The rest of this thesis is organized as follows. Chapter 2 introduces the basic concept of OFDM modulation. In Chapter 3, we will review some typical CP-based frame synchronization techniques presented in the literature, and describe our proposed modified algorithms. Chapter 4 contains the simulation models, simulation results and comparison of the performance between the proposed and the previous synchronization techniques. Finally, we will give a conclusion of the thesis in Chapter 5.
Chapter 2
Overview of OFDM
The basic principle of OFDM is to split a high-rate data stream into a number of lower rate streams that are transmitted simultaneously over a number of subcarriers.
Because the symbol duration increases for the lower rate parallel subcarriers, the relative amount of dispersion in time caused by multipath delay spread is reduced.
ISI is eliminated almost completely by introducing a GI in every OFDM symbol.
This whole process of generating an OFDM signal and the reasoning behind it are described in detail in Section 2.1 to 2.3.
In OFDM system design, a number of parameters are up for consideration, such as the number of subcarriers, guard time, symbol duration, subcarrier spacing, modulation type per subcarrier, and the type of forward error correction coding (FEC).
The choice of parameters is influenced by system requirements such as available bandwidth, required bit rate, tolerable delay spread, and Doppler values. Some requirements are conflicting. These design issues are discussed in Section 2.4.
2.1 OFDM Transmission Basics
OFDM is a MCM technique based on DFT (FFT) and IDFT (IFFT), Figure 2.1 shows the common OFDM system function blocks.
Figure 2.1 Function blocks of OFDM system [25].
An OFDM signal consists of a sum of subcarriers that are modulated by using phase shift keying (PSK) or quadrature amplitude modulation (QAM). The basic structure of OFDM modulator with N subcarriers is shown in Figure 2.2. As shown in Figure 2.2, the OFDM modulator consists of a serial-to-parallel converter (S/P) and an N-subcarrier modulators bank with different subcarrier frequencies. Firstly, the original data symbol streams are fed into the modulator in a serial way and the S/P divides these symbol streams into N parallel subsymbol streams, and then these subsymbols at each branch are used to modulate the different subcarriers. Assume that the original data symbol rate fed into the OFDM modulator is Rs, the reciprocal of the symbol duration Ts. After the S/P, the symbol duration T at each branch is increased to NTs, and the symbol rate is down to Rs/N. Hence, the symbol period is N times longer than that of the symbol in a conventional single carrier communication system. This property benefits an OFDM signal transmitted in a multipath channel environment, because the relative amount of dispersion in time can be reduced. In the following, we will introduce the mathematical description of OFDM signals and discuss some characteristics of OFDM systems.
Figure 2.2 Structure of modulator in an OFDM system with N subcarriers [22].
2.1.1 OFDM Signal Characteristics
In Figure 2.2, we denote x as the transmitted subsymbol at the k ( 1) 2
k+ N + -th
branch, where k is an integer value from 2
−N to 1 2
N − , which is chosen according
to the representation of the subcarrier frequency fk . In OFDM systems, the transmitted subsymbols x usually are PSK or QAM symbols. An OFDM signal k with symbol period T generated by the OFDM modulator can be expressed as follows
2 1
From (2.3), we can see that each subcarrier has exactly an integer number of cycles in the interval T, and the number of cycles between adjacent subcarriers differs by exactly one. This property implies that there is orthogonality among the subcarriers used in OFDM systems. If we multiply the i-th subcarrier φi( )t with the complex conjugated version of another subcarrier φ*j( )t , and integrate the result over the interval of T, we will get
n equation (2.4), the result is zero for all other subcarriers except for i= j, because the frequency difference i j
T
− produces an integer number of cycles within the
integration interval T. As an example, Figure 2.3 shows four subcarriers within one OFDM symbol.
Figure 2.3 Example of four subcarriers within one OFDM symbol [18].
