Abstract:
Wavelet filters computed from a wavelet transform are used as a means of pulse shaping binary data transmitted and received over multiple parallel channels. At the transmitter the data is parsed from a serial bit stream to several parallel streams. Within each of the parallel bit streams symbols are formed. Signals are created from each symbol by up-sampling by inserting zeros between successive symbols. These signals are passed through a bank of low-pass and high-pass filters derived from a wavelet packet transform. The filters are paired: one high-pass with a low-pass. The ordering is alternated to preserve “natural” frequency ordering. These steps are repeated for this set of signals until only one signal remains. The remaining signal is transmitted in the base band of the transmission system or they are transmitted by modulating the carrier of the transmission system. At the receiver the steps are reversed to recover symbols.

Description:
FIELD OF THE INVENTION 
     The present invention relates to methods for encoding and modulating signals, specifically a system and method for the use of wavelet transforms to encode a signal to achieve maximum transmission speed and distance. The encoded signal may be transmitted in the base-band or may be modulated and de-modulated in a multiple-carrier transmission system. 
     BACKGROUND 
     The demand for provision of multi-media and other bandwidth services over telecommunications networks has created a need to transmit high bit rate traffic over copper pairs. This requirement has led to the development of a number of different transmission schemes, such as, ADSL and VDSL. One of the more likely modulation systems for all these transmission schemes is a line code known as DMT (discrete multi-tone), which bears some resemblance to orthogonal frequency division multiplex, and is a spread spectrum transmission technique. 
     In discrete multi-tone transmission, the available bandwidth is divided into a plurality of sub-channels each with a small bandwidth, 4 kHz perhaps. Traffic is allocated to the different sub-channels in dependence on noise power and transmission loss in each sub-channel. Each channel carries multi-level pulses capable of representing up to 11 data bits. Poor quality channels carry fewer bits, or may be completely shut down. 
     Discrete multi-tone transmission (DMT), is disclosed in U.S. Pat. No. 5,479,447 issued December 1995 and in an article entitled “Performance Evaluation of a Fast Computation Algorithm for the DMT in High-Speed Subscriber Loop”, IEEE Journal on Selected Areas in Communications, Vol. 13, No. 9, December 1995 by I. Lee et al. Specifically, U.S. Pat. No. 5,479,447 discloses a method and apparatus for adaptive, variable bandwidth, high-speed data transmission of a multi-carrier signal over a digital subscriber loop. The data to be transmitted is divided into multiple data streams which are used to modulate multiple carriers. These modulated carriers are converted to a single high speed signal by means of IFFT (Inverse Fast Fourier Transform) before transmission. At the receiver, Fast Fourier Transform (FFT) is used to split the received signal into modulated carriers which are demodulated to obtain the original multiple data streams. 
     A DMT system is not entirely satisfactory for use in two-wire subscriber loops, which are very susceptible to noise and other sources of degradation which could result in one or more sub-channels being lost. If only one sub-channel fails, perhaps because of transmission path noise, the total signal is corrupted and either lost or, if error detection is employed, may be retransmitted. It has been proposed to remedy this problem by adaptively eliminating noisy sub-channels, but to do so would involve very complex circuitry. 
     SUMMARY 
     The present invention comprises an apparatus and method of for using wavelet filters computed from a wavelet transform as a means of encoding, transmitting, receiving and decoding information on multiple parallel channels. The method of encoding and decoding has been found to achieve very high transmission rates over great distances. At the transmitter, (1) the data is parsed from a serial bit stream to several parallel streams; the number of bits per symbol need not remain, constant; and there are many means of setting this level in the field of information theory (the most prevalent would be a “water filling” approach). (2) Within each of the parallel bit streams symbols are formed from the bits and consist of the set {0, 1, . . . 2 k  −1} where k is the number of bits per symbol. The symbols are normally Grey coded to decrease the probability of a bit error, which is a common practice in communications systems engineering, however other forms of pre-coding can be used. (3) [a] Signals are created from each symbol by first, up-sampling by inserting zeros between successive symbols (this defines a signal-no longer just an abstraction of the information. [b] These signals are passed through a bank of low-pass and high-pass filters that are derived from a wavelet packet transform (the wavelet and scaling functions, or, equivalently the low-pass and high-pass reconstruction filters). [c] The filters are paired: one high-pass with a low-pass. The ordering is alternated to preserve “natural” frequency ordering. (4) The steps  3 [a],  3 [b] and  3 [c] are recursively repeated log 2  N times for a signal of length N. The resulting signal is transmitted either in the base band or is transmitted by modulating the carrier of the transmission system. At the receiver, Steps 1 through 3 are reversed to recover symbols, with the up-sampled signal down-sampled to remove inserted zeroes. Symbol decisions are made through any one of a number of methods, which increase the probability of a correct decision. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS AND TABLES CONTAINING EXECUTABLE COMPUTER PROGRAMS 
         FIG. 1  shows an exemplary computing environment for the invention. 
         FIG. 2  is a flow diagram of an exemplary method and process used by a transmitter equipped with the invention. 
         FIG. 3  is a flow diagram of an exemplary method and process used by a receiver equipped with the invention. 
     
