Performance evaluation of variable transmission rate OFDM systems via network source coding
- Te-Lung Kung^{1}Email author and
- Keshab K Parhi^{1}
https://doi.org/10.1186/1687-6180-2013-12
© Kung and Parhi; licensee Springer. 2013
Received: 20 April 2012
Accepted: 17 January 2013
Published: 1 February 2013
Abstract
Abstract
Studies related to the network source coding have addressed rate-distortion analysis in both noiseless and noisy channels. However, to the best of authors’ knowledge, no prior work has studied network source coding in the context of orthogonal frequency division multiplexing (OFDM) systems. In addition, the system performance using network source coding schemes also remains unknown. In this article, we first propose two variable transmission rate (VTR) OFDM systems by utilizing network source coding schemes, and then evaluate the system performance of these two proposed VTR-OFDM systems. For the proposed VTR single-input single-output OFDM system, we employ the concept of a network with intermediate nodes to develop a 3-stage encoder/decoder, and the proposed encoder provides three different coding rates from 0.5 to 0.8. As for the proposed VTR multi-band OFDM system, two correlated sources are simultaneously transmitted using the multiterminal source coding schemes, and two sources are encoded by different coding rates from 0.25 to 0.5. Finally, compared with the traditional uncoded OFDM system, the proposed VTR-OFDM systems have at least 1 to 4 dB gain in signal-to-noise ratio to achieve the same symbol error rate in the additive white Gaussian noise channel.
Keywords
1 Introduction
The rapid growth of communications systems for video, voice, and cellular telephone justifies great demands for mobile multimedia [1, 2]. These multimedia services will require high data rate transmissions over broadband radio channels. Due to the limited spectral resource, it is impractical to increase the bandwidth for data transmission. In the case of wired networks, high data rate services are not a problem due to the channel characteristic; however, high data rate transmissions lead to an additional technical consideration for wireless communications networks. Moreover, the further development of the advanced network technologies is constrained by the amount of information that can be sent through the corresponding networks. In wired or wireless network, where bandwidth is strictly limited, it is imperative to use efficient data representations or source codes for optimizing network system performance [3, 4]. In order to transmit data stream over communications networks with limited bandwidth, network source coding can be used to solve this problem.
Network source coding expands the data compression problem for networks beyond the point-to-point network introduced by Shannon [5]. These include networks with multiple transmitters, multiple receivers, decoder side information, intermediate nodes, and any combination of these features. Thus, network source coding attempts to bridge the gap between point-to-point network and recent complicated network environments. Network source coding is a promising technique that provides many advantages, including source dependence, functional demands, and resource sharing [6–12]. By utilizing network source coding, the system performance can be improved by using correlated sources. This leads to an adjustable transmission rate, and the input data streams share the entire network resource during the transmission. If decoders have the information about the relation of transmitted data sources, the input data sources can be reconstructed without any loss in the noiseless channel.
High data rate wireless communications are limited not only by noise but also by the inter-symbol interference (ISI) from the dispersion of the wireless communications channel. Orthogonal frequency division multiplexing (OFDM) systems have been adopted in various wireless communications standards due to their high spectrum efficiency and robustness against the frequency selective fading channels [1, 13]. In addition, OFDM is a prominent technique suitable for high data rate transmissions in multicarrier communications systems, because of the potential to either partially reduce or completely eliminate the adverse ISI effect. In OFDM systems, a wideband channel is converted to a set of narrowband subchannels, and the data is transmitted using all subchannels. Multi-band OFDM (MB-OFDM) systems group all subchannels into multiple subbands, and then transmit data in different subbands simultaneously [14, 15]. Hence, the multi-band signal processing technique can be viewed as either a multiple access scheme that allocates subbands to transmit different users’ data at the same time or an approach to employ the frequency diversities for data transmission in order to improve the system performance.
