Orthogonal signals with jointly balanced spectra: Application to cdma transmissions
- Thierry Chonavel^{1}Email author
https://doi.org/10.1186/1687-1499-2011-176
© Chonavel; licensee Springer. 2011
Received: 20 April 2011
Accepted: 21 November 2011
Published: 21 November 2011
Abstract
This paper presents a technique for generating orthogonal bases of signals with jointly optimized spectra, in the sense that they are made as close as possible. To this end, we propose a new criterion, the minimization of which leads to signals with close energy inside a set of prescribed subbands. Starting with the case of a single subband, we illustrate it by building orthogonal signals with maximum energy concentration in time and in frequency, with the same energy rate outside a fixed frequency interval or a fixed time interval, by resorting to Slepian sequences or Slepian functions, respectively. Then, we present spectrum balancing in a set of frequency intervals. We apply this method to Slepian sequences and Slepian functions, as well as to Walsh-Hadamard codes. On these examples, we point out a number of nice properties of the so-built orthogonal families that are of interest for signaling applications.
PACS: signal processing techniques and tools; modulation techniques
Keywords
1 Introduction
A few studies have been carried out to build orthogonal signals with flat spectrum. Several of these studies are based on invariance property of Hadamard matrices w.r.t. orthogonal transforms.
More specifically, approaches presented in [1] and [2] account for the fact that when collecting orthogonal codes represented by column vectors in a matrix, then any permutation of the lines of the matrix yields columns that represent a new family of orthogonal codes. In [1], this principle is applied to Walsh codes and authors mention the fact that new codes spectra may be more flat than initial Walsh codes. However, permutations are performed randomly, and no criterion is supplied to optimize spectrum flatness. In fact, flatness will occur randomly in generated codes. In [2], the same approach is considered, but spectrum flatness is achieved by changing codes at each data transmission by considering a new random permutation at each time. Thus, flatness is not achieved by each code but only as a mean spectrum property among codes.
Alternatively, for controlling the spectra of the codes, one can generate white noise vectors and then apply amplitude distortion in the Fourier domain to achieve desired spectra. Finally, orthonormality of the codes is achieved by means of a singular value decomposition [3]. Another technique that enables better control of spectral shape consists in splitting code sequences spectra in a set of subbands of interest. In each subband, the Fourier transforms of the sequences are chosen as orthogonal Walsh codes with fixed amplitudes [4]. Proceeding so in each subband yields orthogonal signals in the Fourier domain. Thanks to unitarity of the Fourier transform, orthogonality of sequences is also achieved in the time domain. Note, however, that with these approaches the shape of the signal in the time domain is not controlled.
In a CDMA (Code Division Multiple Access) context [5], users transmit simultaneously and inside the same frequency band. They are distinguished thanks to distinct signaling codes. Often, Walsh codes are considered for multiusers spread spectrum communications. Walsh codes of given length show very variable spectra, and thus, they fail to achieve an homogeneous robustness of all users signaling against multi-path fading that occurs during transmission. Classically, users signals are whitened through the use of a scrambling sequence that consists of a sequence with long period that is multiplied, chip by chip, with users' spread data [6]. Scrambling also enables neighboring basestations insulation in mobile communication networks.
In radiolinks, synchronization of scrambling sequences between basestations and mobiles is not much a problem. Thus, in UMTS (Universal Mobile Telecommunications System) [6], the transmitted chip rate is 3.84Mchips/s and a distance of 1 km represents a propagation delay equivalent to (10^{3}/3 × 10^{8}) × 3.84 × 10^{6} ≈ 13 chips. This shows that scrambling code synchronization search, which is made necessary by transmitter and receiver relative position uncertainty, is not much complicated. On the contrary, in an underwater acoustic CDMA communication, with typical underwater chip rate of only 3.84 kchips/s for communications ranging to a few kilometers [7], a 1 km difference in the distance between both ends of the acoustic link results in a propagation delay equivalent to (10^{3}/1.6 × 10^{3}) × 3.84 × 10^{3} = 2, 400 chips. Thus, it is clear that there are situations where scrambling sequence synchronization can be difficult. In such difficult situations, instead of considering complex scrambling code synchronization, we rather propose to build orthogonal families of codes made of spreading sequences with flat spectra inside the sequences bandwidth. In addition, we would like to be able to build large sets of such signaling bases, for using distinct ones in neighboring basestations and/or to be able to change codes during the communications of a given basestation, for instance for robustness against communication interception.
