a transmission system embodying the present invention is illustrated in fig1 . the system comprises digital input signal source 10 , an encoder 11 , transmission medium 12 , decoder 13 and signal destination 14 . input signal s 1 from signal source 10 is applied to the encoder 11 which encodes it using sub - band filtering and multi - carrier modulation and supplies the resulting encoded signal s o to transmission medium 12 , which is represented by a transmission channel 15 , noise source 16 and summer 17 , the later combining noise with the signal in the transmission channel 15 before it reaches the decoder 13 . although a transmission medium 12 is illustrated , it could be an analogous storage medium instead . the output of the decoder 13 is supplied to the signal destination 14 . the usable bandwidth of channel 15 dictates the maximum allowable rate of a signal that could be transmitted over the channel . a first embodiment of the encoder 11 is illustrated in more detail in fig2 . the input signal s 1 is applied via an input port 20 to analysis filter bank 21 which decomposes it into sub - bands to generate / extract a lowpass sub - band signal y 0 , bandpass sub - band signals y 1 - y n − 2 and a highpass sub - band signal y n − 1 . the sub - band signals y 1 - y n − 2 are supplied to a multi - carrier modulator 22 which uses each sub - band signal to modulate a respective carrier of a selected frequency , as will be explained later . the lowpass sub - band signal y 0 contains more low frequency components than the other sub - band signal , and is used to modulate a low frequency carrier f 0 . the bandpass sub - band signals y 1 - y n − 2 and highpass sub - band signal y n − 1 have more high frequency components than the lowpass wavelet signal y 0 and are therefore used to modulate higher frequency carrier signals f 1 - f n − 1 , respectively , of which the frequencies increase from f 1 to f n − 1 . the modulated carrier signals y ′ 0 - y ′ n − 1 are combined by summer 23 to form the encoded output signal s o which is transmitted via output port 24 to transmission medium 12 for transmission to decoder 13 ( fig1 ). a suitable decoder 13 , for decoding the encoded output signal , will now be described with reference to fig3 . after passing through the transmission medium 12 , the transmitted signal s o may be attenuated and contain noise . hence , as received by the decoder at port 30 it is identified as received signal s ′ o ( the prime signifying that it is not identical to enclosed signal s o ) and supplied to a filter array 31 . each of the filters in the array 31 corresponds to one of the frequencies f 0 - f n − 1 of the multi - carrier modulator 22 ( fig2 ) and recovers the corresponding modulated carrier signals . the recovered modulated carrier signals y ″ 0 - y ″ n − 1 separated by the array then are demodulated by a multi - carrier demodulator 32 to recover the lowpass , bandpass and highpass sub - band signals y * 0 - y * n − 1 corresponding to sub - band signals y 0 - y ( n − 1 ) , respectively , in the encoder 11 . the recovered sub - band signals are supplied to synthesis filter bank 33 which , operating in a complementary and inverse manner to analysis filter bank 21 , produces an output signal s ′ 1 which should closely resemble the input signal s i in fig2 and supplies it to signal destination 14 via output port 34 . usually , the recovered signal s ′ 1 will be equalized using an adaptive equalizer to compensate for distortion and noise introduced by the channel 12 . it should be noted that the highpass subband signal y n − 1 and some of sub - band signals y 0 - y n − 2 in fig2 may not need to be transmitted , if they contain little transmission power as compared with other sub - band signals . when these sub - band signals are not transmitted , the synthesis filter bank 33 shown . in fig3 will insert zeros in place of the missing sub - band signals . the reconstructed signal s ′ 1 would then be only a close approximation to the original input signal s 1 . generally , the more sub - bands used , the better the approximation . preferably , analysis filter 21 is a multiresolution filter bank which implements a discrete wavelet transform ( dwt ). in order to facilitate a better understanding of the embodiments which use dwt , a brief introduction to discrete wavelet transforms ( dwt ) will first be given . dwt represents an arbitrary square integrable function as the superposition of a family of basis functions called wavelets . a family of wavelet basis functions can be generated by translating and dilating the mother wavelet corresponding to the family . the dwt coefficients can be obtained by taking the inner product between the input signal and the wavelet functions . since the basis functions are translated and dilated versions of each other , a simpler algorithm , known as mallat &# 39 ; s tree algorithm or pyramid algorithm , has been proposed by s . g . mallat in “ a theory of multiresolution signal decomposition : the wavelet representations ”, ieee trans . on pattern recognition and machine intelligence , vol . 11 , no . 