Modular digital signal processing system

• US 4,974,187 A
• Filed: 08/02/1989
• Issued: 11/27/1990
• Est. Priority Date: 08/02/1989
• Status: Expired due to Term

• Current Assignee: Aware Incorporated (Crowley Holdings Preferred LLC)
• Original Assignee: Aware Incorporated (Crowley Holdings Preferred LLC)
• Inventors: Lawton, Wayne M.
• Primary Examiner(s): Shaw, Dale M.
• Assistant Examiner(s): Mai, Tan V.
• Application Number:
  US07/388,384
• Time in Patent Office: 482 Days
• Field of Search: 364/728.01, 364/728.03, 364/728.05, 364/715.01, 364/724.12, 382/42
• US Class Current: 708/420
• CPC Class Codes: G06F 17/10 Complex mathematical operations function generation by table look-up G06F1/03; evaluation of elementary functions by calculation G06F7/544
• CPC Class Codes: G06F 17/14 Fourier, Walsh or analogous domain transformations , e.g. Laplace, Hilbert, Karhunen-Loeve, transforms for correlation function computation G06F17/156; spectrum analysers G01R23/16
• CPC Class Codes: G06F 17/15 Correlation function computation including computation of convolution operations arithmetic circuits for sum of products per se, e.g. multiply-accumulators G06F7/5443; digital filters, e.g. FIR, IIR, adaptive filters H03H17/00
• CPC Class Codes: G06F 2218/08 Feature extraction
• CPC Class Codes: G10L 19/0216 using wavelet decomposition

Abstract

A modular digital signal processor system for calculating "wavelet-analysis transformations" and "wavelet-synthesis transformations" of one-dimensional numerical data and multi-dimensional numerical data for solving speech processing and other problems. The system includes one or more "dual-convolver" components, "analyzer-adjunct" components, "synthesizer-adjunct" components, "de-interleaver components" and "interleaver components", and specific configurations of these components for implementing specific functions. Each dual-convolver is capable of loading a finite number of numerical values into its coefficient registers and subsequently performing two convolution operations on input sequences of numerical values to produce an output sequence of numerical values. Two dual-convolvers are configured with an analyzer adjunct or synthesizer adjunct to respectively build a single stage analyzer or synthesizer. Analyzers and synthesizers are configured in conjunction with interleavers and de-interleaver components to build wavelet sub-band processors capable of decomposing one-dimensional sequences of numerical data or multi-dimensional arrays of numerical data into constituent wavelets and to synthesize the original sequences or arrays from their constituent wavelets. Synthesizers are configured with interleavers to build function generators capable of calculating functions including wavelet functions to within any specified degree of detail.

Claims

1. A dual convolver circuit for convolving a sequence of values X(m) wherein m are successive integers, to obtain an output sequence Y(m) according to the equation ##EQU31## the circuit comprising:

a plurality of scalar multipliers e_k, k=0 to L-1, where L is a positive integer greater than 1, for respectively multiplying each value X(m) by a scalar value e_k, each of said scaler multipliers having an output port;

a first adder AD₀ having a first input port connected to the output port of the scalar e₀, a second input port and first and second output ports;

a plurality of adders AD_k, k=1 to L-1, the (k+1)th adder AD_k having a first input port connected to the output port of the scaler multiplier e_k, second and third input ports and first and second output ports;

an Lth adder AD_L-1 having a first input port connected to the output port of the scalar e_L-1, a second input port and first and second output ports;

a plurality of delay and switch means V_k, k=0 to L-3, respectively disposed between the (k+1)th and (k+2)th adders AD_k and AD_k+1, each delay and switch means V_k including a first two sample delay element Z_k1^-2, a second two sample delay element Z_k2^-2, a first switch SW_k1 alternately connecting the first output port of the (k+1)th adder AD_k to an input port of the first two sample delay element Z_k1^-2 in a first state of the circuit and the second input port of the (k+1)th adder AD_k to an output port of the second two sample delay element Z_k2^-2 in a second state of the circuit, and a second switch SW_k2 alternately connecting the third input port of the (k+2)th adder AD_k+1 to an output port of the first two sample delay element Z_k1^-2 in the first state of the circuit and the second outport port of the (k+2)th adder AD_k+1 to an input port of the second two sample delay element Z_k2^-2 in the second state of the circuit;

an (L-1)th delay and switch means V_L-2, disposed between the (L-1)th and Lth adders AD_L-2 and AD_L-1, and including a first two sample delay element Z.sub.(L-2)1^-2 and a second two sample delay element Z.sub.(L-2)2^-2, a first switch SW.sub.(L-2)1 alternately connecting the first output port of the (L-1)th adder AD_L-2 to an input port of the first two sample delay element Z.sub.(L-2)1^-2 in the first state of the circuit and the second input port of the (L-1)th adder AD_L-2 to an output port of the second two sample delay element Z.sub.(L-2)2^-2 in the second state of the circuit, and a second switch SW.sub.(L-2)2 alternately connecting the second input port of the Lth adder AD_L-1 to an output port of the first two sample delay element Z.sub.(L-2)1^-2 in the first state of the circuit and the second output port of the Lth adder AD_L-1 to an input port of the second two sample delay element Z.sub.(L-2)2^-2 in the second state of the circuit; and

