Direct-modulated multiple formats optical OFDM Discrete Wavelet Transform

frequency domain channel equalizer for discrete Wavelet Transform based OFDM systems Discrete wavelet transform based OFDM-IDMA system with AWGN channel
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Published Date:28-11-2017
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DISCRETE WAVELET TRANSFORM BASED OFDM SYSTEM USING CONVOLUTIONAL ENCODING A Dissertation submitted in partial fulfilment of the requirements for the award of Degree of MASTER OF ENGINEERING In WIRELESS COMMUNICATION Submitted By: KARANPREET KAUR Roll No. 801263013 Under the guidance of: ANKUSH KANSAL Assistant Professor, ECED Thapar University, Patiala Department of Electronics and Communication Engineering THAPAR UNIVERSITY, PATIALA July-2014 ABSTRACT The rapidly growing technology has made it possible for the communication systems to transfer data almost everywhere on this planet. But the limited bandwidth allocated to a large number of users restricts the bandwidth availability to the users. This scenario creates a technological challenge to develop the data transmission schemes which are bandwidth efficient. Multicarrier modulation is such a scheme that transmits the data by dividing the serial high data rate streams into a number of low data rate parallel data streams. Orthogonal Frequency Division Multiplexing (OFDM) is a kind of multi-carrier modulation, which divides the available spectrum into a number of parallel subcarriers and each subcarrier is then modulated by a low rate data stream at different carrier frequency. The conventional OFDM systems make use of IFFT and FFT at the transmitter and receiver respectively but DWT-OFDM is an alternative approach to this conventional FFT based OFDM system. Discrete Wavelet Transform (DWT) is broadly considered as an efficient approach to replace FFT in the conventional OFDM systems due to its better time-frequency localization, bit error rate improvement, interference minimization, improvement in bandwidth efficiency and many more advantages. Moreover, Convolutional codes are used in DWT based OFDM system which improves the bit error rate performance of the system. In communication systems, when the signal is transmitted over the channel, noise and unwanted interferences are introduced which leads to the distortion of transmitted signal. Hence, error control coding techniques are used to mitigate the effect of such channel distortions. The original data sequence is appended with redundant bits to increase the reliability of the system by adding cyclic prefix; which also answers the problem of ISI. DWT is an effective tool to study the signals in time frequency joint domain as it is capable of providing simultaneous information about time and frequency, thus gives the time frequency representation of signal. Wavelet based OFDM is employed in order to remove the use of cyclic prefix which decreases the bandwidth wastage and the transmission power is also reduced. The BER performance of the OFDM system had been significantly improved -2.9 by 4 dB at BER of 10 when DWT was used in place of conventional FFT method. Afterwards, all the wavelets were compared to find the optimum wavelet among all. The results achieved had shown that the wavelet „bior5.5‟ outperformed all the other wavelets as viii well as FFT-OFDM system because it makes use of two wavelets, one for decomposition and the other for reconstruction instead of the single one. Finally, the performance of DWT OFDM system using Convolutional codes and without encoding is compared under AWGN as well as Rayleigh channel. The results show that there -2.92 is an improvement of 2.5 dB at BER of 10 when AWGN channel is used. In case of -4.9 Rayleigh channel, an improvement of 3.5 dB has been achieved at BER of 10 . It is because the Convolutional encoding is very effective in removing the burst errors and distortions introduced by the channel. Moreover, the BER performance of a system is affected by the outage probability which occurs when the required data rate is not supported by the specific channel due to variable SNR. Convolutional encoding reduces the outage probability at higher SNR. Thus, DWT based OFDM with encoding performs significantly better at higher SNR. ix CHAPTER 1 INTRODUCTION 1.1 OFDM With the rapid growth in technology, the demand for flexible high data-rate services has also increased. The performance of high data rates communication systems is limited by frequency selective multipath fading which results in intersymbol interference (ISI). In the wireless channels, impairments such as fading, shadowing and interferences