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Figure_09_SpectralEfficiency.m
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Figure_09_SpectralEfficiency.m
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% =====================================================================
% Ronald Nissel, [email protected]
% (c) 2017 by Institute of Telecommunications, TU Wien
% www.nt.tuwien.ac.at
% =====================================================================
% This script plots the time-frequency efficiency of FBMC and f-OFDM.
% See Figure 9 in the paper (slightly changed parameters to improve the
% simulation time)
clear; close all;
%% Parameters
M_L = [1:1:10 12:4:20 20:8:128]; % [1:1:125] % For loop: number of subcarriers
EnergyThreshold = 0.9999; % 0.9999/(1-0.9999)=40dB
K = 1; % Number of complex symbols in time
SubcarrierSpacing = 15e3; % Subcarrier spacing (15kHz, same as LTE)
N_FFT = 1*1024; % 10*1024 % FFT size
OverlappingFactor = 4; % Overlapping factor in FBMC
AddedZeroFactorForPSD = 2; %10 % Increase the number of zeros in time to improve the resolution in frequency => smoother curves. The factor is a multiple T0=1/F.
FOFDM_TF1 = 1+1/14; % Time-Frequency spacing 1 for FOFDM
FOFDM_TF2 = 1+2/14; % Time-Frequency spacing 2 for FOFDM
FOFDM_TF3 = 1+3/14; % Time-Frequency spacing 3 for FOFDM
%% Start Calculation
SamplingRate = SubcarrierSpacing*N_FFT; % Sampling rate
dt = 1/SamplingRate;
for i_L = 1:length(M_L)
L = M_L(i_L); % Number of subcarriers
IntermediateSubcarrier = round(N_FFT/2-L/2);
%% FBMC Object
FBMC_Hermite = Modulation.FBMC(...
L,... % Number subcarriers
2*K,... % Number FBMC symbols
SubcarrierSpacing,... % Subcarrier spacing (Hz)
SamplingRate,... % Sampling rate (Samples/s)
SubcarrierSpacing*IntermediateSubcarrier,... % Intermediate frequency first subcarrier (Hz)
false,... % Transmit real valued signal
'Hermite-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
);
FBMC_PHYDYAS = Modulation.FBMC(...
L,... % Number subcarriers
2*K,... % Number FBMC symbols
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
);
%% f-OFDM Object
FOFDM1 = Modulation.FOFDM(...
L,... % Number subcarriers
K,... % Number FOFDM 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)
AddedZeroFactorForPSD*1/SubcarrierSpacing, ... % Zero guard length (s); Large zero guard to increase the resolution of the power spectral density
(FOFDM_TF1-1)*1/SubcarrierSpacing, ... % Length of the transmit filter (s)
(FOFDM_TF1-1)*1/SubcarrierSpacing, ... % Length of the receive filter (s)
(FOFDM_TF1-1)*1/SubcarrierSpacing... % Length of the additional cyclic prefix (s). Needed to combat ISI and ICI due to the filtering. However, some small ICI and ISI is perfectly fine.
);
FOFDM2 = Modulation.FOFDM(...
L,... % Number subcarriers
K,... % Number FOFDM 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)
AddedZeroFactorForPSD*1/SubcarrierSpacing, ... % Zero guard length (s); Large zero guard to increase the resolution of the power spectral density
(FOFDM_TF2-1)*1/SubcarrierSpacing, ... % Length of the transmit filter (s)
(FOFDM_TF2-1)*1/SubcarrierSpacing, ... % Length of the receive filter (s)
(FOFDM_TF2-1)/SubcarrierSpacing... % Length of the additional cyclic prefix (s). Needed to combat ISI and ICI due to the filtering. However, some small ICI and ISI is perfectly fine.
);
FOFDM3 = Modulation.FOFDM(...
