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5G NR Downlink Carrier Aggregation, Demodulation, and Analysis

This example shows how to generate, aggregate, and demodulate multiple downlink carriers using 5G Toolbox™ features.


This example generates an NR waveform with carrier aggregation (CA). To perform carrier aggregation, the example calculates the frequency offsets for the intraband contiguous CA case, as described in TS 38.104 Section 5.3A. The example also supports customized intraband noncontiguous and interband CA scenarios.

To generate an aggregated downlink waveform, the example configures a standard-compliant downlink fixed reference channel (FRC) for each component carrier. TS 38.101-1 Annex A.3 defines the physical downlink shared channel (PDSCH) FRCs for FR1 and TS 38.101-2 Annex A.3 defines the PDSCH FRCs for FR2. For the FRC waveform generation, you can specify the channel bandwidth, the subcarrier spacing, the modulation, the duplexing mode, and the cell ID. For more information on how to generate DL FRCs, see 5G NR-TM and FRC Waveform Generation.

After generating the component carriers (CCs), the example resamples the waveforms to a common sample rate and combines the CCs to generate the aggregated waveform.

Finally, the example filters and downsamples a selected CC to perform EVM measurements and PDSCH decoding.

Selection of Component Carrier Parameters

The vector ChannelBandwidths specifies the bandwidth for each CC. The length of this vector corresponds to the number of CCs. The elements of ChannelBandwidths must be in the set {5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90 100} MHz for FR1 and in the set {50, 100, 200, 400} MHz for FR2. TS 38.101-1 Section 5.5A.1 Table 5.5A.1-1 and TS 38.101-2 Section 5.5A.1 Table 5.5A.1-1 list the valid combinations of bandwidths for FR1 and FR2 carrier aggregation, respectively.

If the ccSpacings vector is empty, the example calculates the spacings between consecutive carriers, ccSpacings, as described in TS 38.104. To select the spacings of your choice, add the spacings to the ccSpacings vector.

% Configure three carriers for the aggregation. You can select a different
% number of carriers by modifying the number of elements in the
% |channelBandwidths|, |SCSs|, |modulations|, and |nCellIDs| vectors.
frequencyRange = 'FR1'; % (FR1 or FR2)
channelRaster = 100; % Channel raster in kHz (15, 60 or 100)
channelBandwidths = [60 40 40]; % Channel bandwidths in MHz 
                                % (5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 
                                % 90 or 100 for FR1)
                                % (50, 100, 200 or 400 for FR2)
SCSs = [30 30 30]; % Subcarrier spacings in kHz 
                   % (15, 30 or 60 for FR1) 
                   % (60 or 120 for FR2)
modulations = {'QPSK' '64QAM' '256QAM'}; % ('QPSK', '64QAM' or '256QAM' for FR1)
                                         % ('QPSK', '16QAM' or '64QAM' for FR2)
nCellIDs = [1 2 3]; % Cell IDs          
duplexMode = 'TDD'; % Duplex mode ('TDD' or 'FDD')
sv = '15.2.0'; % Standard version ('15.1.0', '15.2.0' or '15.7.0')
numSubframes = 10; % Number of subframes to generate per carrier

% Choose the spacings between consecutive carriers, |ccSpacings|, or leave
% the vector empty for calculating the spacings, as described in TS 38.104
% Section 5.3A
ccSpacings = []; % MHz

% Select the CC to demodulate
CCofInterest = 2;

% Verify the number of CCs, channel bandwidths, subcarrier spacings,
% modulations, and spacings. Additionally, check that the spacings are
% greater than 0.

Component Carrier Configuration

Generate a configuration structure for each CC using hNRReferenceWaveformGenerator. Store the configuration structures for all CCs in a cell array.

% Establish the number of component carriers 
numCC = length(channelBandwidths);

% CC configuration
referenceChannel = cell(1,numCC);
wavegen = cell(1,numCC);
for i = 1:numCC
    % Select a reference channel, FRC, based on the chosen modulation and
    % frequency range
    referenceChannel{i} = strcat('DL-FRC-',frequencyRange,'-',modulations{i});
    % Create a generator object for the above PDSCH FRC reference channel
    wavegen{i} = hNRReferenceWaveformGenerator(referenceChannel{i},...