In order to build such codes, starting from a given othonormal code family, we propose to transform it by means of an orthogonal transform. This orthogonal transform is built by minimizing jointly the mean squared errors among energies of all transformed sequences inside fixed subbands that form a partition of the whole sequences bandwidth.
This technique enables building arbitrarily large number of bases of spectrally balanced orthogonal codes. This is achieved by changing the initialization of the algorithm that we describe in the paper. In particular, distinct bases can be considered for neighboring bases stations in replacement of scrambling sequences. In addition, for a given basestation, it is also possible to change the codes family during transmission. Finally, basestations insulation, spectrum whitness of transmitted signals and data protection that are achieved by scrambling can also be obtained through balanced sequences generation.
To further motivate our search for chip-shaped CDMA sequences rather than more general waveforms, let us recall that CDMA systems employ chip-shaped sequences and that this structure has given rise to specific processing techniques. For instance, in downlink CDMA systems, the emitted signal is made of multiuser chip symbols shaped by the chip waveform at the transmitter output. At the receiver side, chip rate MMSE (Minimum Mean Square Error) equalizers are an efficient tool for downlink CDMA receivers that exploit this data structure [8]. Clearly, chip level equalization cannot be considered for continuously varying signalings such as those considered in [3] and [4]. This motivates our search for chip-shaped sequences.
In this paper, we shall consider balancing of CDMA sequences. Without loss of generality, balancing of CDMA codes will be studied for Walsh codes. We shall see that the corresponding balanced sequences exhibit several nice suitable properties such as low autocorrelation and cross-correlation peaks and good multiuser detection BER performance in asynchronous transmissions.
In order to introduce spectrum balancing, we first study balancing in a single subband and propose an algorithm to perform this task. We use it to supply solutions to the problems of building orthogonal bases of finite time signals with maximally concentrated energy in a frequency bandwidth and its dual that consists in building bases of signals with prescribed bandwidth and maximally concentrated energy in a time interval. This can be achieved by applying the algorithm to Slepian sequences and PSWFs (Prolate Spheroidal Wave Functions), respectively.
Slepian sequences and PSWFs [9, 10] have been used for long in classical areas as varied as spectrum estimation [11] and constantly find applications to new areas such as semiconductor simulation [12] or compressive sensing [13]. In communications, they have been used in particular for subcarriers signaling in OQAM and OFDM digital modulations [14, 15] or channel modeling and estimation [16, 17]. Spectral balancing of Slepian sequences or PSWFs could be of potential interest for some of these areas. It is also of interest for UWB (Ultra Wide Band) communications. Indeed, in UWB, M-ary pulse shape modulation has been proposed and it can be achieved with orthogonal signals such as PSWFs [18]. However, the spectra of Slepian sequences or PSWFs are slightly shifted upward as the sequence order increases. Instead, spectrally balanced pulses have spectra that better occupy the whole bandwidth, thus being more robust against multipath. For this reason, after introducing the spectrum balancing algorithm over a set of frequency intervals, we shall apply it to Slepian sequences and PSWFs balancing.
The remainder of the paper is organized as follows. In Section 2, we show how energies of an orthogonal family of signals can be made equal in a prescribed frequency interval thanks to an orthogonal matrix transform, preserving thus orthogonality of transformed signals. In section 3, we extend this method through the minimization of a criterion intended to jointly equalize energies of signals in a set of frequency subin-tervals. We propose an iterative minimizing algorithm to perform this task, and we apply it to Slepian sequences and PSWFs balancing. In section 4, we consider Walsh code balancing. Simulations show that spectrum whitening achieved by balancing yields good correlation properties of balanced sequences, resulting thus in improved performance of multiuser asynchronous communications.
2 Energy balancing in one frequency band
where S^{ B }is the matrix with general term ${S}_{ba}^{B}={e}^{i\pi \left({f}_{1}+{f}_{2}\right)\left(b-a\right)}\times \frac{sin\pi \left({f}_{2}-{f}_{1}\right)\left(b-a\right)}{\pi \left(b-a\right)}.$.
