Abstract: This study proposes Orthogonal Frequency Division Multiplexing (OFDM), endorsed by encryption and concatenated Error Correction Codes (ECC). OFDM, as the name suggests exploits the carriers’ orthogonality properties and Frequency Division Multiplexing (FDM) caters to the bandwidth requirements for the broadband applications. To reduce the errors due to the channel noise conditions, concatenated codes utilizing Reed Solomon (RS) codes and convolutional codes are used. Its encoded output is encrypted using the chaotic means to espouse discretion before passing through OFDM channel. Error performance of the scheme is analysed using Additive White Gaussian Noise (AWGN) channel. Comparative graphs are plotted for the performance of Bit Error Rate (BER) on different modulation schemes with different inner and outer code rates. For performance measurements, metrics like BER and Correlation values are computed.
INTRODUCTION
The stupendous developments in wireless communication and its smart applications have caused the technology to encroach into variety of fields catering to the demands of its diverse customers. The ever growing demand mandates more and more efficient technology having efficient bandwidth utilisation, reduced errors etc. In order to ply these needs, OFDM was introduced. OFDM is a method in which large information rates over hostile channels are achieved by spreading the data over large number of orthogonal channels that are spaced apart at specific distance (Chang and Gibby, 1968; Chang, 1970; Saltzberg, 1967).
The orthogonality between the carriers allows the overlap of sub carriers and it also prevents the demodulators from seeing frequencies other than their own. This orthogonality is implemented in the symbols by the use of Fast Fourier Transform (FFT) and Inverse Fast Fourier Transform (IFFT) operations. FFT is similar to modulation at the transmitter end and IFFT corresponds to demodulation at the receiver end (Weinstein and Ebert, 1971; Van Nee and Prasad, 2000; Hwang et al., 2009; Peled and Ruiz, 1980). Cyclic Prefix (CP) is used in this technique is used to combat inter channel interference and inter symbol interference.
In the ever-growing wireless communication, transfer of bulk data is involved and hence safety is the primary concern. Error correction and detection is highly responsible for secure digital data transmission over hostile channel. Data in few channels are prone to noise, thus errors are induced during transmission from the transmitter and the receiver. Convolutional codes, cyclic codes, Turbo codes and RS codes are widely used to combat these errors.
Convolution coding comes under error-control coding and is a very special type in which the message bits are in a serial manner, rather than blocks. For this reason, it should possess a memory. Although a convolutional coder takes in a set of symbols and outputs predetermined code symbols, the calculation involved doesn’t depends on the symbols that is being used currently but also on some preceding inputs (Daneshgaran et al., 2004). Hence, a binary convolutional encoder consists of finite number of shift registers with prescribed connections to modulo-2 adders along with a multiplexer. These are used to serialize the outputs from the adders (Haque et al., 2008; Forney, 1970, 1973). The number of inputs in the encoder decides the constraint length. The number of bits in each shift register, inclusive of the current bit inputs is given by the vector elements. If k input bit streams are taken as the input by the encoder (i.e., it’s capable of receiving 2^k possible input symbols) and if it produces n output bit streams (i.e., capable of producing 2^n possible output symbols), then rate of the encoder can be calculated as n/k.
A type of linear block is the RS code and it was proposed in 1950. Its advantage is that its strong to burst errors whereas prone to random errors and so, in fading channel, its performance is good in which more burst errors are present. RS codes are cyclic correction, in accordance with coding theory. Both together brought out this which is capable of structuring codes that can sense a well as accurate numerous symbol errors. When‘t’ symbols are added to the information, then RS code can identify combinations up to ‘t’ incorrect symbols and accurate till ‘t/2’ symbols (Nyirongo et al., 2006; Li and Salehi, 2009). The RS Code can be represented by [n, k, n-k+1] code where n-k+1 represents the hamming distance, n represents the length and k represents the dimension.
Concatenation can be defined as a method by which long powerful codes can be built out of shorter ones. It breaks down computation into small segments which are manageable, hence reducing the decoding complexity. It’s derived from combining a outer code which is non-binary and a binary inner code. The concatenation of the two codes, CC with RS gives rise to benefits in BER performance as they have different characteristic in error handling.CC is adopted for channel induced random errors and is used to correct them and RS codes are designed to handle burst errors.