means for controlling the first and second switches of said delay and switch means V_k, k=0 to L-2 to alternate the first and second states of the circuit with successive value X(m) such that the first state corresponds to values X(m) when m is an even integer and the second state corresponds to values X(m) when m is an odd integer the first through (L-1)th adders AD_k, k=0 to L-2 providing at the second output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said second state, the first through (L-1)th adders AD_k, k=0 to L-2 providing at the second output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said second state, the Lth adder AD_L-1 providing at the second output port thereof the value at the first input port thereof when the circuit is in said second state, the second through (L-

     1)th adders AD_k, k=1 to L-2 providing at the first output port thereof a sum of input values at the first and third input ports thereof when the circuit is in said first state, the Lth adder AD_L, providing at the first output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said first state, the first adder AD₀ providing at the first output port thereof the value at the first input port thereof when the circuit is in said first state.

2. A dual convolver circuit as in claim 1, further comprising an output switch having an output port and means for alternately connecting said second output port of said first adder AD₀ and second output port of said Lth adder AD_L-1 to said output port of said output switch respectively during said first state and said second state of the circuit.

3. A signal processor for processing a sequence of input values X.sup.(1) (m) wherein m are successive integers, comprising:

an analyzer means having a sequence input port and a sequence output port; and

a de-interleaver means having an input port and first and second output ports,

a first de-interleaver connecting means for connecting the input port of the de-interleaver means to the output port of the analyzer means; and

a second de-interleaver connecting means for connecting the first output port of the de-interleaver means to the input port of the analyzer means;

said analyzer means comprising means, responsive to the sequence of input values X.sup.(1) (m) and sequences of input values X.sup.(i) (m) for i=2 to J, J a positive integer greater than one, wherein m are successive integers, at its input port for outputting at its output port sequences of output values Y.sup.(i) (m) for i=1 to J, given by ##EQU32## where N_i for i=1 to J are even positive integers, and k, i=1 to J, k=0 to N_i -1, are numerical constants which, for each i, satisfy for all even integers j the equation ##EQU33## said de-interleaver means having means, responsive to the sequence of values Y.sup.(i) (m), i=1 to J-1, at its input port, for outputting at the first output port a sequence of values X.sup.(i+1) (m) equal to Y.sup.(i) (2m) and for outputting at the second output port a sequence of values S.sup.(i) (m) equal to Y.sup.(i) (2m+1).

4. A signal processor as in claim 3, wherein said second de-interleaver connecting means includes:

a plurality, J in number, of buffer means B_i, where i=1 to J, each having an input port and an output port, for storing the respective sequence of values X.sup.(i) (m);

an analyzer buffer selector means for selectively connecting the output ports of the buffer means B_i, i=1 to J, respectively to the input port of the analyzer means and for selectively connecting the input ports of the buffer means B_i, for i=2 to J to the output port of the de-interleaver means; and

means for providing the sequence X.sup.(1) (m) to the input port of the buffer means B₁.

5. A signal processor as in claim 4, wherein said analyzer means has a parameter input port and is further responsive to the numeral values a.sup.(i)_k at the parameter input port, for outputting the sequence of output values Y.sup.(i) (m) at the sequence output port.

6. A signal processor as in claim 3, wherein the numerical values a.sup.(i)_k and integers N_i are respectively equal to a same even integer N and a same value a_k for all i, i=1 to J.

7. A signal processor as in claim 3, further comprising processing means, having an input port coupled to the output ports of said analyzer means and said de-interleaver means, for processing the sequences S.sup.(i) (m) and outputting at an output port the output sequences T.sup.(i) (m), for each i, i=1 to J and where S.sup.(J) (m) is given by Y.sup.(J) (m).

8. A signal processor as in claim 7, further comprising:

a synthesizer means having an input port and an output port;

an interleaver means, having an output port and first and second input ports;

means for connecting the output port of the interleaver means to the input port of the synthesizer means; and

means for connecting the first input port of the interleaver means to the output port of the synthesizer means, and means for connecting the second input port of the interleaver to the output port of the processing means;

the synthesizer means including means, responsive to a sequence of input values W.sup.(i) (m,n) at its input port for each i, i=1 to J, wherein m represents successive integers, for generating and outputting at its output port a sequence of output values Z.sup.(i) (m) given by ##EQU34## the interleaver means being responsive to the sequence T.sup.(i) (m) at its second input port and to the sequence Z.sup.(i+1) (m) at its first input port for i=1 to J-1, for outputting at the output port the sequence of values W.sup.(i) (m), being equal to Z.sup.(i+1) (m/2) for m even and equal to T.sup.(i) ((m-1)/2) for m odd;

the output port of the processing means being connected to input port of the synthesizer means so that the sequence W.sup.(J) (m) input to the input port of the synthesizer S_J is an output sequence T.sup.(J) (m) of the processing means, whereby if the effect of the processing means is such that the sequences T.sup.(i) (m) are respectively approximately equal to the sequences S.sup.(i) (m), then the sequences Z.sup.(i) (m) will be approximately equal to the respective sequences ##EQU35##