Letting V = [v_{1}, ..., v_{ L }], it comes that the energy of v_{ k }inside [-F, F] is the k th diagonal entry of V^{ H }SV. Now, we whish to transform V = [v_{1}, ..., v_{ L }] into W = [w_{1}, ..., w_{ L }] such that the {w_{ k }}_{k = 1,L}are orthonormal vectors with the same energy inside [-F, F]. This transformation can be expressed as W = VU, where U is some orthogonal matrix of size L. The equal energy constraint amounts to the fact that all diagonal entries of M = U^{ H }(V^{ H }SV)U must be equal. Letting d_{1}, ..., d_{ L }denote the diagonal entries of V^{ H }SV, it is clear that the diagonal entries of U^{ H }(V^{ H }SV)U must all be equal to $\stackrel{\u0304}{d}={L}^{-1}{\sum}_{k=1,L}{d}_{k}$ since orthonormal base changes do not affect the trace.
2.1 Energy balancing algorithm
Finding in a direct way U such that M has equal diagonal entries is unfeasible. Thus, we resort to an iterative procedure to equalize by pairs diagonal entries of M. This is achieved by updating U by means of Givens rotations [19]. In the following, we shall note D = diag(d_{1}, ..., d_{ L }) and R^{(a,b)}(θ) will represent the Givens rotation with angle θ in the subspace of dimension 2 with entry (a, b).
Energy balancing algorithm
- Set U = U_{0}, with ${U}_{0}^{T}{U}_{0}=I$, |
---|
M = U^{ T }DU |
- Iterations: |
while ${\sum}_{i=1,L}\left|{M}_{ii}-\stackrel{\u0304}{d}\right|\ge \epsilon $, |
loop a = 1 → L - 1 |
loop b = a + 1 → L |
$\theta =\frac{1}{2}arctan\left(\frac{{M}_{aa}-{M}_{bb}}{{M}_{aa}+{M}_{bb}}\right)$ |
M = R^{(a,b)}(θ)M R^{(a,b)}(-θ) |
U = U R^{(a,b)}(-θ) |
end loop b |
and loop a |
end while |
W = VU, |
By changing the initialization U_{0} of the matrix U in the algorithm, distinct matrices W are obtained. Thus, there are infinitely many distinct orthonormal families with equal energy inside [-F, F] in the space spanned by the columns of V, obtained by changing U_{0}.
The two following results establish the convergence of the algorithm in Table 1 toward an orthonormal balanced basis. Proofs are supplied in the Appendix.
Theorem 1 Iterations of the balancing algorithm in Table1lead to a sequence of matrices M^{(1)}, M^{(2)}, .... The diagonal part of these matrices converges to$\stackrel{\u0304}{d}I$, where I is the identity matrix.
Let Δ(M) denote the diagonal matrix with i th diagonal entry [Δ(M)]_{ ii }= M_{ ii }, where [P]_{ ab }denotes the entry (a, b) of matrix P. Then, we have
Note that that the proofs of theorems 1 and 2 show that convergence is achieved regardless U_{0}. At convergence, all signals in the columns of W = VU have the same amount of energy inside [-F, F] since these are given by the diagonal entries of M = W^{ T }SW. Furthermore, we have checked on the examples in the next subsection that convergence is fast for any choice of U_{0}.
2.2 Examples
2.2.1 Slepian sequences
For a given time interval, say [0, T], regularly sampled with N samples, and a fixed bandwidth [-F, F], one can ensure that there exists a basis with d sequences of length N that concentrate most of their energy inside [-F, F], provided T ≥ d/(2F). The elements of this basis are named spheroidal wave sequences or Slepian sequences [10].
Slepian sequences of length N are the eigenvectors of the matrix S of size N with general term ${S}_{mn}=\frac{sin\left(2\pi F\left(m-n\right)\right)}{\pi \left(m-n\right)}$. From earlier discussion, it is clear that the eigenvalues of S correspond to the percentage of the energy of the corresponding eigenvectors inside interval [-F, F]. These eigenvectors can be calculated accurately by means of a procedure proposed in [20]. Note that numerically this is not a straightforward task since most eigenvalues are either very close to zero or to one. More precisely, it is well-known that the 2FT largest eigenvalues are close to one and that others show fast decay to zero.
that represents the value of the energy of the sequence W_{ i }lying inside the frequency interval [-F, F]. Thus, all the (W_{ i })_{i = 1,L}have energy outside [-F, F] equal to $1-\stackrel{\u0304}{d}$.
Building 2FT sequences with the same (small) amount of energy outside [-F, F] can be of interest for applications. For instance, this could be interesting for multiuser communications on narrow frequency subbands.
As we can see it, although outband energies are equal, inband spectra remain very different and we will address spectrum equalization of sequences in Section 3.
