DSP-OFDM Modulator Project

DSP-OFDM Modulator ProjectChapter OneIntroduction to the DSP-OFDM Modulator Project1.1 IntroductionThe Orthogonal oftenness Division Multiplexing (OFDM) digital dialogue technique has been attracting a great tinge of researchers every last(predicate) over the world, referable to its unique characteristics. The designers and engineers of mobile radio receiving system communication systems and wireless mul measuredia broadband ar looking forward to curiosityeavour the OFDM to be the air interface of these devices and systems. This exploitation has already been d hotshot with several(prenominal)(prenominal) systems and standards much(prenominal) as Wireless Local Area Ne iirks 802.11a and Digital depiction Broadcast-Terrestrial (DVB-T).The DSP-OFDM Modulator project studies the essential neighborhoods of the OFDM modulator and sensor and utilises the OFDM modulator and demodulator on two sepa send DSP boards. For the OFDM modulator, the project studies the hardwargon DSP slaying of the OFDM modulators diametrical gets such as the QAM mapper and the IFFT. This applies on the OFDM demodulator too. Addition andy, for the OFDM demodulator, the project studies the carrier recuperation issue to recover the OFDM study augur from the carrier direct and the OFDM image timing recovery issue to elucidately pin top dog all(prenominal) OFDM figures boundaries.The Projects involves several aspects of the digital communications and the theoretical and pragmatical DSP and expenditures the MATLAB and the Code Composer Studio (CCS) to tumble and re suffer the designs to be practically implemented.1.2 The Aim and the ObjectivesThe aim of the DSP OFDM Modulator project is to implement OFDM modulator and demodulator on two separate DSP boards. The carrying out is non fastened to any existing OFDM standard such that apply in the DVB-T or other standards.The DSP hardware implantation comprises many DSP and digital communication trading operations to b e implemented with writing the C codes that per strain these operations i.e. the QAM mapping and de-mapping, the IFFT and FFT, the digital IIR filters and the synchronization. in that respectfore, the implementation pass on be stolon mistaken by MATLAB and the Code Composer Studio (CCS) part by part onwards and with the hardware implementation on the DSP boards. The CCS will be utilise to simulate not completely the modulator and demodulator but also the subparts of the hardware implementation such as the FFT and IFFT C codes. For precedent, the C code that will be economic consumption to per course of action N-Point IFFT to a Byzantine array containing N complex elements to sire N placeputs. These N outputs or discrete values will be compared with those N outputs or discrete values obtained from performing N-Point IFFT to the aforesaid(prenominal) N element-complex array in MATLAB in purchase methodicalness to check that this C code will work properly in the DSP rea l fourth dimension implementation of the OFDM modulator.1.3 The Research Background and MotivationsThe good presentation of the theoretical and practical DSP during the taught part of the course encouraged me to tackle this project, as I had not done any practical DSP before I enrolled in the disseminated multiple sclerosis Wireless Communication Systems course. The good understanding of the discrete Fourier transform (DFT) al clinical depressions presenting the conjugated cruciform approach. The design of the Conjugate Symmetric distribution of the subcarrier vectors on the IFFT scuttlebutt indicates makes the IFFT produce a multicarrier maneuver with a real part (In- variety) (I) only in the magazine landed estate, as the imaginary part (Quadrature) (Q) is al miens evolvedness to nobody. It is easier to play and demodulate the OFDM culture signal with a real part only, as the quadrature stirover is no longer required. The Conjugate Symmetric design allows applying the FM flection to potentiometeralize and receive the multicarrier OFDM information signal.1.4 The Thesiss OrganizationThe dissertation consists of louvre chapters. Chapter two is deemed as a literature survey. Chapter two explains the OFDM spectrum and the principles of the OFDM modulator and demodulator. It illustrates how the OFDM information signal carries or represents the digital info small-arms and how the IFFT N outputs (discrete values) are rattling the samples of the OFDM multicarrier information signal for the authentic OFDM attribute being induced. It will be shown how the OFDM figure has longer length than