F-OFDM vs. OFDM Modulation

This example compares Orthogonal Frequency Division Multiplexing (OFDM) with Filtered-OFDM (F-OFDM) and highlights the merits of the candidate modulation scheme for Fifth Generation (5G) communication systems.

Introduction

This example compares Filtered-OFDM modulation with generic Cyclic Prefix OFDM (CP-OFDM) modulation. For F-OFDM, a well-designed filter is applied to the time domain OFDM symbol to improve the out-of-band radiation of the sub-band signal, while maintaining the complex-domain orthogonality of OFDM symbols.

This example models Filtered-OFDM modulation with configurable parameters. It highlights the filter design technique and the basic transmit/receive processing.

s = rng(211);       % Set RNG state for repeatability

System Parameters

Define system parameters for the example. These parameters can be modified to explore their impact on the system.

numFFT = 1024;           % Number of FFT points
numRBs = 50;             % Number of resource blocks
rbSize = 12;             % Number of subcarriers per resource block
cpLen = 72;              % Cyclic prefix length in samples

bitsPerSubCarrier = 6;   % 2: QPSK, 4: 16QAM, 6: 64QAM, 8: 256QAM
snrdB = 18;              % SNR in dB

toneOffset = 2.5;        % Tone offset or excess bandwidth (in subcarriers)
L = 513;                 % Filter length (=filterOrder+1), odd

Filtered-OFDM Filter Design

Appropriate filtering for F-OFDM satisfies the following criteria:

  • Should have a flat passband over the subcarriers in the sub-band

  • Should have a sharp transition band to minimize guard-bands

  • Should have sufficient stop-band attenuation

A filter with a rectangular frequency response, i.e. a sinc impulse response, meets these criteria. To make this causal, the low-pass filter is realized using a window, which, effectively truncates the impulse response and offers smooth transitions to zero on both ends [ 3 ].

numDataCarriers = numRBs*rbSize;    % number of data subcarriers in sub-band
halfFilt = floor(L/2);
n = -halfFilt:halfFilt;

% Sinc function prototype filter
pb = sinc((numDataCarriers+2*toneOffset).*n./numFFT);

% Sinc truncation window
w = (0.5*(1+cos(2*pi.*n/(L-1)))).^0.6;

% Normalized lowpass filter coefficients
fnum = (pb.*w)/sum(pb.*w);

% Filter impulse response
h = fvtool(fnum, 'Analysis', 'impulse', ...
    'NormalizedFrequency', 'off', 'Fs', 15.36e6);
h.CurrentAxes.XLabel.String = 'Time (\mus)';
h.FigureToolbar = 'off';

% Use dsp filter objects for filtering
filtTx = dsp.FIRFilter('Structure', 'Direct form symmetric', ...
    'Numerator', fnum);
filtRx = clone(filtTx); % Matched filter for the Rx

% QAM Symbol mapper
qamMapper = comm.RectangularQAMModulator( ...
    'ModulationOrder', 2^bitsPerSubCarrier, 'BitInput', true, ...
    'NormalizationMethod', 'Average power');

F-OFDM Transmit Processing

In F-OFDM, the sub-band CP-OFDM signal is passed through the designed filter. As the filter's passband corresponds to the signal's bandwidth, only the few subcarriers close to the edge are affected. A key consideration is that the filter length can be allowed to exceed the cyclic prefix length for F-OFDM [ 1 ]. The inter-symbol interference incurred is minimized due to the filter design using windowing (with soft truncation).

Transmit-end processing operations are shown in the following F-OFDM transmitter diagram.

% Set up a figure for spectrum plot
hFig = figure('Position', figposition([46 50 30 30]), 'MenuBar', 'none');
axis([-0.5 0.5 -200 -20]);
hold on;
grid on
xlabel('Normalized frequency');
ylabel('PSD (dBW/Hz)')
title(['F-OFDM, ' num2str(numRBs) ' Resource blocks, '  ...
    num2str(rbSize) ' Subcarriers each'])

% Generate data symbols
bitsIn = randi([0 1], bitsPerSubCarrier*numDataCarriers, 1);
symbolsIn = qamMapper(bitsIn);

% Pack data into an OFDM symbol
offset = (numFFT-numDataCarriers)/2; % for band center
symbolsInOFDM = [zeros(offset,1); symbolsIn; ...
                 zeros(numFFT-offset-numDataCarriers,1)];
ifftOut = ifft(ifftshift(symbolsInOFDM));

% Prepend cyclic prefix
txSigOFDM = [ifftOut(end-cpLen+1:end); ifftOut];

% Filter, with zero-padding to flush tail. Get the transmit signal
txSigFOFDM = filtTx([txSigOFDM; zeros(L-1,1)]);

% Plot power spectral density (PSD)
[psd,f] = periodogram(txSigFOFDM, rectwin(length(txSigFOFDM)), ...
                      numFFT*2, 1, 'centered');
plot(f,10*log10(psd));

% Compute peak-to-average-power ratio (PAPR)
PAPR = comm.CCDF('PAPROutputPort', true, 'PowerUnits', 'dBW');
[~,~,paprFOFDM] = PAPR(txSigFOFDM);
disp(['Peak-to-Average-Power-Ratio for F-OFDM = ' num2str(paprFOFDM) ' dB']);
Peak-to-Average-Power-Ratio for F-OFDM = 11.371 dB

OFDM Modulation with Corresponding Parameters

For comparison, we review the existing OFDM modulation technique, using the full occupied band, with the same length cyclic prefix.

