## Documentation |

This model shows the implementation of a QPSK transmitter and receiver. The receiver addresses practical issues in wireless communications, e.g. carrier frequency and phase offset, timing offset and frame synchronization. The receiver demodulates the received symbols and outputs a simple message to the MATLAB® command line.

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Available Example Implementations |

**Available Example Implementations**

This example includes both MATLAB and Simulink® implementations:

MATLAB script using System objects: commQPSKTransmitterReceiver.mcommQPSKTransmitterReceiver.m

Simulink model using blocks: commqpsktxrx.mdlcommqpsktxrx.mdl

This example model performs all processing at complex baseband to handle a static frequency offset, a time-varying symbol delay, and Gaussian noise. To cope with the above-mentioned impairments, this example provides a reference design of a practical digital receiver, which 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. The example implements some key algorithms in MATLAB, emphasizing textual algorithm expression over graphical algorithm expression. For example, the model uses MATLAB to implement some key processing in the PLL for fine frequency compensation and timing recovery, and some data extraction/control functionalities.

The top-level structure of the model is shown in the following figure, which includes the **Transmitter** subsystem, the channel subsystem, and the **Receiver** subsystem.

The detailed structures of the **Transmitter** subsystem and the **Receiver** subsystem are illustrated in the following figures.

The components are further described in the following sections.

**Transmitter**

**Bit Generation**- Generates the bits for each frame**QPSK Modulator**- Modulates the bits into QPSK symbols**Raised Cosine Transmit Filter**- Uses a rolloff factor of 0.5, and upsamples the QPSK symbols by four

**Channel**

**AWGN Channel with Frequency Offset and Variable Time Delay**- Applies the frequency offset, a variable delay, and additive white Gaussian noise to the signal

**Receiver**

**Raised Cosine Receive Filter**- Uses a rolloff factor of 0.5, and downsamples the input signal by two**Coarse Frequency Compensation**- Estimates an approximate frequency offset of the received signal and corrects it**Fine Frequency Compensation**- Compensates for the residual frequency offset and the phase offset**Timing Recovery**- Resamples the input signal according to a recovered timing strobe so that symbol decisions are made at the optimum sampling instants**Data Decoding**- Aligns the frame boundaries, resolves the phase ambiguity caused by the Fine Frequency Compensation subsystem, demodulates the signal, and decodes the text message

The transmitter includes the **Bit Generation** subsystem, the **QPSK Modulator** block, and the **Raised Cosine Transmit Filter** block. The **Bit Generation** subsystem uses a MATLAB workspace variable as the payload of a frame, as shown in the following figure. Each frame contains 200 bits. The first 26 bits are header bits, a 13-bit Barker code that has been oversampled by two. The Barker code is oversampled by two in order to generate precisely 13 QPSK symbols for later use in the **Data Decoding** subsystem. The remaining bits are the payload. The first 105 bits of the payload correspond to the ASCII representation of 'Hello world ###', where '###' is a repeating sequence of '000', '002', '003',..., '099'. The remaining payload bits are random bits. The payload is scrambled to guarantee a balanced distribution of zeros and ones for the timing recovery operation. The scrambled bits are modulated by the **QPSK Modulator** (with Gray mapping). The modulated symbols are upsampled by four by the **Raised Cosine Transmit Filter** with a rolloff factor 0.5. The output rate of the **Raised Cosine Filter** is set to 200k samples/second.

**AWGN Channel with Frequency Offset and Variable Delay**

The **AWGN Channel with Frequency Offset and Variable Delay** subsystem first applies the frequency offset and a preset phase offset to the transmit signal. Then it adds a variable delay with a choice of the following two types of delay to the signal:

**Ramp delay**- This type of delay is initialized at*DelayStart*samples, and increases linearly at a rate of*DelayStep*samples in each frame. When the actual delay reaches one frame, the delay buffer obtains saturation, and it maintains a delay of one frame.**Triangle delay**- This type of delay linearly changes back and forth between*MinDelay*samples and*MaxDelay*samples at a rate of*DelayStep*samples in each frame

The use of multiple delay characteristics allows you to investigate their effects on receiver performance, particularly on the timing recovery blocks. The delayed signal is processed through an **AWGN Channel**. The diagram of the **AWGN Channel with Frequency Offset and Variable Delay** subsystem is as shown in the following.

The **Raised Cosine Receive Filter** downsamples the input signal by a factor of two, with a rolloff factor of 0.5. It provides matched filtering for the transmitted waveform.