The orthogonality of the different OFDM subcarriers can also be demonstrated in another way. According to (2.1), each OFDM symbol contains subcarriers that are nonzero over an interval T. Hence, the spectrum of a single symbol is a convolution of a group of Dirac pulses located at the subcarrier frequencies with the spectrum of a
square pulse that is one in period T and zero otherwise. The amplitude spectrum of the square pulse is equal to sinc(π fT), which is zero for all frequencies f that are an integer multiple of 1
T. This effect is shown in Figure 2.4, which shows the overlapping sinc spectra of individual subcarriers. At the maximum of each subcarrier spectrum, all other subcarrier spectra are zero. Because an OFDM receiver essentially calculates the spectrum values at those point that correspond to the maxima of individual subcarriers, it can demodulate each subcarrier free from any interference from the other subcarriers.
Figure 2.4 Spectra of individual subcarriers [25].
The orthogonal property is useful for the OFDM demodulator to easily demodulate the subsymbol at any of the subcarriers by using the correlator. The basic structure of the correlator-based OFDM demodulator is illustrated in Figure 2.5.
The correlator output at the j-th branch in the OFDM demodulator is denoted as
yj:
In this way, we can recover the transmitted subsymbols correctly.
Figure 2.5 Structure of correlator-based OFDM demodulator [22].
2.1.2 Implementation Using IFFT / FFT
The complex baseband OFDM signal as defined by (2.1) is in fact nothing more than the inverse Fourier transform (IFT) of N input subsymbols. If we sample s(t) by the sampling period Ts T
= N , then the time discrete equivalent, or IDFT signal model s[n] is given by
In the receiver, the DFT, the reverse operation of IDFT, can be used to recover the transmitted subsymbols at the OFDM subcarriers. The demodulation result of the j-th subcarrier by using the DFT can be expressed as
The demodulation result shown in (2.7) is the same as that in (2.5). Hence, we can
replace the subcarrier oscillators and the correlators used in the OFDM transmitter and receiver by the IDFT and DFT, respectively. In practice, the DFT / IDFT can be implemented very efficiently by the IFFT / FFT. An N point DFT / IDFT require a total of N2 complex multiplications — which are actually phase rotations. The FFT / IFFT drastically reduce the amount of calculations by exploiting the regularity of the operations in the DFT / IDFT. Using the radix-2 algorithm, an N-point FFT / IFFT require only (N 2) log ( )⋅ 2 N complex multiplications [19].
The structure of the OFDM system using the IDFT (IFFT) modulation and the DFT (FFT) demodulation is shown in Figure 2.6 and 2.7, where the block "D/A"
denotes the digital-to-analog converter that converts the discrete time signal to the continuous time signal, and "A/D" is the analog-to-digital converter which performs the reverse operation of the D/A.
Figure 2.6 Structure of transmitter using IDFT (IFFT) [22].
Figure 2.7 Structure of receiver using DFT (FFT) [22].
2.1.3 OFDM Bandwidth Efficiency
For simplicity, we assume that the signal spectra can be band-limited to the bandwidth of its main spectral lobe. In a classical parallel data system, called frequency division multiplexing (FDM), the total signal frequency band is divided into non-overlapping frequency channels. It seems good to avoid spectral overlap of channels to eliminate ICI. As seen in Figure 2.8 (a), the null-to-null bandwidth is
2
B s
W ≅T because the spectrum of the rectangular pulse is represented by the sinc function with first zero at 1
Ts . Since the bit rate is log2
s
R M
= T bits per second, where M is the alphabet, bit-rate-to-bandwidth ratio is 1log2
2
R M
W = . The
conventional multi-band system uses the available bandwidth inefficiently.
OFDM is the overlapping multicarrier modulation scheme, as shown in Figure 2.8 (b), the approximate bandwidth of a N subcarrier becomes B ( 1) 1
s
W N
≅ + NT because the frequency separation between adjacent subcarriers is 1
NTs for the signal
orthogonality. Therefore, the bit-rate-to-bandwidth ratio is log2 1
R N
W = N M
+ .