due to multiple user access highly degrade the system performance. Multicarrier modulation (MCM) is a solution that overcomes these problems in wireless channels. It is the technique of transmitting data that divides the serial high data rate streams into a large number of low data rate parallel data streams 1. Orthogonal Frequency Division Multiplexing (OFDM) is a kind of multi-carrier modulation, which divides the available spectrum into a number of parallel subcarriers and each subcarrier is then modulated by a low rate data stream at different carrier frequency. The conventional OFDM system makes use of IFFT and FFT for multiplexing the signals and reduces the complexity at both transmitter and receiver 2. OFDM is comprised of a blend of modulation and multiplexing. The original data signal is split into many independent signals, each of which is modulated at a different frequency and then these independent signals are multiplexed to create an OFDM carrier. As all the subcarriers are orthogonal to each other, they can be transmitted simultaneously over the same bandwidth without any interference which is an important advantage of OFDM 3. OFDM makes the high speed data streams robust against the radio channel impairments. OFDM is an efficient technique to handle large data rates in the multipath fading environment which causes ISI. With the help of OFDM, a large number of overlapping narrowband subcarriers, which are orthogonal to each other, are transmitted parallel within the available transmission bandwidth. Thus, in OFDM, the available spectrum is utilized efficiently. Fig. 1 Traditional vs. OFDM 1.1.1 Significance With the rapidly growing technology, the demands for high speed data transmission are also increasing. OFDM is a multicarrier modulation technique which has the capability to fulfil this demand for large capacity. OFDM is reliable and economical to handle the processing power of digital signal processors. OFDM is used in many applications such as IEEE 802.11 wireless standard, Cellular radios, GSTN (General Switched Telephone Network), DAB (Digital Audio Broadcasting), DVB-T (Terrestrial Digital Video Broadcasting) 3, HDTV broadcasting, DSL 4 and ADSL modems and HIPERLAN type II (High Performance Local Area Network) 4. 1.1.2 Orthogonality Orthogonality is the property when the signals are mutually independent of each other 5. When the information signals are orthogonal, they can be transmitted simultaneously over the common channel without any interference. In case of loss of orthogonality, the performance of the communication system suffers degradation. The general multiplexing schemes are essentially orthogonal. In Time Division Multiplexing, multiple information signals are transmitted over the single channel but on unique time slots. Only one signal is transmitted during each slot to prevent any kind of interference among different information signals which makes the TDM system orthogonal in nature. Similarly, Frequency Division Multiplexing systems are orthogonal in frequency domain. In OFDM signal, all the subcarriers are well spaced out in frequency to maintain orthogonality among them 6. Orthogonality is achieved by allocating a different subcarrier to each information signal. All the subcarriers are made orthogonal to each other as the baseband frequency of each sub carrier is chosen in such a way that it is an integral multiple of inverse of the symbol period. Orthogonal Frequency Division Multiplexing (OFDM) is a kind of multi-carrier modulation, which divides the available spectrum into a number of parallel subcarriers and each subcarrier is then modulated by a low rate data stream at different carrier frequency which is given in 7 as equation (1.1): T cos(2 f t)cos 2 f t dt n m (1.1)  nm  0  nm where  is the Dirac Delta function. 2 The subcarrier frequency f is defined as given in equation (1.2), n f  n f (1.2) n f 1 s f (1.3) N NT f Here, is the entire bandwidth and 𝑁 is the number of subcarriers. s These subcarriers become orthogonal to each other when two different subcarrier waveforms are multiplied and integrated over symbol period results into zero. 1.1.3 Protection against ISI Intersymbol interference is a very common problem found in the existing high speed communication systems. It occurs when the transmitted data interferes with itself and is not decoded correctly at the receiver. The transmitted signal gets reflected from many large objects; hence more than one copy of the signal reaches the receiver which is known as multipath. These copies of signal reach at the receiver after some delay and interfere with the original signal resulting into ISI 