text:
The invention provides for a method and system for properly tracking, synchronizing and demodulating received packets at a receiver in order to decode data and other informational symbols transmitted by a transmitter. The invention further provides for a method and system for correcting for distortion, phase shift, and frequency offset at a receiver due to variations in the frequencies transmitted by a transmitter. The system and method disclosed herein and employed for acquisition and initial synchronization is effectively immune to channel impairments, such as multi-path.
1. A method for synchronizing a receiver to a transmitter comprising the following steps:
receiving a digital signal from the receiver;
delaying the digital signal by a sample processing interval to produce a delayed signal; and
correlating the digital signal and delayed signal to create a correlator output.
2. The method of
determining a magnitude of the correlator output; and
comparing the magnitude of the correlator output to a preset threshold value wherein when the magnitude exceeds the preset threshold value an incoming packet is detected at the receiver.
3. The method of
determining a magnitude of the correlator output;
monitoring time samples during which the magnitude of the correlator output exceeds a preset threshold value;
determining a sample point at which the magnitude of the correlator output is maximum;
back-biasing by at least one time sample.
4. The method of
determining a phase shift of the correlator output corresponding to a maximum value of the correlator output wherein the phase shift is an estimate of the fractional portion of carrier frequency offset.
5. A method for synchronizing a receiver to a transmitter comprising the following steps:
receiving a digital signal from the receiver;
demodulating long sync symbols from the digital signal; and
correcting for a fractional portion of frequency offset.
6. The method of
7. The method of
8. The method of
9. The method of
10. The method of
11. The method of
12. The method of
correlating the interpolated odd frequency values of the channel transfer function and the actual odd frequency values; and
selecting a correlation value to identify an integer frequency offset number.
13. The method of
correlating the interpolated odd frequency values of the channel transfer function and the actual odd frequency values to create a correlation value;
computing a magnitude of the correlation value; and
selecting the largest magnitude of the correlation value to identify an integer frequency offset number.
14. The method of
associating the largest magnitude of the correlation value with a channel transfer function;
using the channel transfer function to correct data symbols for amplitude and phase shifts.
15. A method for synchronizing a receiver to a transmitter comprising the following steps:
receiving a digital signal from the receiver;
delaying the digital signal by a sample processing interval to produce a delayed signal;
correlating the digital signal and delayed signal to create a correlator output;
determining a phase shift of the correlator output corresponding to a maximum value of the correlator output wherein the phase shift is an estimate of the fractional portion of carrier frequency offset;
extracting long sync symbols from the digital signal;
correcting for a fractional portion of frequency offset;
extracting vectors of modulation values of data sub-carriers with progressive trial integer offsets;
dividing each vector by long sync symbol modulation values to obtain channel transfer functions;
estimating odd frequency values for each of the channel transfer functions;
correlating the interpolated odd frequency values of the channel transfer function and the actual odd frequency values; and
selecting a correlation value to identify an integer frequency offset number.
16. A method for deriving frequency offset correction and sample timing information for symbol number m+1 based on pilot tone information contained in symbol m of a sequence of N data symbols comprising the following steps:
extracting Fourier coefficients of the mth symbol by way of a fast Fourier transform of the receiver;
dividing the Fourier coefficients by a channel response function to correct for amplitude variations and phase shifts during transmission and for phase shifts;
extracting phase shift offsets of pilot tones relative to known phase shifts for the mth symbol;
approximating a straight line of the phase shifts versus frequency;
computing a frequency offset error based on the values of the phase shifts;
combining the frequency offset error with frequency offsets computed for the mth symbol, creating the value of frequency offset to be used for the m+1 symbol; and
combining the slope of the straight line with the phase slope used in the channel response of the mth symbol to create the phase slope of a channel response for the m+1st symbol.
17. The method of
18. A system for synchronizing digital signal at a receiver from a transmitter comprising:
means for delaying the digital signal by a sample processing interval to produce a delayed signal; and
a correlator for correlating the digital signal and delayed signal to create a correlator output.
19. The system of
an integrator for determining a magnitude of the correlator output; and
a comparator means for comparing the magnitude of the correlator output to a preset threshold value wherein when the magnitude exceeds the preset threshold value an incoming packet is detected at the receiver.
20. The system of
an integrator for determining a magnitude of the correlator output;
a means for monitoring samples during which the magnitude of the correlator output exceeds a preset threshold value;
a magnitude detector for determining a sample point at which the magnitude of the correlator output is maximum;
a delay means for back-biasing the received signal by at least one time sample.
21. The system of
a phase shift detector means for determining the phase shift of the correlator output corresponding to a maximum value of the correlator output wherein the phase shift is an estimate of the fractional portion of carrier frequency offset.
[0001] This application claims priority to the provisional patent application Serial No.: 60/250,724, filed on Nov. 30, 2000.
[0002] The present invention relates to a method and apparatus concerning the synchronization of a receiver to a signal to accurately demodulate, decode and retrieve information transmitted across a communication channel.
[0003] Communication systems operate to transmit communication signals having informational content and other characteristics generated at, or applied or provided to, a transmitter upon the communication channel. A receiver receives the transmitted, communication signal and operates to recreate the informational content and other signal characteristics of the communication signal.
[0004] A radio communication system is a communication system in which the communication channel is formed of one or more bands of a frequency spectrum. In a radio communication system, the receiver is typically tuned to frequencies of the communication channel upon which the communication signal is transmitted and includes circuitry for demodulating, decoding and/or converting received signals into lower frequency or baseband signals which permit the informational content and other signal characteristics of the communication signal to be reconstructed. Radio-based communication systems enable communication to be effectuated between remotely-positioned transmitters and receivers without the need to form hard-wired or other fixed connection.
[0005] Distortion is sometimes introduced upon the transmitted signal. The distortion can, for instance, be caused by filter circuitry of the transmitter, or filter circuitry of the receiver, or the communication channel. Some transmission difficulties which distort the communication signal as the communication signal is transmitted by a transmitter to a receiver can sometimes be more readily overcome when digital communication techniques are utilized. Utilization of digital communication techniques is advantageous as communications systems can be efficiently integrated in countries or regions that adopt the standards.
[0006] Advances in communication technologies have permitted communication systems to utilize digital communication techniques. In digital communication systems, a transmitter digitizes an information signal to form a digital signal. Once digitized, the digital signal can be modulated, and once modulated, transmitted upon a communication channel. While some existing communication systems have been converted to permit the utilization of digital communication techniques, other communication systems have been planned, or have been made possible, as a result of technological advancements or the development of national or international standards.
[0007] In November 1999, the IEEE 802.11 standardization committee selected coherent orthogonal frequency division multiplexing (OFDM) as the basis for a 5 GHz wireless local area network (WLAN) standard [1]. This digital communication standard divides the 5150 MHz to 5350 MHz frequency band into eight 20-MHz communication channels. Each of these 20-MHz channels is composed of 52 narrow-band carriers. OFDM sends data in parallel across all of these carriers and aggregates the throughput. The standard supports data rates as high as 54 Mbps in 16.6 MFz occupied bandwidth on 20 MHz channelization.
[0008] The OFDM data symbols are 4 μsecs long and consist of 52 sub-carriers spaced at 312.5 KHz. As shown in FIG. 1, each symbol contains 48 information-bearing sub-carriers and 4 pilot sub-carriers. Assuming a 20 MHz sampling rate, the OFDM symbols can be generated by a length 64 inverse fast Fourier transform (IFFT). The inputs to the IFFT are 48 information bearing modulation values drawn from a BPSK, QPSK, 16-QAM or 64-QAM constellation according to the chosen data rate, 4 known BPSK modulation values prescribed for the pilot sub-carriers and 12 null values [1]. The 64 complex values output from the IFFT are baseband discrete time samples of the sub-carrier multiplex. A 16 sample point cyclic prefix is appended to these 64 sample points as a guard interval to complete the generation of an 80 sample point or 4-μsec duration OFDM data symbol as shown in FIG. 1.
[0009] A WLAN OFDM receiver must be properly synchronized with each received packet in order to decode the data being passed in the OFDM information symbols. The receiver must first detect the arrival of a packet. Further, the receiver must determine and correct for any carrier frequency offset imparted to the sub-carriers due to variation in the nominal values of the in-phase and quadrature (I/Q) modulator and up-converter oscillator frequencies in the transmitter and in the down-converter and I/Q de-modulator oscillator frequencies in the receiver. The receiver must determine the start time of the first OFDM data symbol in the packet. The receiver must determine and remove any amplitude and phase shift that may have been imparted to the sub-carriers during transmission through the multi-path channel. The 20 MHz sampling clock at the receiver must be synchronized with the 20 MHz sampling clock at the transmitter. The preamble and pilot sub-carriers described above and as specified in the IEEE 802.11a standard are provided for these purposes. However, the standard does not provide for methods of implementation of such characteristics. The invention described herein provides a highly practical, yet accurate and robust set of algorithms to synchronize and track packets conforming to the IEEE 802.11 standard and other standards.
[0010] It is in light of this background information related to digital communication systems that the significant improvements of the present invention have evolved.
[0011] The invention provides for a method and system for properly tracking and synchronizing received packets at a receiver in order to decode data and other informational symbols transmitted by a transmitter. The invention further provides for a method and system for correcting for distortion, phase shift, and frequency offset at a receiver due to variations in the frequencies transmitted by a transmitter.
[0012] Additional objects and features of the invention will be more readily apparent from the following detailed description and appended claims when taken in conjunction with the drawings, in which:
[0013]FIG. 1 is a drawing illustrating a WLAN OFDM data symbol.
[0014]FIG. 2 is a drawing illustrating the packet preamble consisting of ten short OFDM sync symbols, and two long OFDM sync symbols with a double length guard interval.