To study convergence speed, we considered 10^{3} Monte Carlo simulations where U_{0} is chosen randomly among orthogonal matrices with uniform distribution. More details about the uniform distribution on orthogonal matrices and how to sample from it can be found in [21]. The value of the stopping parameter has been set to ε = 10^{-10}. In average, convergence is achieved after 8 iterations with best and worst cases of 5 and 10 iterations, respectively. Thus, convergence is very fast when balancing is performed with a single-frequency band for any choice of U_{0}.
2.2.2 PSWFs time energy balancing
Thus, successive PSFWs are supplied by successive eigenvectors of matrix $\stackrel{\u0303}{S}$, starting with the one with largest eigenvalue that represents the PSFW with maximum energy concentration inside [-T/2, T/2].
Hence, looking for energy-balanced PSWFs, that is, PSWFs linear combinations that yield an orthonormal family of functions with the same minimum energy ratio 1 -ρ outside time interval [-T/2, T/2], can be reformulated from our energy balancing framework by replacing matrix S by $\stackrel{\u0303}{S}$.
Thus, we see that the algorithm in Table 1 can be adapted to cope with several problems by changing the scalar product matrix S. Note in particular that convergence theorems 1 and 2 are valid regardless the choice of the scalar product S.
Here again, convergence is fast: for ε = 10^{-10} and 10^{3} Monte Carlo simulations, where U_{0} is chosen randomly among orthogonal matrices with uniform distribution, convergence is achieved after 15 iterations in average. Best and worst convergence cases are obtained for 13 and 16 iterations, respectively.
3 Spectrum balancing of an orthonormal family of signals
Here above, we have introduced an iterative technique for energy balancing inside a prescribed bandwidth. With a view to get orthogonal families of signals with similar spectra in the space spanned by vectors {v_{ k }}_{k = 1,L}, we derive an iterative technique to jointly equalize energies of these vectors in a set of frequency intervals, extending thus the technique proposed in the previous section.
where ${S}_{ab}^{k}$ is a compact form for [S_{ k }]_{ ab }. Although extension to the complex case is straightforward, in this paper, we restrict ourself to the case of real valued signals.
3.1 Balancing algorithm
On another hand, since the term ${\left({M}_{aa}^{k}-{M}_{bb}^{k}\right)}^{2}$ in the denominator of the arctan(.) function in Equation (12) could be a source of unstability and should become close to zero at convergence $\left({M}_{aa}^{k}\approx {M}_{bb}^{k}\right)$, we set it to 0 from the beginning of the algorithm.
Spectrum balancing algorithm
- Set U = U_{0} with ${U}_{0}^{T}{U}_{0}=I$, |
---|
M_{k} = U^{T} (V^{ T }S_{k} V) U, ${M}_{k}^{-}=0\left(k=0,...,K-1\right)$ and $M=\left[{M}_{0},...,{M}_{K-1}\right],{M}^{-}=\left[{M}_{0}^{-},...,{M}_{K-1}^{-}\right]$ |
- Iterations: |
while || M - M^{-} || ≥ ε, (ε ≪ 1) |
${M}_{k}^{-}={M}_{k},\left(k=0,...,K-1\right)$ |
loop a = 1 → L - 1 |
loop b = a + 1 → L |
$\theta =\frac{1}{4}arctan\left(\frac{2{\sum}_{k=0}^{K-1}\left({M}_{ab}^{k}+{M}_{ba}^{k}\right)\left({M}_{aa}^{k}-{M}_{bb}^{k}\right)}{{\sum}_{k=0}^{K-1}{\left({M}_{ab}^{k}+{M}_{ba}^{k}\right)}^{2}}\right)$ |
M_{ k }= R^{(a,b)}(θ) M_{ k }R^{(a,b)}(-θ), (k = 0,..., K - 1) |
U = U R^{(a,b)}, (-θ) |
end loop b |
and loop a |
end while |
W = VU |
3.2 Examples
As already mentioned in the introduction, we can check in Figure 5 that Slepian sequences or PSWFs are slightly shifted upward as order increases while spectrally balanced sequences have spectra that better occupy the whole bandwidth.
More generally, spectrum balancing could be considered for other UWB orthogonal pulses, such as Gaussian, Hermite or Legendre functions, where elements of the family of increasing order tend to have spectra that are centered at increasing frequencies [25].
Achieving spectrum flattening is an interesting property for combatting multipath as we shall see in the next section for another kind of waveform (more specifically balanced Walsh sequences).