those of other digital communication modulation techniques without affecting the entropy rate to be more peppy with distributive dividing lines and many other aspects of the OFDM modulation technique. This project is not tie to any existing OFDM standard. However, it resembles these standards in equipment casualty of the general circumscribetleme nt plots of the OFDM systems and the use of the control carriers, so the employment of the OFDM in the DVB-T and the WLAN 802.11a are set forth briefly in chapter two.Chapter three shows and simulates by using MATLAB the approaches and ideas that will be apply for the hardware DSP implementation. It discusses the (Conjugate Symmetric) proposal that has come out of this project to drive the modulation and demodulation of the OFDM information signal and the use of the squared romaine system to recover the OFDM information signal from the modulated carrier signal. The use of the cyclicalal prefix (CP) to recover the OFDM token timing is also discussed in chapter three.Chapter cardinal presents the hardware implementation of the DSP OFDM modulator and demodulator on two separate DSP boards and shows the variant results of the hardware implantation on the oscilloscopes screen as well as it shows the results of the CCS simulation of the OFDM modulator and demodulator and compar es the OFDM spectrum of the generated OFDM information signal generated by the Conjugate Symmetric approach with that generated from the traditional method.Chapter basketball team is for the conclusion points that be in possession of come out of this project and the further work to be implemented in the future.The accustomed CD contains the real sentence DSP implantation CCS projects of the OFDM modulator (OFDM-TX project) and OFDM demodulator (OFDM-RX project) and the CCS simulation of the OFDM modulator and demodulator (Simulation project) as well as the MATLAB codes and an electronic copy of the thesis.Chapter TwoOFDM rudiments2.1 IntroductionIn the digital communications, the transmitted signal over a wireless channel is more preferred, when the symbolization while is significantly greater than the delay spread (s) of this channel to avoid the intersymbol interference (ISI) due(p) to the cartridge clip dispersion of transmitted symbols. But unfortunately, the symbol da te is reversely relative to the bit rate which means a great constraint when lofty info rate transmission is required over a wireless channel with a relatively high delay spread due to the multi bridle-path environment of that channel 1.The OFDM technique produces the solution to this problem, as it divides the high rate bit blow into (N) very low rate bit streams that are transmitted simultaneously using (N) orthogonal subcarriers for every OFDM symbol. Each of these low rate bit streams modulates an mortal subcarrier. in that respectfore, the symbol m is increased as many as (N) measure without reducing the real bit rate.2.2 The Spectrum of the OFDM Subcarriers token (2-1) y(t) (the dotted curve) is the algebraic total of the 5 sinusoidal wavesFigure (2-2) the spectrum of y(t) in the absolute relative relative frequency domain (tail fin stems or tones)Figure (2-3) the angular function with (?t) era in the judgment of conviction domainFigure (2-4) the spectrom of the rectangular function in the frequency domainFigure (2-5) the spectrum of the OFDM symbol with five subcarriersSuppose y(t) is a signal consisting of the algebraic summation of five sinusoidal waves (subcarriers) in the time domain with five different frequencies (f1, f2, f3, f4 and f5) respectively normal (2-1). Suppose these subcarriers feel the same frequency lay (?f) betwixt for severally one(prenominal)(prenominal)(prenominal) adjacent subcarriers in the frequency domain. The spectrum of y (t) in the frequency domain in monetary value of the magnitude has five stems at f1 to f5 respectively. Each stem ( ace tone) represents one of these five sinusoidal waves or subcarriers figure (2-2).Now, suppose an OFDM symbol (with symbol duration = (?t)) consists of the same five sinusoidal subcarriers mentioned earlier. The spectrum of this OFDM symbol in the frequency domain does not now consist of five stems instead the spectrum is like that one in figure (2-5). The spectru m in figure (2-5) consists of five overlapped sinc functions each of which represents an individual subcarrier.Actually, our OFDM symbol is not identical to y(t). More precisely, it is a ( cut y(t)) with truncation duration equal to the OFDM symbol duration (?t). When a signal is truncated