[0015]FIG. 3 illustrates the QPSK and BPSK modulation values associated with the short and long sync symbol OFDM sub-carriers present in the preamble.
[0016]FIG. 4 is a diagram of the cross-correlator used in the initial iteration of the synchronization algorithm.
[0017]FIG. 5 shows the magnitude of the output of the correlator versus preamble sample point number.
[0018]FIG. 6 is a diagram of the fine frequency correction and sub-carrier demodulation for the second iteration of the synchronization algorithm
[0019]FIG. 7 shows the magnitude of the output of the correlator versus integer frequency shift for an integer frequency offset of p=−1.
[0020]FIG. 8 is diagram of the pilot tone tracking loop showing the error generation and corrections applied to the subsequent OFDM symbol.
[0021] Distortion on a transmission signal can be introduced by filter circuitry at a receiver, transmitter or across a communication channel there between. At the receiver, sub-carriers may have been shifted in frequency up or down by an arbitrary amount. Also, it is not known by the receiver at what sample instant the packet will arrive and most importantly the beginning sample instant of the first and subsequent OFDM data symbols is not known. In order to demodulate and decode the OFDM data symbols, the receiver must shift the sub-carriers to their correct frequencies and commence the demodulation and decoding process for each symbol at its first sample instant. The receiver is assumed to be a digital receiver such that the 20 MHz sample values of the in-phase and quadrature components of the received signal are available for processing by the digital synchronization circuitry.
[0022] Packet detection, symbol timing and carrier frequency offset correction preferably rely on a structured training sequence of special OFDM symbols contained in a packet preamble. The same preamble information may be used to estimate the channel in support of coherent demodulation employed by the receiver. Slow channel variations and residual carrier frequency error may be tracked and removed using pilot sub-carriers with known modulation that are inserted at prescribed slots in each OFDM symbol.
[0023] While the present invention described herein is based on specific specifications, characteristics and techniques based on the 802.11 standard, such specifications, characteristics and techniques are used for purposes of illustrating and describing the present invention. While description and drawings herein represent a preferred embodiment of the present invention, it will be understood that various additions, modifications and substitutions may be made to the specifications, characteristics and techniques of the 802.11 standard without departing from the spirit and scope of the present invention as defined in the accompanying claims. In particular, it will be clear to those skilled in the art that the present invention may be embodied in other specific forms, preamble formats and structures, data formats and structures, arrangements, proportions, and with other elements, materials, and components, without departing from the spirit or essential characteristics thereof. The presently disclosed embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims, and not limited to the foregoing description. Furthermore, it should be noted that the order in which the process is performed may vary without substantially altering the outcome of the process.
[0024] Returning now to FIG. 1, an OFDM data symbol consists of a cyclic prefix of 16 sample points and 64 sample points generated by a 64 point IFFT of the 53 sub-carrier modulation values plus 11 null values. As indicated in FIG. 1, the 53 sub-carrier modulation values consist of 48 data sub-carriers, four pilot sub-carriers and a null value for the center frequency or baseband D.C. term. The sub-carriers are spaced in frequency by an amount Δf=312.5 KHz. Each data sub-carrier is phase and/or amplitude modulated independently. The pilot sub-carriers are BPSK modulated with a known pseudo-random sequence that is removed at the receiver. The length of each data symbol is Ts=ΔT+Tg where ΔT=1/Δf=3.2 μsecs or 64 sample points at 20 MHz sampling rate and is called the OFDM FFT processing interval. Tg=0.8 μsecs or 16 sample points is a short guard interval filled with a cyclic extension that is the last 16 sample points of the signal in the processing interval ΔT and is included to preserve the orthogonality of the sub-carriers over the FFT processing interval in unequalized channels such as the WLAN multi-path channels.
[0025] A training sequence, or preamble, having a duration of 16 μsecs or 320 sample points is illustrated in FIG. 2. FIG. 2 illustrates the packet preamble specified by the standard for synchronization and channel compensation. The sequence is shown consisting of a short OFDM sync symbol 201 of 0.8 μsecs or 16 sample points in duration, which is repeated 9 times, and a long OFDM sync symbol 203 of 3.2 μsecs or 64 sample points in duration, which is repeated once as sync symbol 205. A 1.6 μsec or 32-point duration guard interval 207 (which is just the second half of the points of the long sync symbol) is appended as a cyclic prefix to the long symbol pair. The short symbols may consist of 12 QPSK modulated sub-carriers as indicated in FIG. 3A, and the long symbols may consist of 52 BPSK modulated sub-carriers as indicated in FIG. 3B. Both long and short symbols may be generated using a 64-sample point IFFT with 12 prescribed modulation values and 52 nulls for the short symbols and 52 prescribed modulation values and 12 nulls for the long symbols. However, because the preamble is preferably identical for all packets, the discrete time sample values may be pre-computed and stored at the transmitter.
[0026] Initial Timing and Fine Fractional Frequency Offset Estimates
[0027] The digital synchronization circuitry of a preferred embodiment derives synchronization information from the preamble using an iterative process. Preferably, during a first iteration a digital cross-correlator 401, as shown in FIG. 4A, detects an incoming packet on input 407. The correlator is designed to utilize the maximum available coherent energy in the preamble for detection and to generate a sharp peak for an initial symbol-timing estimate. Carrier frequency offset is measured in terms of sub-carrier frequency spacing Δf (312.5 KHz). The frequency offset consists of an integer value and a fractional value. For example a value of −1.6 corresponds to a carrier frequency offset of −1.6*Δf (−1.6*312.5=−500 KHz).
[0028] In the preferred embodiment, the cross-correlator operates on the incoming sample stream with a 3.2 μsec or 64-sample point delay of one symbol via delay 403. The delayed input 409 is correlated with direct input 407 by correlator 401. The correlation output 411 is aggregated by integrator 405.
[0029] The correlation and integration function is described in more detail in FIGS. 4B-D. The direct input signal is represented in FIG. 4D comprising short sync symbols 415 followed by long sync symbols 420. The delayed input signal is represented in FIG. 4C where the short symbols 425 and long symbols 430 are shown preferably delayed 64 sample points. A preferred integration time is 9.6 μseconds or 192 sample points, preferably consisting of two 96-point intervals separated by 64 sample points and represented in FIG. 4B. When the last sample value of the long sync symbols at sample point 320, or the last point of the preamble sequence, enters the correlator's direct path, the correlation reaches a peak value.
[0030] The short sync symbols are periodic with a period of 16 and the first 96 overlapping symbols integrated by the correlator consist of the first 6 periods of the delayed input and the last six periods of the direct input. The long sync symbols are periodic with a period of 64. However, the two long sync symbols are preceded by the cyclic prefix 430 that consists of the last 32 samples of a long sync symbol. Thus at sample point 320 the last 96 overlapping points integrated in the correlator are 64 sample points of the first period of the long sync 431 in the delayed input and 64 sample points of the second period of the long sync 422 in the direct input plus the second half of the first long sync symbol 420 in the direct input and the cyclic prefix in the delayed input 430 which is identical to the second half of a long sync symbol.
[0031] The expected value of the magnitude of the correlator output is shown in FIG. 5. The correlator has a processing gain of 192 (22.8 dB), the greatest that can be achieved under the WLAN standard. A peak detector can recognize the peak, and the peak's location provides an initial estimate of symbol timing. In a preferred embodiment, to prevent inter-symbol interference, the initial timing estimate is back biased, for example by 2 sample points (100 nsecs), to assure that the symbol sampling interval will commence at the end of the symbol guard interval and not after the beginning of the processing interval.
[0032] The aforementioned cross-correlator 401 preferably utilizes complex numbers to compute correlation. The complex numbers preferably have in-phase sample values as their real parts and quadrature sample values as their imaginary parts. The output of the correlator at each instant consists of a magnitude and a phase value. The aforementioned peak value is in fact a peak in the magnitude of the correlator output. The phase value of the correlator output at the instant of the peak measures the fractional amount of frequency offset of the sub-carriers.
[0033] Integer Frequency Offset Estimates and Channel Estimation
[0034] In a preferred embodiment, a second iteration is now performed, using the sample values from the long sync symbols with the timing of the first sample value determined by the initial timing estimate.
[0035] Reference is made to FIG. 6. The sample values 600 are frequency shifted by the fine fractional frequency estimate 610 obtained in the first iteration so that any residual frequency offset will be an integer multiple of the sub-carrier frequency spacing. The corrected symbol sample values 615 of the two long sync symbols are now demodulated using the receiver's demodulation circuitry preferably consisting of a fast Fourier transform (FFT) 620. The 64 sub-carrier modulation values of the first symbol are averaged with the 64 sub-carrier values of the second symbol for noise reduction. If there is no integer frequency offset, the 53 modulated sub-carriers (52 BPSK modulated carriers plus the DC null value sub-carrier) of this set will correspond to digital frequency numbers −26 thru +26. If there is an integer offset then they are shifted to digital frequency numbers −26 plus the offset thru +26 plus the offset. Sets of 53 sub-carrier modulation values with different offsets may be extracted from the 64 values to test for the integer offset. Each of these sets of 53 sub-carrier modulation values is divided by the known BPSK sub-carrier modulation values of the long sync symbol creating an estimate of the channel transfer function for each offset to be tested.
