Communications System Toolbox

QPSK Transmitter and Receiver

This example shows a digital communications system using QPSK modulation. The example uses Communications System objects to simulate the QPSK transceiver. In particular, this example illustrates methods to address real-world wireless communications issues like carrier frequency and phase offset, timing recovery and frame synchronization.


This example describes the MATLAB implementation of the QPSK transceiver.

MATLAB script using System objects: commQPSKTransmitterReceiver.mcommQPSKTransmitterReceiver.m.

For a Simulink® implementation using blocks, check: commqpsktxrx.mdlcommqpsktxrx.mdl.


The transmitted QPSK data undergoes impairments that simulate the effects of wireless transmission such as addition of Additive White Gaussian Noise (AWGN), introduction of carrier frequency and phase offset, and timing delay. To cope with these impairments, this example provides a reference design of a practical digital receiver. The receiver includes FFT-based coarse frequency compensation, PLL-based fine frequency compensation, timing recovery with fixed-rate resampling and bit stuffing/skipping, frame synchronization, and phase ambiguity resolution.

This example serves three main purposes:

  • To model a general wireless communication system that is able to successfully recover a message, which was corrupted by various simulated channel impairments.

  • To illustrate the use of key Communications System Toolbox™ System objects for QPSK system design, including coarse and fine carrier frequency compensation, closed-loop timing recovery with bit stuffing and stripping, frame synchronization, carrier phase ambiguity resolution, and message decoding.

  • To illustrate the creation of higher level System objects that contain other System objects in order to model larger components of the system under test


The commqpsktxrx_init.mcommqpsktxrx_init.m script initializes simulation parameters and generates the structure prmQPSKTxRx.

prmQPSKTxRx = commqpsktxrx_init % QPSK system parameters

useScopes = true; % true if scopes are to be used
printReceivedData = false; %true if the received data is to be printed
compileIt = false; % true if code is to be compiled
useCodegen = false; % true to run the generated mex file
prmQPSKTxRx = 

                                M: 4
                       Upsampling: 4
                     Downsampling: 2
                               Fs: 200000
                               Ts: 5.0000e-06
                        FrameSize: 100
                     BarkerLength: 13
                       DataLength: 174
                    ScramblerBase: 2
              ScramblerPolynomial: [1 1 1 0 1]
       ScramblerInitialConditions: [0 0 0 0]
                             sBit: [17400x1 double]
                 RxBufferedFrames: 10
           PhaseErrorDetectorGain: 2.0000
                PhaseRecoveryGain: 1
          TimingErrorDetectorGain: 5.4000
               TimingRecoveryGain: -1
           RaisedCosineFilterSpan: 10
                    MessageLength: 105
                       FrameCount: 100
                      PhaseOffset: 47
                             EbNo: 13
                  FrequencyOffset: 5000
                        DelayType: 'Triangle'
    CoarseCompFrequencyResolution: 50
       PhaseRecoveryLoopBandwidth: 0.0100
       PhaseRecoveryDampingFactor: 1
      TimingRecoveryLoopBandwidth: 0.0100
      TimingRecoveryDampingFactor: 1
                          Rolloff: 0.5000
    TransmitterFilterCoefficients: [1x41 double]
       ReceiverFilterCoefficients: [1x41 double]

Code Architecture for the System Under Test

This example models a digital communication system using QPSK modulation. The function runQPSKSystemUnderTest models this communication environment. The QPSK transceiver model in this script is divided into the following four main components.

1) QPSKTransmitter: generates the bit stream and then encodes, modulates and filters it.

2) QPSKChannel: models the channel with carrier offset, timing offset, and AWGN.

3) QPSKReceiver: models the receiver, including components for phase recovery, timing recovery, decoding, demodulation, etc.

4) QPSKScopes: optionally visualizes the signal using time scopes, frequency scopes, and constellation diagrams.

Each component is modeled using a System object. To see the construction of the four main System object components, refer to runQPSKSystemUnderTest.mrunQPSKSystemUnderTest.m.

