Absind together with the filter coefficients Hj ^ are held in a read-only memory (6). The combination of the spectral data into groups is passed through a control part (J) , which also ensures that the groups are fed to a multiplier (8) in the order of their channel index (, -) in order to use the individual filter curves { H ¹ / ¹ ) Throw the appropriate channels weighted. After being temporarily stored in a buffer memory (9), the groups are transformed one after the other with the second Fourier computer (10) back into the time domain where they occur in bandpass scanning. This Fourier calculator implements the IDFT without using the FFT algorithm so that the possible channel widths At- do not have to be limited to powers of two. Because processing overlaps by 50%, At must be an even number. The effort of the second Fourier computer corresponds roughly to that of the first, since (/ £) _ "" ^ N is expediently kept, which compensates for the disadvantage of having to work without the FFT algorithm of a buffer store (11) removed. The recovery of the continuous signals, if desired, can be operated with bandpass filters in the original frequency range or

moderate with low passes. If low-pass filters are used, which requires less effort, the signals are at the same time transposed into the baseband (demodulation).

5 1 sheet of drawings

Claims (7)

Patent claims: 1.Non-recursive digital filter, in which on the input side a scanning device and an analog-to-digital converter are connected to a first Fourier computer which, according to the principle of fast FFT, carries out the discrete Fourier transformation (DFT) of the input values, and in which on the output side the Inverse discrete Fourier transformation (IDFT) of the filtered values is carried out, characterized in that a control part (7) is provided for the filter multiplex operation with different broad band-pass curves which can be spaced from one another, which is based on the output values [Χ /] of the first Fourier computer ( 4) channel-wise, only those spectral values that fall into the bandpass curve of the respective channel are combined into groups N ₁ , with / = running index of the channels, and fed to a multiplier (8) for weighting with the filter coefficients / ß ° of the respective channel, and that a single second Fourier calculator (10) is provided, which discrete the inverse e Fourier transform (IDFT) of the weighted groups is carried out (Fig. 2). 2. Non-recursive digital filter according to claim 1, characterized in that a first memory (3) is provided on the input side following the scanning device (1) and the analog-digital converter (2), in which the digital numbers x _t = .: input values are combined into blocks {x _k \ of length N , followed by the first Fourier computer (4) and a second memory (S) in which the blocks {X _t } of the Fourier transforms are stored, and a third memory (6) is available, in which the characteristic data of each channel are recorded: the filter coefficients Hi ⁰ , a parameter n _h which characterizes the position of the lower channel edge, and a number N ₁ which indicates how many Ijcwrier coefficients (Fourier transforms) are allotted to the channel, that the control part (7) compiles the Fourier coefficients 2u group "according to the channel parameters", · and Λ, and the groups weighted in the multiplier (8) in a fourth Memory (9) is stored, and that a fifth memory (11) is provided following the second Fourier computer (10), at the outputs of which the output values of the individual filter channels can be taken (Fig. 2). 3. Filter according to claim 2, characterized in that the first and the second Fourier calculator after FFT algorithm (FFT, Fast Fourier Transform) work. 4. Filter according to claim 2, characterized in that in the event that K, <N, only the first Fourier computer is an FFT computer. 5. Filter according to one of claims 2 to 4, characterized in that the third memory is a read-only memory and that the first, second, fourth and fifth memories are buffer memories. 6. filter inch one of claims 2 to 5, characterized in that to avoid the cyclic convolution initially the number N which corresponds to the length of the input blocks is selected greater than the number L of the filter parameters in the impulse response, and that the impulse response by appending from NL zeros to the length N of the input ^ blocks, that furthermore the block formation at the filter input takes place in such a way that successive blocks overlap each other in NL + 1 values, and that finally only the last of the Af output values per input block N-L + 1 values are allowed as output values of the filter. 7. Filter according to claim 6, characterized in that the overlap of the input blocks is equal to 50% and that the number L of the filter parameters in the impulse response is chosen to be equal.