    
    
     Table 1 contains MatLab code to covert binary data to Grey code. 
     Table 2 contains MatLab code to return wavelet factors for constructing and deconstructing data using various wavelet filters. 
     Table 3 contains MatLab code to encode a data stream using wavelet filters. 
     Table 4 contains MatLab code to decode an encoded data stream. 
     Table 5 contains the MatLab script used to test the invention. 
     DETAILED DESCRIPTION 
     A system and method is disclosed for encoding a binary data stream, transmitting the stream in the base band or the encoded signal is transmitted by using the encoded stream to modulate a plurality of single carriers, which is de-modulated and decoded at the receiver. 
       FIG. 1  illustrates a generalized example of a suitable computing environment  1000  in which an exemplary embodiment of the invention may be implemented. The computing environment shown in  FIG. 1  is not intended to suggest any limitation as to scope of use or functionality of the invention, as the present invention may be implemented in diverse general-purpose or special-purpose computing environments. 
     With reference to  FIG. 1 , the computing environment  1000  of the apparatus and method of the invention includes at least one processing unit  1200  and memory  1300 . It will be understood that the computing environment may be implemented within a communications system to control a transmitter and receiver, and is implemented in any one of several forms: (a) discrete hardware and software systems; (b) ASICs (application specific integrated circuits, and (c) FPGA (field-programmable gate array. 
     In  FIG. 1 , this most basic configuration  1000  is included within  1100  a dashed line. The processing unit  1200  executes computer-executable instructions and may be a real or a virtual processor. In a multi-processing system, multiple processing units execute computer-executable instructions to increase processing power. The memory  1300  may be volatile memory (e.g., registers, cache, RAM), non-volatile memory (e.g., ROM, EEPROM, flash memory, etc.), or some combination of the two. The memory  1300  stores executable software—instructions and data  1250 —written and operative to execute and implement the software applications required to support the interactive environment of the invention. 
     The computing environment may have additional features. For example, the computing environment  1000  includes storage  1400 , one or more input devices  1550 , one or more output devices  1560 , and one or more communication connections or interfaces  1570 . An interconnection mechanism (not shown) such as a bus, controller, or network interconnects the components of the computing environment. Typically, operating system software (not shown) provides an operating environment for other software executing in the computing environment, and coordinates activities of the components of the computing environment. 
     The storage  1400  may be removable or non-removable, and includes magnetic disks, magnetic tapes or cassettes, CD-ROMs, DVDs, or any other medium which can be used to store information and which can be accessed within the computing environment. The storage  1400  also stores instructions for the software  1250 , and is configured to store data collected and generated during at least one interactive session. 
     The input device(s)  1550  may be a touch input device such as a keyboard, mouse, pen, or trackball, a voice input device, a scanning device, or another device that provides input to the computing environment. For audio or video, the input device(s) may be a sound card, video card, TV tuner card, or similar device that accepts audio or video input in analog or digital form. The output device(s)  1560  may be a display, printer, speaker, or another device that provides output from the computing environment. 
     The communication interface  1570  enable the apparatus and software means to control communication over a communication medium (not shown) with another similar system, for example, the system implements a transmitter that exchange messages with a similarly configured receiver. The communication medium conveys information such as voice signals, video, and data in a modulated data signal. A modulated data signal is a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, the communication media include wired or wireless techniques implemented with an electrical, optical, RF, infrared, acoustic, or other carrier. 
     A Method of Encoding and Decoding a Signal by Filters for Modulation and Demodulation 