To the best of authors’ knowledge, the following properties have not been demonstrated yet [6–12, 16]: (1) the variable transmission rate (VTR) OFDM communications systems using network source coding, and (2) the performance improvement after applying the network source coding to OFDM systems. In this article, two VTR-OFDM communications systems are proposed to evaluate the performance of network source coding schemes. First, a VTR single-input single-output OFDM (VTR-SISO-OFDM) system is proposed. The proposed system generates different transmission rates using a proposed 3-stage encoder. Functions implemented in this 3-stage encoder determine the codebook sizes for storage and the achievable rates for data transmission. The proposed 3-stage encoder consists of three different mapping functions and their corresponding encoders using arithmetic coding scheme. The performance of the VTR-SISO-OFDM system based on the proposed 3-stage encoder is evaluated. Then, a VTR multi-band OFDM (VTR-MB-OFDM) system is proposed. The proposed VTR-MB-OFDM system transmits two correlated sources using the multiterminal source coding scheme [7, 17–19]. The transmission rates are adjusted based on four switch configurations at the transmitter, and 16 switch configurations control the information interchanges during the entire data transmission. The performance of the proposed VTR-MB-OFDM system for these 16 switch configurations is evaluated. We then analyze the variable transmission bit rate for these two proposed systems. The advantages of the proposed systems include availability of different transmission rates, abilities of different users to share the network resource, and robustness in wireless communications environments. Moreover, compared with the traditional uncoded OFDM systems, simulation results show the proposed systems obtain at least 1 to 4 dB gain in signal-to-noise ratio (SNR) to achieve the same symbol error rate (SER) in the additive white Gaussian noise (AWGN) channel. In addition, the rate factors of these two proposed systems vary from 0.55 to 0.96.
The rest of the article is organized as follows. In Section 2, a VTR-SISO-OFDM communications system is proposed. A VTR-MB-OFDM system based on the multiterminal source coding scheme is proposed in Section 3. Performance analysis and simulation results are presented in Section 4. Finally, the article is concluded in Section 5.
2 The VTR-SISO-OFDM system
We propose a VTR-SISO-OFDM system with three different transmission rates in this section. The main goal of the proposed VTR-SISO-OFDM system is to design a variable transmission rate system and to recover the source data with the help of the side information. A 3-stage encoder is proposed in Section 2.1. In Section 2.2, the feasibilities of functions used in the encoder and the variable transmission rates of the proposed VTR-SISO-OFDM system are discussed. Finally, the corresponding decoder is described in Section 2.3.
2.1 The 3-stage encoder of the VTR-SISO-OFDM system
where i _{4} ∈ {1,2,…,E} and p(j _{1}|f(i _{1},j _{1})=i _{2}) is the conditional probability.
Moreover, if g (·) and h (·) are surjective functions, Φ _{2} and Φ _{3} are the corresponding extra information to obtain w _{ i }= g ^{−1}(k _{ i },c _{ i },Φ _{2}) and k _{ i }= h ^{−1}(e _{ i },d _{ i },Φ _{3}), respectively.
From Equations (19) and (22), the proposed 3-stage encoder generates three different transmission rates by selecting different output signals through varying the switch S _{ t }.
In communications systems, two different ways utilized to transmit the data are continuous data transmission and packet data transmission. For the continuous data transmission such as voice, the transmission rate is controlled by the transport layer. In this case, the receiver knows the transmission rate during the entire data flow. As for the packet data transmission, each packet consists of the header and the data, and the information of the switch configuration can be hidden in the header of each packet. Therefore, in order to adjust the transmission rate without notifying the receiver, the transmitter first encodes the switch configuration and then includes the encoded bits in the header. The length of the header depends on the coding rate of the error correcting codes. In this article, we consider continuous data transmission.
After the entire encoding process, we pad the input data, x ^{ T }, with a certain number of zeros (N _{ z }) to maintain the total number of bits (N _{ b }) equal to M × N _{ T }, where ${N}_{z}=M\times {N}_{T}-{\text{Len}}_{{S}_{t}}$, M is the number of bits to represent a symbol, and N _{ T } is the number of total subcarriers. Then, the modified bit stream, $\left[{\mathbf{x}}^{T}\phantom{\rule{1em}{0ex}}{0}_{1\times {N}_{z}}\right]$, is transmitted using a SISO-OFDM system.
