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% =====================================================================
% Ronald Nissel, rnissel@nt.tuwien.ac.at
% (c) 2017 by Institute of Telecommunications, TU Wien
% www.nt.tuwien.ac.at
% =====================================================================
% This script plots the power spectal density of different modulation
% techniques:
% 1) OFDM with CP (worst spectral behaviour)
% 2) FBMC (best spectral properties, complex orthogonality is replaced by real orthogonality)
% 3) WOLA (windowed OFDM. The windowing is done at TX and RX)
% 4) FOFDM (filtered OFDM, sinc + Hann, filtering at TX and RX)
% 5) UFMC (filtered OFDM, subband-wise filtering, Dolph-Chebyshev window, cyclic prefix and zero padding are supported, filtering at TX and RX)
% See Figure 2 in the paper
clear; close all;
L = 24; % Number of subcarriers
K_OFDM = 10; % Number of OFDM symbols in time
K_FBMC = 105; % For FBMC we use a higher number of time-symbols so that the frequency resolution is better. Without that, the figure would look ugly.
NrRepetitions = 200; % Number repetitions for the simulation
QAM_ModulationOrder = 16; % Modulation order, 4,16,64,...
Simulate = true; % Perform also a simulation in order to check the theoretical Power Spectral Density (PSD)
SubcarrierSpacing = 15e3; % Subcarrier spacing (15kHz, same as LTE)
SamplingRate = SubcarrierSpacing*L*14; % We need oversampling (14) in order to see out of band emissions. Furthermore 14 fits the CP length of OFDM.
OverlappingFactor = 4; % Overlapping factor for FBMC. For an overlapping factor of 4 and the PHYDYAS pulse, we see a rectangular filering effect
IntermediateSubcarrier = 50; % Shift frequency (for presentation purposes)
UFMC_ZeroPadding = false; % If true, apply zero padding instead of a cyclic prefix. However, zero padding leads to zero crossing which then dominate the whole picture => for presentation purposes we also use CP in UFMC
% Parameterset 1
TF_FilteredAndWindowedOFDM = 1.09; % Time frequency spacing for UFMC and FOFDM. Chosen so that the interference from filtering is small (the SIR is approximately 65dB).
FilterLengthTXandRX_UFMC = 1/(14*SubcarrierSpacing); % 4.7s, same as the CP in LTE. However, we need filtering at TX and RX => Total length of the filtering is two times larger
FilterLengthTXandRX_FOFDM = 1.18*1/(14*SubcarrierSpacing); % Increased by 1.2 because FOFDM is more robust than UFMC to a larger filter length
% Parameterset 2
% TF_FilteredAndWindowedOFDM = 1.28; % Time frequency spacing for UFMC and FOFDM. Chosen so that the interference from filtering is small (the SIR is approximately 65dB).
% FilterLengthTXandRX_UFMC = 3/(14*SubcarrierSpacing); % 3 times as high as the CP in LTE to get a smaller OOB.
% FilterLengthTXandRX_FOFDM = 1.33*3/(14*SubcarrierSpacing); % Increased by 1.35 because FOFDM is more robust than UFMC to a larger filter length
% Parameterset 3 (perfect orthogonal)
% TF_FilteredAndWindowedOFDM = 1+2/(14);
% FilterLengthTXandRX_UFMC = 1/(14*SubcarrierSpacing);
% FilterLengthTXandRX_FOFDM = 1/(14*SubcarrierSpacing);
% Parameterset 4 (LTE like, interference ~ 55)
% TF_FilteredAndWindowedOFDM = 1+1/(14);
% FilterLengthTXandRX_UFMC = 1/(14*SubcarrierSpacing);
% FilterLengthTXandRX_FOFDM = 1.1*1/(14*SubcarrierSpacing);
ColorFBMC = [0 0 1]*0.5;
ColorOFDM = [1 0 0];
ColorWOLA = [1 0 1];
ColorFOFDM = [1 1 0]*0.7;
ColorUFMC = [0 1 0]*0.5;
CP_LengthFilter = (TF_FilteredAndWindowedOFDM-1)*1/(SubcarrierSpacing);
%% FBMC Object
FBMC = Modulation.FBMC(...
L,... % Number subcarriers
K_FBMC,... % Number FBMC symbols (determines the frequency resolution!)
SubcarrierSpacing,... % Subcarrier spacing (Hz)
SamplingRate,... % Sampling rate (Samples/s)
SubcarrierSpacing*IntermediateSubcarrier,... % Intermediate frequency first subcarrier (Hz)
false,... % Transmit real valued signal
'PHYDYAS-OQAM',... % Prototype filter (Hermite, PHYDYAS, RRC) and OQAM or QAM,
OverlappingFactor, ... % Overlapping factor (also determines oversampling in the frequency domain)
0, ... % Initial phase shift
true ... % Polyphase implementation
);
%% OFDM Object
OFDM = Modulation.OFDM(...