Carrier Aggregation Parameters Calculation

To perform the carrier aggregation, you must calculate the frequency parameters described in TS 38.104, Sections 5.3 and 5.4.

  • Fc_offset is a vector containing the center frequency of each CC at baseband

  • F_offset_low is the frequency offset from the lower Fc_offset to the lower aggregated bandwidth edge

  • F_offset_high is the frequency offset from the upper Fc_offset to the upper aggregated channel bandwidth edge

  • F_edge_low is the lower edge of the aggregated channel bandwidth

  • F_edge_high is the upper edge of the aggregated channel bandwidth

  • BW_channel_CA is the aggregated channel bandwidth of all CCs

Center the lower carrier component at baseband (Fc_offset(1) = 0 Hz) and compute the center frequency for the rest of the CCs. To determine the center frequencies by following the method described in TS 38.104 Section 5.3A, you must calculate the minimum guardbands and the minimum CC spacings. Alternatively, to select your own center frequencies, choose a nonempty ccSpacings vector.

% Get the largest SCS configuration, |SCSConfig|, among the SCS
% configurations supported for each two consecutive channel bandwidths to
% obtain the minimum guardband, as described in TS 38.104 Section
table = hGetGBTable(frequencyRange); % Minimum guardband table
if isequal(frequencyRange,'FR1')
    if any(channelBandwidths(:) == 5)
        SCSConfig = 1; 
        SCSConfig = 2; 
    SCSConfig = 3;

% Calculate the minimum guardband, as described in TS 38.104 Section 5.3.3
% Table 5.3.3-1 for FR1 and Table 5.3.3-1 for FR2, to obtain the minimum CC
% spacings.
minimumGuardBand = zeros(1,numCC);
NDLRB = zeros(1,numCC);
scs = strcat(num2str(2^SCSConfig*15),'kHz'); % Common SCS in kHz to extract 
                                             % minimum guardband 
for i = 1:numCC 
    NDLRB(i) = wavegen{i}.Config.SCSCarriers{1}.NSizeGrid;
    minimumGuardBand(i) = table{{scs},{strcat(num2str(channelBandwidths(i)),...
        'MHz');}}; % kHz
    minimumGuardBand(i) = minimumGuardBand(i)*1e-3; % MHz

% Compute the minimum CC spacings, as defined in TS 38.104 Section
% Use these spacings to calculate the center frequencies and for
% filtering each CC.
minCCSpacings = zeros(1,numCC-1); % Minimum CC spacing
for k = 2:numCC
    minCCSpacings(k-1) = hNRCarrierAggregationChannelSpacing( ...
        channelBandwidths(k-1), channelBandwidths(k), minimumGuardBand(k-1), ...
        minimumGuardBand(k),channelRaster,SCSConfig); % MHz

% Determine the center frequency for each CC with respect to 0 Hz.
% Initially, the lower carrier frequency is at baseband (Fc_offset(1) = 0 Hz).
Fc_offset = zeros(1,numCC);
if isempty(ccSpacings)
    ccSpacings = minCCSpacings;
for k = 2:numCC
    Fc_offset(k) = Fc_offset(k-1) + ccSpacings(k-1); % MHz

Calculate the frequency offsets from the lower and upper center frequencies to the lower and upper aggregated bandwidth edges, respectively, as described in TS 38.104 Section 5.3A.

F_offset_low = (NDLRB(1)*12+1)*(SCSs(1)*1e-3)/2 + minimumGuardBand(1); % MHz
F_offset_high = (NDLRB(end)*12-1)*(SCSs(end)*1e-3)/2 + minimumGuardBand(end); %MHz

Compute the lower and upper edges of the aggregated channel bandwidth, TS 38.104 Section 5.3A.