9. A signal processor as in claim 8, wherein said second interleaver connecting means includes:

a plurality, J in number, of buffer means C_i, where i=1 to J, each having an input port and an output port, for storing the respective sequence of values W.sup.(i) (m);

a synthesizer buffer selector means for selectively connecting the output ports of the buffer means C_i, i=1 to J, to the input port of the synthesizer means and for connecting input ports of the buffer means C_i, for i=1 to J-1, to the output port of the interleaver means; and

means for providing the sequences T.sup.(J) (m) from the output port of the processor means to the input port of the buffer means C_J.

10. A signal processor as in claim 3, wherein said analyzer means

for processing a sequence of input values X.sup.(1) (m) wherein m are successive integers, comprising a plurality, J in number, of successive analyzer circuits A_i, where i=1 to J, each having an input port and an output port;

and said de-interleaver means includes;

J-1 de-interleaver circuits D_i, where i=1 to J-1, each having an input port and first and second output ports, for all i, i=1 to J-1, the input port of the de-interleaver D_i being connected to the output port of the analyzer circuit A_i and the first output port of the de-interleaver D_i is connected to the input port of the analyzer circuit A_i+1 ;

each analyzer circuit A_i comprising means, responsive to the sequence of input values X.sup.(i) (m) for outputting at its output port the sequence of output values Y.sup.(i) (m) given by ##EQU36## where N_i is an even positive integer and a.sup.(i)_k, k=1 to N_i, are numerical constants which, for each i, satisfy for all even integers j the equation ##EQU37## each de-interleaver circuit D_i having means, responsive to the sequence of values Y.sup.(i) (m), for outputting at the first output port the sequence of values X.sup.(i+1) (m) and for outputting at the second output port the sequence of values S.sup.(i) (m).

11. A signal processor as in claim 10, wherein N_i =N for all i, i=1 to J and N is an even positive integer.

12. A signal processor as in claim 10, further comprising a plurality of processing means P_i, J in number, for each value of i, said processing means P_i being coupled to the output port of the de-interleaver D_i for i=1 to J-1 and to the output port of the analyzer circuit A_J, for processing the sequences S.sup.(i) (m) and outputting at an output port the output sequences T.sup.(i) (m).

13. A signal processor as in claim 12, wherein N_i =N for all i, i=1 to J and N is an even positive integer.

14. A signal processor as in claim 12, further comprising a plurality, J in number, of successive synthesizer circuits S_i, where i=1 to J, each having an input port and an output port;

and

J-1 interleaver circuits I_i, where i=1 to J-1, each having an output port and first and second input ports, for all i, i=1 to J-1, the output port of the interleaver circuits I_i is connected to the input port of the synthesizer circuits SS_i, the first input port of the interleaver circuit I_i is connected to the output port of the synthesizer circuit S_i+1, and the second input port of the interleaver I_i is connected to the output port of the processing means P_i ;

each synthesizer circuits SS_i comprising means, responsive to a sequence of input values W.sup.(i) (m) at its input port wherein m represents success integers, for generating and outputting at its output port a sequence of output values Z.sup.(i) (m) given by ##EQU38## each interleaver circuit I_i having means responsive to the sequences T.sup.(i) (m) and Z.sup.(i) (m), for outputting at the output port the sequence of values W.sup.(i) (m), being equal to Z.sup.(i+1) (m/2) for m even and equal to T.sup.(i) ((m-1)/2) for m odd, the output port of the processing means P_J being connected to input port of the synthesizer circuit S_J so that the sequence W.sup.(J) (m) input to the input port of the synthesizer S_J is the output sequence of the processing means P_J, whereby if the effect of the processing means P_i, i=1 to J is such that the sequences T.sup.(i) (m) are respectively approximately equal to the sequences S.sup.(i) (m), then the sequences Z.sup.(i) (m) are approximately equal to the respective sequences ##EQU39##

15. A signal processor as in claim 14, wherein N_i =N for all i, i=1 to J and N is an even positive integer.

16. A signal processor for processing an ordered array of input values X.sup.(1) (m,n) wherein m and n each represent successive integers, comprising:

an analyzer means having an array input port and first and second array output ports; and

a de-interleaver means having an input port and first and second output ports,

a first de-interleaver connecting means for connecting the input port of the de-interleaver means to the first array output port of the analyzer means;

a second de-interleaver connecting means for connecting the first output port of the de-interleaver means to the array input port of the analyzer means;

said analyzer means comprising means, responsive to an array of input values X.sup.(1) (m,n) at its array input port for all i=1 to J, J a positive integer greater than one, wherein m are successive integers, for outputting at its first array output port for i=1 to J-1 and at its second array output port for i=J an array of output values Y.sup.(i) (m,n) given by ##EQU40## where M_i and N_i, i=1 to J, are even positive integers, and α