7 , july 1989 . in this algorithm , the dwt coefficients of one stage can be calculated from the dwt coefficients of the previous stage , which is expressed as follows : w l  ( n , j ) = ∑ m   w l  ( m , j - 1 )  h  ( m - 2  n ) ( 1a ) w h  ( n , j ) = ∑ m   w l  ( m , j - 1 )  g  ( m - 2  n ) ( 1b ) where w ( p , q ) is the p - th wavelet coefficient at the q - th stage , and h ( n ) and g ( n ) are the dilation coefficients corresponding to the scaling and wavelet functions , respectively . for computing the dwt coefficients of the discrete - time data , it is assumed that the input data represents the dwt coefficients of a high resolution stage . equations 1 a and 1 b can then be used for obtaining dwt coefficients of subsequent stages . in practice , this decomposition is performed only for a few stages . it should be noted that the dilation coefficients h ( n ) represent a lowpass filter , whereas the coefficients g ( n ) represent a highpass filter . hence , dwt extracts information from the signal at different scales . the first stage of wavelet decomposition extracts the details of the signal ( high frequency components ) while the second and all subsequent stages of wavelet decompositions extract progressively coarser information ( lower frequency components ). it should be noted that compactly supported wavelets can be generated by a perfect - reconstruction two - channel filter banks with a so - called octave - band tree - structured architecture . orthogonal and biorthogonal filter banks can be used to generate wavelets in these system . a three stage octave - band tree structure for discrete wavelet transformation will now be described with reference to fig4 a and 4b , in which the same components in the different stages have the same reference number but with the suffix letter of the stage . referring to fig4 a , the three decomposition stages a , b and c have different sampling rates . each of the three stages a , b and c comprises a highpass filter 40 in series with a downsampler 41 , and a lowpass filter 42 in series with a downsampler 43 . the cut - off frequency of each lowpass filter 42 is substantially the same as the cut - off frequency of the associated highpass filter 40 . in each stage , the cut - off frequency is equal to one quarter of the sampling rate for that stage . the n samples of input signal s i are supplied in common to the inputs of highpass filter 40 a and lowpass filter 42 a . the corresponding n high frequency samples from highpass filter 40 a are downsampled by a factor of 2 by downsampler 41 a and the resulting n / 2 samples supplied to the output as the highpass wavelet y 3 . the n low frequency samples from lowpass filter 42 a are downsampled by a factor of 2 by downsampler 43 a and the resulting n / 2 samples supplied to stage b where the same procedure is repeated . in stage b , the n / 2 higher frequency samples from highpass filter 40 b are downsampled by downsampler 41 b and the resulting n / 4 samples supplied to the output as bandpass wavelet y 2 . the other n / 2 samples from lowpass filter 42 b are downsampled by downsampler 43 b and the resulting n / 4 samples are supplied to the third stage c , in which highpass filter 40 c and downsampler 41 c process them in like manner to provide at the output n / 8 samples as bandpass wavelet y 1 . the other n / 4 samples from lowpass filter 42 c are downsampled by downsampler 43 c to give n / 8 samples and supplies them to the output as low - pass wavelet y 0 . it should be noted that , if the input signal segment comprises , for example , 1024 samples or data points , wavelets y 0 and y 1 comprise only 128 samples , wavelet y 2 comprises 256 samples and wavelet y 3 comprises 512 samples . instead of the octave - band structure of fig4 a , a set of one lowpass , two bandpass filters and one highpass filter could be used , in parallel , with different downsampling rates . referring now to fig4 b , in order to reconstruct the original input signal , the dwt wavelet signals are upsampled and passed through another set of lowpass and highpass filters , the operation being expressed as : w l  ( n , j ) = ∑ k   w l  ( k , j + 1 )  h ′  ( n - 2   k ) + ∑ l   w h  ( l , j + 1 )  g ′  ( n - 2  l ) ( 2 ) where h ′( n ) and g ′( n ) are , respectively , the lowpass and highpass synthesis filters corresponding to the mother wavelet . it is observed from equation 2 that j - th level dwt wavelet signals can be obtained from ( j + 1 )- th level dwt coefficients . compactly supported wavelets are generally used in various applications . table i lists a few orthonormal wavelet filter coefficients h ( n ) that are popular in various applications , as disclosed by i . daubechies , in “ orthonormal bases of compactly supported wavelets ”, comm . pure appl . math , vol . 41 , pp . 