[0036] In a preferred embodiment, each of these estimates of the channel transfer function is processed in the following manner. First, the values corresponding to even sub-carrier numbers are used to create an interpolated estimate of the values corresponding to the odd sub-carrier numbers. These estimated odd numbered sub-carrier values may be correlated with the actual odd numbered sub-carrier values for each of the channel estimates. With reference to FIG. 7, for the channel estimate corresponding to the correct value of integer offset, very high correlation occurs due to the fact that the channel does not change randomly between adjacent carriers. As a result, it is possible to accurately predict the integer offset value given such values at nearby frequencies. For channel estimates corresponding to incorrect values of integer offset there is no correlation because in the division operation by the known BPSK values of the long sync, sub-carrier values are divided by modulation values corresponding to other sub-carriers. This quotient represents a random noise-like estimate for the channel where the odd numbered and even numbered values are completely uncorrelated. In these cases of random noise-like estimates, the correlation with the actual odd numbered values has an average value of zero. This approach not only reveals which integer carrier frequency offset is the correct one but also estimates the channel transfer function.
[0037] The range of the estimate for integer frequency offset is in principle unlimited. In practice, the range is limited by the IF bandwidth and/or the FFT size. In the preferred embodiment, the range for the integer value of frequency offset is �6 or a maximum carrier frequency offset of �1.875 MHz and is limited by the FFT size of 64. One of the advantages of the algorithm herein disclosed is that the algorithm offers the greatest range of all known carrier frequency offsets. Furthermore, the algorithm provides for maximum accuracy due to the high gain of the correlation operation. Standard carrier frequency offset algorithms use the short sync symbols to extend their range, but only to �2 or a maximum allowed carrier frequency offset of �625 KHz. Also, standard algorithms have less accuracy due to the lower gain in their correlators. The total frequency offset, consisting of fractional plus integer parts, is applied as a correction to the OFDM data symbols in the packet prior to demodulation and decoding.
[0038] The IEEE 802.11 standard specifies coherent demodulation for the OFDM subcarriers at the WLAN receivers. Any phase shift suffered by the sub-carriers in transmission must be corrected at the receiver. Also, because higher data rates use 16-QAM or 64-QAM modulation, amplitude variations introduced in transmission must also be corrected. The channel transfer function is required to provide for the combination of the multi-path propagation channel and all linear filter transfer functions in the WLAN transceiver and any residual symbol timing error. This required channel transfer function is in fact the channel transfer function corresponding to the correct integer frequency offset determined during the processing described above. This estimated transfer function is used to correct the sub-carrier amplitudes and phases following FFT demodulation and prior to decoding.
[0039] In an alternate preferred embodiment of the present invention, the channel transfer function estimate is continually updated during the packet reception using pilot tone information in order to correct for cumulating sampling clock error and any residual frequency offset error as described below.
[0040] Pilot Tone Tracking
[0041] In a preferred embodiment of the present invention, pilot tones are inserted in each OFDM data symbol at sub-carrier numbers �7 (�2.1875 MHz) and �21 (�6.5625 MHz) relative to the RF center frequency. These four sub-carriers are modulated with BPSK modulation values from a known PN sequence so that phase changes from data symbol to data symbol occurs in a prescribed manner known at the receiver. Phase changes from these known values are derived from the demodulation sequences extracted from the FFT outputs at the receiver. Phase changes may be used to track and correct for phase error buildup that may occur during the packet transmission and processing. Phase error may buildup during the packet due to at least three causes: (1) residual error in the carrier frequency offset estimate, (2) error between the sampling clock rates (20 MHz) of the transmitter and receiver and, (3) slow variations in the channel.
[0042] The maximum packet length that is permitted by the OFDM PHY layer WLAN standard is 1365 OFDM symbols (109200 sample points, or 5460 μsecs). In practice, although the first OFDM data symbol in the packet can be decoded with a residual carrier frequency offset error of ��10−2 of the sub-carrier frequency spacing, the error must be less than �10−5 of the sub-carrier frequency spacing in order to correctly decode the final OFDM symbol because of the phase error buildup induced by carrier frequency offset. Similarly, the sampling clock rates (20 MHz) must be equivalent within spacing tolerances less than �0.4�10−6 MHz. Slow variations may occur in the channel although such variations are assumed to be very slow as compared to the length of the OFDM packets because the transmitters and receivers are generally and relatively fixed in their locations during operation (albeit, and not withstanding, the transmitters and receivers may be in the form of portable devices). Therefore, the main purpose of the pilot tone tracking is to eliminate phase error buildup due to very small frequency errors.
[0043] Pilot tones are generally used for synchronization and control purposes. The flow chart in FIG. 8 represents a tracking sequence based on pilot tones. The pilot tone tracking loop represents an estimation of phase change based on known transmitted pilot tone phases. Tracking the phase change based on OFDM symbols as described hereunder can be used to update symbol timing estimates for subsequent OFDM symbols. The pilot tone tracking is preferably represented by a first order digital tracking loop. The phase change of the pilot tones versus pilot tone sub-carrier frequency is obtained for each OFDM symbol from the FFT outputs at step 803 and the known transmitted pilot tone phases at step 805. In one approach, and as shown at step 811, a least squares fit is made to a straight line of phase change versus sub-carrier frequency using the demodulated phase change values of the four pilot sub-carriers. For such a case, the zero order term of this line will be the average phase change of the pilots and provides an updated estimate of any error in carrier frequency offset. A new estimate of frequency offset is obtained from the previous value at step 813 and the error and is applied to the subsequent OFDM symbol at step 815 prior to demodulation. The coefficient of the first order term of the line fit to the data is the average slope of the phase change versus sub-carrier frequency at step 809. This ratio determines any timing error that is accumulating due to mismatch in the sampling clock frequencies between the transmitter and receiver and is used to update the symbol-timing estimate for the subsequent OFDM symbol. In the preferred embodiment the update of the symbol-timing estimate is accomplished by updating the channel frequency response estimate at step 807. In an alternate embodiment, it is accomplished by slipping the OFDM symbol sampling clock initial starting sample.
[0044] The transmission system normally requires automatic gain control (AGC) to bring the signal level within dynamic range of the receiver. The rapidly changing gain of the AGC during the first several short sync symbols may cause the signal detection threshold to be exceeded prematurely. Blocking the signal inputs to the cross-correlator when the gain is changing too rapidly will prevent the signal detection threshold from being exceeded prematurely. A high rate of AGC gain change can be detected by monitoring the AGC error signal.
[0045] Mathematical details and representations of the foregoing are now provided. The short sync signal repeats itself every 16-sample points. A 64-point IFFT of a modulation sequence with non-zero values at every fourth sub-carrier will generate four periods of the short sync. Repeating this sequence 1.5 times generates the ten repetitions of the short sync of 160 sample points. The short sync may be described mathematically by its complex modulation envelope:
xs(n)=Σxs k(n),0≦n≦2.5*N−1,(N=64) (1)
[0046] where
xs k(n)=(2/N)exp[j(2πk(4)n/N+φk)],−6≦k≦6,k≠0 (2)
[0047] and φk are the QPSK phases defined in FIG. 3. Note from (1) and (2) that
xs(n)=xs(n−N/4),N/4≦n≦2.5N−1 (3)
[0048] so that xs(n) repeats ten times in 2.5N=160 sample points.
[0049] The long sync may be described mathematically by its complex modulation envelope
xl(n)=Σxl k(n),−N/2≦n≦2N−1,(N=64) (4)
[0050] where
xl k(n)=(1/N)exp[j(2πkn/N+φ k)],−26≦k≦26,k≠0 (5)
[0051] and φk are the BPSK phases defined in FIG. 3. Note from (4) and (5) that
xl(n)=xl(n−N),N/2≦n≦2N−1,(N=64) (6)
[0052] so that xl(n) repeats 2 times in the 2N points from 0≦n≦2N−1. Furthermore, xl(n) from −N/2≦n≦−1 is identical to xl(n) from N/2≦n≦N−1 and to to xl(n) from 3N/2≦n≦2N−1 . That is, the first 32 points of xl(n) are a cyclic prefix of the basic N point IFFT xl(n).
[0053] The entire preamble may now defined by the 5N=320 sample point sequence
xsync=xs(n)+xl(n−3N),0≦n≦5N−1 (7)
[0054] The initial step of the detection and frequency/timing recovery process is to compute the correlation between the incoming signal samples and the same samples with a delay of N sample points. The integration window of the correlator consists of two intervals. The first integration interval is over the most recent 1.5N=96 points to enter the correlator. This interval is from point n to point n-95. The second portion of the integration interval also consists of 1.5N=96 points but includes those points beginning with the point entering the correlator 160 points earlier. This integration interval is from point n-160 to point n-255, as shown in FIG. 4. Consider this process applied to (7). The cross-correlation obtained at sample point 2.5N−1=159 reaches a local maximum given by
r12(2.5N−1)=72/N (8)
[0055] which is the energy in six periods of the short sync xs(n). This local maximum is succeeded by a global maximum at sample point 5N−1=319 given by
r12(5N−1)=72/N+78/N=150/N (9)
[0056] which is the energy in six periods of the short sync xs(n) plus the energy in 1.5 periods of the long sync xl(n).
[0057] At sample point 7.5N−1=479, the correlator output of a preferred embodiment of the invention reaches another local maximum given by
r12(7.5N−1)=78/N (10)
[0058] which is the energy in 1.5 periods of the long sync. In between the maxima, the correlator output follows a triangular function with a base of 192 sample points (see FIG. 5). A threshold is set halfway between the local maxima and the global maximum with a value r12TH=114/N. Exceeding this threshold provides detection of an incoming packet. The sample point number of the global maximum (sample point 319 in the absence of error) provides the initial estimate for symbol timing.