2.2 The feasibilities of functions and variable transmission rate analysis
Choosing the proper mapping functions in the encoder provides two advantages: (1) the improvement of the system performance and (2) the adjustability of the transmission rate. In order to reduce the size of codebook and to reconstruct the source data perfectly at the receiver, it is necessary to choose the mapping functions that generate smaller cardinalities in the encoder. Also, the help of the extra information provides us much more feasible ways to choose the mapping functions without any constraint.
where Δ f is the subcarrier spacing, $\Delta f=\frac{\mathit{\text{BW}}}{{N}_{T}}$, and BW is the original operation bandwidth.
2.3 The 3-stage decoder of the VTR-SISO-OFDM system
At the receiver, the received bit stream is reconstructed by the demodulation process in the SISO-OFDM system. Once the receiver obtains the estimated bit stream ${\stackrel{~}{\mathbf{x}}}^{T}$, the proposed decoder proceeds to process the data.
where ${f}^{-1}({\stackrel{\u030c}{\mathbf{w}}}^{T},{\mathbf{b}}^{T},{\Phi}_{1})=\{{f}^{-1}({\stackrel{\u030c}{w}}_{i},{b}_{i},{\Phi}_{1}),\forall i\}$ and ${g}^{-1}({\stackrel{\u030c}{\mathbf{k}}}^{T},{\mathbf{c}}^{T},{\Phi}_{2})=\{{g}^{-1}({\stackrel{\u030c}{k}}_{i},{c}_{i},{\Phi}_{2}),\forall i\}$. The entire decoding procedure is listed in Algorithm 1. In Algorithm 1, Dist{·} is the Hamming distance and s(i:j) means from the i th sample to the j th sample in s.
3 The VTR-MB-OFDM system
In this section, a VTR-MB-OFDM system is proposed using the multiterminal source coding scheme. The main goal of the proposed VTR-MB-OFDM system is to design a variable transmission rate system and to evaluate the system performance under different conditions.
Algorithm 1 The decoding algorithm for the 3-stage decoder
3.1 The proposed VTR-MB-OFDM system and transmission rate analysis
From Equations (30) and (33), when S _{1} and S _{2} are equal to one, the lengths of two encoded bit streams, X and Y, are identical; moreover, the allocated bandwidths of these two subbands, $B{W}_{{1}^{\mathit{\text{st}}}}$ and $B{W}_{{2}^{\mathit{\text{nd}}}}$, are also the same.
where n ^{ T } is the AWGN noise. First, the receiver performs the cyclic prefix removal on r ^{ T }, the S/P operation, and the N _{ T }-point FFT operation on r _{ S P }. After the FFT operation, symbols allocated in the specific subbands are extracted, and then demodulated by the corresponding signal demapper. ${\mathbf{x}}_{\text{demod}}^{T}$ is generated using the information from the first $\lceil \frac{X}{M}\rceil $ subcarriers, and ${\mathbf{y}}_{\text{demod}}^{T}$ is also generated using the information from the $(\lceil \frac{X}{M}\rceil +1)$th subcarrier to the $(\lceil \frac{X}{M}\rceil +\lceil \frac{Y}{M}\rceil )$th subcarrier. Then, ${\mathbf{x}}_{\mathit{\text{zr}}}^{T}$ is obtained by removing mod (M− mod (X,M),M) samples at the back of ${\mathbf{x}}_{\text{demod}}^{T}$, and ${\mathbf{y}}_{\mathit{\text{zr}}}^{T}$ is also obtained by removing mod (M − mod (Y,M),M) samples at the back of ${\mathbf{y}}_{\text{demod}}^{T}$. Let us use ${\mathbf{x}}_{\mathit{\text{zr}}}^{T}$ to explain the decoding process, and ${\mathbf{y}}_{\mathit{\text{zr}}}^{T}$ is decoded using the same procedure. X-decoder decodes ${\mathbf{x}}_{\mathit{\text{zr}}}^{T}$ by minimizing the Hamming distances between codewords in the codebook Q _{ x } and the partial bit stream in ${\mathbf{x}}_{\mathit{\text{zr}}}^{T}$ with or without help of the side information through the switch S _{3}, and then the estimated source data ${\widehat{\mathbf{a}}}^{T}$ is obtained. In Algorithm 2, we describe the overall decoding process used in X-decoder and Y-decoder.