L,... % Number subcarriers
K,... % Number FOFDM 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)
AddedZeroFactorForPSD*1/SubcarrierSpacing, ... % Zero guard length (s); Large zero guard to increase the resolution of the power spectral density
(FOFDM_TF3-1)*1/SubcarrierSpacing, ... % Length of the transmit filter (s)
(FOFDM_TF3-1)*1/SubcarrierSpacing, ... % Length of the receive filter (s)
(FOFDM_TF3-1)*1/SubcarrierSpacing... % Length of the additional cyclic prefix (s). Needed to combat ISI and ICI due to the filtering. However, some small ICI and ISI is perfectly fine.
);
% Calculate the expected power over time
[PS_FBMC_Hermite,~] = FBMC_Hermite.PlotTransmitPower;
[PS_FBMC_PHYDYAS,~] = FBMC_PHYDYAS.PlotTransmitPower;
[PS_FOFDM1,~] = FOFDM1.PlotTransmitPower;
[PS_FOFDM2,~] = FOFDM2.PlotTransmitPower;
[PS_FOFDM3,~] = FOFDM3.PlotTransmitPower;
% Calculate the required time according to the threshold
convPS_FBMC_Hermite = conv(PS_FBMC_Hermite,ones(size(PS_FBMC_Hermite)));
convPS_FBMC_Hermite(ceil(end/2)+1:end) = [];
Threshold_FBMC_Hermite_Upper = (convPS_FBMC_Hermite(end)-convPS_FBMC_Hermite(end)*(1-EnergyThreshold)/2);
Threshold_FBMC_Hermite_Lower = (convPS_FBMC_Hermite(end)*(1-EnergyThreshold)/2);
RequiredTime_FBMC_Hermite = sum((convPS_FBMC_Hermite<Threshold_FBMC_Hermite_Upper) & (convPS_FBMC_Hermite>Threshold_FBMC_Hermite_Lower))*(dt);
convPS_FBMC_PHYDYAS = conv(PS_FBMC_PHYDYAS,ones(size(PS_FBMC_PHYDYAS)));
convPS_FBMC_PHYDYAS(ceil(end/2)+1:end) = [];
Threshold_FBMC_PHYDYAS_Upper = (convPS_FBMC_PHYDYAS(end)-convPS_FBMC_PHYDYAS(end)*(1-EnergyThreshold)/2);
Threshold_FBMC_PHYDYAS_Lower = (convPS_FBMC_PHYDYAS(end)*(1-EnergyThreshold)/2);
RequiredTime_FBMC_PHYDYAS = sum((convPS_FBMC_PHYDYAS<Threshold_FBMC_PHYDYAS_Upper) & (convPS_FBMC_PHYDYAS>Threshold_FBMC_PHYDYAS_Lower))*(dt);
convPS_FOFDM1 = conv(PS_FOFDM1,ones(size(PS_FOFDM1)));
convPS_FOFDM1(ceil(end/2)+1:end) = [];
Threshold_FOFDM1_Upper = (convPS_FOFDM1(end)-convPS_FOFDM1(end)*(1-EnergyThreshold)/2);
Threshold_FOFDM1_Lower = (convPS_FOFDM1(end)*(1-EnergyThreshold)/2);
RequiredTime_FOFDM1 = sum((convPS_FOFDM1<Threshold_FOFDM1_Upper) & (convPS_FOFDM1>Threshold_FOFDM1_Lower))*(dt);
convPS_FOFDM2 = conv(PS_FOFDM2,ones(size(PS_FOFDM2)));
convPS_FOFDM2(ceil(end/2)+1:end) = [];
Threshold_FOFDM2_Upper = (convPS_FOFDM2(end)-convPS_FOFDM2(end)*(1-EnergyThreshold)/2);