in the time domain with equal gain over all the truncated points within the period (?t), that means mathematically multiplying this signal with a rectangular function in the time domain with a time duration equal to (?t) figure (2-3). The shape of the spectrum of rectangular function in terms of the magnitude is single sinc wave in the frequency domain abscission the horizontal axis at points equal to the integer multiples of the reciprocal of the time duration (1/?t) figure (2-4). Basically, when any two signals are multiplied in the time domain, the resultant signal of this multiplication has a spectrum in the frequency domain equal to the convolution of the spectrums of the two original signal s. Therefore the spectrum in figure (2-5) represents the resultant of the convolution operation between the five stems of y(t) figure (2-2) and the sinc of the rectangular function figure (2-4) in the frequency domain.Looking at figure (2-5) again, it is easy to notice that the peak of each subcarrier sinc occurs at a point where all other four sincs have magnitudes equal to zero at which. This situation is the condition of the orthogonality between the subcarriers as it ensures the least(prenominal) interference between the subcarriers in the frequency domain. The orthogonality between subcarriers is not achieved, unless the frequency spacing between the subcarriers (?f) is equal to the reciprocal of the OFDM symbol duration (1/(?t)) 2.2.3 The OFDM ModulatorThe OFDM Modulator uses the Quadrature Amplitude Modulation (QAM) Mapper and the Inverse Fast Fourier Transformer (IFFT) to simultaneously generate and modulate the subcarriers of each OFDM symbol. Figure (2-6) shows a general block plot of the OFDM modulator.The OFDM modulator builds and transmits each OFDM symbol consisting of a flesh of subcarriers equal to N as follows. The QAM mapper maps the data bits to (N) QAM vectors. Each of these vectors has real and imaginary components and represents a single subcarrier. The tot of data bits that are mapped to each QAM vector (subcarrier) depends on the QAM gear up (M) as shown in prorogue (2-1). Using QAM mapper with higher order produces higher data rate. However, this will be at the cost of the reaction quality as the conformation of higher order QAM allows higher snap Error Rate (BER) for a givenThe QAM Mapper detail maps data bits to QAM vectors in accordance with the QAM constellation.The successive to Parallel (S/P) buffers the QAM vectors of each OFDM symbol to prepare them for the IFFT operation.The IFFT pointedness modifys the buffered QAM vectors (the subcarriers) from the frequency domain to produce an OFDM symbol term same to the alg ebraic summation of these sinusoidal subcarriers in the time domain to be buffered in the next stage.Guard musical breakup founding and Parallel to accompanying stages add the withstand time musical interval to each buffered OFDM symbol sequence and produces it serially to the next stage.The DSP Low Pass Filter (LPF) and The Digital to one-dimensional convertor (DAC) stages are to smooth the signal and convert the digital sequence into parallel signal.The Up Conversion and The Power Amplification stages.Figure (2-6) general block diagram of the OFDM modulator.The 4-QAM constellation, which is identical to Quadrature Phase Shift Keying (QPSK) constellation, gives the receiver more border to the metamorphoses of the amplitude and signifier of any get QAM vector and allows the receiver to de-map it to the correct 2-bit-combination, as long as it still lies in the same string-circle from which it was originated at the transmitter, whereas 16 and 64 QAM constellations give less tolerance to the change in the phase and amplitude of the received QAM vector due to the noise and interference.Not all subcarriers of an OFDM symbol are utilise to carry the data bits, some of which are used as polisher carriers for the synchronization and channel estimation purposes and for providing the receiver with specific information such as the order of QAM being used by the transmitter.The Serial to Parallel (S/P) stage buffers the N vectors from the QAM stage for each OFDM symbol to produce them in parallel way to IFFT stage. The add together of IFFT points is always greater than the number of the subcarriers (N), so the (S/P) pads the remaining IFFT points, which have not been assigned QAM vectors, with zeroes.The IFFT stage is the heart of the OFDM modulator. It gives the QAM vectors the mathematical ability to be considered as the OFDM subcarriers in the frequency domain and