4 Algorithm 2 The decoding algorithm for multiterminal source decoder of the proposed VTR-MB-OFDM system
4.1 Performance analysis and simulation results
We evaluate the performance of two proposed VTR-OFDM systems, including the VTR-SISO-OFDM system and the VTR-MB-OFDM system. Section 4.1 demonstrates the performance of the proposed VTR-SISO-OFDM system, and then the performance of the proposed VTR-MB-OFDM system is demonstrated in Section 4.2.
Performance evaluation of the VTR-SISO-OFDM system
where z is a random variable, z ∈ {1,2,…,Z}, and ${\sum}_{\lambda =1}^{Z}(Z-\lambda +1)$ is the total number of possible outcomes.
where ${N}_{b}={\sum}_{{j}_{1}=1}^{75}(75-{j}_{1}+1)=2850$, ${N}_{c}={\sum}_{{j}_{2}=1}^{50}(50-{j}_{2}+1)=1275$, and ${N}_{d}={\sum}_{{j}_{3}=1}^{D}(D-{j}_{3}+1)=325$ or 15.
Once the output symbols w ^{ T }, k ^{ T }, and e ^{ T } are generated, we apply arithmetic code to encode each symbol to the specific codeword using the corresponding codebook at each stage. Cardinalities of different codebooks Q _{ t } used to encode symbols at each stage are A × B, W × C, and E. Two bits for rate adjustment are added to the front of the output in order to construct the input bit stream. Then, we use a SISO-OFDM system with 16-QAM signal mapper/demapper, 8192-point IFFT/FFT (N _{ T }= 8192), and N _{ CP }= 256 to transmit the data over the AWGN channel. Assume the operating bandwidth of the communications system is 20 MHz, and each sample is generated by a sampling frequency 5 MHz. The raw transmission bit rate of this SISO-OFDM system is 19.4 Mbps. In addition, the subcarrier spacing, Δ f, is about 2.441 MHz.
System parameters of the proposed VTR-SISO-OFDM system
S _{1} | S _{2} | S _{3} | ||
---|---|---|---|---|
D = 25 | D = 5 | |||
Entropy | 7.4346 | 7.3323 | 11.7240 | 9.4815 |
Encoding rate(bits/sample),${R}_{\text{enc},{S}_{t}}$ | 9 | 8.8570 | 13.2461 | 11.0735 |
Averaged number of used subcarrier,$\u2308\frac{L\xb7{R}_{\text{enc},{S}_{t}}}{M}\u2309$ | 4501 | 4429 | 6624 | 5537 |
Approximately allocatedbandwidth (MHz.),$\u2308\frac{L\xb7{R}_{\text{enc},{S}_{t}}}{M}\u2309\xb7\Delta f$ | 11 | 10.8 | 16.2 | 13.5 |
Transmission bitrate (Mbps), ${R}_{b,{S}_{t}}=$ $\u2308\frac{L\xb7{R}_{\text{enc},{S}_{t}}}{M}\u2309\xb7\frac{{f}_{s}\xb7M}{{N}_{T}+{N}_{\mathit{\text{CP}}}}$ | 10.7 | 10.5 | 15.7 | 13.1 |
4.2 Performance evaluation of the VTR-MB-OFDM system
Assume 10000 packets are transmitted using the proposed VTR-MB-OFDM system and each packet contains 2000 correlated information sequences, a ^{ T } and b ^{ T }. After the correlated sequences are generated, we apply the multiterminal source coding to encode a ^{ T } and b ^{ T } with or without the side information from each other, and then these 2000 correlated sequences are transmitted using two different subbands in MB-OFDM system. Each subband is occupied by 1000 encoded messages. In addition, the bandwidths and the transmission rates of these two subbands are not necessarily the same.