Threshold_FOFDM2_Lower = (convPS_FOFDM2(end)*(1-EnergyThreshold)/2);
RequiredTime_FOFDM2 = sum((convPS_FOFDM2<Threshold_FOFDM2_Upper) & (convPS_FOFDM2>Threshold_FOFDM2_Lower))*(dt);
convPS_FOFDM3 = conv(PS_FOFDM3,ones(size(PS_FOFDM3)));
convPS_FOFDM3(ceil(end/2)+1:end) = [];
Threshold_FOFDM3_Upper = (convPS_FOFDM3(end)-convPS_FOFDM3(end)*(1-EnergyThreshold)/2);
Threshold_FOFDM3_Lower = (convPS_FOFDM3(end)*(1-EnergyThreshold)/2);
RequiredTime_FOFDM3 = sum((convPS_FOFDM3<Threshold_FOFDM3_Upper) & (convPS_FOFDM3>Threshold_FOFDM3_Lower))*(dt);
% Calculate the guard time, T_G
GuardTime_FBMC_Hermite(i_L) = (RequiredTime_FBMC_Hermite-FBMC_Hermite.PHY.TimeSpacing*FBMC_Hermite.Nr.MCSymbols);
GuardTime_FBMC_PHYDYAS(i_L) = (RequiredTime_FBMC_PHYDYAS-FBMC_PHYDYAS.PHY.TimeSpacing*FBMC_PHYDYAS.Nr.MCSymbols);
GuardTime_FOFDM1(i_L) = (RequiredTime_FOFDM1-FOFDM1.PHY.TimeSpacing*FOFDM1.Nr.MCSymbols);
GuardTime_FOFDM2(i_L) = (RequiredTime_FOFDM2-FOFDM2.PHY.TimeSpacing*FOFDM2.Nr.MCSymbols);
GuardTime_FOFDM3(i_L) = (RequiredTime_FOFDM3-FOFDM3.PHY.TimeSpacing*FOFDM3.Nr.MCSymbols);
% Calculate the power spectral density
[PSD_FOFDM1,f_FOFDM1] = FOFDM1.PlotPowerSpectralDensityUncorrelatedData;
[PSD_FOFDM2,f_FOFDM2] = FOFDM2.PlotPowerSpectralDensityUncorrelatedData;
[PSD_FOFDM3,f_FOFDM3] = FOFDM3.PlotPowerSpectralDensityUncorrelatedData;
% Calulate the PSD for FBMC where we add zeros to increase the frequency resolution
ZerosToAdd = zeros(AddedZeroFactorForPSD*FBMC_Hermite.Implementation.FFTSize,1);
PSD_FBMC_Hermite = zeros(FBMC_Hermite.Nr.SamplesTotal+length(ZerosToAdd),1);
PSD_FBMC_PHYDYAS = zeros(FBMC_Hermite.Nr.SamplesTotal+length(ZerosToAdd),1);
for i_l = 1:L
V = zeros(L,2*K);
V(i_l,K)=1;
PSD_FBMC_Hermite = PSD_FBMC_Hermite+abs(fft([FBMC_Hermite.Modulation(V);ZerosToAdd])).^2;
PSD_FBMC_PHYDYAS = PSD_FBMC_PHYDYAS+abs(fft([FBMC_PHYDYAS.Modulation(V);ZerosToAdd])).^2;
end
f_FBMC = (0:length(PSD_FBMC_Hermite)-1)*1/(length(PSD_FBMC_Hermite)*dt);
% Calculate the required frequency resources according to the threshold
convPSD_FBMC_Hermite=conv(PSD_FBMC_Hermite,ones(size(PSD_FBMC_Hermite)));
convPSD_FBMC_Hermite(ceil(end/2)+1:end) = [];
Threshold_FBMC_Hermite_Upper = (convPSD_FBMC_Hermite(end)-convPSD_FBMC_Hermite(end)*(1-EnergyThreshold)/2);
Threshold_FBMC_Hermite_Lower = (convPSD_FBMC_Hermite(end)*(1-EnergyThreshold)/2);
