Opportunistic Error Correction: When Does It Work Best for Ofdm Systems?

The water-filling algorithm enables an energy-efficient OFDM-based transmitter by maximizing the capacity of a frequency selective fading channel. However, this optimal strategy requires the perfect channel state information at the transmitter that is not realistic in wireless applications. In this paper, we propose opportunistic error correction to maximize the data rate of OFDM systems without this limit. The key point of this approach is to reduce the dynamic range of the channel by discarding a part of the channel in deep fading. Instead of decoding all the information from all the sub-channels, we only recover the data via the strong sub-channels. Just like the water-filling principle, we increase the data rate over the stronger sub-channels by sacrificing the weaker sub-channels. In such a case, the total data rate over a frequency selective fading channel can be increased. Correspondingly, the noise floor can be increased to achieve a certain data rate compared to the traditional coding scheme. This leads to an energy-efficient receiver. However, it is not clear whether this method has advantages over the joint coding scheme in the narrow-band wireless system (e.g. the channel with a low dynamic range), which will be investigated in this paper.


Introduction
Wireless communication takes place over multi path fading channels [1][2][3].Typically, the signal is transmitted to the receiver via a multiple of paths with different delays and gains, which induces Inter-Symbol Interference (ISI).To mitigate the ISI effect with a relatively simple equalizer in the wireless receiver, Orthogonal Frequency Division Multiplexing (OFDM) has become a fruitful approach to communicating over such channels [2,4,5].
The key idea of OFDM is to divide the whole transmission band into a number of parallel ISI-free sub-channels, which can be easily equalized by a single-tap equalizer via using scalar division [6,7].The information is transmitted over those sub-channels.Each OFDM sub-channel has its gain expressed as a linear combination of the dispersive channel taps.When the sub-channel has nulls (deep fades), reliable detection of the symbols carried by these faded sub-channels becomes difficult.
With the perfect Channel State Information (CSI) at the transmitter, the maximum data rate of a frequency selective fading channel can be achieved by the water-filling power allocation algorithm [8].This optimal strategy allocates the transmitted power to the subchannels based on its channel condition.In general, the transmitter gives more power to the stronger sub-channels, taking advantage of the better channel conditions, and less or even no power to the weaker ones [2].In other words, the total capacity of a frequency selective channel is increased by sacrificing the weak sub-channels.To achieve a certain data rate over a noisy wireless channel, the water-filling algorithm minimizes the transmitted power.Correspondingly, it gives us an energy-efficient transmitter.However, the water-filling algorithm requires the CSI at the transmitter, which may be unrealistic or too costly to acquire in wireless applications, especially in the rapidness of channel changes.Therefore, we propose a novel coding scheme in this paper to maximize the data rate of OFDM systems without CSI at the transmitter, which is realistic to be applied in practical applications and has the same principle as the water-filling algorithm.
Without CSI at the transmitter, the transmitted power is equally allocated to each sub-channel.To achieve re-liable communication, error correction codes are usually employed in OFDM systems [8][9][10].Over a finite block length, coding jointly yields a smaller error probability than that can be achieved by coding separately over the subchannels at the same rate [2].This theory has been applied in practical OFDM systems like WLAN and DVB systems [11][12][13][14][15].The joint coding scheme utilizes the fact that sub-channels with high-energy can compensate for those with low-energy, but its drawback is that each sub-channel is considered equally important.Consequently, the maximum level of noise floor endured by the joint coding scheme is inversely proportional to the dynamic range 1 .For this par coding scheme, the requirement of the noise floor is even higher to have all received packets decodable.
In a single-user scenario, the noise mainly comes from the hardware, e.g. the RF front and the Analog-to-Digital Converter (ADC) in the receiver.Given a practical wireless system, the noise floor is almost determined.In that case, the maximum data rate of the wireless channel is dependent on the dynamic range of the channel.The higher dynamic range means a lower data rate.Without CSI at the transmitter, we have two approaches to increasing the data rate over a channel with a high dynamic range. One is to reduce the noise floor in the RF front and the ADCs.That leads to the high power consumption in the receiver.For the RF front, its power consumption increases by 3 dB if the noise floor decreases by 3 dB [16].The power consumption in ADCs increases by 6 dB if the quantization noise floor reduces by 3 dB [17].So, this is not a desirable solution to a battery-powered wireless receiver. The other one is to reduce the dynamic range of the channel by discarding a part of the channel in deep fading.Instead of decoding all the information from all the sub-channels, we only recover the data via the strong sub-channels.Just like the water-filling principle, we increase the data rate over the stronger subchannels by sacrificing the weaker ones.In such a case, the total data rate over a frequency selective fading channel can be increased.Correspondingly, the noise floor can be increased to achieve a certain data rate compared to the traditional coding scheme.That leads to an energy-efficient receiver.Without CSI at the transmitter, the joint coding scheme does not allow us to give up any part of the channel as it treats each sub-channel equally important.Therefore, we transmit each packet over a single sub-channel.We take Figure 1 as an example to show the advantage of discarding the weak sub-channels.The whole channel is divided into 16 sub-channels and has a dynamic range of around 19 dB.We assume that a packet is encoded by an error correction code with a rate of 1 R and it can be decoded successfully when its Signal-to-Noise Ratio (SNR) is equal toor larger than 1 .We assume that the maximum noise floor is 1 if we want all the packets to be decoded.In such a case, the total data rate 1 is equal to 1 .However, from this figure, we can see that the weakest sub-channel costs a large part of the dynamic range.By discarding this sub-channel, the dynamic range of the channel is reduced to around 8dB.To compensate for this discarded sub-channel, we use a relatively higher code rate to encode each packet that can be decoded if With this scheme, the total data rate 2 is equal to , the total data rate is increased.Given the same noise floor, the reduced dynamic range (i.e.11 dB in this example).Otherwise, there is no gain from discarding the weak sub-channels.Obviously, 2 is larger than in this example.Given the same data rate (i.e. 2 ), discarding this sub-channel allows us to increase the noise floor in this example.Equivalently, the power consumption in the receiver is decreased.