7. Intersymbol interference can be classified into two types: interchannel ISI and intrachannel ISI. The non-ideal sub band filters leaks the signals from one channel to the adjacent channels which results into interchannel ISI, while intrachannel ISI results from channel dispersion 8. Interchannel ISI is also known as adjacent channel interference or crosstalk. These two are defined as follows: A. Intrachannel ISI It is very easy to mathematically model intrachannel ISI. Consider a channel of length L 9, LL  11 y h x h x h x n k n k 0 n k n k (1.4) kk  01  ISIterm If the channel length is known, ISI can simply be eliminated by upsampling the signal by L before it is passed through the channel and on the receiver side, it is downsampled by L before modulation. But in this case, it is difficult to estimate the length of channel. Moreover, upsampling leads to high bit rate which increases transmission bandwidth requirements. Therefore, this technique is useful only for very short channels in which equalization is easier to implement. Thus, equalization comes out to be the only solution for removing intrachannel interference. 3 B. Interchannel IS1 Interchannel interference or the adjacent channel interference results from the non-ideal sub band filters. Thus, it can be reduced significantly by the use of filters having high spectral containment or simply high stop band attenuation. For this purpose, the long sub band filters are used which cause longer processing delays. Due to the presence of non-ideal sub band filters, every sub channel obtains a random unrelated contribution involving variable amplitudes from all other sub bands. AWGN model is used to approximate these contributions. If the channel is ideal, interchannel ISI does not have any effect due to orthogonality 9. OFDM is very robust against ISI which makes it suitable for the high speed communication systems. When the data transfer speed is increased in the communication systems, the time duration for every single transmission gets shorter. But the delay time caused by multipath makes ISI a limitation in high data rate transmission. OFDM is the solution to this problem as it sends many low speed transmissions simultaneously. Adding a guard interval eliminates the effect of ISI but the guard period must be longer than the delay spread of channel. The remaining effects including amplitude scaling and phase rotation, resulting from ISI are corrected by the channel equalization. 1.1.4 Advantages 1. OFDM allows simultaneous transmission of subcarriers over a common channel thus making the efficient use of the available spectrum. 2. OFDM divides the frequency channel into many narrowband flat fading sub-channels which makes it more resistant to frequency selective fading. 3. OFDM makes use of cyclic prefix which helps in eliminating ISI and ICI. 4. If the symbols are lost due to frequency selectivity of channel, they can be recovered using appropriate channel coding and interleaving. 5. Channel equalization is potentially simpler as compared to the adaptive equalization techniques used in single carrier systems. 6. Maximum likelihood decoding can also be used in OFDM with less complexity. 7. OFDM is comparatively less sensitive to the timing offsets than single carrier systems. 8. FFT and IFFT are used in OFDM for modulation and demodulation in place of arrays of sinusoidal generators which makes it computationally efficient and cost effective. 4 9. It provides better protection against co-channel interference as well as impulsive parasitic noise. 10. OFDM randomizes the burst errors effectively which are caused due to fading and allows proper reconstruction even without forward error correction. 1.1.5 Disadvantages 1. The amplitude of OFDM signal has a very large dynamic range. Therefore, RF power amplifiers are required which possess high peak to average power ratio (PAPR). 2. OFDM systems are more sensitive to the carrier frequency offset and drift as compared to the single carrier systems. 