Joint Sampling Rate and Carrier Frequency Offset Tracking
This example demonstrates joint sampling rate and carrier frequency offset tracking in a WLAN receiver.
In a WLAN radio a single oscillator is typically used to derive clocks for sampling and modulation. The oscillators in the transmitter and receiver radios do not run at the exact same frequency. Due to this mismatch, the sampling instants at the receiver shift relative to the transmitter. Therefore, a sample rate offset (SRO) exists between receiver and transmitter. Similarly, a carrier frequency offset (CFO) exists between the receiver and transmitter due to mismatched carrier frequencies. The inclusion of pilot subcarriers in the IEEE® 802.11™ standard allows for tracking and correction of SRO and CFO impairments.
In OFDM systems SRO manifests itself as a subcarrier and symbol dependent phase rotation and inter-carrier interference (ICI) [ 1 ]. When the SRO is large, and a packet is long, subcarriers far away from DC will experience a substantial impairment. CFO manifests itself as ICI and a symbol dependent phase rotation common to all subcarriers. The phase rotation on subcarriers from one OFDM symbol to the next due to these impairments is illustrated below. is the phase error, is the subcarrier index, is the number of subcarriers. is the SRO, is the carrier frequency offset, is the period of the symbol, is the phase error gradient (PEG), and is the common phase error (CPE). The PEG and CPE can be used to estimate SRO and residual CFO.
In this example, an IEEE 802.11ac™ VHT waveform is generated with fixed SRO and CFO impairments [ 2 ]. The impaired waveform is synchronized and the data portion demodulated and decoded with and without joint timing and phase tracking to correct for SRO and CFO. The equalized constellation with and without tracking is shown to demonstrate the effectiveness of tracking.
Generate a Baseband Waveform
A VHT configuration object is created to parameterize the transmission. Because the impairments are easier to visualize with fewer OFDM symbols, a data payload with only 500 bytes is transmitted using 16-QAM modulation.
% Create a VHT configuration cfgVHT = wlanVHTConfig; cfgVHT.ChannelBandwidth = 'CBW20'; cfgVHT.NumTransmitAntennas = 1; cfgVHT.NumSpaceTimeStreams = 1; cfgVHT.MCS = 4; % 16-QAM and 3/4 coding rate cfgVHT.APEPLength = 500; % Bytes % Create a random PSDU s = rng(10); % Seed the random number generator psdu = randi([0 1],cfgVHT.PSDULength*8,1,'int8'); % Generate a VHT packet tx = wlanWaveformGenerator(psdu,cfgVHT);
Sample rate offset between the transmitter and receiver is modeled by resampling the transmitted waveform. The resample function can be used to model a limited range of large sample rate offsets. In this example a -100 parts per million (PPM) sample rate offset is modeled.
% Model sample rate offset p = 1e4; % Interpolation factor q = 1e4-1; % Decimation factor sro = (1-p/q)*1e6; disp('Impairments:'); disp([' Sample rate offset (SRO): ' num2str(sro,'%.1f') ' PPM']); % Resample the waveform, appending zeros to allow for filter delay N = 100; % Size of filter used for resampling rx = resample([tx; zeros(N,cfgVHT.NumTransmitAntennas)],p,q,N);
Impairments: Sample rate offset (SRO): -100.0 PPM
Residual carrier frequency offset is added to the waveform using the helper function helperFrequencyOffset. In this example we assume the same oscillator is used for sampling and modulation, therefore the CFO will be a function of the SRO and carrier frequency.
fc = 5.25e9; % Carrier frequency, Hertz cfo = (sro*1e-6)*fc; % Carrier frequency offset, Hertz disp([' Carrier frequency offset (CFO): ' num2str(cfo,'%.1f') ' Hz']); fs = wlanSampleRate(cfgVHT); % Baseband sample rate rx = helperFrequencyOffset(rx,fs,cfo); % Add frequency offset
Carrier frequency offset (CFO): -525052.5 Hz
Noise is added to the waveform with 30 dBW variance.
awgnChannel = comm.AWGNChannel('NoiseMethod','Variance','Variance',10^(-30/10)); rx = awgnChannel(rx);
Front-End Synchronization and Receiver Processing
The following processing steps occur to synchronize the packet, in preparation for recovering the data field.
