This model shows the use of a CORDIC (COordinate Rotation DIgital Computer) rotation algorithm in a digital PLL (Phase Locked Loop) implementation for QPSK carrier synchronization. Fixed-Point Designer™ is needed to run this model.
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The structure of a digital PLL is essentially equivalent to that of a continuous-time PLL. A PLL has the following components: a phase error detector (PED), a loop filter, and a controlled oscillator.
In the case of QPSK carrier (phase and frequency) synchronization, implementing the loop filter as a digital P+I (proportional-plus-integrator) filter produces a second order PLL. The controlled oscillator (Phase Accumulator) adjusts the angle of the received QPSK signal via a complex rotation.
You can implement the complex rotation using a variety of approaches, including direct complex multiplication by exp(j*theta). However, such an implementation can be relatively expensive in terms of hardware (e.g., FPGA or ASIC) resources. An alternative approach uses a CORDIC-based rotation algorithm to implement the complex multiplication. This example uses this approach, via the Fixed-Point Designer™ CORDICROTATE function. This results in a multiplier-less complex rotation approximation, where the trade-off is in terms of speed. A small number of CORDIC iterations may often be enough to achieve a good digital PLL response, without the full hardware resource cost of a true complex multiplication.
Tx Data Source
The PN Sequence Generator library block from the Communications System Toolbox™ is the Tx Data Source, generating unsigned 2-bit integer symbols.
The QPSK Modulator Baseband library block from the Communications System Toolbox uses a pi/4 phase offset and binary ordering to compute signed 12-bit fixed-point modulator output values.
Raised Cosine Tx Filter
The Raised Cosine Transmit Filter library block from the Communications System Toolbox performs square root FIR filtering with an upsampling factor of 8.
The Phase/Frequency Offset library block from the Communications System Toolbox simulates the associated transmitter impairments. You can tune the Phase offset and Frequency offset parameter values to see the effect on the PLL Phase Error time scope and the receive signal scatter plot displays.
The AWGN Channel library block from the Communications System Toolbox simulates a noisy channel. You can tune the block Eb/No parameter to see the effect on the PLL Phase Error time scope and the receive signal scatter plot displays.
Raised Cosine Rx Filter
The Raised Cosine Receive Filter library block from the Communications System Toolbox performs square root FIR filtering with a downsampling factor of 8.
CORDIC-Based PLL Subsystem
The CORDIC-Based PLL subsystem consists of a Phase Error Detector (PED), P+I Loop Filter, Phase Accumulator, and CORDICROTATE to form the corrected complex signal output values.
Phase Error Detector
The Phase Error Detector is implemented using a MATLAB function.
P+I Loop Filter
A P+I Loop Filter implements a second order PLL. The loop constants K1 (P gain) and K2 (I gain) are derived from the Normalized loop bandwidth and Damping factor parameters of the masked CORDIC-Based PLL subsystem.
The Phase Accumulator computes the angle Theta.
The MATLAB function CORDICROTATE rotates the complex received signal by Theta using an iterative, multiplier-less, CORDIC-based algorithm.
Use the Phase Error time scope block to view the time-varying PLL Phase Error Detector output values.
Use the Before Carrier Synchronization and After Carrier Synchronization scope blocks to observe the effects of tuning the Transmitter Impairments and AWGN Channel parameters.
To see the effects of transmitter phase and frequency offset impairments, change the Phase offset and Frequency offset parameter values while the model is running. Set the model StopTime to inf and use the PLL Enable/Disable switch to observe changes in the transient response.
To see the effects of a noisy channel, change the Eb/No parameter value while the model is running. Set the model StopTime to inf and use the PLL Enable/Disable switch to observe changes in the transient response.
Vary the PLL Normalized loop bandwidth and Damping factor parameters to tune the underlying P+I Loop Filter behavior while the model is running. Set the model StopTime to inf and use the PLL Enable/Disable switch to observe changes in the transient response.
Note that the phase-locked QPSK receive signal output contains phase ambiguity. For further analysis (e.g., symbol error rate computations), this phase ambiguity may be resolved using one of a number of well known methods, including known training (preamble) signals, varying demodulator phase offsets, constellation re-ordering, etc.
Rice, Michael, "Discrete-Time Phase Locked Loops", Digital Communications: A Discrete-Time Approach, Appendix C, Sec. C.3, Pearson Prentice Hall, 2008.
Andraka, Ray, "A survey of CORDIC algorithm for FPGA based computers", Proceedings of the 1998 ACM/SIGDA Sixth International Symposium on Field Programmable Gate Arrays, 191 - 200, Feb. 22-24, 1998.
Volder, Jack E., "The CORDIC Trigonometric Computing Technique", IRE Transactions on Electronic Computers, Volume EC-8, 330 - 334, September 1959.