     A method and system for encoding and transmitting a signal comprises: (1) generating a recursive wavelet decomposition using analysis wavelet low-pass and high-pass filters selected from the set comprising D-Meyer, Coiflet and Symlet filters; (2) re-ordering the decomposition so that frequency order is maintained; (3) assigning values of an input signal to filters generated by the decomposition; (4) modulating each selected filter by the assigned value, (5) combining the modulated filters, and (5) transmitting the combined filters in base-band or using the combined modulated filters to modulate a carrier. 
     A method and system for decoding a signal encoded and transmitted by the method above comprises: (1) generating a recursive wavelet decomposition using synthesis wavelet low-pass and high-pass filters selected from the set comprising D-Meyer, Coiflet and Symlet filters; (2) re-ordering the decomposition so that frequency order is maintained; (3) using the synthesis filters from the decomposition as matched filters for the received signal; and (4) deriving the input signal assigned value from the filter matching. It will be appreciated that the analysis filters generated by the transmitter and the synthesis filters generated by the receiver are of the same type of wavelet. 
     The method and technique described below and summarized above has been found to provide an optimum method of pulse shaping of binary data for transmission over noisy channels. 
     An exemplary signal encoding process using the invention is described with respect to  FIG. 2  and also with reference to MatLab programming statements shown in Table 1-5, below. 
     With reference Table 1, the binary data stream is converted to Grey code by calling the function “bi2Grey” with the binary data assembled as a binary vector and passed as an argument. A Grey code is a special coding system designed to reduce undetected errors resulting from random perturbations of transmitted binary data. The function “bi2Grey” returns a binary vector with the data passed as an argument returned as a Grey coded binary vector. While a Grey coding has been used in the exemplary embodiment, the exemplary embodiment does not require Grey coding; other coding schemes to reduce transmission errors can be used. 
     With reference to Table 2, the function “wfactors” is called to return factors used in the processes of encoding and decoding a signal using wavelet filter banks. “wfactors” returns (a) “delay”, the beginning index used to down-sample an input signal that is encoded using wavelet filters and (b) the system delay in samples; wherein the input signal is encoded and decoded according to the name of the wavelet, “wname”, passed as an argument to “wfactors.” 
     Table 2 shows delay factors used in an exemplary embodiment of the invention for the wavelet filters: (a) discrete Meyer; (b) Symlet  16 ; (c) Symlet  2 ; (d) Symlet  3 ; (e) Symlet  4 ; (f) Symlet  5 ; (g) Coiflet  2 ; and (h) Coiflet  3 . 
     See  FIG. 2 ,  2000 , a flow diagram of the transmitter encoding process, and Table 3, which contains exemplary MatLab programming statements implementing the process used by the transmitter. The MatLab function “WPconstruct” is called to construct a signal using wavelet filters; “WPconstruct” is passed arguments related to the number of samples, or filter taps, related to filters used (“Nmaster”), and the name of the wavelet filter (“wname”) to use in encoding a binary signal. In  FIG. 2 ,  2100 , the name of the wavelet filter bank used for signal encoding and the number of filter taps are obtained. In Table 3, In lines  13  and  14 , the MatLab function “wfilters” is called to return the low-pass and the high-pass filter corresponding to “wname.” In  FIG. 2 ,  2200 , the low-pass and high-pass filters corresponding to “wname” are generated. 
     In  FIG. 2 ,  2300  and Table 3 lines  16 - 20  a working array is set up to encode a signal. With reference to  FIG. 2 ,  2400 , the source signal is obtained. In lines Table 3, lines  23 - 28 , a random source is created for the purpose of test and illustration; in Table 3, the source signal is Grey encoded by calling the MatLab function “bi2Grey” shown in Table 1. 
     In  FIG. 2 ,  2500 , corresponding to Table 3, lines  27 - 30 , the Grey encoded binary data is aggregated into symbols, and the symbols are zero padded. 
     With reference to  FIG. 2 ,  2600 - 2700  and Table 3, lines  38 - 57 , symbols are first (a) up sampled; (b) encoded using the low-pass and the high-pass filters given by “wname”; and (c) every other filter is switched to maintain natural frequency ordering in the filter-encoded symbols. 
     With reference to  FIG. 2 ,  2800 , the encoded symbols are recursively encoded according the principles of filter-bank encoding, for example given N symbols, log 2 N encoding steps are executed. With reference to  FIG. 2 ,  2900 , the encoded symbols are returned by “WPconstruct.” 
     The encoded symbols are then transmitted in the base band of a transmission system or the encoded symbols are employed to modulate a plurality of carriers, using available modulation techniques. 
     