1.2 CONVENTIONAL OFDM SYSTEM USING FFT Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier modulation technique in which the spectrum of the subcarriers overlap on each other. The frequency spacing among them is selected in such a way that orthogonality is achieved among the subcarriers. The block diagram of a basic OFDM system is shown in Fig. 2. ADD CYCLIC IFFT MODULATOR S/P PREFIX CHANNEL CYCLIC DEMODULATOR P/S FFT PREFIX REMOVAL Figure 1.2: OFDM trans-receiver The inverse transform block can be implemented using either IDFT or IFFT, and forward transform using DFT or FFT. The data generator first generates a serial random data bits stream 10. This serial data stream carrying information is grouped into bits/word according to the modulation scheme used and then each word is converted into parallel bit stream. Each bit stream is used to modulate one of the N orthogonal subcarriers 10. The data is processed using modulator to map the input data into symbols based on the modulation technique used. These symbols are now passed through IFFT block to perform IFFT operation to generate N parallel data streams. Its output in discrete time domain is given by equation (1.5), 5 j2 ni N1 1 N X () n X i e  (1.5) km  N i0 Then the cyclic prefix is added to this transformed data in order to alleviate the ISI effect. Cyclic prefix is usually the last 25% part of the original symbol. The next step is to pass this data through a channel. The channel model can be AWGN, Rayleigh or any other channel. To generate an OFDM symbol, the channel encoding of serial data stream is done followed by modulating the symbol using any modulation scheme. Before modulating the signal using IFFT, these serial data symbols are converted into N parallel data constellation points where N is number of IFFT points. After processing the data through IFFT block, the time domain OFDM modulated symbol is converted back to a serial stream and cyclic prefix which acts as guard interval is added to each OFDM symbol. The basic rectangular pulse shape for the symbols is considered as they have large bandwidth due to its sinc shaped spectrum. Thus, windowing is essential to reduce the out of band energy of the side lobes. Then the symbol stream is converted to analog form for pass-band processing and transmission. The receiver performs the exact opposite of the transmitter 11. To successfully generate OFDM, the relationship among all the carriers must be controlled carefully to sustain the orthogonality of the carriers. Due to this, OFDM symbol is generated choosing the spectrum required firstly, based on the input data, and modulation scheme used. Some data is assigned to each carrier to be produced to transmit. The required amplitude and phase of the carrier is then calculated based on the modulation scheme which is typically differential BPSK, QPSK, or QAM. The multiple orthogonal subcarriers generated, which are overlapped in spectrum mathematically resemble to N-point IDFT of the transmitted symbols. In practice, DFT and IDFT processes are useful for implementing these orthogonal signals. So to generate an OFDM symbol IDFT of modulated signal is computed. Fast Fourier transform and inverse Fast Fourier transform can be used to implement DFT and IDFT, respectively 12. To generate orthogonal sub-carriers, 𝑁 -point IFFT is applied to the transmitted symbols to generate the sum of 𝑁 orthogonal subcarrier signals. Thus an OFDM symbol is generated by computing the IDFT of the complex modulation symbols to be conveyed in each sub-channel. The transformed output is now added with a cyclic prefix before transmission which acts as a guard interval and it is added in order to alleviate ISI effect. It is usually last 25% part of the original OFDM symbol. This symbol is passed through AWGN channel. At the receiver, the exact reverse operation is carried out to obtain the original data back. The Cyclic prefix is 6 eliminated and the data is processed in the FFT block. Finally, it is passed through the demodulator to recover original data. The output of the FFT in frequency domain is given by, j2 ni N1  N UU (i) (n)e (1.6) mk n0 The receiver receives the data after passing through channel which gets corrupted by additive noise. Then, the N-point FFT of the received symbols is taken to obtain the noisy version of transmitted symbols at the receiver. Since all the subcarriers are having finite duration, the spectrum of OFDM signal can also be represented as the summation of the frequency shifted sinc functions taken in the frequency domain. 1.2.1 Mathematical Analysis of FFT-OFDM Consider the transmission system as shown in Figure 1.3. The shape of transmitted spectra is selected so that ICI does not occur; i.e., the spectra of all the individual subchannels are zero at the reamaining subcarrier frequencies. The 𝑁 subcarrier frequencies are modulated by 𝑁 1 serial data elements which are spaced by ∆𝑡 = , where 𝑓 is the symbol rate, and are then 𝑠 𝑓 frequency division multiplexed. To make the system less vulnerable to delay spread impairments, the signalling interval 𝑇 is increased to 𝑁 ∆𝑡 . Moreover, the subcarrier frequencies are chosen such that they are separated by multiples of 1/T which results into zero signal distortion in transmission. If the coherent detection of a signal element is carried out in any one subchannel of the parallel system, it does not give any output for a received element in any other subchannel. The data symbols 𝑑 (𝑛 ) can, be represented as 𝑎 (𝑛 ) + 𝑗 𝑏 (𝑛 ) where 𝑎 (𝑛 ) and 𝑏 (𝑛 ) are real sequences which represent the in-phase and quadrature components respectively and the transmitted waveform can be represented as 5 N1 Dt ancoscos t bnsinsin t (1.7)   nn n0 where 𝑓 = 𝑓 +𝑛 ∆𝑓 and ∆𝑓 = 1/𝑁 ∆𝑡 . The pulse shaping can also be included in place of 𝑛 0 basic rectangular shape to extend the above expression. 