[0059] The present invention accommodates the situation where the received signal has been subjected to an unknown amount of frequency shift offset. For example, assume the sampled frequency shifted signal is
ysync(n)=xsync(n)exp[j2π(p+ε)n/N],0≦n≦5N−1 (11 )
[0060] where
δf=(p+ε)Δf (12)
[0061] is the frequency offset and Δf is the sub-carrier spacing (312.5 KHz). The integer p gives frequency offset to the nearest sub-carrier and
−�≦ε≦� (13)
[0062] is the fractional frequency offset. Returning now to the cross-correlator output, at sample point 5N−1 after the signal enters the receiver, the output is given by
r12(5N−1)=(150/N)exp[j2πε]. (14)
[0063] The magnitude of the output, as in (9), is the peak magnitude of the correlation and provides both detection and an initial estimate of the sample timing whereas the phase of the correlation according to its principal value between −π and π determines the fractional frequency offset ε between −� and �.
[0064] Assume the OFDM packet has been sent through a linear multi-path channel that introduces signal distortion in addition to introducing a frequency offset. This situation will be the case, for example, in WLAN in-door channels. For channels with an impulse response of length Nh sample points, the peak of the expected value of the correlation function output, which is still given by (14), may be shifted to lie between sample point 5N−1 and 5N+Nh−1. In practice it has been demonstrated that for the exponential decaying in-door WLAN channels the actual peak lies no greater than two sample points (100 nsecs) past 5N−1.
[0065] Due to the wide base of the triangular correlation function and the finite length of the channel impulse response, the sample timing offset is subject to an error of one or two samples. This error is normally biased to be greater than the true value due to the channel impulse response as mentioned above. In order to compensate for this delay in the peak, the initial timing estimate is back biased to a smaller value so that the symbol timing estimate for initiating the extraction of the first symbols will never exceed the correct value of, in this case, N=320. An error in the estimate that causes the symbol extraction to begin late, introduces inter-symbol interference (ISI). ISI occurs because the FFT processing interval will overlap the subsequent symbol. However, an error that causes the symbol extraction to begin early does not introduce (ISI) because of the guard interval. The associated timing shift if present is accommodated as part of the channel compensation. A bias of two sample points, say 100 nsecs, has been selected as optimum for the indoor WLAN channels.
[0066] The initial stage of the synchronization process described above has not resolved the integer frequency offset p. The second stage of the frequency/timing recovery process is used to determine p and to obtain an initial estimate of the channel transfer function. In a preferred embodiment and based on the initial timing estimate I, 2N long sync samples are extracted from the stored data stream. Preferably, there are 4N previous samples always stored in memory to support the correlation calculation associated with the initial stage of the processing, so there are no additional requirements for memory imposed by this process. Next, as shown in FIG. 6, these 2N samples are corrected by the estimated value of the fractional frequency offset ε using the algorithm:
ye(n)=y(n)exp[−j2πεn/N],I−2N≦n≦I−1, (15)
[0067] where I (nominally I=5N=320) is the sample number of the first sample in the symbol following the preamble as determined by the initial timing estimate. Now from (11)
ye(n)=xsync(n)exp[j2πpn/N],I−2N≦n≦I−1, (16)
[0068] so that the signal now consists of two periods of the long sync sequence offset by the integer frequency p:
yl(n)=xl(n)exp[j2πpn/N],I−5N≦n≦I−3N−1 (17)
[0069] Comparing (17) with (4) and (5) we see it is composed of the offset set of sub-carriers
y k(n)=(1/N)exp[j(2π(k+p)n/N+φ k)],−26≦k≦26, k≠0. (18)
[0070] Next, and as shown in FIG. 6, the Fourier coefficients are preferably extracted using the N point FFT digital circuitry of the OFDM demodulator on the intervals I−5N≦n≦I−4N−1 and I−4N≦n≦I−3N−1 which, with the exception of timing error I−5N correspond to the two periods of the long sync. In the absence of noise, the N coefficients from both intervals are identical and are given by
Y k=exp(jφ k−p)exp(j2πk(I−5N)/N),−26+p≦k≦26+p,k+p≠0.=0,−N/2≦k≦−26+p−1,26+p+1≦K≦N/2−1,k+p=0. (19)
[0071] In a preferred embodiment, the coefficients from the two intervals are averaged for noise reduction and (19) generates the expected values for the coefficients. Except for the linear phase shift introduced by any residual timing error I−5N, the Fourier coefficient sequence {Yk} of N (N=64) values is the long sync BPSK modulation sequence {exp(jφk)} of 52 values shifted by p in relation to its nominal centered location in the {Yk} sequence.
[0072] In practice, the multi-path channel may introduce additional phase shifts and amplitude variations onto the sub-carriers. Therefore the known BPSK modulation sequence {exp(jφk)} will have been modified by the unknown channel transfer function H(k). Therefore, (19) becomes
Y k =H(k−p)*exp(jφ k−p),−26+p≦k≦26+p,k+p≠0.=0,−N/2≦k≦−26+p−1,26+p+1≦k≦N/2−1,k+p=0. (20)
[0073] where we have incorporated the phase shift due to timing error I−5N into the unknown channel response H(k). Next, the set of P=2pmax+1 shifted test sequences of 52 received modulation values are formed as
Z k,p′ =Y k+p′ =H(k−(p−p′))*exp(jφ k−(p−p′)),−p max ≦p′≦p max,−26≦k≦26,k≠0. (21)
[0074] A channel estimate for each p′ may be derived by multiplying the test sequences by the complex conjugate of the known modulation sequence
H p′(k)=Z k,p′exp(−jφ k)=H(k−(p−p′))*exp(j(φk−(p−p′)−φk)), where −p max ≦p′≦p max,−26≦k≦26,k≠0. (22)
[0075] Clearly when p′=p, Hp′(k)=H(k), the channel impulse response. When p′≠p, then
H p′(k)=H(k−(p−p′))exp(j(φk−(p−p′)−φk))=H(k−(p−p′))exp(jλ k) (23)
[0076] where λk=φk−(p−p′)−φk are uncorrelated and are equally likely to be 0 or π. Now insert a value for the DC term
H p′(0)=[H p′(−1)+Hp′(1)]/2
[0077] in order to obtain 53 sample point sequences for Hp′(k) for 26≦k≦26. The sequences of interpolated odd values of Hp′(k) is as follows:
H p′,odd/int(k)=[H p′(k−1)+H p′(k+1)]/2, k=−25,−23, −21, . . . 23,25 (24)
[0078] where the actual observed odd value sequence is
H p′,odd(k)=H p′(k),k=−25,−23,−21, . . . 23,25 (25)
[0079] Each of the interpolated sequences are correlated with the actual odd value sequences for each value of p′ according to (See FIG. 7):
R p′ =ΣH p′,odd(k)H* p′,odd/int(k). (26)
[0080] First consider the case where p′=p, the correct offset. In this case Hp′(k)=H(k) and Hodd(k)≅Hodd/int(k), as the channel transfer function does not change randomly between adjacent sub-carriers. Accordingly, the channel response at an intermediate frequency can be accurately estimated from the response at adjacent nearby frequencies. In any event, a more accurate interpolation algorithm can be used than (24), if necessary. Therefore
R p =ΣH odd(k)H* odd/int(k).≅ΣH odd(k)H* odd(k). =26|H| 2 avg (27)
[0081] since there are 26 odd frequencies. Now consider the case where p′≠p. In this event Hp′(k)=H(k−(p−p′))exp(jλk). For simplicity, assume that channel transfer function H(k) is unity. Then Hp′(k)=Ik where Ik=exp(jλk) are uncorrelated zero mean random variables with values �1 and variance one. Thus the interpolated odd values are
H p′,odd/int(k)=[H p′(k−1)+H p′(k+1)]/2=[I k−1 +I k+1]/2, where k=−25,−23,−21 . . . 23,25 (28)
[0082] and the actual odd values are
H p′,odd(k)=Ik ,k=−25,−23,−21 . . . 23,25 (29)
[0083] from which one finds that
E{R p′ }=ΣE{I k(I k−1 +I k+1)/2}=0 (30)
[0084] and
Var{R p′}=26/2 (31)
[0085] In a non-unity gain channel, the variance is
Var{R p′}≈(26/2)|H| 4 avg (32)
[0086] so that the signal-sidelobe-ratio of the correlation to determine p is
SNR=R p 2/Var{R p′}=52 (33)
[0087] Having determined the correct value for frequency offset p, the best estimate of the channel transfer function based on the two long sync symbols is simply that corresponding to p, that is
H(k)=H p(k). (34)
[0088] There will be some residual carrier frequency offset due to error in the estimate obtained by processing the preamble as described above. Let m=0,1,2 . . . M−1 designate the OFDM data symbol number in an M symbol packet. Then
p km(n)=exp(j2πnk/N+φ km)*exp(j 2π( n+mN s +N g)εres /N),n=0,1, . . . N−1 (35)
[0089] describes the pilot tone of frequency k (k=−21, −7, 7, 21) during OFDM data symbol number m during its processing interval of N points ( Ns=N+Ng=80 sample points). The BPSK pilot tone phases {φkm} are known at the receiver. Now suppose we have an estimate εm of εres at the beginning of this symbol so we correct the pilot tones and all the sub-carriers in the packet by the estimate such that
pcorrkm(n)=exp[j(2πnk/N+φ km)]*exp[j2π(n+mN s +N g)(εres−εm)/N],n=0,1, . . . N−1. (36)
[0090] The FFT coefficients of the pilot tones for OFDM data symbol m are
P km=exp[jφ km)]*exp [jπ(1+2(mN s +N g)/N−1/N)(εres−εm)]*sin[π(εres−εm)]/{Nsin[π(εres−εm)/N ]}. (37)
[0091] Removing the known pilot tone phases φkm we find that each pilot tone has a phase offset
γkm=π[1+2(mN s +N g)/N−1/N](εres−εm) (38)
[0092] which is independent of sub-carrier number k. Note that without any further correction after the initial correction made during the synchronization process, the phase offset of the data sub-carriers as well as the pilot tones will accumulate with increasing symbol number m during the packet transmission. The phases of the four pilot tones are averaged for noise reduction according to
γm=(�)Σγkm (39)
[0093] and the remaining error in frequency offset is estimated from
errorm=(68 res−εm)=γm/π(1+2mN s /N−1/N). (40)
[0094] We use this error and our previous estimate to generate a new estimate for the m+1st symbol
εm+1=ε m+α*errorm (41)
[0095] which converges exponentially with increasing m to εres for α<1. An ideal value for α has been determined to be 0.707.