       
         
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
             
               
                   
                 function g = bi2Grey( b ) 
               
               
                   
                 % BI2GREY converts a binary code to a Grey code. The most 
               
             
          
           
               
                   
                  1 
                 % significat bit is the left hand side bit. 
               
               
                   
                  2 
                 % When the inpt argument is a binary matrix, each row is 
               
               
                   
                  3 
                 % converted to the Grey code. 
               
               
                   
                  4 
                 % 
               
               
                   
                  5 
                 % See also: GREY2BI 
               
               
                   
                  6 
                 % 
               
               
                   
                  7 
                 % Comments and suggestions to: adrian@ubicom.tudelft.nl 
               
               
                   
                  8 
               
             
          
           
               
                   
                  9 
                 % copy the msb: 
               
               
                   
                 10 
               
               
                   
                 11 
                 [r,c] = size(b); 
               
               
                   
                 12 
                 g = zeros(r,c+1); 
               
               
                   
                 13 
                 %Add a leading zero 
               
               
                   
                 14 
                 g(:,1) = 0;%b(:,1) 
               
               
                   
                 15 
                 b = [zeros(r,1), b]; 
               
               
                   
                 16 
               
               
                   
                 17 
                 for i =1:c, 
               
             
          
           
               
                   
                 18 
                 g(:,i+1) = xor(b(:,i+1),b(:,i) ); 
               
             
          
           
               
                   
                 19 
                 end 
               
               
                   
                 20 
               
               
                   
                 21 
                 g = g(:,2:end); 
               
             
          
           
               
                   
                 22 
                 return; 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
             
             
               
                  1 
                 function [delay, DELTA] = wfactors(wavename) 
               
               
                  2 
                 % returns the appropriates factors for reconstructing/ decomposing 
               
               
                  3 
                 % signals from wavelets at up to the sixth scale through filter banks 
               
               
                  4 
               
               
                  5 
                 % Mark M. Westerman 
               
               
                  6 
                 % WavTech, Inc. 
               
               
                  7 
                 % Richardson, TX 
               
               
                  8 
                 % 9-13-2002 
               
               
                  9 
               
               
                 10 
               
               
                 11 
                 switch lower(wavename) 
               
               
                 12 
                 case ‘dmey’ 
               
             
          
           
               
                 13 
                 DELTA = 61; 
               
               
                 14 
                 delay = [1 1 0 0 0 0]; %dmey 
               
             
          
           
               
                 15 
                 case ‘sym16’ 
               
             
          
           
               
                 16 
                 DELTA = 31; 
               
               
                 17 
                 delay = [1 0 0 0 0 1]; %sym16 
               
             
          
           
               
                 18 
                 case ‘sym2’ 
               
             
          
           
               
                 19 
                 DELTA = 3; 
               
               
                 20 
                 delay = [1 0 1 1 1 1]; %sym2 
               
             
          
           
               
                 21 
                 case ‘sym3’ 
               
             
          
           
               
                 22 
                 DELTA = 9; 
               
               
                 23 
                 delay = [1 1 0 1 1 1]; %sym3 
               
             
          
           
               
                 24 
                 case ‘sym4’ 
               
             
          
           
               
                 25 
                 DELTA = 3; 
               
               
                 26 
                 delay = [1 0 0 1 1 1]; %sym4 
               
             
          
           
               
                 27 
                 case ‘sym5’ 
               
             
          
           
               
                 28 
                 DELTA = 9; 
               
               
                 29 
                 delay = [1 1 1 0 1 1]; %sym5 
               
             
          
           
               
                 30 
                 case ‘coif2’ 
               
             
          
           
               