_j, j=0 to M_i -1 and β

_k, k=0 to N-1, are numerical constants which, for each i, satisfy for all even integers s and t the equations ##EQU41## said de-interleaver means having means, responsive to the array of values Y.sup.(i) (m,n), i=1 to J-1, at its input port, for outputting at the first output port the array of values X.sup.(i+1) (m,n) equal to Y.sup.(i) (2m,2n) and for outputting at the second output port arrays of values S.sup.(i) _OO (m, n), equal to Y.sup.(i) (2m+1,2n+1), (m,n) equal to Y.sup.(i) (2m+1,2n), and S.sup.(i)_EO (m,n) equal to Y.sup.(i) (2m,2n+1).

17. A signal processor as in claim 16, wherein said second de-interleaver connecting means includes

a plurality, J in number, of buffer means B_i, where i=1 to J, each having an input port and an output port, for storing the respective array of values X.sup.(i) (m,n);

an analyzer buffer selector means for selectively connecting the output ports of the buffer means B_i, i=1 to J, respectively to the array input port of the analyzer means and for selectively connecting the input ports of the buffer means B_i, for i=2 to J to the first output port of the de-interleaver means, and

means for providing the array X.sup.(1) (m,n) to the input port of the buffer means B₁.

18. A signal processor as in claim 16, wherein said analyzer means has a parameter input port and is further responsive to application of the numerical constants α

.sup.(i)_j and β

.sup.(i)_k at the parameter input port for outputting the array of output values Y.sup.(i) (m,n) at its array output port.

19. A signal processor as in claim 16, wherein said analyzing means comprises a first dimension analyzer and a second dimension analyzer, said first dimension analyzer having an output port and an input port and means, responsive to the array X.sup.(i) (m,n) for each i, i=1 to J, at its input port, for generating and outputting at its output port for i=1 to J, an array of output values U_A.sup.(i) (m,n) given by ##EQU42## said second dimension analyzer having an output port, an input port connected to the output port of the first dimension analyzer and having means, responsive to the array of values U.sup.(i)_A (m,n) for each i, i=1 to J, at its input port for generating and outputting at its output port for i=1 to J the array of output values Y.sup.(i) (m,n) given by ##EQU43##

20. A signal processor as in claim 16, wherein said de-interleaver means comprises first, second and third de-interleaving means, said first de-interleaving means having an input port and first and second output ports, and means, responsive to the array of values Y.sup.(i) (m,n) for i=1 to J-1 at its input port, for generating and outputting at its first and second output ports respectively the arrays of values U.sup.(i)_D =Y.sup.(i) (m,2n+1) and V.sup.(i)_D =Y.sup.(i) (m,2n), said second de-interleaving means having an input connected to the first output of said first de-interleaving means, first and second output ports, and means, responsive to the array of values U.sup.(i)_D (m,n) at its input port, for generating and outputting at its first and second output ports, respectively, the arrays of values S.sup.(i)_OO (m,n)=U.sup.(i)_D (2m+1,n) at its first output port and the array of values S.sup.(i)_EO (m,n)=U.sup.(i)_D (2m,n) at its second output port, and said third de-interleaving means having an input port connected to the second output port of said first de-interleaving means, first and second output ports, and means, responsive to the array of values V.sup.(i)_D (m,n) at its input port, for generating and outputting at its first and second output ports, respectively, the array of values S.sup.(i)_OE (m,n)=V.sup.(i)_D (2m+1,n) at its first output port and the array of values X.sup.(i+1) (m,n)=V.sup.(i)_D (2m,n) at its second output port.

21. A signal processor as in claim 16, wherein for all i, i=1 to J, N_i is equal to a same even positive integer.

22. A signal processor as in claim 16, further comprising processing means, having an input port coupled to the output ports of said analyzer means and said de-interleaver means, for processing the arrays S.sup.(i)_OO (m,n) and outputting at a first output port, output arrays T.sup.(i)_OO (m,n) T.sup.(i)_OE (m,n) and T.sup.(i)_EO (m,n), for each i, i=1 to J, and for processing the array Y.sup.(i) (m,n) and outputting at a second output port the output array W.sup.(J) (m,n).