7 cos𝜔 𝑡 0 a(0) SERIAL SERIAL sin𝜔 𝑡 0 TO MUX b(0) STREAM 𝑑 𝑛 = 𝑎 𝑛 + 𝑗 𝑏 (𝑛 ) . DATA PARALLEL cos𝜔 𝑡 ENCODER 𝑁 −1 CHANNEL . CONVERTE a(N-1) R b(N-1) sin𝜔 𝑡 𝑁 −1 Fig. 1.3 OFDM Transmitter 5 cos𝜔 𝑡 0 𝑎 (0) PARALLEL 𝑏 (0) DATA TO 𝑑 (𝑛 ) sin𝜔 𝑡 0 cos𝜔 𝑡 𝑁 −1 ENCODER SERIAL CHANNEL 𝑎 (𝑁 − 1) CONVERTER 𝑏 (𝑁 − 1) sin𝜔 𝑡 𝑁 −1 Figure 1.4: OFDM receiver 5 Theoretically, bandwidth efficiency can be achieved by using M-ary digital modulation schemes in OFDM which is defined as bit rate per unit bandwidth, of log 𝑀 bits/s/Hz. If the 2 symbol rate of the serial data stream is given to be 1/∆𝑡 , the bit rate for a corresponding M- ary system is log 𝑀 /∆𝑡 . However, every subchannel transmits at a much lower rate, 2 log 𝑀 /(𝑁 ∆𝑡 ). 2 The total bandwidth of the OFDM system is 5: B f f 2 (1.8) N10 th where 𝑓 is the 𝑛 subcarrier and 𝛿 is the one-sided bandwidth of the sub channel. The 𝑛 subcarriers are uniformly spaced so that 𝑓 −𝑓 = 𝑁 − 1 ∆𝑓 . Since ∆𝑓 = 1/𝑁 ∆𝑡 , due to 𝑁 −1 0 11  the orthogonality constraint, ff (1 . N10 Nt  8 Therefore, the bandwidth efficiency 𝛽 becomes 5: log M 2 (1.9)  1  12t  N  For strictly band-limited spectra having bandwidth ∆𝑓 and orthogonal frequency spacing with 1 𝛿 = ∆𝑓 = 1/2𝑁 ∆𝑡 , 𝛽 = log 𝑀 bits/s/Hz. But the spectra overflow the minimum 2 2 bandwidth by the factor 𝛼 so that 𝛿 = 1 +𝛼 (1/2𝑁 ∆𝑡 ) and the efficiency given by equation (1.9) becomes 5: log M 2  log M (1.10) 2  1 N To attain the highest bandwidth efficiency in an OFDM system, N must be large and 𝛼 must be small. 1.2.2 Implementation of OFDM system using DFT The main drawbacks of using the parallel systems are the hardware complexity required for implementation of the system, and chances of severe mutual interference among the subchannels when the transmission channel distorts the signal. The hardware complexity including filters, modulators, etc. can be minimized by removing any pulse shaping, and by using the DFT to implement the modulation processes 13. The multitone data signal can be effectively implemented by taking the FFT of the original data stream and IFFT can be used to replace the bank of coherent demodulators. The equation (1.7) can be rewritten as: N1  jt n D t Re d n e (1.11)     n0 Assuming 𝑡 = 𝑚 ∆𝑡 , the resultant sampled sequence 𝐷 𝑚 can also be considered as the real part of the discrete Fourier transform of the data sequence 𝑑 (𝑛 ). When the signal is truncated to the interval (0,𝑁 ∆𝑡 ), a frequency response of 𝑠 𝑖 𝑛 𝑥 /𝑥 is achieved on each sub channel having zeros at multiples of 1/𝑇 . However, such spectral shape leads to large side lobes and gives rise to significant interchannel interference in the presence of multipath. The complexity can further be reduced by using the FFT algorithm to implement the DFT if N is large. If the transmission channel is distortion less, the transmitted signals will be received at the receiver without any error due to the orthogonality of the subcarriers. Consider the transmission system in Fig. 1.4. The data to be transmitted is represented by the sequence of 9 𝑁 complex numbers 𝑑 0 ,𝑑 1 ,… , 𝑑 𝑁 − 1 . The data encoder generates these complex numbers from a binary data sequence. Then, DFT is performed on the block of data sequence which generates the transmitted symbols 14 N1  j2/ Nnm D m DFT d n d n e (1.12)   n0 COCHANNEL SERIAL INTERFERER DATA FFT CHANNEL STREAM S/P d(n) D(m) ENCODER R(m) -1 PHASE R(m) DATA 𝐷 (m) P/S FFT 𝑑 (n) CORRECTION DECODER Figure 1.5: OFDM implemented using FFT 5 As a distortionless channel has been assumed, the data sequence received at the output of the IDFT is exactly same as the transmitted data sequence because of the orthogonality among the subcarrier exponentials. The distortions caused due to the transmission medium lead to the impairments in orthogonality. In a non frequency selective flat Rayleigh fading environment, the Rayleigh channel effects can be characterized as the transmitted signal processed multiplicative noise process. This multiplicative process can be represented by a 𝑗 𝜃 (𝑚 ) complex fading envelope having samples 𝑍 (𝑚 ) = 𝐴 (𝑚 )𝑒 where 𝐴 𝑚 are the Rayleigh distribution samples and