[0096] The frequencies of the sampling clocks at the transmitter and receiver may not be exactly the same. Let
Δt′=(1+η)Δt (42)
[0097] where fs=1/Δt is the sampling clock frequency of the transmitter and fs′=1/Δt′ is the sampling clock frequency at the receiver. This error in the sampling clocks creates a cumulative timing error in the pilot tones {k=−21,−7,7,21} so that during the processing interval of data symbol number m
p km(n)=exp(jφ km)*exp[j2πk{n(1+η)+η(mN s +N g)}/N],n=0,1, . . . N−1. (43)
[0098] The FFT coefficients of the pilot tones for OFDM data symbol m are
P km=exp[jφ km)]*exp[jπkη(1+2(mN s +N g)/N−1/N)]*sin(πkη/[N sin(πkη/N)]. (44)
[0099] Removing the known pilot tone phases φkm we find that each pilot tone has a phase offset
βkm =πkη[1+2(mN s +N g)/N−1/N] (45)
[0100] that is linearly dependent on sub-carrier number k and accumulates with increasing OFDM data symbol number m.
[0101] That is,
βkm=μm *k (46)
[0102] where
μm=πη[1+2(mN s +N g)/N−1/N]. (47)
[0103] Consequently, the pilot tones and therefore the data sub-carrier tones are subjected to a total phase shift
Θkm=γm+μm* k (48)
[0104] during OFDM data symbol number m. The pilot tone phases are subject to noise in addition to these systematic phase shift effects due to residual frequency offset error and sampling clock frequency error. Therefore a least squares estimate is obtained for γm and μm using the algorithms
γm=({fraction (1/4)})ΣΘ km (49)
[0105] and
μm=[Σ(Θkm−γm)*k]/[Σk 2]. (50)
[0106] The constant phase offset γm from (49) is used to correct the frequency offset for the subsequent symbol in accordance with (40) and (41). The slope of the phase shifts μm from (50) is used to correct the symbol timing for the subsequent symbol. This can be done in one of two ways. In the preferred embodiment, the channel compensation for the m+1st symbol is corrected from that used for the mth symbol in accordance with the algorithm
H(k)m+1 =H(k)mexp(−jkσ m+1),m=0,1, . . . ,M−1 (51)
[0107] where
σm+1=σm+κ*μm. (52)
[0108] Here H(k)0=H(k) from (34), the initial channel estimate obtained from the long sync signals and σ0=0. Because the slope estimate μm is quite noisy, it is found in practice that a small value of κ≈0.03 is optimum for correcting the sampling clock errors throughout the OFDM WLAN packets assuming sampling clock accuracy of 5 in 104 (20 MHz �10 KHz). Even the least expensive integrated clock circuits easily meet this requirement.
[0109] In an alternate embodiment, the sample timing error is monitored according to
Δn m=σm/(2π).tm (53)
[0110] The timing error is monitored and if |Δnm|>� the first sample point number for the following OFDM data symbol processing interval is slipped forward or backward by one according to whether Δn is negative or positive.
[0111] An advantage of the present invention is that the tracking loop errors depend only on phase change information of the pilot sub-carriers and its operation is independent of any amplitude variations that may occur to the pilots. The loop gain is kept less than one to assure stability in all noise environments.
[0112] The invention disclosed herein has a number of other distinct advantages over other OFDM WLAN synchronization systems and tracking systems. For example, the algorithms and methods described herein constitute an integrated system for initial synchronization and channel compensation using a known preamble. Additionally, the algorithm provides for continuous tracking and correction throughout the duration of the packet using the prescribed pilot tones. The combination of tracking and correction assures that each symbol in the packet is accurately synchronized and compensated prior to data decoding thereby providing a high level quality of service regardless of the packet length.
[0113] The cross-correlator used in the initial iteration of the digital synchronization circuitry has a gain of 22.8 dB. This is the highest gain achievable using the prescribed preamble. This gain is 10.8 dB greater than standard systems using the short sync symbols for detection and coarse carrier frequency offset estimation and 4.8 dB greater than standard systems using the long sync symbols for fine carrier frequency offset estimation. The high correlator gain means increased accuracy of the carrier frequency offset and symbol timing initial synchronization. It also means packet acquisition at 10.8 dB lower input signal-to-noise ratios than alternative techniques.
[0114] The correlator technique disclosed herein and employed for acquisition and initial synchronization is effectively immune to channel impairments, such as multi-path, because both the direct and delayed inputs to the cross-correlator 401 pass through the same channel. This resistance to channel impairments, in combination with the high gain of the correlator, makes the digital synchronization circuitry disclosed herein robust and fully capable of operating accurately and supporting data transmission using the OFDM WLAN Physical layer specified in the 802.11 standard.
[0115] A further advantage of the present invention is that the digital synchronization circuitry has a range for carrier frequency offset correction three times greater than competing techniques. The acquisition range of this circuit is in fact only limited by the size of the FFT in the receiver and the IF bandwidth. The offset correction can be made as large as required by increasing the size of the FFT in the receiver and the IF bandwidth. There is no effective loss of accuracy associated with achieving an increased acquisition range.
[0116] An additional feature of the present invention is that the pilot tone tracking circuitry can adjust each OFDM symbol for residual frequency offset error. The pilot tone tracking circuitry also adjusts each symbol for differences in the transmitter and receiver sampling rates (nominally 20 MHz) and/or residual symbol timing error.
[0117] The digital acquisition, synchronization and tracking circuitry herein disclosed provides robust and accurate synchronization of the carrier frequencies, the symbol sample timing and the sampling frequency clocks of the OFDM WLAN transmitters and receivers. In addition, it provides channel compensation for each OFDM sub-carrier of each symbol facilitating the required coherent demodulation of the OFDM sub-carriers at the receivers.
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We asked to Dalbello Team Rider Tiril Sjåstad Christiansen her injury, the Olympic Games and how is to have a personal snowpark!
- So What happened to your knee?
I was working on a trick i had just done a few times before, but i felt really confident and had a really good day, so i was just really unlucky and twisted my knee in the landing..
-And how is it know?
It´s getting better every day! I had surgery 3 and a half week ago, and it feels good. I just stared work out this week, and it cool to see progression every day.
- You went to the Olympic Games and then you decided to come back it must have been a tough decision…
It´s been the toughest decision i´ve made so far. When i crashed in aspen before christmas i thought i had no chance.. The doctors said 10 % chance and i had two months to get ready. It´s been the hardest, funniest and most emotional months in my life. Even if i was not able to compete in Sochi, it was all worth it, a 100 %.
- What you loved of the Olympic Games?
Everything, it was a really good experience for me. The course was sick, and i wish i was able to ski it more..
- Which athletes impressed u?
Joss Christensen! He haven´t been on top this two past years, but he showed the whole world how hard his been working. It was really fun and motivating to watch
- How is your daily routine now?
I wake up, go to the gym or physio, go to school, then back to the gym and then homework… haha!
- What's the plna of your summer?
I´m gonna work out A LOT, get really strong and healthy, hang out with friends and family and hopefully travel a bit, i miss that…
- And next season?
Ski, ski, ski!!
- Now u have a park, explain us how is born.
Yes!! My grandpa built the resort, and he was also an alpine racer. One of the tracks there are named after him, Asle løypa, and so my mom got a track named after her, Randi løypa. Now they wanted to name the park there after me, Tiril parken. When they asked me I was really honored. My grandpa was my idol and my hero.
- Where is it?
It´s in my home town Geilo.
- Which activities u want to promote there?
Our goal is to build the best park in Norway, and hopefully after a few years, the best park in Scandinavia. We´re trying to arrange camps and other activities through the winter and the summer.
Here Tiril's video of the X Games Tignes Slopestyle back in 2013:

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Actress Alyssa Milano's fashion line is expanding to include NASCAR items for men and women.
The multi-talented star talked about her latest venture and her new role on TV in the series, Mistresses.
Hollywood legend Joan Rivers, 81, remains in critical but stable condition at The Mount Sinai Hospital in Manhattan. The comedian and TV host was undergoing an outpatient procedure at a clinic on the Upper East Side on Thursday when she went into cardiac arrest and was rushed to the hospital. Her daughter, Melissa Rivers, issued a statement saying her mother was "resting comfortably."


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Next, you'll be asked to connect with Twitter. Rest assured, we will never send a tweet from your account
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DMO blog MY TORONTO feat. the city's travel bloggers: http://t.co/vet6zYCPJQ 1st up: @Breathedreamgo & @dobbernation via @TBEXevents
Candid thoughts on #TBEX Toronto from @nerdseyeview (what do YOU think?)
Here's @EnjoyIllinois at travel blogger speed dating in June at #TBEX Toronto. Who's going to TBEX Dublin Oct 3-4?
For #FriFotos #tasty: @Ninutik maple syrup cooled on snow & rolled onto artisanal cheese at #TBEX Toronto
RT @rebekahgrmela: Learning from the blog experts! @katiecook @SheilaS @TourismCollege #ttia14
RT @MackCollier: ONE HOUR till tonight's special #Blogchat with co-host @JoePulizzi discussing content strategy for your blog!