23. A signal processor as in claim 22, further comprising:

a synthesizer means having an array input port, a parameter input port and first and second output ports;

an interleaver means, having an output port and first and second input ports;

means for connecting the output port of the interleaver means to the array input port of the synthesizer means; and

means for connecting the first input port of the interleaver means to the first output port of the synthesizer means, means for connecting the second output port of said processing means to the array input port of said synthesizer means, and means for connecting the second input port of the interleaver to the first output port of the processing means;

the synthesizer means including means, responsive to an array of input values W.sup.(i) (m,n) at its array input port for each i, i=1 to J, wherein m and n represent respective sequences of successive integers, for generating and outputting at its first output port an array for i=1, to J-1 and at its second output port for i=0 of output values Z.sup.(i) (m,n) given by ##EQU44## the interleaver means being responsive to the arrays T.sup.(i)_OO (m,n), T.sup.(i)_OE (m,n) and T.sup.(i)_EO (m,n) at its second input port and to the array Z.sup.(i+1) (m,n) at its first input port for i=1 to J-1, for outputting at the output port the array of values W.sup.(i) (m,n), being equal to Z.sup.(i+1) (m/2,n/2) for m even and n even, equal to T.sup.(i) OO((m-1)/2,n-1)/2) for m odd and n odd, equal to T.sup.(i)_EO (m/2,n-1)/2) for m even n odd, and equal to T.sup.(i)_OE ((m-1)/2,n/2) for m odd and n even;

the second output port of the processing means being connected to input port of the synthesizer means so that the array W.sup.(J) m,n) input to the input port of the synthesizer means is an output array of the processing means, whereby if the effect of the processing means is such that the array Y.sup.(J) m,n) is approximately equal to the array Y.sup.(J) m,n) and the arrays T.sup.(i)_OO (m,n), T.sup.(i)_OE ((m,n) and T.sup.(i)_EO ((m,n) respectively are approximately equal to the arrays S.sup.(i)_OO (m,n), S.sup.(i)_OE (m,n) and S.sup.(i)_EO (m,n) then the arrays Z.sup.(i) m,n) will bee approximately equal to the respective arrays ##EQU45##

24. A signal processor as in claim 23, wherein said second interleaver connecting means includes:

a plurality, J in number, of buffer means C_i, where i-1 t J, each having an input port and an output port, for storing the respective array of values W.sup.(i) (m,n);

a synthesizer buffer selector means for selectively connecting the output ports of the buffer means C_i i-1to J, to the array input port of the synthesizer means and for connecting input ports of the buffer means C_i, for i=1 to J-1, to the output port of the interleaver means; and

means for providing the array W.sup.(J) (m,n) from the second output port of the processor means to the input port of the buffer means C_J.

25. A signal processor as in claim 23 wherein said synthesizing means comprises a first dimension synthesizer and a second dimension synthesizer, said first dimension synthesizer having an input port, an output port and means, responsive to the array of values W.sup.(i) (m,n) for each i, i=1 to J, for generating and outputting at its output port an array of output values V_S.sup.(i) (m,n) given by ##EQU46## said second dimension synthesizer having an output port, an input port connected to the output port of the first dimension synthesizer, and means, responsive to the array of values V_S.sup.(i) (m,n) for i=1 to J, at its input port, for generating and outputting at its output port for i=1 to J, the array of output values Z.sup.(i) (m,n) given by ##EQU47##

26. A signal processor as in claim 23, wherein said interleaver means comprises first, second and third de-interleaving means, said first interleaving means having an output port and first and second input ports, and means, responsive to the arrays of T.sup.(i)_OO (m,n) and T.sup.(i)_EO (m,n), for i=1 to J-1, values respectively at its first and second input port, for generating and outputting at its output port the array of values U.sup.(i)_I (m,n) equal to T.sup.(i)_OO ((m-1)/2,n) for m odd and equal to T.sup.(i)_EO (m/2,n) for m even, said second interleaving means having first and second input ports and an output port and means, responsive to the arrays of values T.sup.(i)_OE (m,n) respectively at its first and second input ports, for generating and outputting at its output port the arrays of values V.sup.(i)_I (m,n) equal to T.sup.(i) ((m-1)/2,n), for m odd and equal to Zⁱ⁺¹ (m/2,n) for m even, and said third interleaving means having a first input port connected to the output port of said first de-interleaving means, a second input port connected to the output port of the second de-interleaver means, and means, responsive to the arrays of values V.sup.(i)_I (m,n) at its first input port, and V.sup.(i)_I (m,n) at its second input port, for generating and outputting at its output port, the array of values W.sup.(i) (m,n) equal to U.sup.(i)_I (m,(n-1)/2) for n odd and equal to V.sup.(i)_I (m,n/2) for n even.

27. A signal processor as in claim 16 wherein said analyzer means includes

a plurality, J in number, of successive two-dimensional analyzer circuits AA_i, where i=1 to J, each having an input port and an output port;

and said de-interleaver means includes

J-1 two-dimensional de-interleaver circuits DD_i, where i=1 to J-1, each having an input port and first and second output ports, for all i, i=1 to J-1, the input port of the de-interleaver DD_i being connected to the output port of the analyzer circuit AA_i and the first output port of the de-interleaver D_i is connected to the input port of the analyzer circuit AA_i+1 ;

each analyzer circuit AA_i comprising means, responsive to a sequence of input values X.sup.(i) (m,n), for outputting at its output port the array of output values Y.sup.(i) (m,n), each de-interleaver circuit DD_i having means, responsive to the array of values Y.sup.(i) (m,n), for outputting at the first output port the values X.sup.(i+1) (m,n) and for outputting at the second output port the arrays of values S.sup.(i)_OO m,n), S.sup.(i)_OE (m,n) and S.sup.(i)_EO (m,n).