So, concatenation is preferred over single coding scheme as it reduces the overall error rate. Through concatenation, long codes are possible which can be decoded using two decoders which are suited for short codes generally. Hence, complexity can be reduced but performance has to be taken care of as the value of attainable error component plays a major role and we find that the value of attainable probability of error decreases exponentially with block lengths for all the rates which are less than the capacity of the channel.
The digital data sent over the wireless channel is an open source and is prone to hacking by the unauthorised users (Abbasi-Moghadam and Vakili, 2012; Payandeh et al., 2006). In an attempt to curb these interruptions, Steganography (Amirtharajan et al., 2012; Amirtharajan and Rayappan, 2012a-c, 2013; Amirtharajan et al., 2013a-c; Ramalingam et al., 2014; Janakiraman et al., 2012; Praveenkumar et al., 2012a, b; Praveenkumar et al., 2013a, b, c, d, e, f, g, h, i, j; Thanikaiselvan et al., 2012; Thanikaiselvan et al., 2013) Cryptography (Praveenkumar et al., 2012a; Rajagopalan et al., 2012; Schneier, 2007), Watermarking came in to existence. Encryption is a class of cryptography, which is done at the transmitter end and decryption at the receiver end is done to safe guard the data. Encryption involves scrambling the original data in an obscure format, thus rendering the cipher text. At the receiving end, the cipher text is broken with the help of a decryption key to recover the original data. In recent times, the chaotic maps are being employed in encryption algorithms (Amin et al., 2010; Belkhouche and Qidwai. 2003; Fridrich, 1997; Chen et al., 2004; Hassan et al., 2013). These maps are mainly based on the permutation and substitution architecture (Hsu, 2004; Mao and Chen, 2005; Wang et al., 2011; Huang and Nien. 2009; Xu et al., 2012; Huang et al., 2013; Borujeni and Eshghi, 2009) which will be repeated numerous times to give up the encrypted image.
Due to the terrific developments in digital communication over a decade, Internet has gone viral these days catering to demands of its diverse users. For sustained benefits, it is necessary that the channel must have broader bandwidth. Further, digital transmission suffers from channel fading and more over the data sent over wireless medium is open to all and the probability of the data being hacked is also more (Khan et al., 2007; Gligoroski et al., 2006).
An extensive study is done on the existing techniques in wireless and various encryption algorithms. This study mandates the devisal of a new method to enhance the efficient channel utilisation by OFDM, control the rate of errors using Concatenated codes and curb hacking by uniquely identifying the copyright of the data by encryption. Substantiation is specified in terms of widely recognized metrics like BER, SNR and correlation values. In the next Section proposed methodology is discussed and the followed Section signifies about the results and discussion. The last section conclusions are drawn.
METHODOLOGY
A linear bit stream of binary bits is given as inputs to the proposed system. Then data’s are encoded using the Forward Error Correction (FEC) codes earlier to transmission. FEC codes determine the ability of the receiver to correct and detect errors with no demand on retransmission, but at a fixed, privileged channel bandwidth. Here FEC used are RS codes as outer codes with code rate 1/7, 3/7 and 5/7, respectively in concatenated with convolutional encoder as inner code using code rate 1/2, 1/3 and 2/3, respectively as in Fig. 1. CP denotes cyclic Prefix and CDP relates to Cyclic De-prefix.
| Fig. 1: | Proposed block diagram |
RS codes as defined by Irving S Reed and Gustavae Solomon, the block length is specified by n = q-1 with message length of K and the distance between codes will be n-k+1.The size of the alphabet must be satisfy q = pm i.e., q must be prime power. RS code is represented by (n, k, n-k+1) and its code rate is given by R = k/n. RS decoder can correct up to t1 symbols in a codeword where 2t1 = n-k. Usually, message is represented using Galois field array. The number of bits in each symbol is considered to be 3 and n = 2 m-1. So, n = 7 and code word length taken into consideration are 1, 3, 5, respectively. So, the code rate used in the proposed system are 1/7, 3/7 and 5/7, respectively.
Contrasting a block coder, convolutional coding is a special case of error-control coding, which uses shift register as memory device. It outputs primarily depends not only on the present input code words but also on the preceding code symbols. It operates with predetermined input and output code words. Constraint length specifies the delay encountered by the encoder. Code generator matrix is a set of octal numbers which denotes the connections in the output of each encoder. The code rate is given by n/k where n represents the number of inputs and k represents the number of outputs. In this study, code rates of 1/2, 1/3 and 2/3 are used. The constraint length and the code generator matrix for the code rates used are given in Table 1.
