The Proceedings of the International Conference on Digital Information Processing, Data Mining, and Wireless Communications, Dubai, UAE, 2015
A Multi-level Spreading Code to Suppress Noise Enhancement in Zero-Forcing Equalization for Direct-Sequence Spread-Spectrum Communication Takahito Hoyagi and Takehiko Kobayashi Wireless Systems Laboratory, Tokyo Denki University 5 Senju-Asahi-cho, Adachi-ku, Tokyo 120-8551, Japan E-mail:
[email protected] Abstract Zero-forcing (ZF) equalization has been proposed to combat frequency selective fading caused by multipath propagation. The ZF equalization is effective when the matrix representing the propagation channel is well-conditioned; when ill-conditioned, however, it causes noise enhancement and, as a result, deteriorates the transmission performance. This paper proposes a new spreading code to suppress the noise enhancement. First, an arbitrary binary spreading code is represented by an N M matrix (M is the length of a transmitting symbol and N the spreading factor). Next, the matrix is singular-value decomposed. Then, all the singular values are replaced by a positive constant, and a reverse operation to the singular-value decomposition generates a new matrix, which is amply well-conditioned. The new matrix represents a multi-level spreading code, which can suppress noise enhancement. Simulation results validated its effectiveness.
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Keywords:
zero-forcing equalization, noise enhancement, DS-SS, multi-level code, multipath
1. Introduction High speed communication is growingly desirable in mobile communications, where multipath propagation is prevailing, causes frequency selective fading, and deteriorates the transmission performance due to inter-symbol interference (ISI). One of the technologies to mitigate this deterioration is a zero-forcing (ZF) equalizer [1], which is realized with relatively simple processing in a time domain. Frequency-domain ZF equalization has been also applied to direct-sequence code division multiple access (DS-CDMA) [2, 3]. The ZF equalizer is
ISBN: 978-1-941968-05-5 ©2015 SDIWC
capable of compensating the ISI when the matrix representing the propagation channel is well-conditioned; when ill-conditioned, however, it even enhances thermal noise, lessening the effect of compensation. This paper proposes a new multi-level spreading code generated from an arbitrary binary code to alleviate noise enhancement. While various multi-level spreading codes have been studied and applied for DS-CDMA [4] and multi-carrier CDMA [5], none has been reported to mitigate noise enhancement when using ZF equalization. Simulation considering multipath propagation validated the effectiveness of the proposed spreading code. 2. Conventional ZF Equalization In direct-sequence spread-spectrum (DS-SS) communication systems, a transmitting SS signal ( N × 1 vector) is given by s = ,
(1)
where d is a data symbol = , , … , , an N ×M matrix representing a spreading code, and N a spreading factor. Received signal r (a N × 1 vector) is given by r = Hs + n,
(2)
where H is an N × N matrix representing the propagation channel and n additive white Gaussian noise. The r is ZF-equalized and despread into a 1 × vector given by = + = + , =
(3)
is an estimated propagation channel where and denotes a generalized inverse matrix. If = ), the estimation is errorless (i.e.
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The Proceedings of the International Conference on Digital Information Processing, Data Mining, and Wireless Communications, Dubai, UAE, 2015
(4)
3. Proposed Spreading Code To suppress noise enhancement, det should be enlarged, since the channel matrix H cannot be artificially modified. The spreading code matrix is singular-value decomposed as = $%& ' ,
(5)
where U and V are unitary matrices, Σ is a diagonal matrix whose components are singular values of C, and ' denotes complex conjugate transposition. To enlarge det C, Σ is replaced by the unit matrix I. Then a new spreading code matrix ( is calculated by ( = $)& ' ,
(6)
where U and V are the same as in Eq. (5). If represents a binary code, the new code given by ( is multi-level. With use of this ( instead of , the noise enhancement denoted by the second term in the righthand side of Eq. (4) can be suppressed. Figure 1 shows the waveforms representing a 64-chip Gold code and a proposed code generated therefrom. While normalized singular values of the Gold code range from 1.00 to 0.02, those of the proposed code are all 1.00, as shown in Fig. 2. The proposed code exhibits auto- and cross-correlation properties nearly the same as those of the Gold code: illustrative examples are shown in Fig. 3. 4. Validation through Simulation Simulation was carried out to validate the effectiveness of the proposed code applied to a DS-SS communication system. At the transmitter, the data symbol was binary phase shift keying
ISBN: 978-1-941968-05-5 ©2015 SDIWC
2 Normalized amplitude
The data symbol d is reproduced even in a multipath propagation environment, if the second term in the righthand side of Eq. (4) is negligibly small compared to the first term. However, when H is ill-conditioned (i.e. det ≈ 0), the receiver noise n is enhanced.
Proposed code Gold code
1 0 -1 -2 0
20 40 Number of point
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Figure 1. Waveforms of a Gold code (binary) and the corresponding proposed code (multi-level).
1 Normalized amplitude
. = +
Proposed code Gold code
0.8 0.6 0.4 0.2 0 0
20 40 Number of point
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Figure 2. Comparison of singular values of spreading code matrices representing a Gold code and the corresponding proposed code with a length of 64.
(BPSK)-modulated, serial-to-parallel converted, spectrum spread with a Gold code or the proposed code, and, after adding a guard interval (= 16 symbols), transmitted. Spreading factor N = 32 or 128. Two models of 10-path channels were assumed: the interval between the 10 paths was equal to that of the symbol; the amplitude of the 10 paths was either (A) uniform (decay factor = 0 dB) or exponentially decayed (decay factor = -2 dB). Channel A represents a severest
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The Proceedings of the International Conference on Digital Information Processing, Data Mining, and Wireless Communications, Dubai, UAE, 2015
0.8
0.8
Cross-correlation
1
Auto-correlation
1
0.6 0.4 0.2 0
-0.2 -0.4
0.6 0.4 0.2 0 -0.2
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0.6 0.4 0.2 0
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-0.2 -0.4
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3.