where OCCUR (x) is the occurrence of x.
where γ is the maximum prime number in 2 · N and p (α,β) is the joint probability function on (α,β). Thus, ${\widehat{f}}_{1}(\xb7)$ and ${\widehat{f}}_{2}(\xb7)$ are injective functions. For the mapping function ${\widehat{f}}_{1}(\xb7)$, we choose N to be 10, because we can achieve better performance using smaller N from the simulation results in Section 4.1. Then, A and B are equal to 10, and the number of distinct possible outcomes (h _{ i },l _{ i }) is 17. After applying the mapping function ${\widehat{f}}_{1}(\xb7)$ to the input pair (a _{ i },b _{ i }) with different information interchanges, the outputs are p(a _{ i },S _{2}· b _{ i }) or p(S _{1}· a _{ i },b _{ i }). As for the mapping function ${\widehat{f}}_{2}(\xb7)$, N is equal to 50, γ is 97, and the outputs of the input pair (a _{ i },S _{2}· b _{ i }) or (S _{1}· a _{ i },b _{ i }) are mod (a _{ i }+ S _{2}· b _{ i },97) or mod (S _{1}· a _{ i }+ b _{ i },97). Therefore, the maximum output of the ${\widehat{f}}_{2}({a}_{i},{b}_{i})$ is 51. The reason we choose γ to be the maximum prime number in 2 · N instead of the maximum prime number in max(h _{ i }+ l _{ i }) is that messages can be decoded when a _{ i }> 47 or b _{ i }> 47 without any loss. In addition, the cardinalities of ${\widehat{f}}_{1}({a}_{i},{b}_{i})$ and ${\widehat{f}}_{2}({a}_{i},{b}_{i})$ are greater than the cardinalities of a ^{ T } and b ^{ T }. Thus, for these two mapping functions ${\widehat{f}}_{1}(\xb7)$ and ${\widehat{f}}_{2}(\xb7)$, the system operates in a higher transmission rate in order to fully utilize the available subcarriers.
Once the output symbols $\widehat{f}({\mathbf{a}}^{T},{S}_{2}\xb7{\mathbf{b}}^{T})$ and $\widehat{f}({S}_{1}\xb7{\mathbf{a}}^{T},{\mathbf{b}}^{T})$, are generated, we apply arithmetic code to encode each symbol to the specific codeword using the corresponding codebook, Q _{ x } and Q _{ y }, at each encoder. Cardinalities of these two codebooks used to encode symbols at each encoder are varying with respect to the mapping functions and N. For the mapping function ${\widehat{f}}_{1}(\xb7)$, the cardinality of each codebook is 27; however, the cardinality of codebooks for the mapping function ${\widehat{f}}_{2}(\xb7)$ with larger N is 51. Then, we use a MB-OFDM system with 16-QAM signal mapper/demapper, 4096-point IFFT/FFT (N _{ T }= 4096), and N _{ CP }= 256 to transmit the input bit stream over the AWGN channel. Assume the operating bandwidth of the communications system is 20 MHz, and each sample is generated by a sampling frequency 5 MHz. The raw transmission bit rate of this MB-OFDM system is 18.8 Mbps. In addition, the subcarrier spacing, Δ f, is about 4.882 MHz.