the 𝜃 (𝑚 ) are uniform distribution samples. The sequence of equation (1.12) is multiplied by these samples to get equation (1.13), R m Z m D m  (1.13) The output data sequence denoted by 𝑑 (𝑘 ) is the IDFT of equation (1.13), N1 1 j2/ N km   d k Z m D m e   N m0  2 NN  11  j m kn   1 N   d n Z m e   N nm  00  10 N1  d n z k n (1.14)   n0 where 𝑧 (𝑛 ) is the IDFT of 𝑍 𝑚 . Clearly seen from equation (1.14), 𝑑 (𝑘 ) is represented as the complex-weighted average of the samples of a complex fading envelope. In the distortion less channel, if 𝑍 𝑚 = 1 for all 𝑚 , 𝑧 (𝑘 −𝑛 ) is the Kronecker delta function given by 𝛿 and 𝑑 𝑘 = 𝑑 (𝑘 ). In the presence of fading, 𝑧 (𝑘 −𝑛 ) ≠ 𝛿 and 𝑘 𝑛 𝑘 𝑛 N1  d k d k z 0 d n z k n (1.15)   n0 nk N1 The second term represents the intersymbol interference caused due to the dn zk n  n0 nk loss of orthogonality among subcarriers. If the fading is not corrected, the output sequence gets corrupted by ISI even in the absence of any co-channel interferer. The solution is to use pilot based correction which provides amplitude and phase reference and it also neutralizes the unwanted effects due to multipath propagation. By definition, in a fading environment coherent detection requires a phase reference but gain correction is also required in an OFDM system to mitigate the intersymbol interference. If the phase correction and gain correction is used in the absence of co-channel interference, in  d k d k equation (1.15), it can be easily shown that  In a cellular mobile system, if the other users are using same carrier frequency, it leads to the dominant transmission impairment assuming the desired signal and undesired co-channel interferer from single user is received simultaneously. It is also assumed that both the signals are modulated by different data sequences having the same signalling rates and are subjected under mutually independent Rayleigh fading 14. Due to the need of enhancing the energy of the cochannel interferer during deep fading of the desired signal, it is not beneficial to use unlimited gain correction in the absence of a cochannel interferer in the received signal. If the unlimited gain correction is done in the presence of a cochannel interferer, it leads to the detrimental effects shown as follows. Let the desired transmitted signal sequence be 𝐷 (𝑚 ) and 𝐼 (𝑚 ) be the cochannel interferer sequence. With the desired fading sequences and interferer complex fading sequences as jm jm d i Z m A m e Z m A m e  and  respectively, the received sequence can dd ii be represented as 11 R m Z m D m Z m I m (1.16)  di −1 where  is the interference-to-signal power ratio 𝑆 𝐼 𝑅 . R(m) is corrected by a complex Z m Z m correction sequence  the complex pilot fading envelope giving cp Rm Zm Zm di  D m D m I m (1.17)  Z m Z m Z m  c p p Taking the IDFT of equation (1.18), the received data sequence becomes N1 Zm  j2/ N km i  d k d n z k n I m e (1.18)   Zm  n0 p where  2 N1 j mkn Zm  1 d N  z k n e   N Z m  m0 p Z m Z m If the unlimited gain correction and phase correction is done, i.e., , pd z k n   , there is no ISI, and equation (1.18) becomes kn N1 Zm 1 i j2/ Nkm  d k d k I m e (1.19)   N Z m  m0 d Here, the cochannel interferer is the only reason for distortion. As 𝑍 𝑚 and 𝑍 𝑚 are 𝑖 𝑑 statistically independent, the average energy of the interferer may get boosted than the desired signal by unlimited gain correction if the desired signal is under fade but the interferer is not 15. A substitute to the unlimited gain correction and phase correction is by having a limited gain correction but the desired signal into deep fades is not followed. However, if the correction of the desired signal is not done perfectly, it leads to increased intersymbol interference. The correction signal in such situation is given in the form of equation (1.20) 15, 𝑗 𝜃 (𝑚 ) 𝑑 𝐴 𝑚 𝑒 𝑤 𝑕 𝑒 𝑛 𝐴 𝑚 𝜖 𝑑 𝑑 𝑍 𝑚 = (1.20) 𝑐 𝑗 𝜃 (𝑚 ) 𝑑 𝜖 𝑒 𝑤 𝑕 𝑒 𝑛 𝐴 𝑚 ≤ 𝜖 𝑑 where 𝜖 is the gain limit. The gain limit is defined in terms of the average value of local field strength. Therefore, if z k n in equation (1.19), it results in intersymbol interference  kn 12 (ISI). As a result, there is a trade-off between the intersymbol interference and energy of the cochannel interference. Moreover, an optimum gain correction factor can be developed as another alternative which considers both distortion effects. The optimum gain correction factor given by 𝐹 (𝑚 ) can be derived by minimizing the mean square distortion between 𝐷 (𝑚 ) and 𝐷 𝑚 . The correction sequence in equation (1.20) becomes, Z m Z m F m  cp 2   Am  i   Zm 1  (1.21)  d  Am  d   It would be more difficult to realize the correction procedure as compared to the gain