28. A signal processor as in claim 27, wherein M_i =M and N_i =N for all i, i=1 to J and M and N are even positive integers.

29. A signal processor as in claim 27, further comprising a plurality of processing means PP_i, J in number, for each value of i, said processing means PP_i being coupled to the output port of the de-interleaver DD_i for i=1 to J-1, for processing the respective arrays S.sup.(i)_OO (m,n), S.sup.(i)_OE (m,n) and S.sup.(i)_EO (m,n) and outputting at a first output port output arrays T.sup.(i)_OO (m,n), T.sup.(i)_OE (m,n) and T.sup.(i)_EO (m,n), and being coupled to the output ort of the analyzer circuit AA_J for i=J for processing the array Y.sup.(J) (m,n) and outputting at a second output port an array W.sup.(J) (m,n).

30. A signal processor as in claim 29, wherein M_i =M and N_i =N for all i, i=1 to J and M and N are even positive integers.

31. A signal processor as in claim 27, further comprising a plurality, J in number, of successive two-dimensional synthesizer circuits SS_i, where i=1 to J, each having an input port and an output port;

and

J-1 interleaver circuits II_i, where i=1 to J-1, each having an output port and first and second input ports, for all i, i=1 to J-1, the output port of the interleaver circuit II_i is connected to the input port of the synthesizer circuit SS_i, the first input port of the interleaver circuit II_i is connected to the output port of the synthesizer circuit SS_i+1, and the second input port of the interleaver circuit II_i is connected to the output port of the processing means PP_i ;

each synthesizer circuit SS_i comprising means, responsive to an array of input values W.sup.(i) (m,n) at its input port wherein m and n represents sequences of successive integers, for generating and outputting at its output port an array of output values Z.sup.(i) (m,n) given by ##EQU48## the interleaver means being responsive to the arrays T.sup.(i)_OO (m,n), T.sup.(i)_OE (m,n) and T.sup.(i) EO (m,n) at its second input port and to the array Z.sup.(i+1) (m,n) at its first input port for i=1 to J-1, for outputting at the output port the array of values W.sup.(i) (m,n), being equal to Z.sup.(i+1) (m/2,n/2) for m even and n even, equal to T.sup.(i)_OO ((m-1)/2, (n-1)/2) for m odd and n odd, equal to T.sup.(i)_EO (m/2,(n-1)/2) for m even and n odd, and equal to T.sup.(i)_OE ((m-1)/2,n/2) for m odd and n even;

the output port of the processing means PP_J being connected to input port of the synthesizer circuit SS_J so that the sequence W.sup.(J) (m,n) input to the input port of the synthesizer SS_J is the output sequence of the processing means PP_J, whereby if the effect of the processing means PP_i, i=1 to J is such that for each i, i=1 to J-1) the arrays T.sup.(i)_OO (m,n), T.sup.(i)_OE (m,n) and T.sup.(i)_EO (m,n) are respectively approximately equal to the arrays S.sup.(i)_OO (m,n), S.sup.(i)_OE (m,n) and S.sup.(i)_EO (m,n) and W.sup.(J) (m,n) is approximately equal to Y.sup.(J) (m,n) then the arrays Z.sup.(i) (m,n) will be approximately equal to the respective arrays ##EQU49##

32. A signal processor as in claim 31, wherein M_i =M and N_i =N for all i, i=1 to J and M and N are even positive integers and wherein for each i, i=1 to J, each j, j=0 to M-1 and each k, k=0 to N-1, α

.sup.(i) j=α

j and β

.sup.(i)_k =α

_k where β

_j, j=0 to M-1 and β

_k, k=0 to N-1 are predetermined numerical values.

33. A single stage wavelet analyzer circuit for processing a sequence of values X(m) wherein m are successive integers, to obtain an output sequence Y(m) according to the equation ##EQU50## where N is a positive even integer, the wavelet analyzer circuit comprising:

first and second dual-convolver circuits and an analyzer-adjunct circuit, said first dual convolver circuit comprising means for convolving the sequence of values X(m) to obtain a first sequence Y₁ (m) at an output port thereof according to the equation ##EQU51## and said second dual convolver circuit comprising means for convolving the sequence of values X(m) to obtain a second sequence Y₂ (m) at an output port thereof according to the equation ##EQU52## said analyzer-adjunct circuit comprising means, having input ports respectively connected to the output ports of said first and second dual convolver circuits, for processing the first and second sequences Y₁ (m) and Y₂ (m) to obtain the sequence Y(m) according to the relation

space="preserve" listing-type="equation">Y(m)=Y.sub.1 (m-2)+Y.sub.2 (m-1) for m even

space="preserve" listing-type="equation">Y(m)=-Y.sub.1 (m)+Y.sub.2 (m-1) for m odd; and

the values β

_k, k=0 to N-1, are numerical constants which satisfy for all even integers j the equation ##EQU53##