Description of the Individual Components


This component generates a message using ASCII characters, converts the characters to bits, and prepends a Barker code for receiver frame synchronization. This data is then modulated using QPSK and filtered with a square root raised cosine filter.


This component simulates the effects of over-the-air transmission. It degrades the transmitted signal with both phase and frequency offset, a time-varying delay to mimic clock skew between transmitter and receiver, and AWGN.


This component regenerates the original transmitted message. It is divided into five subcomponents, modeled using System objects. Each subcomponent is modeled by other subcomponents using System objects.

1) Automatic Gain Control: Ensures that the amplitude of the input of the Coarse Frequency Compensation subsystem is 1/Upsampling Factor , so that the equivalent gains of the phase and timing error detectors keep constant over time. The AGC is placed before the Raised Cosine Receive Filter so that the signal amplitude can be measured with an oversampling factor of four. This process improves the accuracy of the estimate.

2) Coarse frequency compensation: Uses nonlinearity and a Fast Fourier Transform (FFT) to roughly estimate the frequency offset and then compensate for it. The subsystem raises the input signal to the power of four to obtain a signal that is not a function of the QPSK modulation. Then it performs an FFT on the modulation-independent signal to estimate the tone at four times the frequency offset. After dividing the estimate by four, the Phase/Frequency Offset System object corrects the frequency offset.

3) Fine frequency compensation: Performs closed-loop scalar processing and compensates for the frequency offset accurately. The Fine Frequency Compensation subsystem implements a phase-locked loop (PLL) to track the residual frequency offset and the phase offset in the input signal. For more information, see Chapter 7 of [ 1 ]. The PLL uses a Direct Digital Synthesizer (DDS) to generate the compensating phase that offsets the residual frequency and phase offsets. The phase offset estimate from DDS is the integral of the phase error output of the Loop Filter. To obtain details of PLL design, refer to Appendix C.2 of [ 1 ].

4) Timing recovery: Performs timing recovery with closed-loop scalar processing to overcome the effects of delay introduced by the channel. The Timing Recovery subsystem implements a PLL, described in Chapter 8 of [ 1 ], to correct the timing error in the received signal. The NCO Control subsystem implements a decrementing modulo-1 counter described in Chapter 8.4.3 of [ 1 ] to generate the control signal for the Modified Buffer to select the interpolants of the Interpolation Filter. This control signal also enables the Timing Error Detector (TED), so that it calculates the timing errors at the correct timing instants. The NCO Control subsystem updates the timing difference for the Interpolation Filter , generating interpolants at optimum sampling instants. The Interpolation Filter is a Farrow parabolic filter with alpha set o 0.5 as described in Chapter 8.4.2 of [ 1 ]. Based on the interpolants, timing errors are generated by a zero-crossing Timing Error Detector as described in Chapter 8.4.1 of [ 1 ], filtered by a tunable proportional-plus-integral Loop Filter as described in Appendix C.2 of [ 1 ], and fed into the NCO Control for a timing difference update. The Loop Bandwidth (normalized by the sample rate) and Loop Damping Factor are tunable for the Loop Filter. The default normalized loop bandwidth is set to 0.01 and the default damping factor is set to 1 for critical damping. These settings ensure that the PLL quickly locks to the correct timing while introducing little phase noise.

5) Data decoder: Uses a Barker code to perform frame synchronization, phase ambiguity resolution, and demodulation. Also, the data decoder compares the regenerated message with the transmitted one and calculates the BER.


This component provides optional visualization by plotting the following diagrams:

  • a time scope showing the normalized time delay,

  • a spectrum scope depicting the received signal after square root raised cosine filtering,

  • constellation diagrams showing the received signal after receiver filtering, and then after carrier phase and symbol timing recovery.

For more information about the system components, refer to the QPSK Transmitter and Receiver example using SimulinkQPSK Transmitter and Receiver example using Simulink.