When N is large enough, the efficiency of OFDM systems is almost twice as that of FDM systems. Thus, OFDM multi-band systems can more efficiently use the available bandwidth compared to the conventional multi-band systems.
Figure 2.8 Illustration of OFDM bandwidth efficiency: (a) conventional multi-band system, (b) OFDM multi-band system [18].
2.2 Guard Interval and Cyclic Prefix
This section introduces the ideas of guard interval and cyclic prefix, and explains the reason to use CP in OFDM transmission systems.
2.2.1 ISI and ICI Avoiding
One of the most important reasons to do OFDM is the efficient way it deals with multipath delay spread. By dividing the input data stream in N subcarriers, the symbol duration is made N times longer, which also decreases the relative multipath delay spread, relative to the symbol time, by the same factor. In a multipath channel, the delayed replicas of the previous OFDM signal will cause the ISI between
successive OFDM signals as show in Figure 2.9. To eliminate ISI almost completely, a guard interval (GI) is introduced for each OFDM symbol. The GI is chosen larger than the excepted delay spread, such that multipath components from one symbol cannot interfere with the next symbol. A GI consists no signal at all is inserted between successive OFDM signal as shown in Figure 2.10. In this case, however, the problem of ICI would arise. This effect is illustrated in Figure 2.11. In this example, a subcarrier 1 and a delayed subcarrier 2 are shown. When an OFDM receiver tries to demodulate the first subcarrier, it will encounter some interference from the second subcarrier, because within the FFT interval, there is no integer number of cycle differences between subcarrier 1 and 2. At the same time, there will be crosstalk from subcarrier 1 for the same reason.
Figure 2.9 Channel dispersion causes ISI between successive OFDM signals [26].
Figure 2.10 OFDM signals with silent GI [26].
Figure 2.11 A delayed OFDM signal with a silent GI caused ICI on next signal [25].
In 1980, Peled and Ruiz [4] solved the ICI problem with the introduction of a cyclic prefix (CP), a copy of the last part of the OFDM signal attached to the front of the transmitted signal as shown in Figure 2.12. This ensures that delayed replicas of the OFDM symbol always have an integer number of cycles within the FFT interval, as long as the delay is smaller than the GI. As a result, multipath signals with delays smaller than the GI cannot cause ICI.
Figure 2.12 OFDM signal with cyclic prefix [25].
As an example of how multipath affects OFDM, Figure 2.13 shows received signals for a two-ray channel, where the dotted curve is a delayed replica of the solid curve. Three separate subcarriers are shown during three symbol intervals. From the figure, we can see that the OFDM subcarriers are BPSK modulated, which means that there can be 180-degree phase jumps at the symbol boundaries. In this particular example, this multipath delay is smaller than the GI, which means there are no phase transitions during the FFT interval. Hence, an OFDM receiver “sees” the sum of pure sine waves with some phase offsets. This summation dose not destroy the orthogonality between the subcarriers, it only introduce a different phase shift for each subcarrier. The orthogonality does become lost if the multipath delay becomes larger than the GI. In that case, the phase transitions of the delayed path fall within the FFT interval of the receiver.
Figure 2.13 Example of an OFDM signal with three subcarriers in a two-ray multipath channel. The dashed line represents a delayed multipath component [18].
Figure 2.14 shows the digital implementation of the OFDM transmitter which appends the CP in the front of the original OFDM signal s[n], and the structure of the OFDM signal with the guard period L is shown in Figure 2.15. We denote the signal with CP as the complete OFDM signal s n%[ ] and call the original s[n] with length of
In the receiver, we only require the useful part of the complete OFDM signal to perform the FFT demodulation. Hence, we will remove the CP of the complete OFDM signal before the FFT demodulation in the receiver.
Figure 2.14 A digital implementation of appending CP into OFDM signal in transmitter [22].