34. A circuit as in claim 33, wherein said first and second dual-convolvers each includes a plurality of scaler multiplers e_k, k=0 to L-1, L=N/2, for respectively multiplying each value X(m) by a corresponding scaler value, each of said scaler multipliers having an output port;

a first adder AD_O having a first input port connected to the output port of the scaler e_O, a second input port and first and second output ports;

a plurality of adders AD_k, k=1 to L-2, the (k+1)th adder AD_k having a first input port connected to the output port of the scaler e_k, second and third input ports and first and second output ports;

an Lth adder AD_L-1 having a first input port connected to the output port of the scaler e_L-1, a second input port and first and second output ports;

a plurality of delay and switch means V_k, k 32 0 to L-3, respectively disposed between the (k+1)th and (k+2)th adders AD_k and AD_k+1, each delay and switch means V_k including a first two sample delay element Z_k1^-2, a second two sample delay element Z_k2^-2 a first switch SW_k1 alternately connecting the first output port of the (k+1)th adder AD_k to an input port of the first two sample delay element Z_k1^-2 in a first state of the circuit and the second input port of the (k+1) th adder AD_k to an output port of the second two sample delay element Z_k2^-2 in a second state of the circuit, and a second switch SW_k2 alternately connecting the third input port of the (k+2)th adder AD_k+ 1 to an output port of the first two sample delay element Z_k1^-2 in the first state of the circuit and the second outport port of the (k+2)th adder AD_k+1 to an input port of the second two sample delay element Z_K2^-2 in the second state of the circuit;

an (L-1)th delay and switch means V_L-2, disposed between the (L-1)th and Lth adders AD_L-2 and AD_L-1, and including a first two sample delay element Z.sub.(L-2)1^-2 and a second two sample delay element Z.sub.(L-2)2^-2, a first switch SW.sub.(L-2)1 alternately connecting the first output port of the (L-1)th adder AD_L-2 to an input port of the first two sample delay element Z.sub.(L-2)1^-2 in the first state of the circuit and the second input port of the (L-1)th adder AD_L-2 to an output port of the second two sample delay element Z.sub.(L-2)2^-2 in the second state of the circuit, and a second switch SW.sub.(L-2)2 alternately connecting the second input port of the Lth adder AD_L-1 to an output port of the first two sample delay element Z.sub.(L-2)1^-2 in the first state of the circuit and the second output port of the Lth adder AD_L-1 to an input port of the second two sample delay element Z.sub.(L-2)2^-2 in the second state of the circuit; and

means for controlling the first and second switches of said delay and switch means V_k, k=0 to L-2 to alternate the first and second states of the circuit with successive value X(m) such that the first state corresponds to values X(m) when m is an even integer and the second state corresponds to values X(m) when m is an odd integer, the first through (L-1)th adders AD_k, k=0 to L-2 providing at the second output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said second state, the first through (L-1)th adders AD_k, k=0 to L-2 providing at the second output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said second state, the Lth adder AD_L-1 providing at the second output port thereof the value at the first input port thereof when the circuit is in said second state, the second through (L-

     1)th adders AD_k, k=1 to (L-2) providing at the first output port thereof a sum of input values at the first and third input ports thereof when the circuit is in said first state, the Lth adder AD_L, providing at the first output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said first state, the first adder AD₀ providing at the first output port thereof the value at the first input port thereof when the circuit is in said first state;

wherein the respective scalers e_k of the first dual convolver have scaler values equal to a_2k and the respective scalers e_k of the second dual convolver have scaler values equal to a_N-1-2k.

35. A single stage wavelet synthesizer circuit for processing a sequence of values W(m) wherein m designates success integers, to obtain an output sequence Z(m) according to the equation ##EQU54## where N is a positive even integer, the circuit comprising:

first and second dual-convolvers circuits and an analyzer-adjunct circuit, said first dual convolver comprising means for convolving the sequence of values W(m) to obtain a first sequence Z₁ (m) at an output port thereof according to the equation ##EQU55## and said second dual convolver comprising means for convolving the sequence of values W(m) to obtain a second sequence Z₂ (m) at an output port thereof according to the equation ##EQU56## said synthesizer-adjunct circuit comprising means, having input ports respectively connected to the output ports of said first and second dual convolver circuits, for processing the first and second sequences Z₁ (m) and Z₂ (m) to obtain the sequence Z(m) according to the relation

space="preserve" listing-type="equation">Z(m)=Z.sub.1 (m)+Z.sub.2 (m-1) for m even

space="preserve" listing-type="equation">Z(m)=-Z.sub.1 (m-2)+Z.sub.2 (m-1) for m odd

wherein the values β

_k, k=0 to N-1 are numerical constants which satisfy for all even integers j the equation ##EQU57##