The phase error detector gain
of the phase and timing error detectors is proportional to the received signal amplitude and the average symbol energy. To ensure an optimum loop design, the signal amplitude at the inputs of the carrier recovery and timing recovery loops must be stable. The AGC 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, thus improving the accuracy of the estimate. You can refer to Chapter 7.2.2 and Chapter 8.4.1 of [ 1 ] for details on how to design the phase detector gain
.

The **Coarse Frequency Compensation** subsystem corrects the input signal with a rough estimate of the frequency offset. The following diagram shows the **Find Frequency Offset** subsystem in the **Coarse Frequency Compensation** subsystem. This subsystem uses a baseband QPSK signal with a designated phase index
, frequency offset
and phase offset
expressed as
,
. First, the subsystem raises the input signal to the power of four to obtain
, which 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** library block corrects the frequency offset. There is usually a residual frequency offset even after the coarse frequency compensation, which would cause a slow rotation of the constellation. The **Fine Frequency Compensation** subsystem compensates for this residual frequency.

The Fine Frequency Compensation subsystem implements a phase-locked loop (PLL), described in Chapter 7 of [ 1 ], to track the residual frequency offset and the phase offset in the input signal, as shown in the following figure. 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**.

A maximum likelihood **Phase Error Detector (PED)** , described in Chapter 7.2.2 of [ 1 ], generates the phase error. A tunable proportional-plus-integral **Loop Filter** , described in Appendix C.2 of [ 1 ] filters the error signal and then feeds it into the **DDS**. The *Loop Bandwidth* (normalized by the sample rate) and the *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 unity (critical damping) so that the PLL quickly locks to the intended phase while introducing little phase noise.

The **Timing Recovery** subsystem implements a PLL, described in Chapter 8 of [ 1 ], to correct the timing error in the received signal. The input of the **Timing Recovery** subsystem is oversampled by two. On average the **Timing Recovery** subsystem generates one output sample for every two input samples. 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
as described in Chapter 8.4.2 of [ 1 ]. The filter uses an
of 0.5 so that all the filter coefficients become only 1, -1/2 and 3/2, which significantly simplifies the interpolator structure. 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 unity (critical damping) so that the PLL quickly locks to the correct timing while introducing little phase noise.

When the timing error (delay) reaches symbol boundaries, there will be one extra or missing interpolant in the output. The TED implements bit stuffing/skipping to handle the extra/missing interpolants. You can refer to Chapter 8.4.4 of [ 1 ] for details of bit stuffing/skipping.

The timing recovery loop normally generates 100 QPSK symbols per frame, one output symbol for every two input samples. It also outputs a timing strobe that runs at the input sample rate. Under normal circumstances, the strobe value is simply a sequence of alternating ones and zeros. However, this occurs only when the relative delay between Tx and Rx contains some fractional part of one symbol period and the integer part of the delay (in symbols) remains constant. If the integer part of the relative delay changes, the strobe value can have two consecutive zeros or two consecutive ones. In that case, the timing recovery loop generates 99 or 101 QPSK output symbols per frame. However, the downstream processing must use a frame size of 100 symbols, which is ensured by the **Modified Buffer** subsystem.

The **Modified Buffer** subsystem uses the strobe to fill up a delay line with properly sampled QPSK symbols. As each QPSK symbol is added to the delay line, a counter increments the number of symbols in the line. At each sampling instant, the delay line outputs a frame of size 100 to the **Data Decoding** subsystem. However, the **Data Decoding** subsystem runs on its received data only when its enable signal goes high. This occurs when both the counter value reaches 100 and the strobe is high, i.e. each time exactly 100 valid QPSK symbols are present at the **Modified Buffer**.

The **Data Decoding** subsystem performs frame synchronization, phase ambiguity resolution, demodulation and text message decoding. The subsystem uses a QPSK-modulated Barker code, generated by the **Bits Generation** subsystem, to correlate against the received QPSK symbols and achieve frame synchronization. The **Compute Delay** subsystem correlates the data input with the QPSK modulated Barker code, and uses the index of the peak amplitude to find the delay.

The carrier phase PLL of the **Fine Frequency Compensation** subsystem may lock to the unmodulated carrier with a phase shift of 0, 90, 180, or 270 degrees, which can cause a phase ambiguity. For details of phase ambiguity and its resolution, please refer to Chapter 7.2.2 and 7.7 in [ 1 ]. The **Phase Offset Estimator** subsystem determines this phase shift. The **Phase Ambiguity Correction & Demodulation** subsystem rotates the input signal by the estimated phase offset and demodulates the corrected data. The payload bits are descrambled, and the first 105 payload bits are extracted and stored in a workspace variable. All of the stored bits are converted to characters and printed out at the MATLAB command window at the end of the simulation.