C C 
Without CSI at the transmitter, the consequence of discarding the weak sub-channels is the loss of packets that are transmitted over those sub-channels.Two solutions can help us to compensate for it.One is to retransmit the lost packets.If the channel changes fast, this approach becomes not efficient and may cost more than that we gain from sacrificing the weak sub-channels.Also, the feedback channel is required, which is expensive in the wireless system.The other approach is to use erasure codes.In such a case, we treat the lost packets as erasures.With the assistance of a certain erasure code, we can achieve reliable communication with an energy-efficient receiver by discarding part of the channel in deep fading.Hence, we propose an energy-efficient error correction scheme based on erasure codes.To apply it to the OFDM-based wireless system, we divide a block of source bits into a set of packets.By treating each packet as a unit, they are encoded by an erasure code.Each erasure-encoded packet is protected by an error correction code that makes the noisy wireless channel behave like an erasure channel.Afterwards, each packet is transmitted over a sub-channel.Thus, multiple packets are transmitted simultaneously, using frequent division multiplexing.With the CSI at the receiver, the receiver discards the packets that are transmitted over the subchannels in deep fading and only decodes the packets with high energy.Erasure codes assist us to reconstruct the original file by only using the survived packets.Therefore, this scheme is called opportunistic error correction.
As mentioned earlier, the joint coding scheme works better than the separate coding over frequency selective fading channels, but it is not straightforward clear whether the opportunistic error correction can endure the higher level of noise floor than the joint coding.In [18], we have compared both in the simulation, whose results have shown that opportunistic error correction has a better performance than the joint coding over frequency selective fading channels.With the same code rate, it has a SNR 2 gain of around 8.5 dB over Channel Model A [19] compared to the Forward Error Correction (FEC) layer based on the joint coding scheme in current WLAN standards.However, this new method might not perform better than the joint coding scheme over a narrow-band channel (i.e. a flat-fading channel), as all sub-channels suffer the same fading.There is no gain from discarding some sub-channels.To compensate for the redundancy introduced by erasure codes (i.e. the percentage of discarded sub-channels), opportunistic error correction has to employ a relatively higher code rate to encode each erasure-encoded packet with respect to the joint coding scheme.Given the same type of error correction codes, the one with higher code rate always needs higher SNR to decode correctly.If opportunistic error correction utilizes the same type of error correction codes as the joint coding scheme, it will not perform better than the joint coding scheme over the flat-fading channel.This may be applied to the wireless channel with a low dynamic range.Therefore, it is of great interest to investigate the dynamic range of the channel.This new cross coding scheme shows its advantage over the joint coding scheme.This will tell us what kind of communication environment needs this novel approach.In this paper, we evaluate the performance of opportunistic error correction in the WLAN systems for different dynamic ranges of wireless channels.Its performance analysis is based on simulation results and practical measurements.That will give a good insight whether this new algorithm is robust to the imperfections of the real world that are neglected in simulations.
The paper is organized as follows.Opportunistic error correction is first depicted.We explain why this new method is suitable for OFDM systems and how it works.In section IV-A, we describe the system model by showing how we apply this novel scheme in OFDM systems.After that, we compare its performance with FEC layers from WLAN systems over aTGn3 channel [20] in the simulation.Besides, we evaluate its performance in the practical system in section V.The paper ends with a discussion of conclusions.