System parameters of the proposed VTR-MB-OFDM system
prob,${\widehat{\mathit{f}}}_{\mathbf{1}}\mathbf{(}\mathbf{\xb7}\mathbf{)}\mathbf{,}\phantom{\rule{1em}{0ex}}\mathit{N}\mathbf{=}\mathbf{10}$ | |||
---|---|---|---|
x[0] | y[0] | x[1]( y[1]) | |
Entropy | 3.1036 | 3.1036 | 3.8950 |
Encoding rate (bits/sample),${R}_{\text{enc}}={R}_{\text{enc},x\left[{S}_{2}\right]}$ or R _{enc,y [0]} | 4.5091 | 4.5091 | 5.3636 |
Averaged number of usedsubcarrier $\u2308\frac{L\xb7{R}_{\text{enc}}}{M}\u2309$ | 1128 | 1128 | 1341 |
Approximately allocatedbandwidth (MHz) $\u2308\frac{L\xb7{R}_{\text{enc}}}{M}\u2309\xb7\mathrm{\Delta f}$ | 5.5 | 5.5 | 6.5 |
Transmission bit rate (Mbps),${R}_{b}=\u2308\frac{L\xb7{R}_{\text{enc}}}{M}\u2309\xb7\frac{{f}_{s}\xb7M}{{N}_{T}+{N}_{\mathit{\text{CP}}}}$ | 5.18 | 5.18 | 6.16 |
modular, ${\widehat{\mathit{f}}}_{\mathbf{2}}\mathbf{(}\mathbf{\xb7}\mathbf{)}\mathbf{,}\phantom{\rule{1em}{0ex}}\mathit{N}\mathbf{=}\mathbf{50}$ | |||
x [0] | y [0] | x [1]( y [1]) | |
Entropy | 5.3790 | 5.3790 | 6.3344 |
Encoding rate (bits/sample),${R}_{\text{enc}}={R}_{\text{enc},x\left[{S}_{2}\right]}$ or R _{enc,y[0]} | 7.8200 | 7.8200 | 7.8200 |
Averaged number of usedsubcarrier $\u2308\frac{L\xb7{R}_{\text{enc}}}{M}\u2309$ | 1955 | 1955 | 1955 |
Approximately allocatedbandwidth (MHz) $\u2308\frac{L\xb7{R}_{\text{enc}}}{M}\u2309\xb7\mathrm{\Delta f}$ | 9.5 | 9.5 | 9.5 |
Transmission bit rate (Mbps),${R}_{b}=\u2308\frac{L\xb7{R}_{\text{enc}}}{M}\u2309\xb7\frac{{f}_{s}\xb7M}{{N}_{T}+{N}_{\mathit{\text{CP}}}}$ | 8.98 | 8.98 | 8.98 |
For [S _{1} S _{2}] = [0 0], the input pairs are mapped independently. Thus, decoders can decode the estimated bit stream correctly with or without the side information. In the case of [S _{3} S _{4}] ≠ [S _{1} S _{2}] for both mapping functions, decoders decode the estimated bit stream in a way different from the way that encoders generate these encoded bit stream. In addition, for the mapping function ${\widehat{f}}_{2}(\xb7)$, the reason we have the highest SER when [S _{1} S _{2}] = [1 1] is that the decoder can not map a single value back to the original input pair $\{{\xe2}_{i},{\widehat{b}}_{i}\}$ with or without the help of the side information.
5 Conclusion
In this article, two variable transmission rate OFDM systems using network coding schemes are proposed, and the performance of these two proposed systems is evaluated. For the proposed VTR-SISO-OFDM system, the transmission rate is adjusted by selecting the bit stream from the output at different stages with different mapping functions in the encoder; however, how an appropriate mapping function is chosen is important for the system performance. In addition, the cardinalities of the side information also affect the system performance. The larger the codebook size, the less gain in SNR to achieve the same performance as the uncoded communications system. Thus, a trade-off between higher transmission rate and better system performance is necessary. As for the proposed VTR-MB-OFDM system, correlated sources are simultaneously transmitted using different transmission rates. Different switch configurations generate different transmission rates at the transmitter and different system performance at the receiver. Compared with the traditional uncoded OFDM system, simulation results show the proposed VTR-SISO-OFDM system has at least 1 to 3 dB gain in SNR to achieve the same SER, and the proposed VTR-MB-OFDM system has at least 1 to 4 dB gain in SNR to achieve the same SER. In addition, for some conditions, zero SER is achievable for both proposed systems when SNR is greater than 19 dB. Moreover, the rate factors of these two proposed system vary from 0.55 to 0.96.
Declarations
Acknowledgements
The authors would like to thank Emeritus Professor John C. Kieffer and anonymous reviewers for their valuable advice which substantially improved this article.
Authors’ Affiliations
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