                 31 
                 DELTA = 11; 
               
               
                 32 
                 delay = [1 0 1 0 1 1]; %coif2 
               
             
          
           
               
                 33 
                 case ‘coif3’ 
               
             
          
           
               
                 34 
                 DELTA = 17; 
               
               
                 35 
                 delay = [1 1 1 1 0 1]; %coif3 
               
             
          
           
               
                 36 
                 case ‘namez’ 
               
             
          
           
               
                 37 
                 DELTA = []; 
               
               
                 38 
                 delay = [{‘dmey’};{‘sym16’};{‘sym2’};{‘sym3’};{‘sym4’};{‘sym5’};{‘coif2’};{‘coif3’}]; 
               
             
          
           
               
                 39 
                 otherwise 
               
             
          
           
               
                 40 
                 DELTA = 1; 
               
               
                 41 
                 delay = [0 0 0 0 0 0]; 
               
               
                 42 
                 disp(“); 
               
               
                 43 
                 disp(‘------------------CAUTION--------------------------------------’) 
               
               
                 44 
                 disp(‘--------------------------------------------------------------------’) 
               
               
                 45 
                 disp(‘wfactors: unknown case, returning general case’); 
               
               
                 47 
                 disp(‘delay = [0 0 0 0 0 0]’); 
               
               
                 48 
                 disp(‘DELTA = 1’); 
               
               
                 49 
                 disp(‘--------------------------------------------------------------------’) 
               
               
                 50 
                 disp(‘--------------------------------------------------------------------’) 
               
               
                 51 
                 disp(“); 
               
               
                 52 
               
             
          
           
               
                 53 
                 end; 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
             
               
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                 TABLE 3 
               
               
                   
               
             
             
               
                  1 
                 function [sig, syms] = WPconstruct(Nmaster, wname); 
               
               
                  2 
                 % WPConstruct -- constructs a psuedo random signal through wavelet packet construction 
               
               
                  3 
                 % (accomplished through lifting) 
               
               
                  4 
               
               
                  5 
                 %---------------------------------------------------------------------- 
               
             
          
           
               
                  6 
                 % 
                 Wavelet packet lifting - Construction 
               
             
          
           
               
                  7 
                 %---------------------------------------------------------------------- 
               
               
                  8 
                 %---------------------------------------------------------------------- 
               
               
                  9 
               
               
                 10 
                 % Nmaster = 64; 
               
               
                 11 
                 % wname = ‘coif2’; 
               
               
                 12 
               
             
          
           
               
                 13 
                 [lo_r, lo_d] = wfilters(wname,‘l’); 
                 % coefficients for low-pass filter 
               
               
                 14 
                 [hi_r, hi_d] = wfilters(wname,‘h’); 
                 % coefficients for high-pass filter 
               
               
                 15 
               
             
          
           
               
                 16 
                 N = Nmaster; 
               
               
                 17 
                 L = 1000; 
               
               
                 18 
                 K = 0.75*N−1; 
               
               
                 19 
               
             
          
           
               
                 20 
                 symz = zeros(L+16,N); 
                 % create an array of zeros 
               
               
                 21 
               
             
          
           
               
                 22 
                 %Add greycode functionality 
               
             
          
           
               
                 23 
                 symz(1:L,2:K+1) = randsrc(L,K,[1:16]); 
                 % creates a matrix representing a random source 
               
             
          
           
               
                 24 
                 % 
               
               
                 25 
                 % 
               
               
                 26 
                 % 
               
               
                 27 
                 map = [0; bi2de((bi2Grey(de2bi((0:15)’,‘left-msb’))),‘left-msb’)*2–15]; 
               
               
                 28 
                 syms = map(symz+1); 
               
               
                 26 
               
               
                 27 
                 %4 bits per *channel* per symbol 
               
               
                 28 
                 % syms(1:L,IND) = randsrc(L,K,[1]); 
               
               
                 29 
               
               
                 30 
                 %Pad with some zeros to flush the filters 
               
               
                 31 
                 L = L+16; 
               
               
                 32 
               
               
                 33 
                 sig = syms; 
               
               
                 34 
                 % plot(sig) 
               
               
                 35 
                 % pause(3) 
               