Figure 2.15 Structure of a complete OFDM signal with CP [22].
s[N-L] s[N-1] s[0]
L
s[1] s[N-1]LL
2.2.2 Linear Convolution Equivalent
In addition to avoid the ICI and ISI introduced by channel dispersion, the CP used in the OFDM system has another special purpose. As s n%[ ] is transmitted through the channel, the received complete OFDM signal r n%[ ] is the linear convolution of s n%[ ] and the impulse response of the channel:
[ ] [ ] [ ] 0 h 2 (2.9) r n% =s n% ∗h n ≤ ≤ + + −n N L L
where ∗ denotes the linear convolution and h[n] denotes the impulse response of the channel with the length L . We assume that h L is smaller than the guard period Lh here. As mentioned, the CP is the last part of the original OFDM signal s[n], so the result of the linear convolution described in (2.9) for n=L,L,L+ −N 1 is the N-point circular convolution of s[n] and h[n] given by
[ ] [ ] N [ ] (2.10) r n =s n ⊗ h n
for n=0,L,N−1, where ⊗N denotes the N-point circular convolution. Figure 2.16 shows the relation between (2.9) and (2.10), respectively. After removing the CP in the receiver, we use the DFT demodulation to recover the subsymbols in the received OFDM signal r[n]. According to the discrete time linear system theory [19], the DFT of r[n] in (2.10) is equivalent to multiplying the frequency response of the OFDM signal s[n] with that of the channel h[n], and the result is given by
DFT( [ ]) DFT( [ ] [ ]) [ ] (2.11)
j N j
y = r n = s n ⊗ h n =x H j
for j=0,L,N−1, where H j[ ] is the frequency response of the channel at the frequency of the j-th subcarrier. Notes that H j[ ] in (2.11) is the DFT of channel impulse response h[n]. From (2.11), we see that the demodulation result at the j-th subcarrier is the product of the original data subsymbol xj and the frequency response of the channel at the same frequency, H j[ ]. This property states that the receiver in the OFDM system does not require the complex adaptive channel equalization technique used in conventional single carrier systems. In the OFDM systems, the
subsymbol xj can be recovered in the receiver by dividing the demodulation result yj by simply the weight equal to H j[ ]. This is the reason why the CP is a copy of the last part of the original rather than a copy of any part of the signal.
Figure 2.16 Relation between linear convolution and circular convolution of and OFDM signal and channel impulse response [22].
2.3 Windowing
Looking at an example OFDM signal like in Figure 2.13, sharp phase transitions caused by the modulation can be seen at the symbol boundaries. Essentially, an OFDM signal like the one depicted in Figure 2.13 consists of a number of unfiltered QAM subcarriers. As a result, the out-of-band spectrum decreases rather slowly, according to a sinc function. As an example of this, the spectra for 16, 64, and 256 subcarriers are plotted in Figure 2.17. For larger number of subcarriers, the spectrum goes down more rapidly in the beginning, which is caused by the fact that
the sidelobes are closer together. However, even the spectrum for 256 subcarriers has a relatively small -40 dB bandwidth that is almost four times the -3 dB bandwidth.
Figure 2.17 Power spectral density (PSD) without windowing for 16, 64, and 256 subcarriers [18].
2.3.1 Common Used Window Type
To make the spectrum go down more rapidly, windowing can be applied to the individual OFDM symbols. Windowing an OFDM symbol makes the amplitude go smoothly to zero at the symbol boundaries. A commonly used window type is the raised cosine window, which is defined as
we allow adjacent OFDM symbols to partially overlap in the roll-off region and the transition between the consecutive symbol intervals are smoothed. The time
structure of the OFDM signal now looks like Figure 2.18.
Figure 2.18 OFDM cyclic extension and raised cosine windowing. Ts is the symbol time, T the FFT interval, Tg the guard time, Tprefix the preguard interval, Tpostfix the postguard interval, and β is the roll-off factor [18].