Then the concatenated coded outputs are passed over chaotic encryption block.
Chaotic mapping: This is primarily used in secured transmission, the variables act as the encryption keys. In the encrypting process, diffusion process will be carried out and the first stage in the decryption process will be diffused decryption is done in order to get the original value using the three dimensional sequence which are complex.
| Table 1: | Constraint length and code generator matrix for the code rates used |
The initial conditions used in encryption are used as decryption key and same confusion key is used to get back the position of the data:
| • | For encryption, generate numbers randomly from 0 to 255 |
| • | Then XOR operation is done on the randomly generated bits with the transformation equation to produce new set of bits |
| • | The chaotic transformation is given by Eq. 1-2: |
| (1) |
| (2) |
where, u represents the bit location in a row and v represents the bit location in column:
| • | u(n+1) and v(n+1) represents the next bit locations in row and column, respectively |
| • | Then the horizontal, vertical and diagonal correlation values are computed to upgrade the encryption algorithm used |
Horizontal correlation: Read the binary input. For each row, correlation between adjacent pixel values are found out using the equation:
xi = c(i,j); yi = c(i, j+1); xi1 = xi1+xi; yi1 = yi1+yi
where, xi and yi are adjacent values. xi1 is the sum of all individual values. yi1 is the sum of all individual values adjacent to xi:
zi = xi*yi; zi1 = zi1+zi; xi2 = xi*xi; yi2 = yi*yi;
xi3 = xi3+xi2; yi3 = yi3+yi2
where, xi1, yi1, zi1, xi3, yi3 are initialized to zero. Now horizontal correlation is found by Eq. 3:
| (3) |
Where:
a1 = (n*zi1)-(xi1*yi1)
b1 = ((n*xi3)-(xi1*xi1))
b2 = ((n*yi3)-(yi1*yi1))
b3 = sqrt(double(b1*b2))
where, n is m*(n-1), m, n denotes the row number and column number, respectively.
Diagonal correlation: Read the binary input. Find the correlation between each value and its diagonal value by using:
xi = c(i, j); yi = c(i+1,j+1); xi1 = xi1+xi; yi1 = yi1+yi
where, xi and yi are diagonal element values. xi1 is the sum of all individual values and yi1 is the sum of all individual values diagonal to xi. Then diagonal values are multiplied and squared separately:
zi = xi*yi; zi1 = zi1+zi; xi2 = xi*xi; yi2 = yi*yi;
xi3 = xi3+xi2; yi3 = yi3+yi2
xi1, yi1, zi1, xi3, yi3 are initialized to zero. Now diagonal correlation is found by Eq. 4:
| (4) |
Where:
a1 = (n*zi1)-(xi1*yi1)
b1 = ((n*xi3)-(xi1*xi1))
b2 = ((n*yi3)-(yi1*yi1))
b3 = sqrt(double(b1*b2))
where, n is (m-1)*(n-1), m, n denotes the row number and column number, respectively.
Vertical correlation: Read the input binary. For each column find the correlation between each value with that in the next row. The values of adjacent rows in a column are given by:
xi = c(i, j); yi = c(i+1, j); xi1 = xi1+xi; yi1 = yi1+yi
where, xi and yi are diagonal values, xi1 is the sum of all individual values and yi1 is the sum of all individual values diagonal to xi. Then adjacent values are multiplied and squared separately:
zi = xi*yi; zi1 = zi1+zi; xi2 = xi*xi; yi2 = yi*yi;
xi3 = xi3+xi2; yi3 = yi3+yi2
where, xi1, yi1, zi1, xi3 and yi3 are initialized to zero. Now vertical correlation is found by Eq. 5:
| (5) |
where:
a1 = (n*zi1)-(xi1*yi1)
b1 = ((n*xi3)-(xi1*xi1))
b2 = ((n*yi3)-(yi1*yi1))
b3 = sqrt(double(b1*b2)
where, n is (m-1)*(n), m, n denotes the row number and column number, respectively.