converts them to the time domain to form the multi-subcarrier information signal. In ot her words, as all the (N) QAM vectors of each OFDM symbol are the parallel inputs of the IFFT operation, the IFFT stage considers these QAM vectors as tones or stems in the frequency domain and converts them into correspondent subcarriers in the time domain for the given OFDM symbol duration. Each QAM vector has a specific phase and amplitude which corresponds to the bit combination this vector represents in accordance with the QAM constellation. The IFFT coverts each QAM vector into a correspondent sinusoidal subcarrier in the time domain with amplitude and phase directly related to those of that vector and a frequency that is directly proportional with the sequence of IFFT point, to which the vector has been assigned. That means if a QAM vector with sequence (n) (assigned to an IFFT point with sequence n) generates a subcarrier with frequency equal to (f), the vector with sequence (n-1) generates a subcarrier with a frequency equal to (f ?f) and the vector with sequence (n+1) gen erates a subcarrier with a frequency equal to (f + ?f). The IFFT stage can simultaneously produce all the N-subcarriers for each OFDM symbol as it performs the conversion from the frequency domain to the time domain for N (QAM vectors) in one parallel operation for each OFDM symbol. The OFDM symbol signal in the time domain represents the algebraic summation of all subcarriers of that symbol. Now, it is obvious how the OFDM modulator divides the high rate bit stream into (N) take down rate bit streams which are simultaneously transmitted over (N times higher OFDM symbol duration) without reducing the actual bit rate.The Guard Interval Insertion stage appends a concord period at the starting line of each OFDM. The Guard Interval (GI) (also called the Cyclic Prefix (CP)) makes a dissolution between the consecutive OFDM symbols to contribute in the ISI reduction and to eliminate the Intercarrier dissonance (ICI) between the subcarriers. The guard interval essential be greater tha n the highest path difference duration. As a result, multipath signals with delay smaller than the GI cannot evidence ICI 3. The guard interval is generally equal to or less than the quarter symbol duration 4. Practically, the guard interval is generated by taking an use up copy of the end part of the OFDM symbol and adding it to the starting signal of the symbol. The guard interval (GI) can be used by the receiver to adjust the beginning and end of each received OFDM symbol through the cross correlativity operation.Now, the sequence of the OFDM symbol is converted into serial sequence. The Guard Interval Insertion and the Parallel to Serial (P/S) stages are shown as one stage in figure (2-6). The DSP LPF smoothes the information signal.The Digital to Analogue Convertor (DAC) converts the incoming digital sequence into line of latitude signal.Finally, the Up Conversion and Power Amplification stage mixes the information signal with a locally generated carrier and boosts the re sulted signal to be transmitted.The input data bits to the OFDM modulator in figure (2-6) may be first scrambled for the security purposes, encoded for the Forward Error Correction (FEC) purposes and interleaved (to randomize the bursts of misplay 5). Therefore, scrambler, encoder and interleaver blocks may precede the other stages to provide the OFDM modulator with scrambled, encoded and interleaved coded bits 6.It is also possible to up convert the signal whilst it is still in the digital signal bear upon domain before converting it to the analogue form.The Carrier Recovery and the Down Converting stage recovers the information signal from the carrier signal.The Sample and Hold circuit and the Analogue to Digital Convertor (ADC) stage converts the information signal from the analogue form to produce the digital sequence for the DSP processing.The Guard Interval Removal and the Serial to Parallel (S/P) stage removes the cyclic prefix (CP) and produces all the useful samples of th e current OFDM symbol being processed to the FFT stage simultaneously.The FFT stage converts the subcarriers of the OFDM symbol from the time domain to the frequency domain and produces them to the QAM De-mapper as vectors through the (P/S) buffer. One water tapdance Equalizer can be used to equalize the vector constellation after the FFT stage.The Parallel to Serial (P/S) stage buffers the vectors of each OFDM symbol to produce them serially to the QAM De-mapper.The QAM De-mapper assigns each vector to the