906 - 966 , 1988 . these wavelets have the property of having the maximum number of vanishing moments for a given order , and are known as “ daubechies wavelets ”. an embodiment of the invention in which the higher sub - bands are not transmitted , and which uses discrete wavelet transforms for encoding a digital signal , will now be described with reference to fig5 . in the transmitter / encoder 11 ′ of fig5 the input signal s i , is supplied to an input port 20 of analysis filter means comprising an octave - band filter bank 51 for applying discrete wavelet transform as illustrated in fig4 a to the signal s i to generate lowpass sub - band wavelet signal y 0 , two bandpass sub - band wavelet signals , y 1 and y 2 , and the highpass sub - band wavelet signal y 3 . in this implementation , only sub - band wavelet signals y 0 , y 1 , and y 2 will be processed . highpass sub - band wavelet signal y 3 is discarded . interpolator means 52 interpolates sub - band wavelet signals y 0 , y 1 and y 2 by factors 2m , 2m and m , respectively , where m is an integer , typically 8 to 24 , such that the three sub - band wavelet signals ( y 0 , y 1 , and y 2 ) have equal sample rates . thus , within interpolator 52 , the sub - band wavelet signals y 0 , y 1 and y 2 are upsampled by upsamplers 53 0 , 53 1 and 53 2 , respectively , which insert zero value samples at intervals between actual samples . the upsampled signals then are filtered by three raise - cosine filters 54 0 , 54 1 and 54 2 , respectively , which insert at each upsampled “ zero ” point a sample calculated from actual values of previous samples . the raise - cosine filters are preferred so as to minimize intersymbol interference . the three interpolated sub - band wavelet signals are supplied to double side - band ( dsb ) multi - carrier modulator 55 which uses them to modulate three separate carrier signals f 0 , f 1 and f 2 , where f 0 & lt ; f 1 & lt ; f 2 provided by carrier generator 56 . the modulator 55 comprises multipliers 57 0 , 57 1 and 57 2 which multiply the carrier signals f 0 , f 1 and f 2 by the three interpolated wavelet signals y u 0 , y u 1 and y u 2 , respectively . the resulting three modulated carrier signals y ′ 0 , y ′ 1 and y ′ 2 are added together by a summer 58 to form the encoded signal s o for transmission by way of port 24 to transmission medium 12 . at the corresponding decoder 13 ′ shown in fig6 the signal s ′ o received at port 30 is supplied to each of three bandpass filters 61 0 , 61 1 and 61 2 which recover the modulated carrier signals y ″ 0 , y ″ 1 and y ″ 2 . the recovered modulated carrier signals y ″ 0 , y ″ 1 and y ″ 2 are demodulated using multi - carrier double sideband ( dsb ) demodulator 62 . a carrier generator 63 generates carrier signals having frequencies f 0 , f 1 and f 2 , which are supplied to multipliers 64 0 , 64 1 and 64 2 within the demodulator 62 and which multiply the carrier signals f 0 , f 1 and f 2 by the recovered modulated carrier signals y ″ 0 , y ″ 1 and y ″ 2 , respectively . the dsb demodulator 62 comprises lowpass filters 65 0 , 65 1 , and 65 2 for filtering the outputs of the multipliers 64 0 , 64 1 and 64 2 , respectively , as is usual in a dsb demodulator . the demodulated signals from the filters 65 0 , 65 1 and 65 2 are decimated by 2m , 2m and m , respectively , by decimators 66 0 , 66 1 and 66 2 of a decimator unit 66 and the resulting recovered sub - band signals y * 0 , y * 1 and y * 2 each supplied to a corresponding one of four inputs of a synthesis filter bank 67 which applies to them an inverse discrete wavelet transform ( idwt ) as illustrated in fig4 b to recover the signal s ′ i which corresponds to the input signal s i . the highpass sub - band wavelet signal y 3 , which was not transmitted , is replaced by a “ zero ” signal at the corresponding “ highest ” frequency input 68 of the synthesis filter bank 67 . the resulting output signal s ′ i from the synthesis filter bank 67 is the decoder output signal supplied via output port 34 , and is a close approximation of the input signal s i supplied to the encoder 11 ′ of fig5 . the bandwidth of the transmitted signal s o is wider than that of the original signal s i because each sub - band has upper and lower sidebands . a bandwidth reduction can be achieved by using single sideband ( ssb ) modulation . to do so , the encoder 11 ′ of fig5 would be modified by replacing each of the multipliers 57 0 , 57 1 and 57 2 by a ssb modulator . fig7 a , 7 b and 7 c illustrate operation of the encoder using very simplified signals and , for convenience of illustration , ssb modulation . fig7 a shows the frequency spectrum of a much - simplified input signal s i occupying a bandwidth bw centered at frequency f c . as shown in fig7 b , after analysis filtering and interpolation , the input signal s i has been partitioned into three interpolated sub - band signals , y u 0 , y u 1 and y u 2 . it should be noted that , for complex input signals , the sub - band signals y 0 , y 1 and y 2 prior to interpolation have a very wide spectrum . after upsampling and filtering by the interpolator 52 ( fig5 ), sub - band signals y u 0 , y u 1 and y u 2 each have a frequency spectrum that is much narrower than the frequency spectrum of the original signal s i . following modulation by the dsb multi - carrier modulation means 55 , the bandwidths bw 0 , bw 1 , and bw 2 of the corresponding modulated carriers y ′ 0 , y ′ 1 and y ′ 2 are determined by the sampling rate