A new multiple access scheme, Waveform Division Multiple Access (WDMA) based on the orthogonal wavelet function, is presented. After studying the correlation properties of different categories of single wavelet functions, the one with the best correlation property will be chosen as the foundation for combined waveform. In the communication system, each user is assigned to different combined orthogonal waveform. Demonstrated by simulation, combined waveform is more suitable than single wavelet function to be a communication medium in WDMA system. Due to the excellent orthogonality, the bit error rate (BER) of multiuser with combined waveforms is so close to that of single user in a synchronous system. That is to say, the multiple access interference (MAI) is almost eliminated. Furthermore, even in an asynchronous system without multiuser detection after matched filters, the result is still pretty ideal and satisfactory by using the third combination mode that will be mentioned in the study.
There are two definitions of UWB system: the bandwidth of signal spectrum exceeds 25% of the center frequency or the −10 dB bandwidth of signal is over 500 MHz [
In recent years, multiple access UWB communication has been a hot research topic as the development of UWB technologies. In UWB communication systems, the multiple access approaches are commonly divided into two paths: time hopping (THUWB) and direct sequence (DSUWB) [
Among all the techniques in WDMAUWB communications systems, it is of great importance to choose the analog waveform to denote a message symbol. In UWB systems, Rayleigh pulse and Gaussian monocycle [
In this study, orthogonal wavelet is another example that can satisfy the requirement of orthogonality [
The paper is organized as follows. In Section
Assume that there are
The traditional receiver of a WDMAUWB system consists of a pulse demodulator and a matched filter corresponding to each user. The signal arriving at the matched filters can be described as follows:
The matched filter part of the receiver is shown in Figure
Matched filter part in the receiver.
Let vector
Wavelets consist of a group of functions that satisfy the following formula:
The constant of (
An orthogonal wavelet is a function
According to the definition of orthogonality, orthogonal wavelet has the following property:
In this study,
There are four common categories of orthogonal wavelets used in WDMA system, Daubechies wavelet, Symlets wavelet, Coiflets wavelet, and Meyer wavelet. In each category, one kind of wavelets is chosen to be studied through the simulation. In the following simulation, three wavelet functions
In the category of Daubechies, set db8 wavelet as an example. The waveforms, selfcorrelation properties, and mutual correlation properties of the three wavelet functions are shown in Figures
Waveform of db8 functions and their correlation property.
Waveforms
Selfcorrelation
Mutual correlation
From Figure
Similar performances and the same conclusions can be obtained from the simulations of other three orthogonal wavelets, sym8 wavelet, coif5 wavelet, and Meyer wavelet.
In addition, the theoretical calculation values of mutual correlation of three wavelet functions in each category are shown in Table
Theoretical calculation values of mutual correlation.
Two wavelet functions  db8 wavelet value of mutual correlation  sym8 wavelet value of mutual correlation  coif5 wavelet value of mutual correlation  Meyer wavelet value of mutual correlation 


1.7497 × 10^{−13}  4.3351 × 10^{−14}  1.2654 × 10^{−10}  2.1762 × 10^{−16} 

7.7623 × 10^{−7}  −5.5633 × 10^{−5}  6.8283 × 10^{−7}  −4.2253 × 10^{−4} 

−8.7764 × 10^{−8}  9.3128 × 10^{−6}  −3.8248 × 10^{−8}  5.8239 × 10^{−4} 
From Table
Synchronous system is an ideal condition. In the actual communication system in outer space, the signal transmitted by each user at the same time may not reach the receiver corresponding to each user at the same time. So there is great need to study the correlation property of single orthogonal wavelet in the asynchronous system.
In order to research the mutual correlation property of two wavelet functions in an asynchronous way, fix one wavelet function and translate the other wavelet function point by point. At each point, calculate the value of mutual correlation of the two functions. At last, compute the statistical mean, variance, and maximum of all the values of mutual correlation.
Take db8 orthogonal wavelet as an example. Fix
Asynchronous mutual correlation values of three db8 wavelet functions.
As is known above, the value of selfcorrelation of each wavelet function is one. From Figure
Comparing Figures
Similar waveforms and the same conclusions can be obtained from the simulations of other three orthogonal wavelets, sym8 wavelet, coif5 wavelet, and Meyer wavelet. The statistical mean, variance, and maximum of all the values of mutual correlation are shown in Tables
Statistics of asynchronous mutual correlation values for db8 wavelet.
Statistics data 




Mean  0.0927  0.0307  0.0307 
Variance  0.0401  0.0050  0.0050 
Maximum  1.0000  0.3467  0.3467 
Statistics of asynchronous mutual correlation values for sym8 wavelet.
Statistics data 




Mean  0.0927  0.0303  0.0303 
Variance  0.0401  0.0051  0.0051 
Maximum  1.0000  0.3523  0.3523 
Statistics of asynchronous mutual correlation values for coif5 wavelet.
Statistics data 