               
                 36 
               
               
                 37 
                 %Now construct the signal 
               
               
                 38 
                 for j = 1:log2(N) 
               
             
          
           
               
                 39 
                 N = N/2; 
               
               
                 40 
                 L = L*2; 
               
               
                 41 
                 old_sig = sig; 
               
               
                 42 
                 sig = zeros(L,N); 
               
               
                 43 
               
               
                 44 
                 ind = 1; 
               
               
                 45 
                 for k=0:2:2*N−1 
               
               
                 46 
               
             
          
           
               
                 47 
                 %To maintain natural frequency ordering, we must switch every other pair 
               
               
                 48 
                 %of HP/LP filters 
               
               
                 49 
                 k1 = mod(ind+1,2)+1; 
               
               
                 50 
                 k2 = mod(ind,2)+1; 
               
               
                 51 
               
               
                 52 
                 sig(:,ind) = filter(lo_r,1,upsample(old_sig(:,k+k1)))‘ + ... 
               
             
          
           
               
                 53 
                 filter(hi_r,1,upsample(old_sig(:,k+k2)))’; 
               
               
                 54 
               
             
          
           
               
                 55 
                 ind = ind+1; 
               
             
          
           
               
                 56 
                 end; 
               
             
          
           
               
                 57 
                 end; 
               
               
                 58 
               
               
                 60 
                 syms = symz; 
               
               
                   
               
             
          
         
       
     
     With reference to  FIG. 3 , the receiver employs a corresponding exemplary process to decode received symbols. The exemplary process is further shown by a coding example using MatLab in Table 4 that follows. Table 4 illustrates coding used to decompose the symbols of the received signal and to convert the symbols back into the original binary stream processed by the transmitter. 
     
       
         
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
             
               
               
             
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
             
             
               
                  1 
                 function [dec, DELTA] = WPdecompose(sig, Nmaster, 
               
             
          
           
               
                   
                 wname); 
               
             
          
           
               
                  2 
                 %WPdecompose -- decomposes a specially made sequence from 
               
             
          
           
               
                  3 
                 WPconstruct 
               
             
          
           
               
                  4 
                 %----------------------------------------------------------------- 
               
             
          
           
               
                  5 
                 % 
                 Wavelet packet lifting -- Deconstruction 
               
             
          
           
               
                  6 
                 %----------------------------------------------------------------- 
               
               
                  7 
                 %----------------------------------------------------------------- 
               
               
                  8 
               
               
                  9 
                 %delay determines the beginning index of downsampling 
               
               
                 10 
                 %DELTA is the delay (in samples) of the system 
               
             
          
           
               
                 11 
                 [delay, DELTA] = 
                 % obtain delay for wavelet used 
               
             
          
           
               
                   
                 wfactors(wname); 
               
             
          
           
               
                 12 
               
             
          
           
               
                 13 
                 N = Nmaster; 
               
               
                 14 
                 [lo_r, lo_d] = wfilters(wname,‘l’); 
               
               
                 15 
                 [hi_r, hi_d] = wfilters(wname,‘h’); 
               
               
                 16 
               
               
                 17 
                 Nprime = 1; 
               
               
                 18 
                 L = length(sig); 
               
               
                 19 
               
               
                 20 
                 dec = sig; 
               
               
                 21 
                 swap = 1; 
               
               
                 22 
               
               
                 23 
                 while Nprime &lt; N 
               
             
          
           
               
                 24 
                 Nprime = 2*Nprime; 
               
               
                 25 
                 L = L/2; 
               
               
                 26 
                 old_dec = dec; 
               
               
                 27 
                 dec = zeros(L,Nprime); 
               
               
                 28 
                 ind = 1; 
               
               
                 29 
               
               
                 30 
                 for k = 1:2:(Nprime) 
               
               
                 31 
               
             
          
           
               
                 32 
                 tl = filter(lo_d,1,old_dec(:,ind)); 
               
               
                 33 
                 th = filter(hi_d,1,old_dec(:,ind)); 
               
               
                 34 
               
               
                 35 
                 %Once again, we must swap the LP/HP sections 
               
               
                 36 
                 k1 = mod(ind+1,2); 
               
               
                 37 
                 k2 = mod(ind,2); 
               