L,... % Number subcarriers
K_OFDM,... % Number OFDM Symbols
SubcarrierSpacing,... % Subcarrier spacing (Hz)
SamplingRate,... % Sampling rate (Samples/s)
SubcarrierSpacing*IntermediateSubcarrier,... % Intermediate frequency first subcarrier (Hz)
false,... % Transmit real valued signal
1/(14*SubcarrierSpacing), ... % Cyclic prefix length (s)
0 ... % Zero guard length (s)
);
%% Windowed OFDM (WOLA)
WOLA = Modulation.WOLA(...
L,... % Number subcarriers
K_OFDM,... % Number OFDM Symbols
SubcarrierSpacing,... % Subcarrier spacing (Hz)
SamplingRate,... % Sampling rate (Samples/s)
SubcarrierSpacing*IntermediateSubcarrier,... % Intermediate frequency first subcarrier (Hz)
false,... % Transmit real valued signal
0, ... % Cyclic prefix length (s)
0, ... % Zero guard length (s)
CP_LengthFilter/2, ... % Window overlap length at TX (s)
CP_LengthFilter/2 ... % Window overlap length at RX (s)
);
%% Filtered OFDM
FOFDM = Modulation.FOFDM(...
L,... % Number subcarriers
K_OFDM,... % Number OFDM Symbols
SubcarrierSpacing,... % Subcarrier spacing (Hz)
SamplingRate,... % Sampling rate (Samples/s)
SubcarrierSpacing*IntermediateSubcarrier,... % Intermediate frequency first subcarrier (Hz)
false,... % Transmit real valued signal
0, ... % Cyclic prefix length (s)
0, ... % Zero guard length (s)
FilterLengthTXandRX_FOFDM, ... % Filter length at TX (s)
FilterLengthTXandRX_FOFDM, ... % Filter length at RX (s)
CP_LengthFilter ... % Addional cyclic prefix for the filtering (s)
);
%% UFMC
UFMC = Modulation.UFMC(...
L,... % Number subcarriers
K_OFDM,... % Number OFDM Symbols
SubcarrierSpacing,... % Subcarrier spacing (Hz)
SamplingRate,... % Sampling rate (Samples/s)
SubcarrierSpacing*IntermediateSubcarrier,... % Intermediate frequency first subcarrier (Hz)
false,... % Transmit real valued signal
0, ... % Cyclic prefix length (s)
0, ... % Zero guard length (s)
FilterLengthTXandRX_UFMC, ... % Filter length at TX (s)
FilterLengthTXandRX_UFMC, ... % Filter length at RX (s)
CP_LengthFilter, ... % Addional cyclic prefix for the filtering (s)
UFMC_ZeroPadding ... % TRUE for zero padding; FALSE for a conventional cyclic prefix
);
%% Get TX and RX matrices to calculate SIR for FOFDM and UFMC due to ISI/ICI
G_TX_FOFDM = sparse(FOFDM.GetTXMatrix);
G_RX_FOFDM = sparse(FOFDM.GetRXMatrix);
G_TX_UFMC = sparse(UFMC.GetTXMatrix);
G_RX_UFMC = sparse(UFMC.GetRXMatrix);
D_FOFDM = G_RX_FOFDM*G_TX_FOFDM;
D_UFMC = G_RX_UFMC*G_TX_UFMC;
% Inband Signal to interference ratio due to filtering (the filter length is longer than the CP to improve filter characteristics)
SIR_dB_FOFDM = 10*log10(sum(abs(diag(D_FOFDM)).^2)/(sum(sum(abs(D_FOFDM-diag(diag(D_FOFDM))).^2,2),1)));
SIR_dB_UFMC = 10*log10(sum(abs(diag(D_UFMC)).^2)/(sum(sum(abs(D_UFMC-diag(diag(D_UFMC))).^2,2),1)));
%% Information
fprintf('===================================================\n');
fprintf(' |(complex)TF-Spacing| Bandwidth(LF)| \n');
fprintf('OFDM (with CP) |%17.2f |%8.2f MHz | \n', OFDM.PHY.TimeSpacing*OFDM.PHY.SubcarrierSpacing , OFDM.PHY.SubcarrierSpacing*OFDM.Nr.Subcarriers/1e6);
fprintf('FBMC |%17.2f |%8.2f MHz | \n', FBMC.PHY.TimeSpacing*FBMC.PHY.SubcarrierSpacing*2 , FBMC.PHY.SubcarrierSpacing*FBMC.Nr.Subcarriers/1e6);