F_edge_low = Fc_offset(1) - F_offset_low; % MHz
F_edge_high = Fc_offset(end) + F_offset_high; % MHz

Calculate the aggregated channel bandwidth, TS 38.104 Section 5.3A

BW_channel_CA = F_edge_high - F_edge_low; % MHz
fprintf('BW_channel_CA: %0.4f MHz\n',BW_channel_CA);
BW_channel_CA: 141.0800 MHz

Determine the frequency shift to center the aggregated channel bandwidth at baseband (0 Hz).

shiftToCenter = -1*(BW_channel_CA/2 + F_edge_low);

% Center aggregated bandwidth at baseband
Fc_offset = Fc_offset + shiftToCenter;
F_edge_low = Fc_offset(1) - F_offset_low;
F_edge_high = Fc_offset(end) + F_offset_high;

% Display frequency band edges
fprintf('F_edge_low:  %0.4f MHz\n',F_edge_low);
F_edge_low:  -70.5400 MHz
fprintf('F_edge_high: %0.4f MHz\n',F_edge_high);
F_edge_high: 70.5400 MHz
fprintf('F_offset_low:  %0.4f MHz\n',F_offset_low);
F_offset_low:  30.7050 MHz
fprintf('F_offset_high: %0.4f MHz\n',F_offset_high);
F_offset_high: 20.6750 MHz
% Display carrier frequencies
for i = 1:numCC
    fprintf('Component Carrier %d:\n',i);
    fprintf('   Fc: %0.4f MHz\n', Fc_offset(i));
Component Carrier 1:
   Fc: -39.8350 MHz
Component Carrier 2:
   Fc: 9.9650 MHz
Component Carrier 3:
   Fc: 49.8650 MHz

Oversampling Rate Calculation

Calculate the required oversampling factors for each component carrier, OSRs, to use a common sampling rate for the aggregated waveform.

% Obtain sample rates of the component carriers
CCSR = zeros(1,numCC);
carriers = cell(1,numCC);
for i = 1:numCC
    carriers{i} = nrCarrierConfig;
    carriers{i}.NCellID = nCellIDs(i);
    carriers{i}.NSizeGrid = NDLRB(i);
    carriers{i}.SubcarrierSpacing = SCSs(i);
    carriers{i}.CyclicPrefix = wavegen{i}.Config.BandwidthParts{1}.CyclicPrefix;
    info = nrOFDMInfo(carriers{i});
    CCSR(i) = info.SampleRate; % Hz

% Calculate the oversampling ratio for the largest BW CC to ensure the
% waveform occupies a maximum of 85% of the total bandwidth, |bwfraction|
bwfraction = 0.85; % Bandwidth utilization of 85%
OSR = (BW_channel_CA/bwfraction)/(max(CCSR)/1e6);

% To simplify the resampling operation, choose an oversampling ratio which
% is a power of 2: calculate the next power of two above OSR
OSR = 2^ceil(log2(OSR));

% Calculate the overall sample rate for the aggregated waveform
SR = OSR*max(CCSR); % Hz
fprintf('\nOutput sample rate: %0.4f Ms/s\n\n',SR/1e6);
Output sample rate: 245.7600 Ms/s
% Calculate the individual oversampling factors for the component carriers

Waveform Generation and Carrier Aggregation

Call the generateWaveform function from the hNRReferenceWaveformGenerator helper file to generate the waveform for each CC. Resample each carrier to a common sample rate. Finally, use comm.MultibandCombiner to frequency modulate the carriers to the appropriate center frequency, and aggregate the CCs to form the combined waveform.

% Generate and aggregate the component carriers
tmwaveinfo = cell(1,numCC);
waveInfo = cell(1,numCC);
resourcesInfo = cell(1,numCC);
clear waveform
for i = numCC:-1:1
    % Generate each CC
    [wf,waveInfo{i},resourcesInfo{i}] = generateWaveform(wavegen{i},...
    % Resample the CCs so that they have the same sample rate  
    waveform(:,i) = resample(wf,OSRs(i),1)/OSRs(i);    

% Aggregate all CC. comm.MultibandCombiner applies the frequency offsets
% and combines the resulting signals together.
carrierAggregator = comm.MultibandCombiner(InputSampleRate = SR,...
    FrequencyOffsets = Fc_offset*1e6,...
    OutputSampleRateSource = 'Property',...
    OutputSampleRate = SR);

waveform = carrierAggregator(waveform);

Plot Carrier Aggregated Spectrum

Display the spectrum of the carrier aggregated signal by using the hNRCarrierAggregationPlotSpectrum helper function. The spectrum plot shows the individual carrier bandwidths. This example does not upconvert the waveform to radio frequency (RF), the center of the aggregated bandwidth is at baseband (0 Hz).

specPlot = hNRCarrierAggregationPlotSpectrum(waveform,SR,...
    'Power Spectrum of Carrier Aggregation',{'Signal spectrum'});

Figure Spectrum Analyzer contains an axes object and other objects of type uiflowcontainer, uimenu, uitoolbar. The axes object with title Power Spectrum of Carrier Aggregation contains an object of type line. This object represents Signal spectrum.

