WO1987007053A1

WO1987007053A1 - Accumulating recursive computation

Info

Publication number: WO1987007053A1
Application number: PCT/GB1987/000297
Authority: WO
Inventors: David John Spreadbury; Thomas Edgar Curtis
Original assignee: Gec Avionics Limited; Secretary Of State For Defence In Her Britannic Ma
Priority date: 1986-05-07
Filing date: 1987-05-06
Publication date: 1987-11-19
Also published as: GB2191316A; GB8611124D0; JPH01500779A; EP0273921A1; GB8710679D0

Abstract

A processor computes Xk? from a series of data xo?..xn?..xN-1? for 0k<N where Xk? = (I) in a case where W?nk = W?nk N so that Xk? = W?Q(..W?Q (W?Q xN-1? + xN-2?) + xN-3?)...)+x1?) +xo? where Q is any non-zero integer relatively prime to N but where N need not be relatively prime to k. Selected data xn? are fed in one by one and summed in an accumulator (1). Each sum required for a given Xn? is fed to a recursive ''kernel'' (3) where it is multiplied by W the required number of times. While the kernel (3) calculates Xn? the accumulator (1) is concurrently operating on the next data.

Description

Accumulating Recursive Computation.

Frequency analysis of a waveform is a basic technique in signal processing since certain information carried by a signal can be very di fficult or impossible to extract otherwise than in the frequency domain. The range of applications of frequency analysis is very wide, and includes, for exampl e, sonar, radar and image processing. In the majority of applications the Di screte Fourier Transform (DFT) is used to compute the frequency domain matrix [X_k] from the time domain series [x_n] according to the equation for an N-point DFT:

where W^nk = e

x_n is the value of the n^th (complex) waveform sample and X_k is the amplitude of the k^th frequency component.

The expression (1), and hence also the present invention, is not limited to applications in frequency analysis, but the primary use of the expression is to perform a DFT.

The DFT may be calculated by various methods, but digital techniques are preferable where hi gh precision is requi red. The necessary hardware is simplified greatly if the DFT is calculated by a recursive prime radix transform, as described by T.E. Curtis and J.E.Wickenden in I .E.E. Proceedings Vol .130, Part F, No.5, pages 424-425. Equation (1) may be written in the recursi ve form:

X_k = W^k(... (W^k(W^kx_{(N-1 )} + x_(N-2)) + x_(N_₃₎ )... )+x₁)+x₀ (2)

and in the case that N and k are relatively prime and nk can be written modulo N i.e. W^nk = w(^nk±N), the terms W^k in equation (2) can be replaced by W^Q where Q is any non-zero value containing no prime factors common to N that are not common to all values of k. The advantages of this approach are that the DFT can be performed using only add/rotate operations rather than multiply operations which are ouch more complex to implement digital ly. The value Q can be chosen for ease of implementation and it is fixed for al l k (except k=0), whereas equation (1) requi res a different rotation for each value of k. N, which is the transform length, will typical ly be several hundred or more, and, in general , the transform wi ll be calculated for all k between 0 and N-1. To make N and k relati vely prime therefore means essential ly that N must be prime. This excludes the use of 'easy ' numbers such as N=500 which has factors 2,4,5,10 and so on, in addition to 1 and 500. Convenient binary numbers, such as 512 are also excluded since these will also have many factors.

Another drawback of this prime radix technique is that the D.C. component X_o can only be computed either by re-setting Q to zero, or in a separate accumulating device.

An object of this invention is to remove the condition that N be relatively prime to k.

According to one aspect of the invention a processor for computing a value X_k from a series of input data x_o..x_n..x_N-1 for selected values of k in the range 0≤ k<N where

in a case where W^nk = W^{nk ± N} so that x_k =W^Q(..W^Q (W^Qx_N-1 +x_N-2)+x_N-3)...)+x₁) +x_o

where Q is any non-zero integer containing no prime factors common to N that are not common to all values of k, the processor being arranged to calculate and store sequential current values, each current value being the sum o a product component and a sum component, the sum component being he sum of a group of selected data, the number in each group, whi may be one, being derived by multiplying together those prime factors of N which are common to k, and the product component being the product of W^Q and the preceding current value, the arrangement being such that a value of X_k is obtained for any non-zero Integral value of N.

The processor may include a kernel arranged to calculate the current values, an accumulator arranged to operate concurrently with the kernel to sum Input data, and a selector adapted to select data to be loaded into the kernel from raw input data and summed data from the accumulator.