fprintf('WOLA |%17.2f |%8.2f MHz | \n', WOLA.PHY.TimeSpacing*WOLA.PHY.SubcarrierSpacing , WOLA.PHY.SubcarrierSpacing*WOLA.Nr.Subcarriers/1e6);
fprintf('FOFDM |%17.2f |%8.2f MHz | \n', FOFDM.PHY.TimeSpacing*FOFDM.PHY.SubcarrierSpacing , FOFDM.PHY.SubcarrierSpacing*FOFDM.Nr.Subcarriers/1e6);
fprintf('UFMC |%17.2f |%8.2f MHz | \n', UFMC.PHY.TimeSpacing*UFMC.PHY.SubcarrierSpacing , UFMC.PHY.SubcarrierSpacing*UFMC.Nr.Subcarriers/1e6);
fprintf('===================================================\n');
fprintf('SIR [dB] for FOFDM: %2.2f \n', full(SIR_dB_FOFDM));
fprintf('SIR [dB] for UFMC: %2.2f \n', full(SIR_dB_UFMC));
fprintf('SIR [dB] for FBMC: %2.2f \n', FBMC.GetSIRdBDoublyFlat); % Interference due to imperfect prototype filter
%% Calculate the Power Spectral Density (PSD)
[PSD_FBMC_Theory,f_FBMC] = FBMC.PlotPowerSpectralDensityUncorrelatedData;
[PSD_OFDM_Theory,f_OFDM] = OFDM.PlotPowerSpectralDensityUncorrelatedData;
[PSD_WOLA_Theory,f_WOLA] = WOLA.PlotPowerSpectralDensityUncorrelatedData;
[PSD_FOFDM_Theory,f_FOFDM] = FOFDM.PlotPowerSpectralDensityUncorrelatedData;
[PSD_UFMC_Theory,f_UFMC] = UFMC.PlotPowerSpectralDensityUncorrelatedData;
% Normalize over energy
PSD_FBMC_Theory = PSD_FBMC_Theory/(sum(PSD_FBMC_Theory)*SamplingRate/FBMC.Nr.SamplesTotal);
PSD_OFDM_Theory = PSD_OFDM_Theory/(sum(PSD_OFDM_Theory)*SamplingRate/OFDM.Nr.SamplesTotal);
PSD_WOLA_Theory = PSD_WOLA_Theory/(sum(PSD_WOLA_Theory)*SamplingRate/WOLA.Nr.SamplesTotal);
PSD_FOFDM_Theory = PSD_FOFDM_Theory/(sum(PSD_FOFDM_Theory)*SamplingRate/FOFDM.Nr.SamplesTotal);
PSD_UFMC_Theory = PSD_UFMC_Theory/(sum(PSD_UFMC_Theory)*SamplingRate/UFMC.Nr.SamplesTotal);
% Normalize to 0dB
NormalizationFactor_Theory = max([PSD_FBMC_Theory]);
PSD_FBMC_Theory = PSD_FBMC_Theory/NormalizationFactor_Theory;
PSD_OFDM_Theory = PSD_OFDM_Theory/NormalizationFactor_Theory;
PSD_WOLA_Theory = PSD_WOLA_Theory/NormalizationFactor_Theory;
PSD_FOFDM_Theory = PSD_FOFDM_Theory/NormalizationFactor_Theory;
PSD_UFMC_Theory = PSD_UFMC_Theory/NormalizationFactor_Theory;
figure();
plot([-12 -12], [-150,20],'color', [0.8 0.8 0.8]);
hold on;
plot([12 12], [-150,20],'color', [0.8 0.8 0.8]);
plot(f_OFDM/OFDM.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_OFDM_Theory]),'Color', ColorOFDM);
plot(f_WOLA/WOLA.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_WOLA_Theory]),'Color', ColorWOLA);
plot(f_UFMC/UFMC.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_UFMC_Theory]),'Color',ColorUFMC);
plot(f_FOFDM/FOFDM.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_FOFDM_Theory]),'Color',ColorFOFDM);
plot(f_FBMC/FBMC.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_FBMC_Theory]),'Color', ColorFBMC);
ylim([-100 4]);
xlim([-50 50]);
xlabel('Normalized Frequency, f/F');
ylabel('Power Spectral Density [dB]');
set(gca,'XTick',[-50:10:50]);
if Simulate
%% Simulate to Check Theory
tic
QAM = Modulation.SignalConstellation(QAM_ModulationOrder,'QAM');
PAM = Modulation.SignalConstellation(sqrt(QAM_ModulationOrder),'PAM');
PSD_FBMC_Simulation = zeros(FBMC.Nr.SamplesTotal,1);