limiting procedure. The frequency selective fading can also be present in addition to the distortions caused by ISI and cochannel interference. This makes the received signal envelopes decorrelated at different frequencies which reduce the effectiveness of the pilot correction procedure as there is possibility that a data point which is being corrected may be decorrelated from the corresponding pilot complex fading envelope. Eventually, OFDM technique has a major advantage of averaging the impairments which makes the bursty Rayleigh channel less bursty 14. The correlation among the samples of the complex fading envelope decides the extent to which the averaging process approaches a Gaussian channel. Also, more the value of N, more the independent fades are averaged which enables the randomization of the burst errors thus helps in bit error correction. The simulation results make this property more evident which indicate that the curves for the BER fall between the exponential Gaussian channel curves and the linear Rayleigh channel curves. If the value of N is very large, the BER curve approaches that for a Gaussian channel 15. 1.3 PROBLEM DEFINITION Wavelet based OFDM is found to be an efficient method to replace FFT based OFDM systems as wavelets has many advantages as compared to FFT-OFDM. DWT based OFDM has the potential to decrease the hardware complexity because Cyclic Prefix is not required in this case and proposed system gives nearly perfect reconstruction. DWT is an effective tool to study the signals in time frequency joint domain as it has the ability to provide simultaneous information about time and frequency, thereby gives the time frequency representation of the signal. It has been found that wavelets have compact localization in both time domain and frequency domain and have better orthogonality. DWT 13 based OFDM has the ability to combat the narrowband interference as the wavelets possess high spectral containment properties; making the system more robust against inter-carrier interference as compared to FFT realization. As cyclic prefix is not used in DWT OFDM, the data rates are better than that of FFT OFDM systems 16. Wavelet based OFDM is employed in order to remove the use of cyclic prefix which decreases the bandwidth wastage and the transmission power is also reduced by the use of wavelet transform. The spectral containment of the channels in DWT-OFDM is also better than the FFT-OFDM. Discrete wavelet transform is a type of wavelet transform which is found to be an alternative approach to replace IFFT and FFT in OFDM systems. In Wavelet transform, the desired signal is decomposed into set of basis waveforms, known as wavelets, which provide the way for analyzing the signals by investigating the coefficients of wavelets. DWT is used in several applications and has become very popular among engineers, technologists and mathematicians. The basis functions of wavelet transform are localized both in time and frequency and possess different resolutions in both domains which makes the wavelet transforms a powerful tool in various applications. Different resolutions correspond to analyze the behaviour of the process and the power of the transform. Due to these properties, the wavelets and wavelet transform find their applications in various fields such as data compression, image compression, radar, computer graphics and animation, astronomy, human vision, nuclear engineering, acoustics, biomedical engineering, music, seismology, turbulence, magnetic resonance imaging, fractals and pure mathematics. Since wavelet transform has many advantages such as flexibility, lesser sensitivity against channel distortion and interference as well as better utilization of spectrum, it has been proposed to design the sophisticated wireless communication systems 16. Wavelets are beneficial in various aspects such as channel modelling, data representation, transceiver design, and source and channel coding, data compression, interference minimization, energy efficient networking and signal de-noising in wireless communication systems. A low pass filter and high pass filter is employed to operate as QMF and satisfies perfect reconstruction and ortho-normal properties. In wavelet based OFDM, the modulated signal is transmitted using zero padding and vector transposing. DWT is known as a flexible and highly efficient method for decomposition of signals. 