- The packet is detected
- Coarse carrier frequency offset is estimated and corrected
- Symbol timing synchronization is established
- Fine carrier frequency offset is estimated and corrected
- The L-LTF is OFDM demodulated and noise estimation is performed
- The VHT-LTF is OFDM demodulated and channel estimation is performed
% Generate field indices ind = wlanFieldIndices(cfgVHT); % Packet detection tOff = wlanPacketDetect(rx,cfgVHT.ChannelBandwidth); % Coarse frequency offset correction lstf = rx(tOff+(ind.LSTF(1):ind.LSTF(2)),:); coarseCFOEst = wlanCoarseCFOEstimate(lstf,cfgVHT.ChannelBandwidth); rx = helperFrequencyOffset(rx,fs,-coarseCFOEst); % Symbol timing synchronization nonhtPreamble = rx(tOff+(ind.LSTF(1):ind.LSIG(2)),:); symOff = wlanSymbolTimingEstimate(nonhtPreamble,cfgVHT.ChannelBandwidth); tOff = tOff+symOff; % Fine frequency offset correction lltf = rx(tOff+(ind.LLTF(1):ind.LLTF(2)),:); fineCFOEst = wlanFineCFOEstimate(lltf,cfgVHT.ChannelBandwidth); rx = helperFrequencyOffset(rx,fs,-fineCFOEst); % Noise power estimation lltf = rx(tOff+(ind.LLTF(1):ind.LLTF(2)),:); lltfDemod = wlanLLTFDemodulate(lltf,cfgVHT); noiseEst = helperNoiseEstimate(lltfDemod,cfgVHT.ChannelBandwidth,cfgVHT.NumSpaceTimeStreams); % Channel estimation vhtltf = rx(tOff+(ind.VHTLTF(1):ind.VHTLTF(2)),:); vhtltfDemod = wlanVHTLTFDemodulate(vhtltf,cfgVHT); chanEst = wlanVHTLTFChannelEstimate(vhtltfDemod,cfgVHT);
Recovery Without Sample Rate Offset or Residual CFO Tracking
The coarse and fine frequency offset estimation and correction removes the majority of CFO, but residual CFO remains due to the presence of impairments in the waveform. This must be tracked and corrected by the receiver.
disp('Front-end impairment correction:'); frontEndCFOEst = coarseCFOEst+fineCFOEst; disp([' Estimated CFO: ' num2str(frontEndCFOEst,'%.1f') ' Hz']); residualCFO = cfo-frontEndCFOEst; disp([' Residual CFO after initial correction: ' num2str(residualCFO,'%.1f') ' Hz']);
Front-end impairment correction: Estimated CFO: -524764.3 Hz Residual CFO after initial correction: -288.2 Hz
The helper function trackingVHTDataRecover recovers the VHT data field with optional pilot tracking to correct for timing and phase errors due to SRO and CFO. Pilot tracking is controlled using the helper object trackingRecoveryConfig.
The data field is first recovered without pilot tracking. The data field is extracted from the waveform using the start and end sample indices of the field at the baseband rate. If the receiver sampling rate is higher than the transmitter rate, the receiver requires more samples than the transmitter produces. To allow for this, Ne additional samples are extracted from the waveform and passed to the recovery function. The maximum number of additional samples required is a function of the expected SRO, the baseband sampling rate, and the maximum packet duration.
% Recovery configuration with pilot tracking disabled cfgRec = trackingRecoveryConfig; cfgRec.PilotTracking = 'None'; % Extract data field with Ne additional samples to allow for negative SRO maxDuration = 5.484e-3; % Maximum packet duration in seconds maxSRO = 120; % PPM Ne = ceil(fs*maxDuration*maxSRO*1e-6); % Number of extra samples dataInd = tOff+(ind.VHTData(1):ind.VHTData(2)+Ne); dataInd = dataInd(dataInd<=length(rx)); % Only use indices within waveform data = rx(dataInd,:); % Perform demodulation and decoding [rxPSDUNoTrack,~,eqSymNoTrack] = trackingVHTDataRecover(data,chanEst,noiseEst,cfgVHT,cfgRec);
The equalized constellation is plotted which shows a rotation of all constellation points caused by residual CFO, and a spreading of constellation points due to SRO. Despite the modest AWGN added to the waveform, the impairments cause bit errors within the decoded PSDU.