CC Demodulation and Filtering

Choose a CC, then demodulate, filter and downsample the CC of your choice, following these steps.

  • Bring the CC to baseband (0 Hz)

  • Calculate the passband and stopband frequencies of the filter

  • Filter out the neighboring CCs by using the designed filter and downsample the CC.

% Verify carrier of interest ID
if CCofInterest > numCC || CCofInterest <= 0 || mod(CCofInterest,1) ~= 0
        'Cannot demodulate CC number %d, choose an integer number that falls between 1 and %d\n',...
        CCofInterest, numCC) ;
fprintf(1,'Extracting CC number %d: \n', CCofInterest);
Extracting CC number 2: 
% Define downsampling filter order
filterOrder = 201;

% Precalculate the filter passband and stopband values for all CC
firPassbandVec = (NDLRB*12-1).*(SCSs*1e-3)/2 / (SR/1e6/2);
firStopbandVec = hNRCarrierAggregationStopband(minCCSpacings,NDLRB,SR,SCSs);

% Choose the passband and stopband values for the carrier of interest
firPassband = firPassbandVec(CCofInterest);
firStopband = firStopbandVec(CCofInterest);

% Pad signal with zeros to consider filter transient response length
waveform = [waveform; zeros(filterOrder*2,size(waveform,2))];

% Center the carrier of interest at 0 Hz
frequencyShifter = comm.PhaseFrequencyOffset(SampleRate = SR,...
    FrequencyOffset = -Fc_offset(CCofInterest)*1e6);
demodulatedCC = frequencyShifter(waveform);

% To ease the filter design requirements, apply the downsampling in two
% stages if necessary. Use a downsampling factor of 4 in the initial stage.
% If the quality of the resulting signal is not as required, consider a
% different filter design in this initial stage.
if (firStopband < 0.1)
    % Downsample by 4 in the initial stage
    SRC = 4;
    demodulatedCC = resample(demodulatedCC,1,SRC);
    % Update passband and stopband values
    firPassband = firPassband * SRC;
    firStopband = firStopband * SRC;
    % Do not apply an extra downsampling 
    SRC = 1;

% Design the lowpass filter to filter out the CC of your choice
frEdges = [0 firPassband firStopband 1];
firFilter = dsp.FIRFilter;
firFilter.Numerator = firpm(filterOrder,frEdges,[1 1 0 0]);

% Display the response of the designed filter

Figure Filter Visualization Tool - Magnitude Response (dB) and Phase Response contains an axes object and other objects of type uitoolbar, uimenu. The axes object with title Magnitude Response (dB) and Phase Response contains an object of type line.

% Filter the signal to extract the component of interest
rxWaveform  = firFilter(demodulatedCC);

% Plot the demodulated and filtered spectra
filteredSpecPlot = ...
    hNRCarrierAggregationPlotSpectrum([demodulatedCC, rxWaveform],SR,...
            'Demodulated and Filtered Waveform Power Spectrum',...
            {'Carrier aggregated signal' 'Filtered signal'});

Figure Spectrum Analyzer contains an axes object and other objects of type uiflowcontainer, uimenu, uitoolbar. The axes object with title Demodulated and Filtered Waveform Power Spectrum contains 2 objects of type line. These objects represent Carrier aggregated signal, Filtered signal.

% Downsample the filtered carrier to its baseband rate
rxWaveform = downsample(rxWaveform,OSRs(CCofInterest)/SRC);

EVM Measurements

The hNRPDSCHEVM helper function returns the PDSCH EVM by performing synchronization, OFDM demodulation, channel estimation, and equalization. The function displays the EVM for each slot and frame and the overall EVM averaged over the entire input waveform. The function also plots these graphs: EVM per OFDM symbol, slot, subcarrier, and overall EVM.