According to another aspect of the invention a processor is arranged to compute a value X_k from a series of input data x_o..x_n..x_N_₁ for selected values of k in the range 0≤k<N where

in a case where W^nk = W^{nk ± N} so that

x_k =W^Q(..W^Q (W^Qx_N-1+x_N-2) +x_N-3)...)+x₁) +x₀

where Q is any non-zero integer containing no prime factor coomon to which is not common to all selected values of k, the processor being arranged to calculate X_k by

1) summing those terms x_n for which |nk|_N = m for n = 0 to N - 1, for each volume of m between 0 and N - 1 in steps of Q ii) multiplying the summed x_n terms for each m by W ^{l mQl}N to produce product terms iii) summing the product terms over all m to produce X_k. According to another aspect of the Invention a processor is arranged to compute a value X_k from a series of input data x_o..x_n..x_N-1 for selected values of k in the range 0≤ k<N where

in a case where W^nk = W^{nk ± N} so that

X_k=W^Q(..W^Q- (W^Qx_N-1+x_N-2) +x_N_₃)...)+x₁) +x₀

where Q is any non-zero integer containing no prime factor common to N which is not common to all selected values of k, the processor being arranged to calculate X_k by i) setting m = N-1,

11 ) multiplying by W⁰- to produce a product, iii ) summing those terms x_n for which|nk| _N m for n= 0 to N - 1, iv) adding said sum and said product to produce a new stored current value, v) setting m = m - 1 vi) repeating steps (ii) to (v) until m = 0, the stored current value then being equal to X_k.

One embodiment of the invention will now be described by way of example with reference to the accompanying drawings of which:-

Figure 1 shows matrix expansions of the X_k series for N = 5 and for various values of Q;

Figure 2 shows the case for N = 6 in matrix rotation;

Figure 3 is a block diagram of the elements of the processor; and

Figure 4 is a table showing the operation of the processor for the case N = 6.

Referring to the drawings, Figure 1a is a direct expansion of Equation (1) for the case N = 5. In Figure lb the powers of W have been written modulo N i.e. as the remainder after dividing by N. In this case, N happens to be prime, and therefore relatively prime to all k, and it will be observed that except for k = 0, each power of W appears once in each row of the W matrix. Figures 1c, d and e show matrix expansions of the recursive form of the series with Q = 1, 2 and 3 respectively. The same terms are being evaluated, but in a different order. The rate at which the power of W increases down the W column matrix is equal to Q.

Figure 2(a) shows the case of N = 6 in matrix notation, with the powers of W written modulo N. N is not prime and consequently powers of W occur more than once in rows of the W matrix, so it cannot be re-expressed directly in the simple recursive form of Figures 1c-e. Instead, the elements of the x square matrix shown 1n Figure 2b for Q = 1 are sums of a number (F) of x_n terms spaced along each row by F = 1 zero elements. The number of terms, F, is found by selecting those prime factors of N which are common to the prevailing k. For example, prime factors of N = 6 are 1, 2 and 3. For k = 2, say, factor 3 is not common, which leaves the factors 1 and 2, which, multiplied together give F = 2. Thus F = 2 for N = 6, k = 2, Q = 1. When N and k are relatively prime, as with N = 5, then F = 1 for all k and the matrix structure reverts to that of Figures 1c-e.

This more general matrix structure based on sums of F terms automatically incorporates the case for X_o which fits, into the pattern with F = N. (In this case of k - 0, it should be noted that all numbers are factors of zero). Despite the apparent complexity of the x matrix, there is no extra data to be handled, it is just that some data has to be summed before processing. The case for N = 6 is illustrated for the sake of simplicity, but in general N will be much larger -typically of the order of hundreds of thousands; the same rules apply however for all N. N is the transform length , a fundamental parameter that often directly affects system performance. The removal of the requirement that N be prime (which would previously have been necessary to achieve the recursive matrix form of Figures 1 c, d or e) allows greater choice and wider and simpler application of the control algorithm.

Figure 3 shows the computing structure. This could be in software or analogue or digital hardware but for convenience of description it will be treated as digital hardware. There are three components, an accumulator 1, a selector 2 and a kernel 3. The kernel constitutes the means for computing the terms W^QX and the final expressions for X_k. The kernel and accumulator operate concurrently.

The input data x_n are time samples of some signal which is to be analysed in the frequency domain, for example Doppler signals from a radar or vibration from an engine. The corresponding frequency components X_n are normally computed for all of 0 k N although a subset only may be required in certain applications. N and k are chosen by the user according to the application in hand. Q is also chosen by the user. Its value is always modulo N, and must not be zero or contain any prime factors common to N that are not also common to each and every value taken by k. (Illegal values of Q do not allow a solution to the recursive form equation (2), since the powers of W required in the direct form, equation (1), cannot be generated). The particular value of Q may be chosen for simplicity (e.g.Q = 1), or to achieve some specific implementation advantage.