Opportunistic Error Correction
OFDM enables a relative easy implementation of wireless receivers over frequency selective fading channels [6], but it does not guarantee reliable communications over such channels.Therefore, error correction codes have to be employed in wireless channels.In OFDM systems, coding is performed in the frequency domain.Whether source bits are encoded jointly or separately over all the sub-channels depends on the transmission mode.There are two modes to transmit an encoded packet [21]:  Mode I is to transmit a packet over a single subchannel.In this case, the coding is done separately over all the sub-channels. Mode II is to transmit a packet over all the subchannels.With this method, the coding is performed jointly over all the sub-channels.Both transmission modes have advantages and disadvantages.Using Mode I, the receiver can predict whether the received packet is decodable since each sub-channel is modeled as a flat-fading channel.The packets trans-mitted over the sub-channel with low energy can be discarded without going through the whole receiving chain.Correspondingly, the processing power can be reduced.This is a desirable feature for a battery-powered receiver, which cannot be achieved by using Mode II.But Mode I endures a lower Noise Floor (NF) than Mode II to achieve the same quality of communication.As stated earlier, lower NF means higher power consumption in the wireless receiver which is not favorable by a battery-powered receiver.
To have a receiver with both energy-efficient features (i.e. a low processing power from Mode I and a high noise floor from Mode II), we propose opportunistic error correction which combines the separate coding scheme and the joint coding scheme together.Opportunistic error correction is a cross coding scheme.Via erasure codes, source bits are encoded jointly over all the sub-channels; then, each erasure-encoded packet is encoded individually over a sub-channel by error correction codes.This is different from the traditional coding scheme (i.e. the separate coding scheme or the joint coding scheme).
Opportunistic error correction is specially designed for OFDM systems.It is based on erasure codes.Any erasure codes can be applied in it.In this paper, we use fountain codes [22].Fountain codes are a kind of rateless erasure codes.In [23], MacKay describes the encoder of a fountain coder as a metaphorical fountain that produces a stream of encoded packets.Anyone who wishes to receive the encoded file holds a bucket under the fountain and collects enough packets to recover the original data.It does not matter which packet is received, only a minimum amount of packets have to be received correctly [24].In other words, with the help of fountain codes, each transmitted packet becomes independent with respect to each other.This allows us to discard some parts of wireless channel with deep fading by transmitting one fountain-encoded packet over a single sub-channel, leading to a reduction of processing power.
Figure 2 shows how opportunistic error correction works.With a fountain code, the transmitter can generate an in-principle infinite sequence of fountain-encoded packets.In this paper, the transmitter generates t number of fountain-encoded packets.Then, each packet is encoded by an error correction code to make wireless channels behave like an erasure channel.Afterwards, each packet is transmitted over a single sub-channel.

N
At the receiver side, the channel is first estimated.With the channel knowledge, the receiver makes a decision about which packets are to be decoded.We assume that fountain-encoded packets can go through the error correction decoding.Packets only survive if they succeed in the error correction decoder.The fountain decoder can reconstruct the original file by col- required at the receiver is slightly larger than the number of source packets [23]: where  is the percentage of extra packets and is called the overhead.For high throughput,  is expected to be as small as possible.However, fountain codes (e.g. Luby-Transform (LT) codes [25]) require a large  for small block size by only using the message-passing algorithm to decode.For example, the practical overhead of LT codes is 14% when , which limits its application in the practical system [26].In [27], we have shown that the overhead is reduced to 3% by combining the message passing algorithm and Gaussian Elimination to decode LT codes for .
The performance of opportunistic error correction depends on its parameters (i.e. the rate of erasure codes and error correction codes, the number of discarded subchannels).Given a set of parameters, whether it performs better than the traditional coding scheme depends on the dynamic range of the channel, which will be analyzed in  the next section.