36. A circuit as in claim 35, wherein said first and second dual-convolvers each includes a plurality of scalers e_k, k=0 to L-1, where L is a positive integer greater than 2, for respectively multiplying each value X(m) by a corresponding scaler value, each of said scalers having an output port;

a first adder AD₀ having a first input port connected to the output port of the scaler e₀, a second input port and first and second output ports;

a plurality of adders AD_k, k=1 to L-2, the (k+1)th adder AD_k having a first input port connected to the output port of the scaler e_k, second and third input ports and first and second output ports;

an Lth adder AD_L-1 having a first input port connected to the output port of the scaler e_L-1, a second input port and first and second output ports;

a plurality of delay and switch means V_k, k=0 to L-3, respectively disposed between the (k+1)th and (k+2)th adders AD_k and AD_k+1, each delay and switch means V_k including a first two sample delay element Z_k1^-2, a second two sample delay element Z_k2^-2, a first switch SW_kl alternately connecting the first output port of the (k+1)th adder AD_k to an input port of the first two sample delay element Z_K1^-2 in a first state of the circuit and the second input port of the (k+1)th adder AD_k to an output port of the second two sample delay element Z_k2^-2 in a second state of the circuit, and a second switch SW_k2 alternately connecting the third input port of the (k+2)th adder AD_k+ 1 to an output port of the first two sample delay element Z_k1^-2 in th first state of the circuit and the second outport port of the (k+2)th adder AD_k+ 1 to an input port of the second two sample delay element Z_k2^-2 in the second state of the circuit;

an (L-1)th delay and switch means V_L-2 disposed between the (L-1)th and Lth adders AD_L-2 and AD_L-1, and including a first two sample delay element Z.sub.(L-2)1^-1 and a second two sample delay element Z.sub.(L-2)2^-2, a first switch SW.sub.(L-2)1 alternately connecting the first output port of the (L-1)th adder AD_L-2 to an input port of the first two sample delay element Z.sub.(L-2)1^-2 in the first state of the circuit and the second input port of the (L-1)th adder AD_L-2 to an output port of the second two sample delay element Z.sub.(L-2)2^-2 in the second state of the circuit, and a second switch SW.sub.(L-2)2 alternately connecting the second input port of the Lth adder AD_L-1 to an output port of the first two sample delay element Z.sub.(L-2)1^-2 in the first state of the circuit and the second output port of the Lth adder AD_L-1 to an input port of the second two sample delay element Z.sub.(L-2)2^-2 in the second state of the circuit; and

means for controlling the first and second switches of said delay and switch means V_k, k=0 to L-2 to alternate the first and second states of the circuit with successive value X(m) such that the first state corresponds to values X(m) when m is an even integer and the second state corresponds to values X(m) when m is an odd integer the first through (L-)th adders AD_k, k=0 to L-2 providing at the second output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said second state, the first through (L-1)th adders AD_k, k=0 to L-2 providing at the second output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said second state, the Lth adder AD_L-1 providing at the second output port thereof the value at the first input port thereof when the circuit is in said second state, the second through (L-1)th adders AD_k, k=1 to L-2 providing at the first output port thereof a sum of input values at the first and third input ports thereof when the circuit is in said first state, the Lth adder AD_L, providing at the first output port thereof a sum of input values at the first and second input ports thereof when the circuit is in said first state, the first adder AD₀ providing at the first output port thereof the value at the first input port thereof when the circuit is in said first state;

wherein the respective scalers e_k of the first dual convolver have scaler values equal to a_N-2-2k and the respective scalers e_k of the second dual convolver have scaler values equal to a_N-1-2k.

37. A function generator, comprising:

a source of null sequences;

a plurality, J in number, of successive synthesizer circuits S_i, where i=1 to J, each having an input port and an output port; and

J-1 interleaver circuits I_i, where i=1 to J-1, each having an output port and a data input port and a null input port, for all i, i=1 to J-1, the output port of the interleaver I_i is connected to the input port of the synthesizer circuit S_i, the data input port of the interleaver I_i is connected to the output port of the synthesizer circuit S_i+1, and the null input port of the interleaver I_i is connected to the output port of the processing means P_i ;

each synthesizer circuit S_i comprising means, responsive to a sequence of input values W.sup.(i) (m) at its input port wherein m represents success integers, for generating and outputting at its output port a sequence of output values Z.sup.(i) (m) given by ##EQU58## each interleaver circuit I_i having means for outputting at the output port the sequence of values W.sup.(i) (m), being equal to Z.sup.(i+1) (m/2) for m even and equal to T.sup.(i) ((m-1)/2) for m odd, the output port of the processing means P_J being connected to input port of the synthesizer circuit S_J so that the sequence W.sup.(J) (m) input to the input port of the synthesizer S_J is the output sequence of the processing means P_J, whereby if the effect of the processing means P_i, i=1 to J is such that the sequences T.sup.(i) (m) are respectively approximately equal to the sequences S.sup.(i) (m), then the sequences Z.sup.(i) (m) are approximately equal to the respective sequences ##EQU59##