of the input signal s i . the total bandwidth bw 0 + bw 1 + bw 2 + 2g may be greater than the bandwidth bw if all sub - bands are used , but may be less if only two are used . the output signal s o from the summing means 58 has a spectrum which , as shown in fig7 c , has three lobes , namely a lower frequency lobe centered at frequency f 0 , a middle frequency lobe centered at frequency f 1 and an upper frequency lobe centered at frequency f 2 . the three lobes are separated from each other by two guard bands g to avoid interference and ensure that each carries information for its own sub - band only . simplified versions of the input signal s i , sub - band wavelet signals y 0 , y 1 , y 2 and y 3 , sub - band wavelet modulated carriers y ′ 0 , y ′ 1 and y ′ 2 , and the transmitted signal s o , which are similar in the encoders of fig2 and 5 , are shown in fig8 - 10 . fig8 shows the simplified input signal s o , ( which is not the same as that illustrated in fig7 a ). fig9 a , 9 b , 9 c and 9 d illustrate the sub - band wavelet signals y o , y 1 , y 2 and y 3 obtained by dwt processing of the input signal s i . fig1 a , 10 b and 10 c illustrate the corresponding modulated carrier signals y ′ 0 , y ′ 1 and y ′ 2 obtained by modulating the carrier signals f 0 , f 1 , and f 2 with the sub - band wavelet signals y 0 , y 1 and y 2 , respectively . because the waveform of the simplified input signal is so smooth , the wavelet signal y 2 is interpolated by a factor of 2 only , and the wavelet signal y 0 and y 1 by a factor of 4 only . this is , of course , for illustration only ; in practice the interpolator may typically range from 1 : 8 to 1 : 24 . fig1 shows the encoded signal s o and fig1 shows its frequency spectrum which comprises the spectrum components of y ′ 0 , y ′ 1 and y ′ 2 centered at frequencies of 1000 hertz , 3000 hertz and 5000 hertz , respectively . for a message rate of 750 hertz . the asymmetric distribution of transmission power between the lower and high frequency carriers should be noted . it should be appreciated that these simplified signals are for illustration only and that real signals would be much more complex . fig1 a , 13 b and 13 c illustrate the recovered modulated carrier signals y ″ 0 , y ″ 1 , and y ″ 2 , and fig1 a , 14 b and 14 c illustrate the recovered sub - band wavelet signals y * 0 , y * 1 and y * 2 . finally , fig1 illustrates the reconstructed signal s ′ i which can be seen to be a close approximation of the input signal s i shown in fig8 . in the above - described embodiment , the highpass sub - band signal y 3 is not used , on the grounds that it probably contains negligible energy . if it has significant energy , however , it could be used , and the encoder and decoder modified appropriately . while similar implementations using more than two sub - bands and carriers are possible , and might be desirable in some circumstances , for most applications , and especially communication of digital signals via twisted wire subscriber loops , they would be considered complex without significant improvement in performance . it should be appreciated that other kinds of modulation might be used to modulate the sub - band signals , for example , narrow - band frequency modulation , and so on . it should also be appreciated that the signal source 10 and encoder 11 could be parts of a transmitter having other signal processing means . likewise , the decoder 13 and signal destination 14 could be parts of a corresponding receiver . although the above - described embodiments of the invention use three or more of the sub - band signals , it is envisaged that other applications , such as deep space communications , might use only one or two of the wavelets . an advantage of embodiments of the present invention , which use sub - band signals to modulate carriers , is that transmission is reliable because the impairment of one sub - band in the system would cause the transmission system to degrade only gently . also , the decoder bandpass filters can be easily designed because there are only a few frequency bands used . moreover , in applications involving data transmission , data synchronization and clock recovery can be easily achieved in the decoder . it should be noted that the present invention is not limited to transmission systems but could be used for other purposes to maintain signal integrity despite noise and attenuation . for example , it might be used in recording of the signal on a compact disc or other storage medium . the storage medium can therefore be equated with the transmission medium 12 in fig1 . it should be appreciated that the encoders and decoders described herein would probably be implemented by a suitably programmed digital signal processor or as a custom integrated circuit . although embodiments of the invention have been described and illustrated in detail , it is to be clearly understood that the same are by way of illustration and example only and not to be taken by way of the limitation , the scope of the present invention being limited only by the appended claims . 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