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Introduction to the DSP-OFDM Modulator Project
The Orthogonal Frequency Division Multiplexing (OFDM) digital communication technique has been attracting a great concern of researchers all over the world, due to its unique characteristics. The designers and engineers of mobile wireless communication systems and wireless multimedia broadband are looking forward to exploit the OFDM to be the air interface of these devices and systems. This exploitation has already been done with several systems and standards such as Wireless Local Area Networks 802.11a and Digital Video Broadcast-Terrestrial (DVB-T).
The DSP-OFDM Modulator project studies the essential parts of the OFDM modulator and demodulator and implements the OFDM modulator and demodulator on two separate DSP boards. For the OFDM modulator, the project studies the hardware DSP implementation of the OFDM modulator’s different parts such as the QAM mapper and the IFFT. This applies on the OFDM demodulator too. Additionally, for the OFDM demodulator, the project studies the carrier recovery issue to recover the OFDM information signal from the carrier signal and the OFDM symbol timing recovery issue to correctly pinpoint each OFDM symbol’s boundaries.
The Projects involves several aspects of the digital communications and the theoretical and practical DSP and uses the MATLAB and the Code Composer Studio (CCS) to analyze and simulate the designs to be practically implemented.
1.2 The Aim and the Objectives
The aim of the DSP OFDM Modulator project is to implement OFDM modulator and demodulator on two separate DSP boards. The implementation is not tied to any existing OFDM standard such that used in the DVB-T or other standards.
The DSP hardware implantation comprises many DSP and digital communication operations to be implemented through writing the C codes that perform these operations i.e. the QAM mapping and de-mapping, the IFFT and FFT, the digital IIR filters and the synchronization.
Therefore, the implementation will be first simulated by MATLAB and the Code Composer Studio (CCS) part by part before and with the hardware implementation on the DSP boards. The CCS will be used to simulate not only the modulator and demodulator but also the subparts of the hardware implementation such as the FFT and IFFT C codes. For example, the C code that will be used to perform N-Point IFFT to a complex array containing N complex elements to produce N outputs. These N outputs or discrete values will be compared with those N outputs or discrete values obtained from performing N-Point IFFT to the same N element-complex array in MATLAB in order to check that this C code will work properly in the DSP real time implementation of the OFDM modulator.
1.3 The Research Background and Motivations
The 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 MSc Wireless Communication Systems course. The good understanding of the discrete Fourier transform (DFT) allows presenting the Conjugate Symmetric approach. The use of the Conjugate Symmetric distribution of the subcarrier vectors on the IFFT input points makes the IFFT produce a multicarrier signal with a real part (In-phase) (I) only in the time domain, as the imaginary part (Quadrature) (Q) is always set to zero. It is easier to modulate and demodulate the OFDM information signal with a real part only, as the quadrature modulation is no longer required. The Conjugate Symmetric proposal allows applying the FM modulation to transmit and receive the multicarrier OFDM information signal.
1.4 The Thesis’s Organization
The thesis consists of five chapters. Chapter two is considered 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 data bits and how the IFFT N outputs (discrete values) are actually the samples of the OFDM multicarrier information signal for the current OFDM symbol being generated. It will be shown how the OFDM symbol has longer duration than those of other digital communication modulation techniques without affecting the data rate to be more resilient with dispersive channels and many other aspects of the OFDM modulation technique. This project is not tied to any existing OFDM standard. However, it resembles these standards in terms of the general block diagrams of the OFDM systems and the use of the pilot carriers, hence the employment of the OFDM in the DVB-T and the WLAN 802.11a are described briefly in chapter two.
Chapter three shows and simulates by using MATLAB the approaches and ideas that will be used for the hardware DSP implementation. It discusses the (Conjugate Symmetric) proposal that has come out of this project to facilitate the modulation and demodulation of the OFDM information signal and the use of the squared cosine method to recover the OFDM information signal from the modulated carrier signal. The use of the cyclic prefix (CP) to recover the OFDM symbol timing is also discussed in chapter three.
Chapter four presents the hardware implementation of the DSP OFDM modulator and demodulator on two separate DSP boards and shows the different results of the hardware implantation on the oscilloscope’s screen as well as it shows the results of the CCS simulation of the OFDM modulator and demodulator and compares the OFDM spectrum of the generated OFDM information signal generated by the Conjugate Symmetric approach with that generated from the traditional method.
Chapter five is for the conclusion points that have come out of this project and the further work to be implemented in the future.
The attached CD contains the real time 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.
In the digital communications, the transmitted signal over a wireless channel is more preferred, when the symbol duration is significantly greater than the delay spread (s) of this channel to avoid the intersymbol interference (ISI) due to the time dispersion of transmitted symbols. But unfortunately, the symbol duration is reversely proportional to the bit rate which means a great constraint when high data rate transmission is required over a wireless channel with a relatively high delay spread due to the multipath environment of that channel .
The OFDM technique produces the solution to this problem, as it divides the high rate bit stream 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 individual subcarrier. Therefore, the symbol duration is increased as many as (N) times without reducing the actual bit rate.