correspondent bit combination to produce the data bits.Figure (2-7) general block diagram of the OFDM demodulator.2.4 The OFDM DemodulatorThe OFDM modulation operation is completely reversed in the demodulator. At first, the information signal moldiness be recovered from the carrier. This is done by the carrier recovery and down converting stage. Figure (2-7) shows a general block diagram of the OFDM demodulator.The analogue to digital convertor (ADC) converts the information s ignal into a digital sequence.The guard interval removal stage removes the inserted guard interval or cyclic prefix from the beginning of each OFDM symbol. The OFDM demodulator could use the cyclic prefix at the beginning of each OFDM symbol to pinpoint the beginning and end of each symbol, as the cyclic prefix at the beginning of each OFDM symbol is identical to the end part of that symbol within a duration equal to the cyclic prefix duration.Now, the digital sequence of each OFDM symbol, which represents the algebraic summation of the subcarriers signals in the time domain, is simultaneously presented to the FFT stage to convert these subcarriers into their correspondent vectors in the frequency domain. The parallel presentation of the symbols digital sequence to the FFT stage involves the idea of serial to parallel conversion of this sequence.The subcarriers may also be equalized before being presented to the QAM de-mapper using a one tap equalizer.The QAM de-mapper assigns each vector in the frequency domain to the correspondent binary program bit combination in accordance with the QAM constellation being used in the transmitter and receiver.The serial sequence of the received coded bits must be de-interleaved and then decoded and descrambled, if the scrambling, encoding and the interleaving are applied in the transmission side.The number of data bits per each OFDM symbol can be easily measured by multiplying the number of subcarriers that are used to carry the data bits (Payload subcarriers) by the number of bits represented by the QAM vector in accordance with the QAM constellation table (2-1). The carrier recovery operation can also be done after the sample and hold stage within the digital signal processing unit.2.5 Digital Video Broadcasting-Terrestrial (DVB-T)The DVB-T employs the OFDM due to its excellent public presentation in the multipath environments which are common in the terrestrial broadcasting, as the OFDM distributes a high bit stream o ver a high number of orthogonal subcarriers, each of which carries a low bit rate stream simultaneously, which makes the symbol duration much higher than the delay of the indirect paths 7.The DVB-T has two regularitys 2K and 8K. As 2K and 8K rooms have the same data rata, selecting which mode should be used depends on the requirements. The 2K mode has about(predicate) 250 S symbol duration and 4 kc spacing between its subcarriers, whereas the 8K mode has about 1 m S symbol duration and 1 KHz spacing between its subcarriers. These characteristics make the 8K mode with its higher symbol duration more resilient with multipath situations and channels with a high delay spread but the 2K mode resists better the shift in the frequency caused by Doppler effects due to the relative mobility between the transmitter and receiver, as it has higher frequency spacing between its subcarriers. The DVB-T has (FEC) similar to that of the DVB-S (Satellite) 8. It has the following code rates (1/2, 2/3, 3/4, 5/6 and 7/8). Not all subcarriers are used as onus carriers to carry the coded bits (data bits + redundant bits) some subcarriers are used for channel estimation and correction. These subcarriers are the pilot carriers which have vectors lying on the I (In-phase) axis of the QAM constellation with angles equal to either 0 degrees or clxxx degrees, hence they have only real components unlike the freight vectors which have real and imaginary components in order to recognize between them. The mapping of the pilot carriers to be delivered as vectors to the IFFT stage in the OFDM modulator is achieved through the BPSK modulation which uses the I (in-phase) axis of the constellation. Figure (2-8) shows the locations of DVB-T subcarriers on the 4-QAM constellation.The locations of the loading carriersThe locations of the revenant and sprinkling pilot carriersThe locations of the TPS pilot carriersFigure (2-7) general block diagram of the OFDM demodulator.The DVB-T uses 4, 1 6 or 64 QAM to modulate the coded bits to be represented as consignment subcarrier vectors, thitherfore each shipment