To show it's safe there, Thailand's tourism industry turned to social media & travel blogs via @4Hoteliers
No DMO too small to partner w/ @BrandUSA says @AEylon: they need your pics, videos, content, research data. #DMAI14
RT @SparksSocial: @MJManzanares @TourismCurrents @TripTelevision DMAI Social Networking Rockstars! #dmos #hotelmarketing
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The rationality of EIA forecasts under symmetric and asymmetric loss
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Bibliographic InfoPaper provided by University of California at Berkeley, Department of Agricultural and Resource Economics and Policy in its series CUDARE Working Paper Series with number 1009.
Length: 28 pages
Date of creation: 2005
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Handle: RePEc:are:cudare:1009
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Related researchKeywords: consumption; energy forecasting;
Other versions of this item:
- Auffhammer, Maximilian, 2007. "The rationality of EIA forecasts under symmetric and asymmetric loss," Resource and Energy Economics, Elsevier, vol. 29(2), pages 102-121, May.
ReferencesReferences listed on IDEAS
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- Tom Doan, . "ERSTEST: RATS procedure to perform Elliott-Rothenberg-Stock unit root tests," Statistical Software Components RTS00066, Boston College Department of Economics.
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8905, Michigan State - Econometrics and Economic Theory.
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CitationsCitations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
Cited by:
- Fischer, Carolyn & Herrnstadt, Evan & Morgenstern, Richard, 2009.
"Understanding errors in EIA projections of energy demand,"
Resource and Energy Economics,
Elsevier, vol. 31(3), pages 198-209, August.
- Fischer, Carolyn & Herrnstadt, Evan & Morgenstern, Richard D., 2008. "Understanding Errors in EIA Projections of Energy Demand," Discussion Papers dp-08-54, Resources For the Future.
- Pierdzioch, Christian & Rülke, Jan-Christoph & Stadtmann, Georg, 2012.
"Oil price forecasting under asymmetric loss,"
Discussion Papers
314, European University Viadrina Frankfurt (Oder), Department of Business Administration and Economics.
- Christian Pierdzioch & Jan-Christoph R�lke & Georg Stadtmann, 2013. "Oil price forecasting under asymmetric loss," Applied Economics, Taylor & Francis Journals, vol. 45(17), pages 2371-2379, June.
- Christian Pierdzioch & Jan C Rülke & Georg Stadtmann, 2012. "Forecasting the Dollar/British Pound Exchange Rate: Asymmetric Loss and Forecast Rationality," Economics Bulletin, AccessEcon, vol. 32(3), pages A213.
- Gholam Hossein Hasantash & Hamidreza Mostafaei & Shaghayegh Kordnoori, 2012. "Modelling the Errors of EIA's Oil Prices and Production Forecasts by the Grey Markov Model," International Journal of Economics and Financial Issues, Econjournals, vol. 2(3), pages 312-319.
- Pierdzioch, Christian & Rülke, Jan-Christoph & Stadtmann, Georg, 2012. "On the loss function of the Bank of Canada: A note," Economics Letters, Elsevier, vol. 115(2), pages 155-159.
- Mamatzakis, E. & Koutsomanoli-Filippaki, A., 2014. "Testing the rationality of DOE's energy price forecasts under asymmetric loss preferences," Energy Policy, Elsevier, vol. 68(C), pages 567-575.
- Andrea Bastianin & Matteo Manera & Anil Markandya & Elisa Scarpa, 2011. "Oil Price Forecast Evaluation with Flexible Loss Functions," Working Papers 2011.91, Fondazione Eni Enrico Mattei.
- Christoph Jeßberger, 2011. "Multilateral Environmental Agreements up to 2050: Are They Sustainable Enough?," Ifo Working Paper Series Ifo Working Paper No. 98, Ifo Institute for Economic Research at the University of Munich.
- Wilkerson, Jordan T. & Cullenward, Danny & Davidian, Danielle & Weyant, John P., 2013. "End use technology choice in the National Energy Modeling System (NEMS): An analysis of the residential and commercial building sectors," Energy Economics, Elsevier, vol. 40(C), pages 773-784.
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text: Please advise how to overclock Kingston HyperX T1 KHX8500D2T1K2/4G? Tags: Memory Overclocking Last response: June 30, 2011 5:45 PM in Overclocking Share midarkmind June 29, 2011 3:52:58 PM Hello, I am a pretty much a noob when it comes to overclocking RAM. I recently figured out how to overclock my AMD Phenom II X4 945 with FSB tweaking, so I thought that I would give this a try too. I am worried about voltage and timings primarily. I have no problem with trial and error, but when it comes to raising voltages I don't want to take many risks. So I hope there is a guru out there that can give me some advise. I have only one gaming rig right now and I don't want to accidentally burn sh$% up! I have tried to simply overclock them by raising the RAM multiplier in the bios, however they have never responded well to this and run unstable or causes the system not to post at all. I then have to tear out my cross-fire setup to get to the reset pin on the mobo. Not fun. =( Anyone know a concrete or abstract guide that will basically walk through this? Or am I hoping for too much? My PC specs: AMD Phenom II X4 945 Deneb (Overclocked 3.6GHz) Socket AM3 125W Quad-Core Processor GIGABYTE GA-MA785G-UD3H AM3/AM2+/AM2 AMD 785G HDMI ATX AMD Motherboard COOLER MASTER V8 RR-UV8-XBU1-GP 120mm Rifle CPU Cooler Intel Core i7 compatible 2 Cross-fired MSI R5770 Hawk Radeon HD 5770 1GB 128-bit GDDR5 PCI Express 2.1 x16 HDCP Ready CrossFireX Support Video Card 2 Western Digital Caviar Black WD1001FALS 1TB 7200 RPM SATA 3.0Gb/s 3.5" Internal Hard Drive -Bare Drive Kingston HyperX T1 Series 8GB 240-Pin DDR2 SDRAM DDR2 1066 (PC2 8500) Dual Channel Kit Desktop Memory Model KHX8500D2T1K2/4G XION Power Real XON-1250P14HE 1250W ATX 12V v2.2 / EPS 12V v2.91 / SSI v2.92 SLI 3 ways SLI supported. More about : advise overclock kingston hyperx khx8500d2t1k2 venjhammet a b K Overclocking June 30, 2011 4:51:16 PM Well as for my opinion it would be safe just overclock the cpu and never the ram. midarkmind June 30, 2011 5:45:28 PM Ok thanx. I think I figured it out with the timing and voltages. Its not all that complicated. I just need to do trial and error, until it works. I guess I am unsure if I can even overclock because I have 8GB. I don't know if its possible because once you add 8GB it automatically down clocks the ram from 1066mhz with 4GB to 800mhz with 8GB. I guess it never hurts to try. I think it could be possible. However it gets complicated because CPU has a locked multiplier. As soon as I change timings and increase voltage I will need to increase the speed with FSB to raise the speed of the RAM. Hope it goes well. Can't find your answer ? Ask ! Publish Related resources how to overclock KINGSTON HYPERX PC3500 512MB? Forum More resources Read discussions in other Overclocking categories CPUs Motherboards Memory Chipsets Heatsinks Water Cooling Intel AMD !


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MORE than 100 golfers took part in the Sandhurst Cricket Club Golf Day on Sunday.
The four-person ambrose event was held at the Bendigo Golf Club.
Craig Howard, Shane Hogan, Kevin Bowe and Shane Coutts won with a score of 64-10 1/2- 53 1/2.
Jeremy Darmody, Richard Graham, Darren McGregor and Colin Runnalls came in second with a score of 63-8 7/8-54 1/8.
The Gross winners were Craig Burn, Darren Hunt, Lawrence Thomas and Geoff Fankhauser.
The team scored 62 on a count back.
Gross runners-up were Mark Holland, Dale Oddy, Robbie Wells and Aaron McLean.
Nearest to pins were Shane Hogan on the fifth hole, Jamie Polinelli on the 12th hole and Ben Deacon on the 17th.
The straightest drive was Peter Fyffe.
The $1000 dragon draw winner was Ross Johnson.


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GaveKal is a financial services firm that offers institutional investors and high net worth individuals three different services: fund management, independent research on global macro-economic trends and events, and independent advisory work on China and its impact on the global economy.
After receiving his bachelor's degree from Duke University and studying Mandarin at Nanjing University, Louis-Vincent joined the French Army where he served as a second lieutenant in a mountain infantry battalion. After a couple of years, Louis-Vincent left the army and joined Paribas Capital Markets where he worked as a financial analyst first in Paris, then in Hong Kong. Louis-Vincent left Paribas in 1999 to launch GaveKal Research with Charles and Anatole in London. In 2002, Louis-Vincent left the London office and returned to Hong Kong; the idea at the time was that Asia was set to become an ever more important factor in global growth and that, consequently, GaveKal needed to offer its clients more information, and more ideas, relating to Asia. Louis-Vincent contributes frequently to the research and was the main author of our books Our Brave New World and The End is Not Nigh. Louis-Vincent is CEO of Gavekal and MW GaveKal. Louis-Vincent is a French national. He speaks French and English and is rusty in Spanish and Mandarin. He is married with two sons and one daughter.
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This document records the king's reconstructive work at the cities of Sippar, Ururk, Ur, Borsippa, Larsa, and Dilbat. It also commemorates the king's repairing of the temple of Ninkarrak (a form of the healing Goddess of Gula) at the city of Babylon. From Babylon (modern Babel Governorate), neo-Babylonian era, 604-562 BCE, Mesopotamia, Iraq. (The British Museum, London).