As indicated in Figure 3, the accumulator 1 operates to Load, Sum or Hold. The Load operation is used when it is necessary to reinitialise the accumulator, i.e. at the start of each X_k computation sequence. The current input value x_n at A is loaded and held. 'Sum' adds the next x_n to the stored value. 'Hold' simply holds the current stored value, with no loading. The data x_n are input under the control of some control means (not shown) in the order required for the values of N, k and Q in use.

The selector 2 operates to select Zero, Accumulate or Raw for the input to the kernel 3. If Accumulate is selected the total from the accumulator 1 is fed into the kernel. If Raw is selected, the Raw input value x_n is read off path 4, which is a bypass around the accumulator. The facility for selecting Raw data enables parallel processing to take place, as will be described below.

The kernel 3 also requires reinitialising before the start of each X_n computation sequence, and this is achieved by a 'Load' operation. Otherwise, the kernel Computes i.e. multiplies the value it holds (which will initially be zero) by W^Q and adds the value selected by the Selector. The sum thus formed provides the new stored value.

The value at each point A,B,C,D in the processor is shown in the data table of Figure 4, which also shows the mode of operation of the three processor elements at each step in the computation, which is illustrated for N = 6, Q = 1.

At step 1, the accumulator is reinitialised (to zero) then loads the first input value, x₅. At step 2, the accumulator adds the next input x₃ to the stored value x₅ and so on as shown. Steps 1 to 6 compute X₃, with F = 3, the accumulator and kernel are then reinitialised (Load) and steps 7 to 12 compute X₂, with F = 2. At Steps 13 to 18 the accumulator continues to load and sum the Input data x_n, but now the selector selects Raw, that is the data loaded Into the kernel is the raw data x_n from the bypass path 4. The kernel thus computes X₁, for which F = 1 so no accumulation is required and the accumulator meanwhile accumulates the data for the X_o term, for which F = 6. This accumulation is completed at Step 18, as the computation of X₁ is completed by the kernel. The accumulator 'Holds' its value until Step 19 when it is loaded Into the reinitialised kernel and can be output directly at D as X_Q. This parallel computation of X₀ and X₁ Increases the speed of computation.

Claims

1. A processor arranged to compute a value X_k from a series of Input data x_o..x_n..X_N-1 for selected values of k in the range 0≤k<N where

In a case where W^nk = W^{nk ± N} so that

X_k=W^Q(..W^Q (W^Qx_N-1 +x_N_₂) +x_N-3)...)+x₁) +x₀

where Q is any non-zero integer containing no prime factor common to N which is not common to all selected values of k, the processor being arranged to calculate and store sequential current values, each current value being the sum of a product component and a sum component, the sum component being the sum of a group of selected data, the number in each group, which may be one, being derived by multiplying together those prime factors of N which are common to k, and the product component being the product of W^Q and the preceding current value, the arrangement being such that a value of X_k is obtained for any non-zero integral value of N.

2. A processor according to Claim 1 comprising a kernel arranged to calculate said current values, an accumulator arranged to operate concurrently with said kernel to sum input data, and a selector adapted to select data to be loaded into said kernel from raw input data and summed data from said accumulator.

3. A processor arranged to compute a value X_k from a series of input data x_o..x_n..x_N-1 for selected values of k in the range 0≤k<N where

in a case where W^nk = W^{nk ± N} so that

X_k =W^Q(..W^Q (W^Qx_N-1 +x_N-2) +x_N-3)...)+x₁) +x₀

where Q is any non-zero integer containing no prime factor common to N which Is not common to all selected values of k, the processor being arranged to calculate X_k by

1) summing those terms x_n for which

= m for n = 0 to N - 1, for each volume of m between 0 and N - 1 in steps of Q ii) multiplying the summed x_n terms for each m by W

to produce product terms iii) summing the product terms over all m to produce X_k.

4. A processor arranged to compute a value X_k from a series of input data x₀..x_n..x_N-1 for selected values of k in the range 0≤k<N where

In a case where W^nk = W^{nk ± N} so that

X_k = W^Q(..W^Q (W^Qx_N-1 +x_N-2) +x_N-3)...)+x₁) +x₀

1) setting m = N-1, ii) multiplying by W^Q to produce a product, iii) summing those terms x_n for which|nk|_N = m for n = 0 to

N - 1, iv) adding said sum and said product to produce a new stored current value, v) setting m = m - 1 vi) repeating steps (11) to (v) until m = 0, the stored current value then being equal to X_k.

5. A processor according to Claim 4 wherein two or more of steps (ii) to (v) are performed concurrently.

6. A processor substantially as hereinbefore described with reference to the accompanying drawings.