Mean  0.0507  0.0155  0.0155 
Variance  0.0239  0.0024  0.0024 
Maximum  1.0000  0.3074  0.3074 
Statistics of asynchronous mutual correlation values for Meyer wavelet.
Statistics data 




Mean  0.1087  0.0281  0.0281 
Variance  0.0427  0.0019  0.0019 
Maximum  1.0000  0.1685  0.1685 
The first statistical data, mean, is for the average value of mutual correlation. The second one, variance, is for the stability of mutual correlation following with the change of the translation points. And the last one, maximum, is for the maximum possible interference between users in WDMAUWB system.
From Tables
According to the research mentioned above, Meyer wavelet is chosen among the four categories as the foundation for designing combined waveform. In this study, there are three kinds of combination. To distinguish the differences between three kinds of combination, suppose there are two users.
Combination 1:
user 1:
user 2:
Combination 2:
user 1:
user 2:
Combination 3:
user 1:
user 2:
The waveforms of three kinds of combination modes are shown in Figures
Waveforms of three combination modes.
Combination 1
Combination 2
Combination 3
Take combination 1 for example, to prove the orthogonality between two users. Consider
In synchronous system, the correlation properties, especially the selfcorrelation values of the combined orthogonal wavelet and the theoretical value of mutual correlation between two users, of these three combinations, are arranged in Figures
Theoretical values of mutual correlation of 3 kinds of combinations.
Combination mode  Value of mutual correlation between two users 

Combination 1  −3.9328 × 10^{−16} 
Combination 2  −5.3033 × 10^{−4} 
Combination 3  −2.6554 × 10^{−4} 
Selfcorrelation values of three combination modes.
Combination 1
Combination 2
Combination 3
From Table
The method of studying the mutual correlation of combined waveform in an asynchronous way is the same with that of single wavelet function. Fix the waveform of user 1 and translate the waveform of user 2 point by point. The results of three combinations are shown in Figures
Asynchronous mutual correlation values of three combinations.
Combination 1
Combination 2
Combination 3
Additionally, the statistical mean, variance, and maximum of all the values of mutual correlation for three combination modes are shown in Table
Statistics of asynchronous mutual correlation values of 3 combination modes.
Statistics data  Combination 1  Combination 2  Combination 3 

Mean  0.1806  0.1582  0.0128 
Variance  0.0887  0.0565  2.5405 × 10^{−4} 
Maximum  1.1673  1.4087  0.0560 
In the research of single wavelet function, the selfcorrelation value of single wavelet function is one. And in the simulation of combined waveform including two wavelet functions, the selfcorrelation value of one combined waveform is two. From Table
From Table
Suppose that it is a synchronous WDMAUWB system and there are 10 users in this system. The waveform of each user is given below. Each user transmits 32 bits of information continuously each time and repeats transmitting for 8000 times. Then, calculate bit error rate (BER) with the signal noise ratio (SNR) from 0 dB to 10 dB.
Waveform of each user using combination 1:
user 1:
user 2:
user 3:
user 4:
user 5:
user 6:
user 7:
user 8:
user 9:
user 10:
Waveform of each user using combination 2:
user 1:
user 2:
user 3:
user 4:
user 5:
user 6:
user 7:
user 8:
user 9:
user 10:
Waveform of each user using combination 3:
user 1:
user 2:
user 3:
user 4:
user 5:
user 6:
user 7:
user 8:
user 9:
user 10:
The performance of each combination mode in a synchronous WDMAUWB system is shown in Figure
Performance of each combination mode in a synchronous system.
From Figure
With the same simulation parameters, research the BER of three combination modes for multiuser in an asynchronous WDMAUWB system. The result and differences can be seen in Figure
Performance of each combination mode in an asynchronous system.
Seen from Figure
The multiple access interference is the main reason that limits the UWB system capacity. In this study, a new multiple access scheme, WDMA based on the orthogonal wavelet function, is presented. After studying the correlation property of different categories of single wavelet functions, Meyer wavelet with the best correlation property is chosen to be the foundation for combined waveform. Each user is assigned to different combined orthogonal waveform. Proved by simulation, combined waveform is more suitable than single wavelet function to be a communication medium in WDMA system. Due to the excellent orthogonality, the BER of multiuser with combined waveforms is so close to that of single user in a synchronous system and even in an asynchronous system without multiuser detection after matched filters in the receivers, by using the third combination mode. In future work, multiuser detection technology will be studied with the purpose of reducing the BER of utilizing combinations 1 and 2 as UWB pulses in an asynchronous WDMA system.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research is supported by “the National Natural Science Foundation of China” (Grant no. 61102084), “the Foundation of China Academy of Space Technology (CAST),” and “the China Postdoctoral Science Foundation” (Grant no. 2011M500665).