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Let's Learn From the Past: 1971 PiratesAugust 8, 2013 12:00 AM
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From the Negro League dominance of the Pittsburgh Crawfords and Homestead Grays to the Pittsburgh Courier's unparalleled coverage of African-American athletes, Western Pennsylvania has been home to many significant achievements by minorities in sports.
In 1971, the Pittsburgh Pirates continued this groundbreaking tradition by producing the first all-minority starting lineup in Major League Baseball, more than 24 years after Jackie Robinson broke the color barrier.
Before the Pirates' home game against Philadelphia on Sept. 1, Pirates manager Danny Murtaugh filled in a starting lineup card that consisted of nine African-American and Latino players.
The lineup included future Hall of Famers Roberto Clemente and Willie Stargell as well as All-Stars such as catcher Manny Sanguillen and pitcher Dock Ellis.
The all-minority lineup fared well for the Pirates, as they defeated the Phillies, 10-7, after trailing early in the game.
The game didn't get much recognition at the time in Pittsburgh, mostly because the Pittsburgh newspapers were on strike. However, in Philadelphia, the Evening Bulletin ran a story headlined, "Pirates Starters All Black" and quoted Murtaugh as saying, "When it comes to making out the lineup, I'm color blind and my athletes know it."
The Pirates finished the 1971 regular season with a 97-65 record and defeated the San Francisco Giants three games to one to win the National League Championship and a trip to the World Series for the first time since 1960.
Pittsburgh took on the defending champion Baltimore Orioles, who had just finished their third consecutive season with more than 100 wins to cement their third straight trip to the World Series.
The Pirates, led by many of the same players from the all-minority lineup in September, upset the Orioles, 2-1, in the decisive Game 7 in Baltimore.
Clemente was named Most Valuable Player of the World Series, the first Latin-American to win the award. He batted .414 and equaled his performance in the 1960 World Series with at least one hit in all seven of the series' games.
Today, the 1971 Pittsburgh Pirates, whose roster included 14 whites, six African-Americans and seven Latin-Americans, are considered trailblazers in the sport as their success encouraged the signings of more minority players in Major League Baseball.
Visitors to the History Center can see memorabilia from the 1971 Pittsburgh Pirates championship team, including a glove used by pitcher Steve Blass in Game 7 of the World Series, in the Western Pennsylvania Sports Museum. For more information, visit www.heinzhistorycenter.org.lifestyle


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English. 41052 members
English. 275 members
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It's a hunter hut.
It seems to me someone wanted to show typical russian live) - with drinking vodka, playing the
I have a lot of hobbies. I go to the gym and a swimming pool, play the guitar and read books.
I can see a multycoloured cap standing on a plate - it's colors is rather funny. On the second picture I can see a
It can seem very strange but in this moment I am doing this exercise
The shopping is very impotant part of my life - I do exerciese and I need a lot of good fresh products so I do it


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A collection of news and information related to Breckin Meyer published by this site and its partners.
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Displaying items 23-33
Wednesday's TV Highlights: 'Raising Fame' on TLC
TV Picks: 'The Fall,' Nik Wallenda, 'Homegoings,' legal dramedies
Mark-Paul Gosselaar, Breckin Meyer get naked for Heather Locklear
Heather Locklear shakes up 'Franklin & Bash'
Star Trek doc is a family enterprise
Cubs' Ryan Dempster puts teammates to work at casino-themed benefit, Charlie Sheen coming to Chicago, Tom Morello vs. Rahm Emanuel
TCA Press Tour 2011: No More Bad Hair Days for Mark-Paul GosselaarStory | Jan 6, 2011 | 4:34 PMWhen Mark-Paul Gosselaar starred in his last series, 2008's "Raising the Bar," it wasn't his character of a driven public defender that attracted the most notice. It was his hair--an unkempt, grungy mullet that many viewers said looked greasy and......
TCA Press Tour 2011 Day 2: Dr. Drew, Mike Tyson and OprahStory | Jan 7, 2011 | 4:35 AMThe winter 2011 edition of the Television Critics' Assn. press tour got busy today in Pasadena with presentations from Turner Communications' networks, Playboy, BET, and Discovery Communications. Rob Corddry took the stage with the cast (including Megan...
TNT gears up for summer, announces upcoming premieresStory | Mar 10, 2011 | 4:35 PMTNT announces its summer schedule....
Wednesday's TV Highlights: 'Haunted Collector' on SyfyStory | May 31, 2011 | 9:37 PMClick here to download TV listings for the week of May 29 - June 4 in PDF format TV listings for the week of May 29 - June 4 in PDF format (from latimes.com/entertainment/news/tv) Weekly TV Listings and more can......
This week: NBA Finals start; 'America's Got Talent' returns; 'Franklin & Bash' debutsStory | May 30, 2011 | 4:37 PMReality television is the main staple this week. NBC had great ratings success with "America's Got Talent" last summer, and the talent contest starts a new season with two hours at 8 p.m. Tuesday. "The Voice," the NBC show drawing the most attention these...


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Saturday 16 Mar 2013 9:08p.m.
The Warriors produced a stirring comeback with three unanswered tries but fell agonisingly short to go down 16-14 to the Sydney Roosters in Auckland.
Down 16-0 with half an hour to go in the NRL clash at Eden Park on Saturday night, the Warriors scored through Shaun Johnson, Pita Godinet and Bill Tupou.
But Johnson mishit his sideline conversion of Tupou's 75th-minute try to leave the Warriors short and keep them winless after two rounds.
Much of the pre-match attention had been on Sonny Bill Williams and his second match back in the NRL.
Williams was again solid and did produce one magic late offload that led to centre Michael Jennings almost scoring what would have been the victory-sealing try.
Williams was also involved in a couple of skirmishes with young centre Konrad Hurrell that brought the crowd alive.
Early in the first half, he took exception to being pushed as he was running back into position for a restart.
The part-time boxer turned and shaped up to Hurrell, only to be held back by Warriors prop Sam Rapira.
Minutes later, when Hurrell ran at him, Williams pulled off what seemed like a fair strip of the ball, only for the Roosters to be penalised for a two-man tackle.
But two Hurrell errors early in the second half were to prove decisive.
A play-the-ball penalty gave the Roosters the field position to send substitute Mitchell Aubusson over with some nice footwork.
Hurrell then lost possession in a tackle deep in his own territory and the visitors again capitalised with centre Shaun Kenny-Dowall touching down for a 16-0 lead.
That was the signal for a Warriors fightback.
They received a break deep in defence when Roosters fullback Anthony Minichiello couldn't hold a pass and Johnson scored from an 80-metre run.
Godinet then produced a piece of individual magic to force his way over from dummy-half, with Johnson's conversion making it a six-point game.
Johnson was the architect when the home side scored again, his kick-ahead chased down by winger Bill Tupou.
Earlier, the Roosters split the game open in the fifth minute thanks to a superb sideline break by Jennings.
The ball was worked to Minichiello, who finished in fine style despite some frantic cover defence from Warriors skipper Simon Mannering.
James Maloney's conversion made it 6-0 and heralded a period of Roosters dominance.
The Warriors held out and then began to get into the contest, but they struggled to make their attack click.
Three times, they were in strong field position and three times, they were stopped by errors.
While the match attracted 32,740, the figure was lower than the attendances for the two previous Warriors fixtures at Eden Park, with the record being 38,412 two years ago.
Bronze for Snyders at PanPacs
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And I get it, a programming language is just a tool. Totally. But, at some point, that language we're comfortable with turns into a broken hammer and you're sitting there telling somebody you don't need a fucking screwdriver, man, because you can set this claw on the back of this hammer at a 45 degree angle and turn.
from Jeff Lembeck via The Pastry Box


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Sarah Jessica Parker plays an extreme multitasker in this mild comedy based on Allison Pearson's novel. A Boston-based businesswoman, wife and mother, the central character finds herself overwhelmed ... all the more so when she lands a big new account that gives her a new peer (Pierce Brosnan) who's both interesting and interested.
Greg Kinnear plays her spouse in a cast that also includes Kelsey Grammer, Christina Hendricks ("Mad Men") and "Saturday Night Live" veterans Seth Meyers and Jane Curtin. DVD extra: interview with Pearson. ** (PG-13: AS) (Also on Blu-ray and Movies on Demand)
Buy 'I Don't Know How She Does It' on DVD.
DVD review
'I Don't Know How She Does It'
Christina Hendricks (The Weinstein Company)
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Food Matters – Hawaiian 'Vacation'
When my colleagues at All Points West found out I was headed to Hawaii for a little vacation this week, they were a little jealous. So they decided to make me do some work. But it's the kind of work … Continue reading
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The Wall Street Journal describes how online retailers are racing each other in a costly effort to make same-day delivery a reality.
The pope will be joining Twitter with the handle @pontifex, but don't expect him to follow you, Reuters reports.
The taxi app Uber is facing a broad regulatory crackdown from governments across the country, The New York Times says.
Apple has started selling "unlocked" iPhone5 smartphones, which are sold without carrier subsidies, according to The Washington Post.
The Internet in Syria was up and running again on Sunday after a two-day outage, possibly caused by the embattled government, CNET reports.
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Happy Sunday! I hope you are having a lovely weekend. The weather here is perfect, so I've been trying to get outside and enjoy it. This pretty pink flower is from my Camellia. I love the vibrant color and that it's just starting to bloom now, unlike so many other things in the garden. It's evergreen, so it looks good all year round.
This shot is of a Snapdragon I treated myself to in order to add some fall color. The flower shape is so unique!
What's happening in your garden now that it's fall? Are you overwhelmed with squashes? Planting cool season crops like lettuce, broccoli, and beets? (I did!)