subcarrier can carry 2, 4 or 6 coded bits every OFDM symbol respectively. The DVB-T uses a guard interval length equal to (1/4, 1/8, 1/16 or 1/32) of the OFDM symbol duration 8.2.5.1 The DVB-T OFDM SubcarriersThe DVB-T 2K mode has 2048 subcarriers, but it only uses 1705 subcarriers and sets the rest to zero. The 1705 carriers are numbered from 0 to 1704. It uses 1512 subcarriers as payload carriers and the remaining 193 subcarriers as pilot carriers. There are three types of the pilot carriers the continual pilots, pick pilots and the (Transmission Parameter Signaling) (TPS) pilots. The continual pilots have frozen(p) positions in the OFDM symbol spectrum. For example the sequences 0, 48, 969, 1683 and 1704 in the couch (0 1704) are reserved as positions for the continual pilots. The continual pilots are used by the receiver to estimate the amount of phase rota tion of the received QAM vectors. Every group of 12 subcarrier vectors has only one decompose pilot. The pass around pilots do not have fixed positions. Among each 12 carriers positions there is one variable position for one scatter pilot. The position of each scatter pilot regularly varies from symbol to symbol by parachuting 3 positions forward with respect to its position in the previous symbol. The scatter pilots are used to estimate the channel too. The TPS pilot carriers have fixed positions and are used by the transmitter to inform the receiver about the transmission parameters such as.The DVB-T mode (2K or 8K)Modulation type of the payload subcarrier vectors (4, 16, or 64) QAMFEC code rate (1/2, 2/3, 3/4, 5/6 or 7/8)Length of the guard interval (1/4, 1/8, 1/16 or 1/32)Like the continual and scatter pilots, the TPS pilot carriers lie on the I (in-phase) axis. Each OFDM symbol in the 2K mode has 17 TPS pilot carriers with fixed positions. Within the same symbol all the 17 T PS pilots are either at 0 degrees or 180 degrees. The receiver determines the state of TPS pilots whether the TPS pilots of the received symbol are at 0 degrees or 180 degree based on the majority voting rule. Through the TPS pilots, the transmitter sends the receiver 67 information bits every OFDM kind. The OFDM frame consists of 68 OFDM symbols. The TPS pilots are Differential Bi-Phase Shift Keying (DBPSK) modulated. That means the receiver considers receiving an information bit = (0), if the state of the TPS pilots change from the previous symbol to the current symbol and considers receiving an information bit = (1), if the phase or state of the TPS pilots does not change from the previous symbol to the current symbol. 68 OFDM symbols are required to transmit the 67 information bits, as the first symbol is used to determine the initializing state of the TPS pilots. The 67 bits inform the receiver about the transmission parameters, for exampleBits 26 and 27 represent the QAM ord er (00=4, 01=16, 10=64)Bits 31, 32 and 33 represent the code rate (000=1/2, 001=2/3, 010=3/4, 011=5/6, 100=7/8)The DVB-T 8K mode has 6817 subcarrier per each OFDM symbol. The subcarriers of the 8K have the same principles and use of those of 2K with difference in their numbers only. circumvent (2 2) shows the different subcarriers of both 2K and 8K modes.The scatter pilot carriers have two different numbers of the subcarriers, as the scatter pilot carriers coincide with fixed locations of the continual pilot carriers due to their jump 8.2.6 WLAN 802.11aWireless Local Area Networks (WLANs) 802.11a employ OFDM as a digital communication technique for reliable and high data rate transmission. Each OFDM symbol is expressed by 64 subcarriers, but the actual used subcarriers are (52) (64 52 =12 subcarriers are set to zero). There are 48 payload carriers to carry the coded bits (data and redundancy bits) and 4 pilot carriers. The frequency spacing between the subcarriers is (?f = 312.5 KHz). The required channel bandwidth can be calculated by multiplying the total number of subcarriers by the frequency spacing = 312.5 K * 64 = 20 MHz. To achieve the orthogonality between the subcarriers the OFDM symbol duration (?t) must be equal to the reciprocal of (?f) (?t = 1/ ?f), hence ?t = 1/312.5 KHz = 3.2 s. 802.11a appends a guard interval (GI) equal of (1/4) the OFDM symbol duration at the beginning of each OFDM symbol (GI = 0.25 * 3.2 s = 0.8 s), therefore each OFDM symbol occupies (3.2 s + 0.8 s = 4 s) time interval. That means a wireless device transmits 250,000 OFDM symbol per second. 