ConstNoTrack = comm.ConstellationDiagram; ConstNoTrack.Title = 'Equalized symbols with no pilot tracking'; ConstNoTrack.ReferenceConstellation = helperReferenceSymbols(cfgVHT); ConstNoTrack(eqSymNoTrack(:)); [~,berNoTrack] = biterr(rxPSDUNoTrack,psdu); disp('Bit error rate:'); disp([' Without tracking: ' num2str(berNoTrack)]);
Bit error rate: Without tracking: 0.086558
Recovery With Sample Rate Offset Tracking and Residual CFO Tracking
Now the data field is recovered with joint timing and phase pilot tracking to correct for SRO and residual CFO.
The tracking algorithm used in this example estimates absolute values of and per OFDM symbol, and applies a per subcarrier and symbol phase correction to the demodulated symbols to reverse the phase errors caused by SRO and CFO. The phase error between each received pilot subcarrier and the expected value is calculated per symbol and averaged over PilotTrackingWindow OFDM symbols. From this, least-square estimates of and are calculated per symbol. These estimates are used to apply a phase correction to each symbol and subcarrier [ 3, 4 ].
% Recovery configuration with pilot tracking enabled cfgRec = trackingRecoveryConfig; cfgRec.PilotTracking = 'Joint'; % Joint timing and phase tracking cfgRec.PilotTrackingWindow = 9; % Averaging window in OFDM symbols % Perform demodulation and decoding [rxPSDU,~,eqSymTrack,cpe,peg] = trackingVHTDataRecover(data,chanEst,noiseEst,cfgVHT,cfgRec);
The equalized constellation is plotted which shows a clear 16-QAM constellation with no spreading or rotation. There are no bit errors.
ConstTrack = comm.ConstellationDiagram; ConstTrack.Title = 'Equalized symbols with joint pilot tracking'; ConstTrack.ReferenceConstellation = helperReferenceSymbols(cfgVHT); ConstTrack(eqSymTrack(:)); [~,berTrack] = biterr(rxPSDU,psdu); disp([' With tracking: ' num2str(berTrack)]);
With tracking: 0
The function trackingVHTDataRecover returns measurements from which the residual CFO, and SRO can be estimated:
- cpe - The common phase error (radians) per symbol
- peg - The phase error gradient (radians per subcarrier) per symbol
The SRO and residual CFO are estimated from these measurements using a linear least-square fit of the rate of change. The measurements are plotted using the helper function trackingPlotSROCFOEstimates.
[residualCFOEst,sroEst] = trackingPlotSROCFOEstimates(cpe,peg,cfgVHT); % Display estimated SRO, residual CFO and total CFO fprintf('Tracked impairments:\n'); fprintf(' Estimated residual CFO: %3.1f Hz (%.1f Hz error)\n', ... residualCFOEst,residualCFOEst-residualCFO); fprintf(' Estimated SRO: %3.1f PPM (%.1f PPM error)\n',sroEst,sroEst-sro); cfoEst = frontEndCFOEst+residualCFOEst; % Initial + tracked CFO estimate fprintf('Estimated CFO (initial + tracked): %.1f Hz (%.1f Hz error)\n',cfoEst,cfoEst-cfo); rng(s); % Restore the state of the random number generator
Tracked impairments: Estimated residual CFO: -260.5 Hz (27.8 Hz error) Estimated SRO: -101.2 PPM (-1.2 PPM error) Estimated CFO (initial + tracked): -525024.8 Hz (27.8 Hz error)
The example shows how sample rate and carrier frequency offsets can be tracked and corrected for while recovering the data field of a WLAN waveform.
Data field recovery functions with joint pilot tracking for VHT, HT-MF and non-HT formats are provided, along with an object to configure the recovery algorithms:
This example uses the following helper functions:
- Speth, Michael, et al. "Optimum receiver design for wireless broad-band systems using OFDM." IEEE Transactions on communications 47.11 (1999): 1668-1677.
- IEEE Std 802.11ac™-2013 IEEE Standard for Information technology - Telecommunications and information exchange between systems - Local and metropolitan area networks - Specific requirements - Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications - Amendment 4: Enhancements for Very High Throughput for Operation in Bands below 6 GHz.
- Chiueh, Tzi-Dar, Pei-Yun Tsai, and I-Wei Lai. Baseband receiver design for wireless MIMO-OFDM communications. John Wiley & Sons, 2012.
- Horlin, P.F. and Bourdoux, A. Digital Compensation for Analog Front-Ends: A New Approach to Wireless Transceiver Design. Wiley, 2008.