Then the encrypted data bits are sent to the signal mapper block, which is a modulator that can modulate the encoded, encrypted data in to complex constellation points by making use of BPSK, QPSK, 8, 16, 32 and 64 QAM, respectively. Then, serial-to-parallel conversion takes place for grouping the bits to Inverse Fast Fourier Transform (IFFT). This is done to insert orthogonal sub-carriers into the symbols obtained by modulation, to obtain OFDM symbols. Cyclic prefix, also known as guard intervals of 1/4th of the total OFDM symbol are inserted to combat inter channel interference and inter symbol interference. Then digital data is scaled into analog form and is transmitted through AWGN channel.
The channels are the models of real-world phenomena like scattering, dispersion, fading etc. AWGN is used to add uniform white Gaussian noise. Then the received information is again converted back to digital form for further processing. Then the guard interval inserted earlier is removed. Then Fast Fourier Transform (FFT) operation is carried out to separate all the carriers of the OFDM signal and the parallel data stream is converted into serial bits, then chaotic decryption is done to de scramble the encrypted data bits. Depending on the type of code rate adopted in the outer and the inner concatenated codes, decoding is carried out. After the decoding, the original information bits can be retrieved. The output of the system can be compared with the input to check its accuracy and reliability.
RESULTS AND DISCUSSION
Sample input data bits of 18432 bits with RS codes as outer code and convolutional encoder as inner code and the encoded outputs are encrypted using chaotic means and then it is passed over OFDM system considering AWGN channel. BER graphs are plotted for various inner and outer code combinations using BPSK, QPSK, 8, 16 and 64 QAM.
The system is evaluated using convolutional codes of rate 1/2, 1/3 and 2/3, respectively without making use of concatenated codes. From the evaluation its observed that, for a BER of 10^-2, utilizing the code rate of 1/2, BPSK is 4.3 dB which is 9.28 dB improvement over QPSK which in turn provides an improvement of 5 dB with respect to 8 QAM, which provides an improvement of 9.28 dB with respect to 16 QAM, which overruns 32 QAM by 8 dB which is heightened by 7 dB with respect to 64 QAM which is analogous to code rates 1/3 and 2/3. Compared to all the three, convolutional encoder with code rate 1/3 is good at this regard.
Figure 2-4 shows the BER using outer code rate of 1/7 and inner code rates of 1/2, 1/3 and 2/3, respectively in OFDM systems.
| Fig. 2: | BER using code rates of 1/7 and 1/2 for RS and convolutional encoder, respectively in OFDM system |
| Fig. 3: | BER using code rates of 1/7 and 1/3 for RS and convolutional encoder, respectively in OFDM system |
| Fig. 4: | BER using code rates of 1/7 and 2/3 for RS and convolutional encoder, respectively in OFDM system |
| Fig. 5: | BER using code rates of 3/7 and 1/2 for RS and convolutional encoder, respectively in OFDM system |
From Fig. 2, BPSK is 2.45 dB which is 1.85 dB progressed over QPSK which in turn provides an improvement of 2.97 dB with respect to 8 QAM, which provides an improvement of 4.13 dB with respect to 16 QAM, which overruns 32 QAM by 4.6 dB which is heightened by 4.87 dB with respect to 64 QAM which is analogous to code rates 1/3 and 2/3. Compared to all the three, Concatenated codes with outer code rate of 1/7 and inner code rate 2/3 is providing better BER compared to other two.
As concatenation is used, for a BER of 10^-2, BPSK provides 2.45 dB and without concatenation its about 4.3 dB, results in an enhancement in Eb/No of 2 dB. For the same BER, if 64 QAM is considered without concatenation Eb/No is about 42.3 dB and with concatenation its about 20.876 providing an improvement of about 21dB.
Figure 5-7 shows the BER using outer code rate of 3/7 and inner code rate of ½, 1/3 and 2/3, respectively in OFDM systems.
For the outer code rate of 3/7 and inner code rate of 1/2 from Fig. 5, BPSK is 2.82 dB which is 1.42 dB improvement over QPSK which in turn provides an improvement of 3.5 dB with respect to 8 QAM, which provides an improvement of 3.6 dB with respect to 16 QAM, which overruns 32 QAM by 5.1 dB which is heightened by 5.2 dB with respect to 64 QAM which is analogous to code rates 1/3 and 2/3.
| Fig. 6: | BER using code rates of 3/7 and 1/3 for RS and convolutional encoder, respectively in OFDM system |
| Fig. 7: | BER using code rates of 3/7 and 2/3 for RS and convolutional encoder, respectively in OFDM system |
Compared to all the three, Concatenated codes with outer code rate of 3/7 and inner code rate 1/3 is providing superior BER.