802.11a allows wireless devices to have (8) transmission data rates or modes (6, 9, 12, 18, 24, 36, 48 and 54) M bits/sec. 802.11a uses (BPSK, QPSK, 16-QAM or 64-QAM) to modulate the payload carriers and uses (1/2, 2/3 or 3/4) code rate for the FEC in accordance with transmission data rate being used.The different (8) modes use different modulation types and different code rates as sho wn in table (2-3) 6. 802.11a uses BPSK modulation to modulate the payload carries in modes 1 and 2 unlike the DVB-T which only uses QAM modulation to modulate the payload carriers. For each mode, the OFDM symbol has the same total duration (4 s) (250,000 OFDM symbol/Sec) and the same channel bandwidth (20 MHz), as it has the same number of subcarriers (48 payload carriers and 4 pilot carriers).Looking back at table (2-3) (Mode (8) 54 Mbps), as the 64-QAM modulation is used to modulate the payload carriers, each payload carrier in the OFDM symbol carries (log2 (64) = 6 coded bits). Each OFDM symbol carries (48 payload carriers/OFDM symbol * 6 coded bits/payload carrier = 288 coded bits / OFDM symbol). The number of data bits per each OFDM symbol = 288 * (code rate = 3/4) = 216 data bits / OFDM symbol. There are 250,000 OFDM symbols / Sec, hence the data bit rate = 216 * 250,000 = 54 Mbps.Chapter ThreeThe MATLAB Analyses for the Hardware capital punishment Approaches3.1 IntroductionT hroughout this chapter the ideas and approaches that will be used for the DSP hardware implementation of the OFDM modulator and demodulator on two separate DSP boards will be discussed and copy by using the MATLAB.There are mainly three approaches.The use of the (Conjugate Symmetric) with the carrier vectors which are the inputs of the IFFT stage in the OFDM modulator to produce an OFDM information signal in the time domain with a real part only for easier modulation and demodulation, which is the proposal that has come out from this project.The use of the squared cosine to recover the OFDM information signal from the carrier signal in the receiver (the synchronization of the carrier frequency signal).The make use of the guard interval (GI) or the cyclic prefix (CP) for the synchronization of the OFDM symbol (i.e. The Symbol clock Recovery) to allow the receiver to know the correct boundaries of each received OFDM symbol to set the FFT window at the correct positions of the receive d OFDM signal.3.2 The Mathematical summary of a Multicarrier SignalTo understand the idea of the (Conjugate Symmetric) and the role of the IFFT and FFT in the OFDM system, lets consider y(t) as a continuous multicarrier signal with a real part only in the time domain consisting of the algebraic summation of five sinusoidal waves or subcarriers which have the following frequencies (1, 2, 3, 4 and 5) KHz and phase shifts (p4,,3p4, 5p4, 7p2, 9p4) respectively with equal amplitude = (28) for each. For our y(t), each two adjacent subcarriers (in the frequency domain) have 90 degrees phase shift. y(t) can be expressed in the time domain as in Eq. (3-1)It is not necessary for the five subcarriers forming y(t) to have the same magnitude. It is just to simplify this discussion.Now, if y(t) is sampled with sampling frequency (Fs). Fs must be greater than (2 * 5 KHz = 10 KHz), where 5 KHz is the highest frequency of y(t) according to the Nyquist-Shannon theorem. Nyquist-Shannon theorem stipul ates that the sampling rate or frequency must be at least two times greater than the highest frequency of the sampled signal to avoid the aliasing which prevents providing the DSP system with a right copy of the sampled signal 9. When a continuous signal in the time domain is sampled, a sample is taken at every (t = n * Ts). Ts is the sampling interval (Ts = 1 / Fs) and n is zero or positive integer number representing the sequence of the sample. y(t) is no longer continuous. Now, y(t) represents a sequence of discrete values. In Eq. (3-1), y(t) is replaced by y(n) in the left hand side and (t) is replaced by (n * Ts) or (n / Fs) in the right hand side as in Eq. (3-2).If Fs is set to 16 KHz (16 KHz 10 KHz) and 16-point FFT operation is performed to y(n) to produce y(n)s spectrum in the frequency domain in order to study it.The 16 point FFT operation needs 16 discrete values or samples of y(n) for n = 0, 1, 2,13, 14, 15. The results of 16 point FFT operation are 16 complex vectors i n the frequency domain. The 16 discrete values (samples) of y(n) are the inputs of the FFT in the time domain and the outputs are 16 complex vectors of Y(m) which represents y(n)s spectrum in the frequency domain. Table (3-1) lists the 16 discrete values of y(n) in the time domain and table (3-2) lists the 16 vectors