% Parameterize the channel estimator configuration using the structure |cfg|
cfg = struct();
cfg.Evm3GPP = true; % To measure EVM as defined in TS 38.104, Annex B(FR1) 
                    % / Annex C(FR2) set |Evm3GPP| to |true|. 
cfg.TargetRNTIs = [];
cfg.PlotEVM = true;
cfg.DisplayEVM = true;
cfg.Label = wavegen{CCofInterest}.ConfiguredModel{1};

% Perform EVM measurements and plot results
[evmInfo,eqSym,refSym] = hNRPDSCHEVM(wavegen{CCofInterest}.Config,...
EVM stats for BWP idx : 1
Low edge RMS EVM, Peak EVM, slot 1: 3.111 8.587%
High edge RMS EVM, Peak EVM, slot 1: 2.795 8.020%
Low edge RMS EVM, Peak EVM, slot 2: 3.008 8.550%
High edge RMS EVM, Peak EVM, slot 2: 2.774 7.533%
Low edge RMS EVM, Peak EVM, slot 3: 3.156 9.854%
High edge RMS EVM, Peak EVM, slot 3: 2.796 7.604%
Low edge RMS EVM, Peak EVM, slot 4: 2.984 8.644%
High edge RMS EVM, Peak EVM, slot 4: 2.784 7.600%
Low edge RMS EVM, Peak EVM, slot 5: 2.968 8.779%
High edge RMS EVM, Peak EVM, slot 5: 2.774 7.659%
Low edge RMS EVM, Peak EVM, slot 6: 3.106 9.877%
High edge RMS EVM, Peak EVM, slot 6: 2.783 7.265%
Low edge RMS EVM, Peak EVM, slot 10: 3.129 10.707%
High edge RMS EVM, Peak EVM, slot 10: 2.779 8.086%
Low edge RMS EVM, Peak EVM, slot 11: 3.105 9.934%
High edge RMS EVM, Peak EVM, slot 11: 2.767 7.837%
Low edge RMS EVM, Peak EVM, slot 12: 3.071 8.785%
High edge RMS EVM, Peak EVM, slot 12: 2.780 7.284%
Low edge RMS EVM, Peak EVM, slot 13: 3.010 8.784%
High edge RMS EVM, Peak EVM, slot 13: 2.792 7.980%
Low edge RMS EVM, Peak EVM, slot 14: 3.005 8.365%
High edge RMS EVM, Peak EVM, slot 14: 2.789 7.437%
Low edge RMS EVM, Peak EVM, slot 15: 2.944 8.279%
High edge RMS EVM, Peak EVM, slot 15: 2.791 7.852%
Low edge RMS EVM, Peak EVM, slot 16: 3.057 8.974%
High edge RMS EVM, Peak EVM, slot 16: 2.788 7.686%
Averaged low edge RMS EVM, frame 0: 3.051%
Averaged high edge RMS EVM, frame 0: 2.784%
Averaged RMS EVM frame 0: 3.051%

Figure contains 3 axes objects. Axes object 1 with title EVM vs OFDM symbol, BWP idx : 1 contains 2 objects of type line. These objects represent rms EVM, peak EVM. Axes object 2 with title EVM vs Slot, BWP idx : 1 contains 2 objects of type line. These objects represent rms EVM, peak EVM. Axes object 3 with title EVM vs Subcarrier, BWP idx : 1 contains 2 objects of type line. These objects represent rms EVM, max EVM.

Figure EVM (%) contains an axes object. The axes object with title EVM Resource Grid, BWP idx : 1 contains an object of type surface.

Figure contains an axes object. The axes object with title Equalized Symbols Constellation, BWP idx : 1 contains 2 objects of type line.

Averaged overall RMS EVM: 3.051%
Overall Peak EVM = 10.7065%

PDSCH Decoding

Decode the PDSCH of the recovered signal and check the resulting CRC for errors.