               
                 38 
               
               
                 39 
                 dec(:,k + k1) = tl(1+delay(swap):2:end); 
               
               
                 40 
                 dec(:,k + k2) = th(1+delay(swap):2:end); 
               
               
                 41 
                 ind = ind+1; 
               
             
          
           
               
                 42 
                 end; 
               
               
                 43 
                 swap = swap+1; 
               
               
                 44 
                 % keyboard 
               
             
          
           
               
                 45 
                 end; 
               
               
                   
               
             
          
         
       
     
     With reference to  FIG. 3 , a flow diagram  3000  of the receiver decoding process is shown. With reference to the MatLab programming statements in Table 4, the receiver decoding logic “WPdecompose” is called when the receiver acquires a signal. With reference to Table 4, line  1 , “WPdecompose” is called by passing arguments: (1) the signal acquired; (2) the number of filter taps used for encoding filters; and (3) type of wavelet filter used to encode the signal. 
     With reference to  FIG. 3 ,  3100 - 3200  and Table 3, lines  14 - 15 , decoding or reconstruction filters are derived or obtained—the decoding filters generated according to the requirements of quadrature mirror filter perfect reconstruction. 
     With reference to  FIG. 3 ,  3400 ,- 3500  and Table 3, lines  30 - 45 , the encoded symbols are decoded using the reconstruction filters. At each iteration of the reconstruction, filter coefficients that were swapped in order to achieve natural frequency ordering are swapped again so that the coefficients will be in proper order for reconstruction of the signal encoded by the transmitter. Zeroes inserted by the transmitter up-sampling are removed by down-sampling the signal during each iteration of the decoding. 
     With reference to  FIG. 3 ,  3300 , the reconstruction is recursively executed until the original encoded symbols are decoded. The reconstructed or decoded signal is returned from the receiver  3600 . 
     A MatLab Script Demonstrating the Combined Use of the MatLab Programming Examples 
     With reference to Table 5 a MatLab script is shown that uses the MatLab functions shown in previous tables. The script calls the MatLab functions used by the transmitter and the receiver to encode, and decode a signal. The script simulates modulation and signal corruption. 
     
       
         
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
             
           
               
                 TABLE 5 
               
               
                   
               
             
             
               
                 %A script for evaluating the performance of the OFDM/DMT wavelet 
               
               
                 varient 
               
               
                 Nmaster = 16; 
               
               
                 wname = ‘coif2’; 
               
               
                 [sig,syms] = WPconstruct(Nmaster,wname); 
               
               
                 n = 0.75*Nmaster − 1; 
               
               
                 SNR = 25; 
               
               
                 noise_lev= 10{circumflex over ( )}(−SNR/10)*var(sig); 
               
               
                 sym_errs = 0; 
               
               
                 bit_errs = 0; 
               
               
                 num_bits = 0; 
               
               
                 while bit_errs&lt;20 
               
             
          
           
               
                   
                 noise = noise_lev*randn(size(sig)); 
               
               
                   
                 [dec, DELTA] = WPdecompose(sig + noise, Nmaster, wname); 
               
               
                   
                 map = [0 1 3 2 7 6 4 5 15 14 12 13 8 9 11 10 ]’ + 1; 
               
               
                   
                 [INDX,QUANT] = quantiz(dec(DELTA:DELTA+999,[2:n+1]), 
               
               
                   
                 [−14:2:14], map); 
               
               
                   
                 Q = map(reshape(INDX,1000,n)+1); 
               
               
                   
                 errs = symerr(Q,syms(1:1000,2:n+1)); 
               
               
                   
                 sym_errs = sym_errs + errs; 
               
               
                   
                 if errs %Only compute BER if there are symbol errors 
               
             
          
           
               
                   
                 biterrs = biterr(Q, syms(1:1000,2:n+1)); 
               
             
          
           
               
                   
                 else 
               
             
          
           
               
                   
                 biterrs = 0; 
               
             
          
           
               
                   
                 end; 
               
               
                   
                 num_bits = num_bits + 4000*n; 
               
               
                   
                 bit_errs = bit_errs + biterrs; 
               
               
                   
                 fprintf(‘(%f, %f, %f)\n’,errs, biterrs, bit_errs/num_bits); 
               
             
          
           
               
                 end;