Welcome! I'm so glad you're here! A few links to get you started: Food Bloggers of the World, Unite! - How I Bake - Refresh & Rethink. Then, learn a little more about me & my bakery adventures & scroll through the Recipe Index or monthly favorites above.
Questions? Comments? I love them! Leave a comment or send me an email.
Sunday, October 10, 2010
Sunday Flowers
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I tweeted your blog on twitter today. (say that 3 times real fast!) :)ReplyDelete
Let's get together and do some holiday baking at my house this season ok? mmm.... chocolate dipped pretzel rods and rolled in crushed candy cane? Is that really baking? maybe not, but it's yummy and looks nice.
I love your site, It is a pleasure to visit.
I have added your site to my site.
Please link my site to your site.
Thank you!
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Eesti in brief - 2006-02-15
Feb 15, 2006
Vice Admiral Tarmo Kouts, commander of the nation's defense forces, left for Iraq to meet with Estonian officers currently serving in the war. Kouts was scheduled to meet with members of the multinational coalition. The commander is accompanied by acting commander of the Estonian Army Lt. Col. Meelis Kiili and chief of the general staff's operations unit Major Sten Reimann. There are currently 38 Estonian soldiers serving in Iraq.
The Supreme Court discussed the restitution or compensation of property to people who resettled in Germany in 1941 and said it would pass a ruling on the matter in April. A court spokesperson said the Supreme Court handling the matter in a full session would pass a ruling in two months, whereas usually it rules in one month. Parties reclaiming their property and tenants' representatives attended the court session, as well as representatives of Parliament, the government, the legal chancellor and Justice Minister Rein Lang.
Parliament scrapped a bill that would have given Estonians an extra day-off if a public holiday falls on a Saturday or Sunday. The bill, which was a second-attempt, was removed from Parliament's agenda with 47 votes against, seven in favor and one abstention. Speaking on behalf of Parliament's social affairs committee, which opposed the bill, Reform Party MP Leino Magi cited the government's opinion that the explanatory letter accompanying the bill did not analyze the proposal's economic impact, nor did it include word from the employers' central association. The government also said that, while the proposal would help improve the well-being of employees and uphold social values, the matter should be solved between the employee and employer.
The number of brothels in Tallinn has declined substantially over the past couple of years, the Linnaleht newspaper reported. If as recently as a few years ago there were a little over 40 so-called "clubs" where intimate services could be bought, then there are less than 10 now. "And even of these, many are on the verge of extinction," North Prefecture spokesperson Tuuli Harson was quoted as saying. She added that the number of so-called apartment brothels had declined as well. In 2004, 43 criminal investigations were launched into alleged cases of pimping, and 37 investigations in 2005. Prostitution is not illegal, but pimping is a criminal offense punishable by up to five years' imprisonment.
Parliament's national defense committee discussed whether the presidential residence needed a fence around it. Kalev Timberg, the interior ministry general secretary, and Police Chief Raivo Aeg briefed the committee on the outcome of a disciplinary probe conducted at the Police Board, as well as on planned security measures for the presidential palace. According to the evaluation, a barrier should surround the square in front of the palace since as it is now anyone can approach the premises unhindered. But MPs said that a security fence would not complement the historic Kadriorg park, and that instead options that would better suit the park's environment should be weighed.
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I have a confession to make. I, BreezyK, am a serial eavesdropper. While some might consider this behavior rude, sneaky, or generally unbecoming, I prefer to think of myself as suffering from a rare form of chronic, insatiable curiosity. Eavesdropping, as it happens, is the only known cure. Fortunately for me, a big city like … Continue reading
Tag Archives: Television
Friday Five
It's been a long week, amiright guys? Thanks for sticking it out with me for another week of daily posts. Just to recap, I: Ate Everything In New York Read and Reviewed A Few Books Embarrassed Myself In An Art Gallery; and Admitted to Wearing a Lot of Really Ridiculous Stuff in High School. If … Continue reading
Friday Five
So it's only my fourth day of consecutive posting, and I'm already exhausted. Remind me again why I signed up for this?? Since writing a post about one coherent topic feels like A LOT right now, here instead are a bunch of random things I've been thinking about lately, all wrapped up under one alliterative … Continue reading
You Had Me At "Showstopper"
I bought something from The Shopping Channel the other day. I'm not proud of it. Actually, f*&k that, I'm totally proud. It was a sweet-ass deal. The incident occurred during a very frequent rare night in. I was sipping my Night Wine and flipping between "Guiliana and Bill" and "Dancing With The Stars" when something … Continue reading
10 Inventions From Pop Culture I Wish Were Real
Few films have left such a lasting impact on my life as the 1999 cult classic Office Space. Not only are its characters hilarious and its lines highly quotable, its tongue-in-cheek (yet painfully accurate) depiction of corporate culture continues to haunt me in my day-to-day life. Anyway, while catastrophizing about my life the other day, … Continue reading
The Fantasy Boyfriend Draft
It's that time of year again, folks- when body paint, beer and tailgate parties abound, and the men in your life fall into a state of temporary insanity for the next few months. That's right it's football season! How do I, a girl whose only knowledge of football stems from the movie The Water Boy, … Continue reading
My Life Through Instagram Vol. 2
What up, homies? I know my blog has been about as active as a Giant Panda on Valium lately, and for that I apologize. Side note: did you know Giant Pandas spend approximately 16 hours a day eating?? We have so much in common. Anyway, it's been a busy few weeks for your girl BreezyK … Continue reading


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Click here to watch CRI Weekly News for August 22, 2014
United Way of Mahaska County's third annual "Stuff the Bus" initiative successfully wrapped up on Tuesday.
Oskaloosa residents might soon have to pay a higher fee after consuming gas and electricity. Each month properties classified as commercial, residential, public entity and industry, have to pay a franchise fee on their MidAmerican Energy bill.
Watch CRI's Weekly Newscast
Oskaloosa held an open house this Saturday at the Musco Technology Center to celebrate Iola Cadwallader's 90th birthday, and to celebrate the cast and crew of the musical "Georgiana."
Mahaska County could soon have a 1,100-mile oil pipeline running across a part of the county. A company from Texas wants to build an underground oil pipeline from North Dakota to Illinois.
William Penn University's transportation needs received a boost this past weekend. WPU's Booster Club held its third annual summer fundraising.
Another topic the city council took up this week involved possible changes to an ordinance involving dangerous dogs.
Oskaloosa's High School Softball team made it to the state tournament for the first time in program history.
The Oskaloosa City Council received a presentation from the Mahaska County Recreation Foundation (MCRF) and unanimously approved revising the city's policy regarding "dangerous dogs" and "potentially dangerous dogs" on Monday night.
The summer musical "Georgiana" made its triumphant return to Oskaloosa after a 34-year hiatus. The cast of the musical is made up of more than 80 Oskaloosa community members.
Main Street's annual "Sweet Corn Serenade" is set to make an appearance at the downtown square on Thursday (July 31) from 12 p.m. to 9 p.m.
William Penn's Booster Club is busy putting some finishing touches on it's third annual summer fundraiser this Saturday, July 26th.
The Oskaloosa Baseball team also had its post season dreams dashed. The Indians blanked Mount Pleasant during the first round of substate 4-0.
On Monday morning, the Mahaska County Board of Supervisors unanimously approved a letter of support for the Skilled Iowa Initiative. The initiative allows employers to determine the skill level of potential employees.
The dispute is ongoing between the Oskaloosa Water Department, which includes one of its trustees Errin Keltner, and the City of Oskaloosa and Mayor David Krutzfeldt. The dispute is over Keltner's removal from the Board.


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Practical and stylish, our dark wood folding TV table can be folded down when not in use to save valuable space in your home. • Dark wood finish• Folds away when not in use• Easy to assemble and disassemble Perfect for using your laptop, completing paperwork or eating dinner, this stylish dark wood TV table provides a strong and steady surface to sit at, and can be collapsed after use to save space. Featuring an elegant dark wood finish, our collapsible table has a sophisticated look that will complement any home, and benefits from a strong construction that can easily hold the weight of crockery, laptops and books. Easy to assemble and disassemble, simply store away in a cupboard, behind the sofa or under the stairs when not in use.
The delivery options available to you at the checkout may vary according to other items you purchase within the same order. Unfortunately, you are unable to select more than one delivery method per order.In some instances, we will despatch your order in several parts, you will be notified of this when you checkout.
We hope you love this product, but if you decide it's not right for you can return it for FREE:
Store return
If you live near one of our stores and can easily carry your item, this is the fastest way to return or exchange a product. Locate your nearest store here.
Collect+ return
For products that weigh less than 10kg and are smaller than 50x30x30cm, you can return your item at thousands of collection points around the country. Complete the returns form provided with your delivery and enclose with your parcel. Locate your local collect+ point here. If your parcel weighs more than 10kg and is larger than 50cm x 30cm x 30cm, please contact us to arrange a return.
Exceptions apply, see our full returns policy here.
(based on 9 reviews)
of respondents would recommend this to a friend.
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Reviewed by 9 customers
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(1 of 1 customers found this review helpful)
By reubenmuriel
from scarborough
About Me Midrange Shopper
use this item to have my dinners on
Bottom Line Yes, I would recommend this to a friend
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(2 of 2 customers found this review helpful)
Good buy
By Bigcolblue
from Troon
Good for TV dinners
Perfect for armchair eating and laptop
By Liz Flitwick
from Flitwick
About Me Budget Shopper
Perfect for my armchair bound parent and compact
TV table
By Mummybear
from Reading
It enables me to do things like jig saws and letters etc from my arm chair
Its what I wanted and suited my needs .
By Gerry
from Newbury
I needed it to put my lap top and it works very well good size not in everyone's one.