3.3 Convergence
where C and β are positive constants. A straightforward consequence of Equation (15) is that the sequence of matrix (M^{(n)})_{n≥0}converges and convergence is achieved at geometric rate.
4 Walsh codes balancing
As discussed in the introduction, in a CDMA context we are looking for signals that are constant over chip intervals, a natural approach is to search them in the space spanned by the orthogonal Walsh-Hadamard basis. Then, if the sampled signals of this basis are given columnwise in a matrix form, any new orthogonal basis of the vector space is achieved by applying an orthogonal matrix transform on the right-hand side. Note that instead, references [1] and [2] in the introduction apply matrix permutations on the left-hand side of the matrix of code sequences.
As in the case of continuous signals discussed in the examples of Sections 2 and 3, the algorithm works by starting from an orthonogonal basis and successively transform it into new orthonogonal bases of the same vector space. Of course, some specific properties of the initial family such as constant absolute amplitude in the case of Walsh codes are not preserved by orthogonal transforms, while others such as the chip structure of codes (signal constant over chip durations) are preserved because this property is shared by all signals in the vector space spanned by initial Walsh codes.
4.1 Spectral balancing of Walsh sequences
4.2 Convergence and balanced codes properties
4.2.1 convergence
For codes of length 256, convergence becomes very slow. This is because when using K subbands for codes of K chips, the main loop of the algorithm requires about K^{3} operations. However, we checked that stopping the algorithm after about 100 iterations already yields quite good mixing in terms of spectral shapes (see Figure 11) and, as we shall see it in section 4.3, BER performance.
4.2.2 Amplitude
We have just seen that the amplitudes of balanced codes can have continuous values. Thus, using the proposed codes instead of classical binary codes such as Walsh codes results in slight increase of complexity of the system, mainly at the receiver side where sequence matched filtering will involve multiplications instead of sign shifts of sampled received signals. Although chip amplitude of balanced codes can have continuous values, in practical systems they should be rounded to remain in some discrete alphabet and thus facilitate digital processing. We have checked that 8 bits encoding of the chips of balanced codes is enough to avoid BER performance degradation for codes of length 32 and codes of length 256. In other words, rounded balanced codes yield no noticeable difference in the BER curves of balanced codes, as it will be shown in section 4.3.
4.2.3 Correlation and cross-correlation of sequences
Correlation and cross-correlation properties of codes dictate the performance of a multiuser communication system at high SNR [26]. For simple receivers based on single-user matched filter, correlation properties are important in particular for receiver synchronization, while in asynchronous systems, cross-correlations of codes limits performance. Thus, we are going to consider these properties and compare them between Walsh codes and balanced codes.
Since Walsh codes are not considered as good codes in terms of correlation and cross-correlation, we also made a comparison with brute force codes considered in [28]. These codes are obtained by means of an exhaustive search algorithm among codes with good cross-correlation properties. Figure 15 shows that these codes achieve quite poor correlation performance, even when removing the constant code autocorrelation (the one with triangular shape).
As far as cross-correlations are considered, the second line in Figure 15 shows that both balanced and brute force codes achieve good performance, unlike Walsh codes.
Finally, above results advocate in favor of multilevel balanced codes that can achieve higher correlation performance, at the expense of relaxation of the constant amplitude property.
4.3 Asynchronous transmission
Of course, for a fixed spreading code length, the matched filter receiver performs worse as the number of interfering users increases and decorrelator or MMSE detectors would lead to improved BER curves [5]. However, here we only considered the simpler matched filter receiver to focus on code properties rather than on receiver performance.
5 Conclusion
We have proposed a general purpose procedure for deriving spectrally balanced bases of signals in a given signal subspace, approximated as a subspace of ℝ^{ N }. As examples, we have shown how this procedure enables building spectrally balanced families of signals with maximum time and spectral concentration from Slepian sequences and Slepian functions. We have also shown how it is possible to build efficient signalization sequences for CDMA multiuser communications that show performance similar in terms of BER to brute force optimized binary sequences. Large numbers of such families of codes can be built thanks to the relaxation upon the constant amplitude constraint, but codes maximum amplitude remains acceptable for most applications. Clearly, using balanced signals in applications such as synchronization or for designing radar waveforms is promising, due in particular to nice correlation properties and the wide variety of waveforms that can be generated.
Appendix
Proof of theorem 1
that is, Δ(M^{(k)}) tends to $\stackrel{\u0304}{d}I$.
Proof of theorem 2
Declarations
Authors’ Affiliations
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