% Perform time synchronization on the input waveform
offset = nrTimingEstimate(carriers{CCofInterest},rxWaveform,...
rxWaveform = rxWaveform(1+offset:end,:);

% Perform OFDM demodulation
rxGrid = nrOFDMDemodulate(carriers{CCofInterest},rxWaveform);

% Get the allocated slots and OFDM symbols per slot
allocatedSlots = zeros(1,size(resourcesInfo{CCofInterest}.WaveformResources.PDSCH.Resources,2));
for i=1:length(allocatedSlots)
    allocatedSlots(i) = resourcesInfo{CCofInterest}.WaveformResources.PDSCH.Resources(i).NSlot;
L = carriers{CCofInterest}.SymbolsPerSlot; % OFDM symbols per slot

% Create a DLSCH decoder System object
decodeDLSCH = nrDLSCHDecoder;
decodeDLSCH.MultipleHARQProcesses = false;
decodeDLSCH.TargetCodeRate = wavegen{CCofInterest}.Config.PDSCH{1}.TargetCodeRate;
decodeDLSCH.LDPCDecodingAlgorithm = 'Normalized min-sum';
decodeDLSCH.MaximumLDPCIterationCount = 6;

for NSlot=1:length(allocatedSlots)
    % Extract slot
    SlotID = allocatedSlots(NSlot);
    rxSlot = rxGrid(:,(1:L)+(SlotID*L),:);
    refSlot = waveInfo{CCofInterest}.ResourceGridBWP(:,(1:L)+(SlotID*L),:);
    % Perform channel estimation
    [estChannelGrid,noiseEst] = nrChannelEstimate(carriers{CCofInterest},...
    % Get PDSCH resource elements from the received grid and channel estimate
    pdschIndices = resourcesInfo{CCofInterest}.WaveformResources.PDSCH.Resources(NSlot).ChannelIndices;
    [pdschRx,pdschHest] = nrExtractResources(pdschIndices,rxSlot,estChannelGrid);
    % Perform equalization
    [pdschEq,csi] = nrEqualizeMMSE(pdschRx,pdschHest,noiseEst);
    % Perform layer demapping, symbol demodulation, and descrambling
    modulation = wavegen{CCofInterest}.Config.PDSCH{1}.Modulation;
    RNTI = wavegen{CCofInterest}.Config.PDSCH{1}.RNTI;
    [dlschLLRs,rxSymbols] = nrPDSCHDecode(pdschEq,modulation,...
    % Scale LLRs by CSI
    csi = nrLayerDemap(csi); % CSI layer demapping
    NumCW = size(resourcesInfo{CCofInterest}.WaveformResources.PDSCH.Resources(NSlot).Codeword,2);
    for cwIdx = 1:NumCW
        Qm = length(dlschLLRs{cwIdx})/length(rxSymbols{cwIdx}); % Bits per symbol
        csi{cwIdx} = repmat(csi{cwIdx}.',Qm,1);   % Expand by each bit per symbol
        dlschLLRs{cwIdx} = dlschLLRs{cwIdx} .* csi{cwIdx}(:);   % Scaled symbols
    % Calculate the transport block sizes for the codewords in the slot
    trBlkSize = resourcesInfo{CCofInterest}.WaveformResources.PDSCH.Resources(NSlot).TransportBlockSize;
    % Decode the DL-SCH transport channel
    decodeDLSCH.TransportBlockLength = trBlkSize;
    NLayers = wavegen{CCofInterest}.Config.PDSCH{1}.NumLayers;
    RVSequence = wavegen{CCofInterest}.Config.PDSCH{1}.RVSequence;
    [decbits,crc] = decodeDLSCH(dlschLLRs,modulation,NLayers,RVSequence);
    % Display the CRC status
    if crc
        disp(['Slot ' num2str(SlotID) ': CRC failed']);
        disp(['Slot ' num2str(SlotID) ': CRC passed']);
Slot 1: CRC passed
Slot 2: CRC passed
Slot 3: CRC passed
Slot 4: CRC passed
Slot 5: CRC passed
Slot 6: CRC passed
Slot 10: CRC passed
Slot 11: CRC passed
Slot 12: CRC passed
Slot 13: CRC passed
Slot 14: CRC passed
Slot 15: CRC passed
Slot 16: CRC passed


  1. 3GPP TS 38.104 "NR; Base Station (BS) radio transmission and reception" 3rd Generation Partnership Project; Technical Specification Group Radio Access Network.

  2. 3GPP TS 38.101-1 "NR; User Equipment (UE) radio transmission and reception; Part 1: Range 1 Standalone." 3rd Generation Partnership Project; Technical Specification Group Radio Access Network.

  3. 3GPP TS 38.101-2 "NR; User Equipment (UE) radio transmission and reception: Part 2: Range 2 Standalone." 3rd Generation Partnership Project; Technical Specification Group Radio Access Network.

See Also


Related Topics