When you run the simulation, it displays bit error rate and numerous graphical results.

These plots are shown in the following, starting with the power spectrum of the **Raised Cosine Transmit Filter** output.

In the following is the actual delay applied to the input signal.

In the following is the power spectrum of the received signal, with its unknown delay, a frequency offset, and AWGN.

In the following are the scatter plot and the power spectrum of the **Raised Cosine Receive Filter** output. Note that the frequency range of the power spectrum plot is reduced by a factor of two, due to the downsampling at the **Raised Cosine Receive Filter**.

In the following are the power spectrum of the received signal raised to the power of four and the estimated frequency offset by the **Coarse Frequency Compensation** subsystem. In the power spectrum of the signal raised to the fourth power, you can identify a peak around 20 kHz, corresponding to the tone at four times the frequency offset.

In the following are the estimated phase difference and a scatter plot at the output of the **Fine Frequency Compensation** subsystem.

In the following are the estimated (fractional) timing difference and a scatter plot at the output of the **Timing Recovery** subsystem. The estimated timing difference
is fed to the **Interpolation Filter** for resampling.

As shown in the following scatter plot, the QPSK constellation is evident only after carrier recovery and bit timing recovery.

The example allows you to experiment with multiple system capabilities to examine their effect on bit error rate performance. For example, you can view the effect of changing the frequency offset, delay type and on the various displays.

You can tune the *FFT Size* and *Number of Spectrum Averages* for the **Coarse Frequency Compensation** subsystem to see the effect of the estimation accuracy and the tolerance to a high noise level. The resolution of the estimate is the frequency spacing between two adjacent FFT points, i.e. *100 kHz/FFT Size*. There is a speed versus accuracy tradeoff when choosing the value of *FFT Size*. To get a more accurate frequency estimate usually requires a larger *FFT Size*. However, a larger *FFT Size* also incurs a higher computational burden. If the resolution of the **Coarse Frequency Compensation** subsystem is low, then the **Fine Frequency Compensation** subsystem must have a wider frequency tracking range.

Due to the existence of noise and zero padding of the input, the FFT output might have some outliers in the estimation results. To ease the effect of these bad estimates, you can adjust the *Number of Spectrum Averages* to average the FFT result across multiple frames. The larger *Number of Spectrum Averages* improves the robustness of the coarse frequency estimation, but this also incurs a greater computational burden. Also, the fourth-power operation can correctly estimate an offset only if the offset satisfies the following inequality:

, or

, or

.

Also, this FFT-based **Coarse Frequency Compensation** subsystem is designed for a scenario with a static frequency offset. In practice, the frequency offset might vary over time. This model can still track a time-varying frequency drift by the **Coarse Frequency Compensation** subsystem. However, the coarse frequency estimates take on discrete values, separated by the frequency resolution of the subsystem. You might observe jumps between frequency estimates. You can also implement coarse frequency compensation with a filter to get a smoother estimation output.

You can adjust the PLL design parameters such as *Loop Bandwidth* and *Damping Factor* in both **Fine Frequency Compensation** and **Timing Recovery** subsystems to see their effect on pull-in range, convergence time and the estimation accuracy. With a large *Loop Bandwidth* and *Damping Factor*, the PLL can acquire over a greater frequency offset range. However a large *Loop Bandwidth* allows more noise, which leads to a large mean squared error in the phase estimation. "Underdamped systems (with Damping Factor less than one) have a fast settling time, but exhibit overshoot and oscillation; overdamped systems (with Damping Factor greater than one) have a slow settling time but no oscillations." [ 1 ]. For more detail on the design of these PLL parameters, you can refer to Appendix C in [ 1 ].

The **Timing Recovery** subsystem relies on a stable constellation which does not rotate over time. So this requires an accurate frequency offset compensation. In this model, if the actual frequency offset exceeds the maximum frequency offset that can be tracked by the current coarse compensation subsystem, you can increase its tracking range by increasing the oversampling factor. Another way to adjust the tracking range is to implement a rotationally-invariant timing error detector (e.g., Gardner timing error detector described in Chapter 8.4.1 of [ 1 ]) first and correct the rotation afterwards.

1. Michael Rice, "Digital Communications - A Discrete-Time Approach", Prentice Hall, April 2008.

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