Auto- and cross-correlation functions of: (a) a Gold code and (b) the proposed cod
multipath environment, while Channel B a milder one. At the receiver, the guard interval was first subtracted from the received signal. Then the channel matrix was estimated with use of pilot symbols included in the received signal. Errorless estimation was assumed. Next, is or ′ as in Eq. (3) multiplied by and then demodulated. Table 1 lists the major simulation parameters. Figure 4 shows constellations of the despread signal with use of a Gold code and that the proposed code, when signals-to-noise ratio (SNR) = 20 dB. A Gold code (Fig. 4(a)) yielded widely scattered constellation received from the Channel
ISBN: 978-1-941968-05-5 ©2015 SDIWC
A, indicating noise enhancement, which was suppressed by the proposed code (Fig. 4(b)). Meanwhile , the constellations scattered less, when received from Channel B, indicating a smaller degree of noise enhancement, as shown in Fig. 4(c). which was also suppressed by the proposed code, as shown in Fig. 4(d). Error vector magnitude (EVM) has been widely used to quantitatively represent the spread of constellation. Figure 5 depicts the EVM of the ZF-equalized and despread signals received from Channels A and B with use of the two codes (N = 128). The proposed code significantly reduced EVM, particularly when SNR was low, as shown in Fig. 5.
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The Proceedings of the International Conference on Digital Information Processing, Data Mining, and Wireless Communications, Dubai, UAE, 2015
Table 1. Parameters
Simulation parameters.
Values BPSK Gold code or the proposed code 32 and 128 32 and 128 16 symbols
Channel A Channel B
10 paths, uniform amplitude 10 paths, exponentially - decayed amplitude Interval between the 10 paths = symbol interval
References
Quadratue component
0 -1 -2 -2
-1
0
1
2 1 0 -1 -2 -2
2
In-phase component
-1
(a)
1 0 -1
-1
0
1
2
(b)
2
-2 -2
0
In-phase component
Quadratue component
Conventional ZF equalization has the drawback of enhancing thermal noise when a propagation channel matrix is ill-conditioned. To suppress noise enhancement, a new multi-level spreading code has been proposed for DS-SS communication systems. A conventional binary code was represented by a matrix, which was then singular-value decompositioned. While the singular values range widely in general, the proposed method replaced them with a positive constant. Then a reverse operation to the singular value decomposition yielded a new spreading code matrix, which corresponded to a new code. Simulation results showed use of the new multi-level code suppressed noise enhancement and provided lower bit error rates in a DS-SS system. A 6-bit ADC was found almost sufficient to generate the multi-level code at the receiver.
1
Quadratue component
5. Conclusion
2
1
2 1 0 -1 -2 -2
2
In-phase component
-1
0
1
2
In-phase component
(c)
(d)
Figure 4. Constellations of despread BPSK signals received from Channel A: (a) a gold code and (b) the proposed code; and from Channel B: with use of (c) a gold code and (d) the proposed code. N = 64. In the figures, ○ and × represent 1 and -1 symbols, respectively. Gold code (Channel A) Gold code (Channel B) Proposed code (Channel A) Proposed code (Channel B)
80 60
EVM (%)
The bit error rates of the system outperformed the conventional system, as shown in Fig 6. Figure 7 shows the influence of the number of bits of the analog-to-digital converter (ADC) generating the proposed multi-level code at the receiver on bit error rates when received from Channel A. Figure 7 revealed that a 6-bit ADC almost suffices.
Quadratue component
Modulation Spreading code Spreading factor, N Symbol length, M Guard interval
40 20
[1] R. W. Lucky, “Automatic equalization for digital communication,” Bell Syst. Tech. J., pp. 547-588, Apr. 1965. [2] F. Adachi, D. Grag S. Takaoka, and K. Takeda, “Broadband CDMA techniques,” IEEE Wireless Commun., vol. 12, no. 2, pp. 8-18, Apr. 2005. [3] K. Takeda, T. Itagaki and F. Adachi, “Joint
，
ISBN: 978-1-941968-05-5 ©2015 SDIWC
10 0
5
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25
SNR [dB]
Figure 5. Error vector magnitude of despread BPSK signals with use of a Gold code and the proposed code. N = 128.
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The Proceedings of the International Conference on Digital Information Processing, Data Mining, and Wireless Communications, Dubai, UAE, 2015
Bit Error Rate
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-3
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1 bit 3 bit 4 bit 6 bit 64 bit
5
10 SNR [dB]
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Figure 7. Influence of the number of bits of the ADC generating the proposed multi-level code at the receiver on bit error rates in environment channel.
-2
-3
-4
0
5
10 SNR [dB]
15
20
(b) Figure 6. Simulation results of bit error rates of a DS-SS system using a Gold code and the proposed code: (a) Channel A and (b) Channel B. frequency-domain equalization and antenna diversity combining for orthogonal multicode DS-CDMA signal transmissions in a frequency-selective fading channel,” IEICE Trans. Commun., vol. E87-B, no. 7, Jul. 2004. [4] K. Usha and K. J Sankar, “New multi level spreading codes for DS CDMA Communication,” in Conference on Advances in Communication and Control Systems 2013 (CAC2S 2013), pp 154-159, 2013. [5] J. Zhang and D. W. Matolak. “Multiple level orthogonal codes and their application on MC-CMA systems,” Computer Communications, vol. 32, pp 492-500, 2009.
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