System Model
OFDM system with Consider a single-user s N equally spaced orthogonal sub-channels shown in Figure 3.In the system, k X is the symbol to be transmitted over the -th k sub-cha l, n nne x is the -th n transmitted symbol time domain n is the channel output, n y is the -th n received symbol and k Y is the receiv symbol at e -th k sub-channel.As mentioned earlier, the channel nois ainly comes from the hardware in the transmitter and receiver.For simplicity, we assume a perfect transmitter which does not generate any noise to disturb the transmitted signal.However, the discussion below holds more generally.
The channel output n r can in the , r -th n ed th e m be expressed as: where is the number of channel taps, is the  .Due to the additional cyclic prefix in each OFDM sy bol, the linear convolution in Equation ( 2) can be considered as a cyclic convolution [2].So, after the OFDM demodulation, we can write k Y as:  .Thus, each sub-channel has the same noise floor, but its SNR is different: where is the energy of the dB k E by: -th k sub-channel and defined dB 10 and is defined by: dB Error correcting codes can be app effect lied to mitigate the of deep fades.Different coding scheme requires different level of NF (i.e.dB N ) to decode successfully.Assume that K source packets are encoded by a coding scheme then transmitted over the system as shown in Figure 3.Each packet consists of k source bits.We encode K k  source bits by the following coding schemes, respectively:  Coding I is to encode them by a Low-Density Parity Check (LDPC) code [8] with a rate of R .Each en- coded packet is transmitted over a single sub-channel.So,Coding I is a separate coding scheme. Coding II is to encode them by the same LDPC code as Coding I.But Coding II is a joint coding scheme as each packet is transmitted over all the sub-channels. Coding III is to encode them by opportunistic error correction based on LT codes.We define the rate of LT codes as LT R K N  . Each fountain-encoded packet is protected by a LDPC code with a rate of LDPC R and transmitted over a single sub-channel.To have the same rate as Coding I and II, the number of discarded sub-channels ds N can be expressed as: where 0 For the convenience in the analysis, we sort the sub-channels by its energy: The dynamic range of a wireless channel is defined as: ing I: To have all the packets decodable, the maximum NF for Coding I should be: 1) Cod 2) Coding II: The maximum NF for Coding I is not as straightforward as Coding I.As the jo employs the fact that the strong sub-c th int coding scheme hannels can help e weak sub-channels, we use S to classify the weak and strong sub-channels.In such a case, The key i N ber of sub-chanby using a relaely higher rate of error correction codes can be compensated by the reduced dynamic range, opportunistic error correction (i.e.Coding III) does not perform worse than the traditional coding schemes (i.e.Coding I and II).

Performance Analysis in Simulation
In this section, we analyze the performance of opportunistic error corr lts.
ection in the simulation.In [18] and [27], er over coding rr which have been explained in the above d cross layer can be applied in any s systems.In this paper, the IEEE by the fountain encoder.Then, a CRC checksum is ad we have shown that this new approach works bett Channel Model A [19] than the traditional joint scheme from WLAN standards.In this paper, we choose the TGn channel [20] as the channel model.Before checking its overall performance in the TGn channel, let uslook at the statistical characteristics of TGn channels' dynamic range D at different transmission bandwidths (BW).Figure 4 shows the cumulative probability of D for TGn channels at 5 MHz, 10 MHz and20 MHz.Although they have different BW, their D mainly distributes in the rang of 0 ~ 40 dB (i.e. at a probability of 99%).In this section, we analysis the performance of opportunistic error correction over the TGn channel model at different D and its overa performance at different BW.