Figure 8-10 shows the BER using outer code rate of 5/7 and inner code rate of 1/2, 1/3 and 2/3, respectively in OFDM system. For the outer code rate of 3/7 and inner code rate of 1/2, BPSK is 2.82 dB which is 1.48 dB improvement over QPSK which in turn provides an improvement of 2.98 dB with respect to 8 QAM, which provides an improvement of 4.1 dB with respect to 16 QAM, which overruns 32 QAM by 5.2 dB which is heightened by 5.1 dB with respect to 64 QAM which is analogous to code rates 1/3 and 2/3. Compared to all the three, Concatenated codes with outer code rate of 5/7 and inner code rate 1/3 is providing better BER compared to other two. Compared to all the concatenated codes discussed above, outer code rate of 1/7 and inner code rate of 1/3 is ideal.
Table 2 provides the number of errors and BER adopting inner codes 1/2, 1/3 and 2/3, respectively with outer code rate of 1/7 using various modulation schemes after chaotic encryption.
| Fig. 8: | BER using code rates of 5/7 and 1/2 for RS and convolutional encoder, respectively in OFDM system |
| Fig. 9: | BER using code rates of 5/7 and 1/3 for RS and convolutional encoder, respectively in OFDM system |
| Table 2: | Number of errors and BER for inner codes 1/2, 1/3 and 2/3, respectively with outer code rate of 1/7 after chaotic encryption |
Number of errors is the difference between the original and the encrypted image. As the number of errors increases, there exist no similarity between the original and the encrypted data. From Table 1, for code rate of 1/3, the number of errors is more indicates that there is no similarity. BER is a validation which assess about the error formed in the methodology between original and encrypted data. For the entire code rate used, BER is more than 0.5 which signifies that almost 50% error rate thereby declaring the encrypted output is extremely stochastic.
| Fig. 10: | BER using code rates of 5/7 and 2/3 for RS and convolutional encoder, respectively in OFDM system |
| Table 3: | Correlation values and SNR for inner codes 1/2, 1/3 and 2/3, respectively with outer code rate of 1/7 after chaotic encryption |
Pixel correlation is one of the important parameters for consideration of encryption. Every encryption algorithm aims at reducing this correlation to literally null value to justify the performance. Correlation value of 1 illustrate that high correlation exists among bits and 0 indicates that there is no correlation between the original and the encrypted data bits. If the value is 0, then the methodology offers only obscurity and uncertainty. The correlation coefficient is given by Eq. 6:
| (6) |
σj, σk standard deviations of j and k, respectively
Covariance (j, k) = Covariance of j and k
Covariance (j, k) is expressed as Eq. 7:
| (7) |
Here, μj and μk are j and k’s mean, respectively.
Table 3 provides the correlation and SNR values adopting inner codes 1/2, 1/3 and 2/3, respectively with outer code rate of 1/7 using various modulation schemes after chaotic encryption. SNR is the metric to confirm the superiority of the encrypted data. For 2/3 as inner code rate SNR deteriorates for the other two code rates the values are satisfactory. The pixel correlation to a zero which is what most needed in an encryption routine. This is so because, the lowest the correlation, then highest is the obscurity and in turn protect against security. Code rate of 1/3 for various modulation schemes provides almost zero correlation values and 32 QAM provides least correlation value of -4.5209e-004 adopting 1/3 inner code rate.
CONCLUSION
OFDM makes the most of orthogonality that exist between carriers which is the most substantive requisite for broadband applications as they oblige privileged bandwidth. Here, the concatenated codes of RS codes blended with convolutional encoders with diverse code rates are utilised to prevent channel noise. Then, by chaotic means encryption of concatenated outputs are done which provide robustness and then it’s transmitted over OFDM system. Graphs have been plotted for BER amongst the various modulation schemes adopting concatenated codes having assorted inner and outer code rates. From the BER graphs, concatenated codes provides better BER compared to the one without concatenation. The outer code rate of 1/7 and inner code rate of 1/3 is best suited when compared to all. The methods proposed here provide strong proof in terms of better and satisfactory encryption metrics such as BER, correlation values and SNR. The 32 QAM has the least correlation value of -4.5209e-004 using 1/3 as inner code rate. OFDM when mixed with encryption will be suitable solution to all the wireless threats and problems.