System Under Test

The main loop in the system under test script processes the data frame-by-frame. Set the MATLAB variable compileIt to true in order to create generated code; this can be accomplished by using the codegen command provided by the MATLAB Coder™ product. The codegen command translates MATLAB® functions to a C++ static or dynamic library, executable, or to a MEX file, producing a code for accelerated execution. The generated C code runs several times faster than the original MATLAB code. For this example, set useCodegen to true to use the code generated by codegen instead of the MATLAB code.

The inner loop of runQPSKSystemUnderTest uses the four System objects previously mentioned. There is a for-loop around the system under test to process one frame at a time.

for count = 1:prmQPSKTxRx.FrameCount
    transmittedSignal = step(hTx);
    corruptSignal = step(hChan, transmittedSignal, count);
    [RCRxSignal,coarseCompBuffer, timingRecBuffer,BER] = step(hRx,corruptSignal);
    if useScopes
        stepQPSKScopes(hScopes,RCRxSignal,coarseCompBuffer, timingRecBuffer);

Execution and Results

To run the System Under Test script and obtain BER values for the simulated QPSK communication, the following code is executed. When you run the simulations, it displays the bit error rate data, and some graphical results. The figures displayed are, respectively:

1) Scatter plot of the Raised Cosine Receive Filter output.

2) Power spectrum of the Raised Cosine Receive Filter output.

3) Scatter plot at the output of the Fine Frequency Compensation subcomponent.

4) Estimated (fractional) timing difference at the output of the Timing Recovery subcomponent.

if compileIt
    codegen -report runQPSKSystemUnderTest.m -args {coder.Constant(prmQPSKTxRx),coder.Constant(useScopes),coder.Constant(printReceivedData)} %#ok
if useCodegen
   BER = runQPSKSystemUnderTest_mex(prmQPSKTxRx, useScopes, printReceivedData);
   BER = runQPSKSystemUnderTest(prmQPSKTxRx, useScopes, printReceivedData);
fprintf('Error rate = %f.\n',BER(1));
fprintf('Number of detected errors = %d.\n',BER(2));
fprintf('Total number of compared samples = %d.\n',BER(3));
Error rate = 0.007191.
Number of detected errors = 74.
Total number of compared samples = 10290.

Alternate Execution Options

As already mentioned in the section Run System Under Test, by using the global variables at the beginning of the example, it is possible to interact with the code to explore different aspects of System objects and coding options.

By default, the variables useScopes and printReceivedData are set to true and false, respectively. The useScopes variable enables MATLAB scopes to be opened during the example execution. Using the scopes, you can see how the simulated subsystems behave and also obtain a better understanding of how the system functions in simulation time. When you set this variable to false, the scopes will not open during the example execution. When you set printReceivedData to true, you can also see the decoded received packets printed in the command window. The other two variables, compileIt and useCodegen, are related to speed performance and can be used to analyze design tradeoffs.

When you set compileIt to true, this example script will use MATLAB Coder™ capabilities to compile the script runQPSKSystemUnderText for accelerated execution. This command will create a MEX file (runQPSKSystemUnderTest_mex) and save it in the current folder. Once you set useCodegen to true to generate the mex file, the example is able to run the system implemented in MATLAB much faster. This feature is essential for implementation of real-time systems and is an important simulation tool. To maximize simulation speed, set useScopes to false and useCodegen to true to run the mex file.

For other exploration options, refer to the QPSK Transmitter and Receiver example using SimulinkQPSK Transmitter and Receiver example using Simulink.


This example utilizes several System objects to simulate digital communication over an AWGN channel. It shows how to model several parts of the QPSK system such as modulation, frequency and phase recovery, timing recovery, and frame synchronization. It measures the system performance by obtaining BER with the error rate measurement System object. It also shows that the generated C code runs several times faster than the original MATLAB code.


This example uses the following script and helper functions:


1. Rice, Michael. Digital Communications - A Discrete-Time Approach. 1st ed. New York, NY: Prentice Hall, 2008.