System Setup
The opportunistic e or correction layer is based on fountain codes e ll section.This propose OFDM-based wireles 802.11a system is taken as an example of OFDM systems.
In Figure 5, the proposed new error correction scheme is depicted.The key idea is to generate additional packets by the fountain encoder.First, source packets are encoded ded to each fountain-encoded packet and the LDPC encoding is applied.On each sub-channel, a fountainencoded packet is transmitted.Thus, multiple packets are transmitted simultaneously, using frequency division multiplexing.At the receiver side, we assume that synchronization ld, the received fountain-encoded packet will go through the LDPC decoding, otherwise it will be discarded.This means that the receiver is allowed to disca low-energy sub-channels (i ackets) to lower the processing power consumption.After the LDPC decoding, the CRC checksum is used to discard the erroneous packets.As only packets with a high SNR are processed by the receiver, this will not happen often.When the receiver has collected enough fountain-encoded packets, it starts to recover the source data.

Simulation Results
re encoded by FEC I, II wards, they are mapped into OFDM modulation.signed by using parameters and channel estimation are perfect in the simulation.If the SNR of the sub-channel is equal to or above the thresho rd .e. p In this section, we compare three FEC schemes in simution as follows: la  FEC I: LDPC codes at 0.5 R  with interleaving from the IEEE 802.11n standard [12]   648 n  . FEC II: fountain codes with the (175,255) LDPC code [28] plus 7-bit CRC using the transmission Mode I, which is the opportunistic error correction layer. FEC III: fountain codes with the (175,255) LDPC code plus 7-bit CRC using the transmission Mode II.Three FEC schemes are simulated as function of the dynamic range D and/or the bandwidth BW by transmitting 1000 bursts of data (i.e.around 100 million bits) over the TGn channel.Each burst consists of 583 source packets with a length of 168 bits.With the same code rate of 0.5 R  , source packets a and III, respectively.After QAM-16 symbols before the For the case of FEC II and III, each burst is encoded by a LT code (de 0.03, c  0.3   [23]) and decoded by the message-passing algorithm and Gaussian elimination together.From [27], we of the same code rate (i.e.know that 3% overhead is required to recover the source data successfully.To each fountain-encoded packet, a 7-bit CRC is added, then the (175,255) LDPC encoder is applied.Under the condition 0.5 R  ), we are allowed to discard 21%  (10,20]dB to (20,30]   Therefore, we can conclude that fountain codes make error correction coding schemes more robust to the variation of As mentioned before, the key point of opportunistic of n r that ere is no benefit to have this tradeoff when the dynamic ran fo D , fountain codes still can recover the original data when the other part of fountain-encoded packets is lost less than expected in the channel with 2 D .However, this does not apply to FE

Practical Evaluation
The C++ simulation results in the above section have shown the performance of opportunistic error correction in comparison with the joint coding scheme (i.e.FEC I and III) over the TG n channel with different D and BW, respectively.C++ simulation, with its highly accurate double-precision numerical environment, is on the one hand a perfect tool for the investigation of the algorithms.On the other hand, many imperfections of the real-world are neglected (e.g.perfect synchronization and channel estimation are assumed in Section IV, which does not happe in the real-world).So, simulation may sh istic receiver performance.In this secn, we evaluate its performanc ice to estigate whether opportunistic error correction is more roal- GHz by a Quadrature Modulator (AD8346) and transmitted using ntenna.
antenna, and the reverse chain for the receiver.In the receiver, there is no power amplifier and band-pass RF filter before the down-converter but a low-pass base nd fi

The Transmitter
The data is generated offline in C++.The generation consists of the random source bits selection, the FEC encoding and the digital modulation as we depict in Section IV-A.The generated data is stored in a file.A server software in the transmit PC uploads the file to the Ad link PCI-7300Aboard 5 which transmits the data to DAC (AD9761) 6 via the FPGA board.After the DAC, the base band analog signal is up converted to 2.

The Receiver
The reverse process takes place in the receiver.The received RF signal is first down converted by a Quadrature Demodulator (AD8347) 8 , then filtered by the 8th order low-pass Butterworth analog filter to remove the aliasing.The base band analog signal is q 9 ceiver should synchronize h the transmitter and estimate the channel using the preambles and the pilots, which are defined in [11].Timing and frequency synchronization is done by the Schmidl & Cox algorithm [30] and the channel is estimated by the zero forcing algorithm.In addition, the residual carrier frequency offset n start as we describe in Section IV-A.sitions in Figure 9), while the receiver ante e bits.Different channel bits can go through the same random However, itdoes not apply co wit is estimated by the four pilots in each OFDM symbol [31].After the synchronization and the channel estimation, decoding ca