RequiredBandwidth_FBMC_Hermite = sum((convPSD_FBMC_Hermite<Threshold_FBMC_Hermite_Upper) & (convPSD_FBMC_Hermite>Threshold_FBMC_Hermite_Lower))*(f_FBMC(2)-f_FBMC(1));
convPSD_FBMC_PHYDYAS=conv(PSD_FBMC_PHYDYAS,ones(size(PSD_FBMC_PHYDYAS)));
convPSD_FBMC_PHYDYAS(ceil(end/2)+1:end) = [];
Threshold_FBMC_PHYDYAS_Upper = (convPSD_FBMC_PHYDYAS(end)-convPSD_FBMC_PHYDYAS(end)*(1-EnergyThreshold)/2);
Threshold_FBMC_PHYDYAS_Lower = (convPSD_FBMC_PHYDYAS(end)*(1-EnergyThreshold)/2);
RequiredBandwidth_FBMC_PHYDYAS = sum((convPSD_FBMC_PHYDYAS<Threshold_FBMC_PHYDYAS_Upper) & (convPSD_FBMC_PHYDYAS>Threshold_FBMC_PHYDYAS_Lower))*(f_FBMC(2)-f_FBMC(1));
convPSD_FOFDM1=conv(PSD_FOFDM1,ones(size(PSD_FOFDM1)));
convPSD_FOFDM1(ceil(end/2)+1:end) = [];
Threshold_FOFDM1_Upper = (convPSD_FOFDM1(end)-convPSD_FOFDM1(end)*(1-EnergyThreshold)/2);
Threshold_FOFDM1_Lower = (convPSD_FOFDM1(end)*(1-EnergyThreshold)/2);
RequiredBandwidth_FOFDM1 = sum((convPSD_FOFDM1<Threshold_FOFDM1_Upper) & (convPSD_FOFDM1>Threshold_FOFDM1_Lower))*(f_FOFDM1(2)-f_FOFDM1(1));
convPSD_FOFDM2=conv(PSD_FOFDM2,ones(size(PSD_FOFDM2)));
convPSD_FOFDM2(ceil(end/2)+1:end) = [];
Threshold_FOFDM2_Upper = (convPSD_FOFDM2(end)-convPSD_FOFDM2(end)*(1-EnergyThreshold)/2);
Threshold_FOFDM2_Lower = (convPSD_FOFDM2(end)*(1-EnergyThreshold)/2);
RequiredBandwidth_FOFDM2 = sum((convPSD_FOFDM2<Threshold_FOFDM2_Upper) & (convPSD_FOFDM2>Threshold_FOFDM2_Lower))*(f_FOFDM2(2)-f_FOFDM2(1));
convPSD_FOFDM3=conv(PSD_FOFDM3,ones(size(PSD_FOFDM3)));
convPSD_FOFDM3(ceil(end/2)+1:end) = [];
Threshold_FOFDM3_Upper = (convPSD_FOFDM3(end)-convPSD_FOFDM3(end)*(1-EnergyThreshold)/2);
Threshold_FOFDM3_Lower = (convPSD_FOFDM3(end)*(1-EnergyThreshold)/2);
RequiredBandwidth_FOFDM3 = sum((convPSD_FOFDM3<Threshold_FOFDM3_Upper) & (convPSD_FOFDM3>Threshold_FOFDM3_Lower))*(f_FOFDM3(2)-f_FOFDM3(1));
% Calculate the guard frequency F_G
GuardFrequency_FBMC_Hermite(i_L) = (RequiredBandwidth_FBMC_Hermite - FBMC_Hermite.PHY.SubcarrierSpacing * FBMC_Hermite.Nr.Subcarriers);
GuardFrequency_FBMC_PHYDYAS(i_L) = (RequiredBandwidth_FBMC_PHYDYAS - FBMC_PHYDYAS.PHY.SubcarrierSpacing * FBMC_PHYDYAS.Nr.Subcarriers);
GuardFrequency_FOFDM1(i_L) = (RequiredBandwidth_FOFDM1 - FOFDM1.PHY.SubcarrierSpacing * FOFDM1.Nr.Subcarriers);
GuardFrequency_FOFDM2(i_L) = (RequiredBandwidth_FOFDM2 - FOFDM2.PHY.SubcarrierSpacing * FOFDM2.Nr.Subcarriers);