In practice, the OFDM signal is generated as follows: first, Nc input QAM values are padded with zeros to get N input samples that are used to calculate an IFFT.
Then, the last Tprefix samples of the IFFT output are inserted at the start of the OFDM symbol, and the first Tpostfix samples are appended at the end. The OFDM symbol is then multiplied by a raised cosine window w(t) to more quickly reduce the power of out-of-band subcarriers. The OFDM symbol is then added to the output of the previous OFDM symbol with a delay of Ts, such that there is an overlap region of
βTs.
2.3.2 Choice of Roll-Off Factor
Figure 2.19 shows spectra for 64 subcarriers and different values of the roll-off factor β. It can be seen that a roll-off of 0.025－so the roll-off region is only 2.5%
of the symbol interval－already makes a large improvement in the out-of-band spectrum. Larger β improve the spectrum further, at the cost, however, of a decreased delay spread tolerance. The latter effect is demonstrated in Figure 2.20, which shows the signal structure of an OFDM signal for a two-ray multipath channel.
The receiver demodulates the subcarriers between the dotted lines. Although the relative delay between the two multipath signals is smaller than the GI, ICI and ISI are introduced because of the amplitude modulation in the gray part of the delayed OFDM symbol. The orthogonality between subcarriers holds when amplitude and phase of the subcarriers are constant during the entire T-second interval. Hence, a roll-off factor of β reduces the effective GI by βTs.
Figure 2.19 Spectral of raised cosine windowing with roll-off factor of 0 (rectangular window), 0.025, 0.05, and 0.1 [18].
Figure 2.20 OFDM symbol windows for a two-ray multipath channel, showing ICI and ISI, because in the gray part, the amplitude of the delayed subcarrier is not constant [18].
2.3.3 Decision between Windowing and Filtering
Instead of windowing, it is also possible to use conventional filtering techniques to reduce the out-of-band spectrum. Windowing and filtering are dual techniques, multiplying an OFDM signal by a window means the spectrum is going to be a convolution of the spectrum of the window function with a set of impulse at the subcarrier frequencies. When using filters, care has to be taken not to introduce ripping effects on the envelope of the OFDM symbols over a timespan that is larger than the roll-off region of the windowing approach. Too much rippling means the undistorted part of the OFDM envelope is smaller, and this directly translates into less delay spread tolerance. The windowing technique is more feasible because a digital filter requires at least a few multiplications per sample, while windowing only requires a few multiplications per symbol, for those samples which fall into the roll-off region, windowing is an order of magnitude less complex than digital filtering.
2.4 Choice of OFDM Parameters
The choice of OFDM parameters is a tradeoff between various, often conflicting requirements. Usually, there are three main requirements to start with: bandwidth, bit rate, and delay spread. Generally, the following condition should be satisfied:
1 (2.13)
d
N B f τ ≤ ≤
where τ is the channel r.m.s delay spread, f is the maximum Doppler frequency d
spread, N is the number of subcarriers, and B is the total occupied bandwidth in the system. Therefore, the procedures of selecting system parameters are described in this section.
2.4.1 Guard Time and Symbol Duration
The delay spread directly dictates the time duration of the GI since the CP length must exceed the maximum delay spread. As a rule, the guard time Tg should be about two to four times the r.m.s delay spread of the channel τ . Tg also depends on the type of coding and QAM modulation. Higher order QAM (like 64-QAM) is more sensitive to ICI and ISI than QPSK, while heavier coding obviously reduces the sensitivity to such interference.
Now the guard time has been set, the symbol duration Ts can be fixed. To minimize the SNR loss caused by the GI, it is desirable to have Ts much larger than the guard time. It cannot be arbitrarily large because a larger Ts means more
Now the guard time has been set, the symbol duration Ts can be fixed. To minimize the SNR loss caused by the GI, it is desirable to have Ts much larger than the guard time. It cannot be arbitrarily large because a larger Ts means more