Measurement Setup
Measurements are carried out in the corridor of Signals and Systems Group, located at the 9th floor of Building Hogekamp in University of Twente, the Netherlands.The measurement setup is shown in Figure 9.The transmitter (TX) was positioned in front of the elevator (i.e. one of the circle po nna (RX) was in the left side of the corridor (i.e. the cross positions in Figure 9).89 measurements were done inth is scenario with a non-line-of-sight situation.The average transmitting power is around −10 dB m and the distance between the transmitter and the receiver is around 6 ~ 52.5 meters.The measurements were conducted at 2.3 GHz carrier frequency and 20 MHz bandwidth.
In the simulation depicted in section IV, these FEC schemes can be compared by using the same sourc frequency selective channel. in the real environment.The wireless channel is timevariant even when the transmitter and the receiver are stationary (e.g. the moving of elevator with the closed door can affect the channel).Hence, we should compare them by using the same channel bits.
Because not every stream of random bits is a codeword of a certain coding scheme, it is not possible to derive its corresponding source bits from any sequence of random bits, especially for the case of FEC II and FEC III.Fortunately, the decoding of FEC I is based on the parity check matrix.Any stream of random bits can have its unique sequence of source bits with its corresponding syndrome matrix.The receiver can decode the received data based both on the parity check matrix and the syndrome matrix.So, FEC I can use the same channel bits with FEC II.In such a case, they can be compared under the same channel condition (i.e.channel fading, channel noise and the distortion caused by the hardware.).Therefore, we only compare the joint coding scheme from the IEEE 802.11n standard (i.e.FEC I) with opportunistic error correction (i.e.FEC II) in there al world.
In the measurements, FEC I and II are compared with the same code rate (i.e.0.5 R  ).More than 600 blocks of source packets are transmitted over the air.Each block nsists of 97944bits.Source bits are encoded by FEC II.
The encoded bits are shared by FEC I as just explained.Afterwards, they are mapped into QPSK symbols 10 before the OFDM modulation.Each measurement corresponds to the fixed position of the transmitter and the receiver.It is possible that some measurements might fail in decoding.Due to the lack of a feedback channel in the testbed, no retransmission can occur.In this paper, we assume that the measurement fails if the received data per measurement has a BER higher than 3 10  by using FEC I.For the case of FECII, if the packet loss is more than 21% as expected, we assume that the measurement fails.

Measurement Results
In total, 89 measurements have been done.There are 7 blocks of data transmitted in each measurement.The estimated D of the channel over those 89 measurements distributes in the range of around 50% of the measurements have dB.FEC II succeeds in all the measurements but that does not happen to FEC I. Figure 10 shows the percentage of the successful measurements for each D .With FEC I, the probability of the successful measurements decreases as D increases.In the simulation, FEC I works better than FEC II at R range in all measurements, so we evaluate their practical performance by analyzing the statistical characteristics of meas- 10 The choice for QPSK instead of QAM-16 in the measurements is due to the noise floor of the testbed, whose noise floor is around -2 (i.e.SNR 20  dB). Figure 6 shows that the required SNR for 0 dB  

20,30 D 
dB should be at least 20 dB for FEC I to achieve a BER of 4 10  or lower.With the non-perfect synchronization and channel estimation, a higher SNR is expected in the real world than in the simulation to achieve the same order of BER.Therefore, we choose a lower order of modulation scheme to have more successful measurements to compare FEC I and II in the real world.[11,16]

Figure 1 .
Figure 1.An example to show the advantage of discarding the weak sub-channels.In this example, each packet is transmitted over a single sub-channel.By discarding the weakest sub-channel, the dynamic range of the channel is reduced by around 11dB.(a) No sub-channel is discarded.(b) 1 sub-channel is discarded.

