sdr.ClosedLoopPLL

class sdr.ClosedLoopPLL

A class that defines the performance of a closed-loop PLL.

Note

This class is meant for performance analysis only.

Notes¶

Closed-Loop PLL Block Diagram¶

            bb[n]    phase_err[n]
        +---+    +-----+    +----+
x[n] -->| X |--->| PED |--->| LF |---+
        +---+    +-----+    +----+   |
          ^                          | phase_est[n]
          |      +-----+             |
   lo[n]  +------| NCO |<------------+
                 +-----+

x[n] = Input signal
bb[n] = Baseband signal
phase_err[n] = Phase error signal
phase_est[n] = Phase estimate signal
lo[n] = Local oscillator signal
PED = Phase error detector
LF = Loop filter
NCO = Numerically-controlled oscillator

The transfer function of the 2nd order, proportional-plus-integrator loop filter is

H_{L F} (z) = K_{1} + K_{2} \frac{1}{1 - z^{- 1}} = \frac{(K_{1} + K_{2}) - K_{1} z^{- 1}}{1 - z^{- 1}} .

The transfer function of the NCO is

H_{N C O} (z) = K_{0} \frac{z^{- 1}}{1 - z^{- 1}} .

The closed-loop transfer function of the PLL is

H_{P L L} (z) = \frac{K_{p} K_{0} (K_{1} + K_{2}) z^{- 1} - K_{p} K_{0} K_{1} z^{- 2}}{1 - 2 (1 - \frac{1}{2} K_{p} K_{0} (K_{1} + K_{2}) z^{- 1} + (1 - K_{p} K_{0} K_{1}) z^{- 2}} .

References¶

Michael Rice, Digital Communications: A Discrete-Time Approach, Appendix C: Phase Locked Loops.

Examples¶

See the Phase-locked loops example.

Constructors¶

ClosedLoopPLL(noise_bandwidth: float, damping_factor: float, ...): Creates a closed-loop PLL analysis object.

Methods¶

phase_lock_time() → float: Returns the phase lock time of the PLL.

frequency_lock_time(freq_offset: float) → float: Returns the frequency lock time of the PLL.

lock_time(freq_offset: float) → float: Returns the lock time of the PLL.

phase_error_variance(cn0: float) → float: Returns the variance of the phase error of the PLL in steady state.

Properties¶

property sample_rate : float: The sample rate $f_{s}$ of the PLL in samples/s.

property BnT : float: The normalized noise bandwidth $B_{n} T$ of the PLL.

property Bn : float: The noise bandwidth $B_{n}$ of the PLL in Hz.

property zeta : float: The damping factor $ζ$ of the PLL.

property K0 : float: The NCO gain $K_{0}$ .

property Kp : float: The phase error detector (PED) gain $K_{p}$ .

property K1 : float: The proportional gain $K_{1}$ of the loop filter.

property K2 : float: The integral gain $K_{2}$ of the loop filter.

property iir : IIR: The IIR filter that represents the closed-loop transfer function of the PLL.

property omega_n : float: The natural frequency $ω_{n}$ of the PLL in radians/s.

property omega_3dB : float: The 3-dB bandwidth $ω_{3 dB}$ of the PLL in radians/s.