GuardFrequency_FOFDM3(i_L) = (RequiredBandwidth_FOFDM3 - FOFDM3.PHY.SubcarrierSpacing * FOFDM3.Nr.Subcarriers);
% Calculate the time-frequency efficiency
TimeFrequencyEfficiency_FMBC_Hermite(i_L) = K*L/(RequiredTime_FBMC_Hermite*RequiredBandwidth_FBMC_Hermite); % K for complex symbols! For real symbols K we need to include the factor 0.5
TimeFrequencyEfficiency_FMBC_PHYDYAS(i_L) = K*L/(RequiredTime_FBMC_PHYDYAS*RequiredBandwidth_FBMC_PHYDYAS); % K for complex symbols! For real symbols K we need to include the factor 0.5
TimeFrequencyEfficiency_FOFDM1(i_L) = K*L/(RequiredTime_FOFDM1*RequiredBandwidth_FOFDM1);
TimeFrequencyEfficiency_FOFDM2(i_L) = K*L/(RequiredTime_FOFDM2*RequiredBandwidth_FOFDM2);
TimeFrequencyEfficiency_FOFDM3(i_L) = K*L/(RequiredTime_FOFDM3*RequiredBandwidth_FOFDM3);
TimeFrequencyEfficiency_Kinfinity_FMBC_Hermite(i_L) = 0.5*L/(FBMC_Hermite.PHY.TimeSpacing*RequiredBandwidth_FBMC_Hermite);
TimeFrequencyEfficiency_Kinfinity_FMBC_PHYDYAS(i_L) = 0.5*L/(FBMC_PHYDYAS.PHY.TimeSpacing*RequiredBandwidth_FBMC_PHYDYAS);
TimeFrequencyEfficiency_Kinfinity_FOFDM1(i_L) = L/(FOFDM1.PHY.TimeSpacing*RequiredBandwidth_FOFDM1);
TimeFrequencyEfficiency_Kinfinity_FOFDM2(i_L) = L/(FOFDM2.PHY.TimeSpacing*RequiredBandwidth_FOFDM2);
TimeFrequencyEfficiency_Kinfinity_FOFDM3(i_L) = L/(FOFDM3.PHY.TimeSpacing*RequiredBandwidth_FOFDM3);
disp([int2str(i_L/length(M_L)*100) '%']);
end
ColorFBMC = [0 0 1]*0.5;
ColorFBMC_Hermite = [0 0 1];
ColorFOFDM = [1 1 0]*0.7;
figure();
plot(M_L,TimeFrequencyEfficiency_FMBC_Hermite,'Color',ColorFBMC_Hermite);hold on;
plot(M_L,TimeFrequencyEfficiency_FMBC_PHYDYAS,'Color',ColorFBMC);hold on;
plot(M_L,TimeFrequencyEfficiency_Kinfinity_FMBC_Hermite,'Color',ColorFBMC_Hermite);hold on;
plot(M_L,TimeFrequencyEfficiency_Kinfinity_FMBC_PHYDYAS,'Color',ColorFBMC);hold on;
% plot(M_L,TimeFrequencyEfficiency_FOFDM1,':','Color',ColorFOFDM);hold on;
% plot(M_L,TimeFrequencyEfficiency_FOFDM2,':','Color',ColorFOFDM);hold on;
% plot(M_L,TimeFrequencyEfficiency_FOFDM3,':','Color',ColorFOFDM);hold on;
plot(M_L,TimeFrequencyEfficiency_Kinfinity_FOFDM1,'Color',ColorFOFDM);hold on;
plot(M_L,TimeFrequencyEfficiency_Kinfinity_FOFDM2,'Color',ColorFOFDM);hold on;
plot(M_L,TimeFrequencyEfficiency_Kinfinity_FOFDM3,'Color',ColorFOFDM);hold on;
xlim([0 max(M_L)]);
xlabel('Number of Subcarriers L');
ylabel('Time-Frequency Efficiency $\rho$');
title('See Fig. 9');