1.4 WAVELETS A wavelet is a small waveform that has effectively limited duration having an average value of zero. Wavelets have limited duration and tend to be asymmetric and irregular. The wavelet 14 analysis consists of breaking up a signal into scaled and shifted versions of the original signal or mother wavelet. Wavelets are a class of functions used to localize a given function in both space and scaling. A family of wavelets can be constructed from a function ψ (x), sometimes known as a "mother wavelet," which is confined in a finite interval 17. Daughter 𝑎 ,𝑏 wavelets 𝜓 (𝑥 ), are then formed by translation (𝑏 ) and contraction (𝑎 ). Wavelets are especially useful for compressing image data, since a wavelet transform has properties which are in some ways superior to a conventional Fourier transform. An individual wavelet can be defined by 17, xb 1/2 ab ,  xa (1.22)   a  Then,  1 tb  W f a,b f t dt  (1.23)    a a   where 𝑎 is the scaling factor and 𝑏 is the translation factor. 1.4.1 Orthonormal Wavelets The orthonormal wavelets can be given in terms of convolution equation as follows 18: n n n  h n h g .. g . g n  k k k 1 k j  2 22  n n n  hn g g .. g . gn (1.24) M1 M21Mk j  2 2 2 kj , 0,1,......,M 2 M where kj is a non-negative integer for and is the number of subcarriers. The sequences 𝑔 (𝑛 ) and 𝑕 (𝑛 ) correspond to the discrete impulse responses of the low pass and high pass filters of a Quadrature Mirror Filter bank with perfect reconstruction. They are used to construct the orthonormal wavelets, 𝑕 𝑛 , from the tree- 𝑘 structured QMF bank. The high pass filter can be found from the low pass filter by the n h n1 g L1 n , relationship  where L is the length of the sequence. The n j orthonormality among the wavelets is given as: h n,2 h n m k j m   kj 15 . where denotes the inner product, 𝛿 is the Kronecker delta function and 𝑛 is a positive 𝑗 integer. 1.4.2 Quadrature Mirror Filters An important feature of the discrete wavelet transform is the relationship between the impulse responses of the high pass (analysis) and low pass (scale) filters. The relationship is stated below 19: n h n1 g L1 n , (1.25)  where 𝑔 𝑛 and 𝑕 𝑛 are the impulse responses of the high pass and low pass filters, and 𝐿 is the filter length. Filters satisfying this condition are commonly used in signal processing, and they are known as the Quadrature Mirror Filters (QMF). The two filtering and sub-sampling operations can be expressed by the expressions given in (1.26), y k x n g 2k n  , high n y k x n h 2k n (1.26)  low n The reconstruction in this case is easy since the half-band filters form orthonormal bases. The above procedure is followed in a reverse order for the reconstruction. The signals at every level are upsampled by two, passed through the synthesis filters 𝑔 ′𝑛 , and 𝑕 ′𝑛 (highpass and lowpass, respectively), and then added up. A nice feature to note here is that the impulse responses of the analysis and synthesis filters are conjugate time reversed versions of one another i.e.  h'n g nand g n h n  Therefore, the reconstruction formula for each layer is given as:  x n y k g22 k n y k h k n (1.27)   high low k 1.4.3 Properties of Wavelets The most important properties of wavelets are the admissibility and the regularity conditions and these are the properties which gave wavelets their name. The square integrable function Ѱ(t) satisfying the admissibility condition, 16 2   (1.28) d   can be used to first analyze and then reconstruct the signal without loss of information. In the above equation Ѱ(𝜔 ) is the Fourier Transform of Ѱ(𝑡 ).The admissibility condition implies that the Fourier Transform of Ѱ(𝑡 ) vanishes at zero frequency, i.e. 2  0 (1.29)  0 This means that wavelets must have a band-pass like spectrum. A zero at the zero frequency also means that the average value of the wavelet in time domain must be zero, i.e.,  t dt 0   Therefore it must be oscillatory. That is Ѱ 𝑡 must be a wave. A. Regularity The wavelet transform of a one-dimensional function is two-dimensional. The time- bandwidth product of the wavelet transform is the square of the input signal and for most practical applications this is not a desirable property. Therefore, one imposes some additional conditions on the wavelet functions in order to make the wavelet transform decrease quickly with decreasing scale s. These are the regularity conditions and they state that the wavelet function should have some smoothness and concentration in both time and frequency domains 20. B. Vanishing moments If we expand the wavelet transform into the Taylor series at 𝑡 = 0 until order 𝑛 (let 𝜏 = 0 for simplicity) we get N p  1 tt  p  X s,0 x 0 dt R n1  (1.30)    ps s  p0  th 𝑝 Here 𝑥 stands for the 𝑝 derivative of 𝑥 and 𝑅 (𝑛 + 1) means the rest of the expression. Now if we define the moments of the wavelet by p N t () t dt (1.31) p  Then we can write the equation (31) into the finite development 17