Figure 2 .
Figure 2. Pictural diagram of opportunistic error correction for OFDM systems.lecting enough packets.The number of fountain-encoded packets

Figure 3 .
Figure 3. System model showing the transmission over one tion.sub-channel in the OFDM system with ideal synchroniza- dBSNR k S  means the weak sub-channels and dB SNR k S  means the strong sub-channels.Besides, we assume that Coding II can decode the received packets correctly ( fountain-encoded packet can be received correctly if its SNR is not than S III is to exchange the code rate of error correction codes with the num nels to be discarded.If the price paid tiv and if .That might hold for LDPC ds n be discarded, the maximum NF for Coding III is expressed as: , t istic error correction , we will search here is in ppo wireless app tions.In the nex  in the simu- lation resu

Figure 4 .Figure 5 .
Figure 4.The cumulative distribution curves of the dynamic range for the TGn channel at 5 MHz, 10 MHz and 20 MHz.D

Figure 6 .
Figure 6.Performance comparison in the simulation between FEC I, II and III at R 0.5  ceives en over the TGn channel at different ranges.FEC II and III can achieve error-free when the fountain decoder re ough number of fountain-encoded ckets.We represent BER = 0 by in this figure.

2 . 2 .
SNR gain increases with BW.With respect to FEC III at the error-free quality, FEC II gains a SNR of 3 dB at BW = 5 MHz, 5 dB at BW = 10 MHz and 20 MHz.C II an hind is as follows.Due to the variation of the channel, a burst data encounters several channels with different C I. n ow a too optim tio e in pract inv bust to the re world's imperfections.i.e.FEC II) is to exchange the code rate the used error correction codes with the number of discarded sub-channels.Simulatio esults conclude In general, FE d III performs better than FEC I at BW = 5 MHz, 10 MHz and 20 MHz.The reason beth ge of the channel D is within 10 dB.The profit starts 10 D  dB and increases with D .r Channels at Different Bandwidth In this part, we compare them over the TGn channel with different bandwidth: 5 MHz, 10 MHz and 20 MHz.Fig- ure 4 has presented that different bandwidth has different probability distribution of D .The average D increases with the cha nel bandwidth.Simulation results are shown in Figure 7, where e e that FECII sc han t C (i.e.FEC I and III) at any BW.The perfor I, II and III degrades when BW increases.FEC I loses around 3 dB when BW doubles.When BW changes from 5 MHz to 10 MHz, there is a SNR loss of around 2 dB in FEC II and around 4 dB in FE III.Both FEC II and III lose 1 dB when BW increases from 10 MHz to 20 MHz.In a word, FEC II is less sensitive to the variation of BW than FEC I and III, because the performance of FEC II is more robust to the increase of D than FEC I and III.Comparing with FEC I at BER of 5 10  or lower, FEC II has a SNR gain of around 11 dB at BW = 5MHz, around 12.5 dB at BW = 10 MHz and around 14.5 dB at BW = For the case of FEC II and III, if some part of fountain-encoded packets are lost more than expected ina channel with 1

Figure 7 . 8 .
Figure 7. Performance comparison between FEC I, II and III at R 0.5  rror-free d over the TGn channel at different bandwidths (i.e. 5 MHz, 10 MHz and 20 MHz).FEC II and III can achieve e ecoding when the fountain decoder receives enough fountain-encoded packets.We represent BER = 0 by 8 10 in this

Figure 9 .
Figure 9. Measurement Setup: antennas are 0.9 m above the concrete floor.The measurements are done in the corridor of the Signals and Systems Group.The receiver is positioned at the left side of the corridor (i.e. the cross positions) and the transmitter is at the gray part as shown in the figure.The room contains one coffee machine, one garbage bin and one glass cabin.

Figure 10 .
Figure 10.The comparison between FEC I and FEC II in the probability of successful measurements for each range over 89 measurements.For FEC I, successful meas urement means .For FEC II, measurem succeeds only if -free quality.

Figure 11 10 
Figure 11.Histogram of and I dB SNR ,

4 . 4 BER
Figure 11(a) 4 1.4 10   .Th has a SNR gain n comparison at taps and x is the transmitted symbo s N   are mutual independent.From the central limit theorem, n r can be modeled as a Gaussian-distributed random riable with zero mean and a variance of va

Figure 6, we
4of the transmitted packets.In FEC II, we transmit one packet per can see that the performance of FEC I degrades (i.e. a SNR loss of around s D increases by 10 dB.That does not apply to FEC II.FEC II is more robust to the variation of D .Only e dynamic range of the channel D changes from dB, FEC II loses around 2 dB in SNR to achieve the error-free quality.From dB, there is no performance loss as D increases.
1R is the code rate of LT codes (i.e. 1 1.
dB while around lies in the same range.