% Plot power spectral density (PSD) for OFDM signal
[psd,f] = periodogram(txSigOFDM, rectwin(length(txSigOFDM)), numFFT*2, ...
                      1, 'centered');
hFig1 = figure('Position', figposition([46 15 30 30]));
plot(f,10*log10(psd));
grid on
axis([-0.5 0.5 -100 -20]);
xlabel('Normalized frequency');
ylabel('PSD (dBW/Hz)')
title(['OFDM, ' num2str(numRBs*rbSize) ' Subcarriers'])

% Compute peak-to-average-power ratio (PAPR)
PAPR2 = comm.CCDF('PAPROutputPort', true, 'PowerUnits', 'dBW');
[~,~,paprOFDM] = PAPR2(txSigOFDM);
disp(['Peak-to-Average-Power-Ratio for OFDM = ' num2str(paprOFDM) ' dB']);
Peak-to-Average-Power-Ratio for OFDM = 9.721 dB

Comparing the plots of the spectral densities for CP-OFDM and F-OFDM schemes, F-OFDM has lower side lobes. This allows a higher utilization of the allocated spectrum, leading to increased spectral efficiency.

Refer to the comm.OFDMModulator System object™ which can also be used to implement the CP-OFDM modulation.

F-OFDM Receiver with No Channel

The example next highlights the basic receive processing for F-OFDM for a single OFDM symbol. The received signal is passed through a matched filter, followed by the normal CP-OFDM receiver. It accounts for both the filtering ramp-up and latency prior to the FFT operation.

No fading channel is considered in this example but noise is added to the received signal to achieve the desired SNR.

% Add WGN
rxSig = awgn(txSigFOFDM, snrdB, 'measured');

Receive processing operations are shown in the following F-OFDM receiver diagram.

% Receive matched filter
rxSigFilt = filtRx(rxSig);

% Account for filter delay
rxSigFiltSync = rxSigFilt(L:end);

% Remove cyclic prefix
rxSymbol = rxSigFiltSync(cpLen+1:end);

% Perform FFT
RxSymbols = fftshift(fft(rxSymbol));

% Select data subcarriers
dataRxSymbols = RxSymbols(offset+(1:numDataCarriers));

% Plot received symbols constellation
switch bitsPerSubCarrier
    case 2  % QPSK
        refConst = qammod((0:3).', 4, 'UnitAveragePower', true);
    case 4  % 16QAM
        refConst = qammod((0:15).', 16,'UnitAveragePower', true);
    case 6  % 64QAM
        refConst = qammod((0:63).', 64,'UnitAveragePower', true);
    case 8  % 256QAM
        refConst = qammod((0:255).', 256,'UnitAveragePower', true);
end
constDiagRx = comm.ConstellationDiagram( ...
    'ShowReferenceConstellation', true, ...
    'ReferenceConstellation', refConst, ...
    'Position', figposition([20 15 30 40]), ...
    'EnableMeasurements', true, ...
    'MeasurementInterval', length(dataRxSymbols), ...
    'Title', 'F-OFDM Demodulated Symbols', ...
    'Name', 'F-OFDM Reception', ...
    'XLimits', [-1.5 1.5], 'YLimits', [-1.5 1.5]);
constDiagRx(dataRxSymbols);

% Channel equalization is not necessary here as no channel is modeled

% Demapping and BER computation
qamDemod = comm.RectangularQAMDemodulator('ModulationOrder', ...
    2^bitsPerSubCarrier, 'BitOutput', true, ...
    'NormalizationMethod', 'Average power');
BER = comm.ErrorRate;

% Perform hard decision and measure errors
rxBits = qamDemod(dataRxSymbols);
ber = BER(bitsIn, rxBits);

disp(['F-OFDM Reception, BER = ' num2str(ber(1)) ' at SNR = ' ...
    num2str(snrdB) ' dB']);

% Restore RNG state
rng(s);
F-OFDM Reception, BER = 0.00083333 at SNR = 18 dB

As highlighted, F-OFDM adds a filtering stage to the existing CP-OFDM processing at both the transmit and receive ends. The example models the full-band allocation for a user, but the same approach can be applied for multiple bands (one per user) for an uplink asynchronous operation.

Refer to the comm.OFDMDemodulator System object™ which can be used to implement the CP-OFDM demodulation after receive matched filtering.

Conclusion and Further Exploration

The example presents the basic characteristics of the F-OFDM modulation scheme at both transmit and receive ends of a communication system. Explore different system parameter values for the number of resource blocks, number of subcarriers per blocks, filter length, tone offset and SNR.

Universal Filtered Multi-Carrier (UFMC) modulation scheme is another approach to sub-band filtered OFDM. For more information, see the UFMC vs. OFDM Modulation example. This F-OFDM example uses a single sub-band while the UFMC example uses multiple sub-bands.

F-OFDM and UFMC both use time-domain filtering with subtle differences in the way the filter is designed and applied. For UFMC, the length of filter is constrained to be equal to the cyclic-prefix length, while for F-OFDM, it can exceed the CP length.

For F-OFDM, the filter design leads to a slight loss in orthogonality (strictly speaking) which affects only the edge subcarriers.

Selected Bibliography

  1. Abdoli J., Jia M. and Ma J., "Filtered OFDM: A New Waveform for Future Wireless Systems," 2015 IEEE® 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, 2015, pp. 66-70.

  2. R1-162152. "OFDM based flexible waveform for 5G." 3GPP TSG RAN WG1 meeting 84bis. Huawei; HiSilicon. April 2016.

  3. R1-165425. "F-OFDM scheme and filter design." 3GPP TSG RAN WG1 meeting 85. Huawei; HiSilicon. May 2016.