2.2 The Spectrum of the OFDM Subcarriers
Figure (2-1) y(t) (the dotted curve) is the algebraic summation of the 5 sinusoidal waves
Figure (2-2) the spectrum of y(t) in the frequency domain (five stems or tones)
Figure (2-3) the rectangular function with (?t) duration in the time domain
Figure (2-4) the spectrom of the rectangular function in the frequency domain
Figure (2-5) the spectrum of the OFDM symbol with five subcarriers
Suppose 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 figure (2-1). Suppose these subcarriers have the same frequency spacing (?f) between each adjacent subcarriers in the frequency domain. The spectrum of y (t) in the frequency domain in terms of the magnitude has five stems at f1 to f5 respectively. Each stem (single 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 spectrum 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 (truncated 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 cutting 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 signals. 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 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)) .
2.3 The OFDM Modulator
The 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 diagram of the OFDM modulator.
The OFDM modulator builds and transmits each OFDM symbol consisting of a number 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 number of data bits that are mapped to each QAM vector (subcarrier) depends on the QAM order (M) as shown in table (2-1). Using QAM mapper with higher order produces higher data rate. However, this will be at the cost of the reception quality as the constellation of higher order QAM allows higher Bit Error Rate (BER) for a given
- The QAM Mapper stage maps data bits to QAM vectors in accordance with the QAM constellation.
- The Serial to Parallel (S/P) buffers the QAM vectors of each OFDM symbol to prepare them for the IFFT operation.
- The IFFT stage converts the buffered QAM vectors (the subcarriers) from the frequency domain to produce an OFDM symbol sequence equivalent to the algebraic summation of these sinusoidal subcarriers in the time domain to be buffered in the next stage.
- Guard Interval Insertion and Parallel to Serial stages add the guard 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 Analogue Convertor (DAC) stages are to smooth the signal and convert the digital sequence into analogue 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 tolerance to the changes of the amplitude and phase of any received 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 quadrant 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 used to carry the data bits, some of which are used as pilot 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 number 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 other 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) generates 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) lower 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 guard period at the beginning of each OFDM. The Guard Interval (GI) (also called the Cyclic Prefix (CP)) makes a separation between the consecutive OFDM symbols to contribute in the ISI reduction and to eliminate the Intercarrier Interference (ICI) between the subcarriers. The guard interval must be greater than the highest path difference duration. As a result, multipath signals with delay smaller than the GI cannot cause ICI . The guard interval is generally equal to or less than the quarter symbol duration . Practically, the guard interval is generated by taking an exact copy of the end part of the OFDM symbol and adding it to the beginning of the symbol. The guard interval (GI) can be used by the receiver to determine the beginning and end of each received OFDM symbol through the cross correlation 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 analogue signal.
Finally, the Up Conversion and Power Amplification stage mixes the information signal with a locally generated carrier and boosts the resulted 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 error ). Therefore, scrambler, encoder and interleaver blocks may precede the other stages to provide the OFDM modulator with scrambled, encoded and interleaved coded bits .
It is also possible to up convert the signal whilst it is still in the digital signal processing 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 the 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 tap 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 Demodulator
The OFDM modulation operation is completely reversed in the demodulator. At first, the information signal must 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 signal 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 symbol’s 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 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 calculated 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 performance in the multipath environments which are common in the terrestrial broadcasting, as the OFDM distributes a high bit stream over 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 .
The DVB-T has two modes 2K and 8K. As 2K and 8K modes have the same data rata, selecting which mode should be used depends on the requirements. The 2K mode has about 250 µ S symbol duration and 4 KHz 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) . It has the following code rates (1/2, 2/3, 3/4, 5/6 and 7/8). Not all subcarriers are used as payload 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 180 degrees, hence they have only real components unlike the payload 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 payload carriers
- The locations of the continual and scatter pilot carriers
- The locations of the TPS pilot carriers
Figure (2-7) general block diagram of the OFDM demodulator.
The DVB-T uses 4, 16 or 64 QAM to modulate the coded bits to be represented as payload subcarrier vectors, therefore each payload 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 .
2.5.1 The DVB-T OFDM Subcarriers
The 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, scatter pilots and the (Transmission Parameter Signaling) (TPS) pilots. The continual pilots have fixed positions in the OFDM symbol spectrum. For example the sequences 0, 48, 969, 1683 and 1704 in the range (0 – 1704) are reserved as positions for the continual pilots. The continual pilots are used by the receiver to estimate the amount of phase rotation of the received QAM vectors. Every group of 12 subcarrier vectors has only one scatter pilot. The scatter 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 jumping 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) QAM
- FEC 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 TPS 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 frame. 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 example:
- Bits 26 and 27 represent the QAM order (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. Table (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 jumping .
2.6 WLAN 802.11a
Wireless 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 µs. 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 µs = 0.8 µs), therefore each OFDM symbol occupies (3.2 µs + 0.8 µs = 4 µs) 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 shown in table (2-3) . 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 µs) (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.
The MATLAB Analyses for the Hardware Implementation Approaches
Throughout 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 simulated 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 received OFDM signal.
3.2 The Mathematical Analysis of a Multicarrier Signal
To understand the idea of the (Conjugate Symmetric) and the role of the IFFT and FFT in the OFDM system, let’s 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 stipulates 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 . 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 in 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
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