PSD_OFDM_Simulation = zeros(OFDM.Nr.SamplesTotal,1);
PSD_WOLA_Simulation = zeros(WOLA.Nr.SamplesTotal,1);
PSD_FOFDM_Simulation = zeros(FOFDM.Nr.SamplesTotal,1);
PSD_UFMC_Simulation = zeros(UFMC.Nr.SamplesTotal,1);
for i_rep = 1:NrRepetitions
x_OFDM = QAM.SymbolMapping(randi(QAM_ModulationOrder,L,K_OFDM));
x_PAM = PAM.SymbolMapping(randi(sqrt(QAM_ModulationOrder),L,K_FBMC));
s_FBMC = FBMC.Modulation(x_PAM);
s_OFDM = OFDM.Modulation(x_OFDM);
s_WOLA = WOLA.Modulation(x_OFDM);
s_FOFDM = FOFDM.Modulation(x_OFDM);
s_UFMC = UFMC.Modulation(x_OFDM);
PSD_FBMC_Simulation = PSD_FBMC_Simulation + abs(fft(s_FBMC)).^2;
PSD_OFDM_Simulation = PSD_OFDM_Simulation + abs(fft(s_OFDM)).^2;
PSD_WOLA_Simulation = PSD_WOLA_Simulation + abs(fft(s_WOLA)).^2;
PSD_FOFDM_Simulation = PSD_FOFDM_Simulation + abs(fft(s_FOFDM)).^2;
PSD_UFMC_Simulation = PSD_UFMC_Simulation + abs(fft(s_UFMC)).^2;
TimePassed = toc;
if mod(i_rep,100)==0
disp(['Realization ' int2str(i_rep) ' of ' int2str(NrRepetitions) '. Time left: ' int2str(TimePassed/i_rep*(NrRepetitions-i_rep)/60) 'minutes']);
end
end
% Normalize over energy
PSD_FBMC_Simulation = PSD_FBMC_Simulation/(sum(PSD_FBMC_Simulation)*SamplingRate/FBMC.Nr.SamplesTotal);
PSD_OFDM_Simulation = PSD_OFDM_Simulation/(sum(PSD_OFDM_Simulation)*SamplingRate/OFDM.Nr.SamplesTotal);
PSD_WOLA_Simulation = PSD_WOLA_Simulation/(sum(PSD_WOLA_Simulation)*SamplingRate/WOLA.Nr.SamplesTotal);
PSD_FOFDM_Simulation = PSD_FOFDM_Simulation/(sum(PSD_FOFDM_Simulation)*SamplingRate/FOFDM.Nr.SamplesTotal);
PSD_UFMC_Simulation = PSD_UFMC_Simulation/(sum(PSD_UFMC_Simulation)*SamplingRate/UFMC.Nr.SamplesTotal);
% LS Estimation to normalize to 0dB
NormalizationFactor_Simulation = 1./mean((PSD_FBMC_Simulation'*PSD_FBMC_Theory)./(PSD_FBMC_Simulation'*PSD_FBMC_Simulation));
PSD_FBMC_Simulation = PSD_FBMC_Simulation/NormalizationFactor_Simulation;
PSD_OFDM_Simulation = PSD_OFDM_Simulation/NormalizationFactor_Simulation;
PSD_WOLA_Simulation = PSD_WOLA_Simulation/NormalizationFactor_Simulation;
PSD_FOFDM_Simulation = PSD_FOFDM_Simulation/NormalizationFactor_Simulation;
PSD_UFMC_Simulation = PSD_UFMC_Simulation/NormalizationFactor_Simulation;
figure();
plot(f_OFDM/OFDM.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_OFDM_Theory PSD_OFDM_Simulation]),'Color', ColorOFDM);
hold on;
plot(f_WOLA/WOLA.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_WOLA_Theory PSD_WOLA_Simulation]),'Color', ColorWOLA);
plot(f_UFMC/UFMC.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_UFMC_Theory PSD_UFMC_Simulation]),'Color', ColorUFMC);
plot(f_FOFDM/FOFDM.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_FOFDM_Theory PSD_FOFDM_Simulation]),'Color', ColorFOFDM);
plot(f_FBMC/FBMC.PHY.SubcarrierSpacing-IntermediateSubcarrier-ceil(L/2)+1/2,10*log10([PSD_FBMC_Theory PSD_FBMC_Simulation]),'Color', ColorFBMC);
plot([-12 -12], [-150,20],'color', [0.8 0.8 0.8]);
plot([12 12], [-150,20],'color', [0.8 0.8 0.8]);
ylim([-100 4]);
xlim([-50 50]);
xlabel('Normalized Frequency, f/F');
ylabel('Power Spectral Density [dB]');
set(gca,'XTick',[-50:10:50]);
title('Theory vs Simulation');
end
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