# Digital PLL's -- Part 1

## 1. Introduction

Figure 1.1 is a block diagram of a digital PLL (DPLL). The purpose of the DPLL is to lock the phase of a numerically controlled oscillator (NCO) to a reference signal. The loop includes a phase detector to compute phase error and a loop filter to set loop dynamic performance. The output of the loop filter controls the frequency and phase of the NCO, driving the phase error to zero.

One application of the DPLL is to recover the timing in a digital demodulator. In this case, the phase detector computes the phase error from the complex I/Q signal. Another application would be to lock to an external sine wave that is captured by an A/D converter.

A basic DPLL model is shown in Figure 1.2. In this model, the reference signal and the NCO output are both phases. The phase error is simply the difference of the reference phase and the NCO phase. This is the DPLL model we will use here. (As an example of a reference phase, if a reference sine wave were applied to a Hilbert transformer, the phase of the Hilbert I/Q output could be used to generate the reference phase: phi_ref= arctan(Q/I) ).

We will use Matlab to model the DPLL in the time and frequency domains (Simulink is also a good tool for modeling a DPLL in the time domain). Part 1 discusses the time domain model; the frequency domain model will be covered in Part 2. The frequency domain model will allow us to calculate the loop filter parameters to give the desired bandwidth and damping, but it is a linear model and cannot predict acquisition behavior. The time domain model can be made almost identical to the gate-level system, and as such, is able to model acquisition.

**Figure 1.1 Digital PLL**

**Figure 1.2 Digital PLL model using phase signals**

## 2. Components of the DPLL Time domain model

As shown in Figure 1.2, the DPLL contains an NCO, phase detector, and a loop filter. We now describe these blocks for a 2^{nd} order PLL [1, 2]. To keep things simple, all blocks use the same sample frequency. This discussion deals with the phase of sampled sine waves. For background, see Appendix B.

### The NCO

The phase of a sine wave is given by:

phi= 2*pi*mod(f*n*Ts,1)

Here f is the frequency, n is sample index, and T_{s} is sample time. To create an NCO, we just implement the above equation. But rather than generate phase in radians, we remove the 2*pi to give phase in cycles. Then the range of phase (renamed u) is 0 to 1.

u = mod(f*n*Ts,1); % cycles

The implementation in Figure 2.1 is an accumulator that rolls over when x >1, with input f*T_{s} = f/f_{s}. The output u is phase in cycles. The output y is a sine wave at frequency = f. The cosine function is generated using a lookup table or CORDIC algorithm.

For our DPLL we make two modifications, as shown in Figure 2.2. We add input Vtune and scaling K_{nco }to allow an offset from the center frequency, f. Typically, K_{nco} << 1. Also, we removed the sinusoidal output; only the phase output is needed in the model. With the sine output removed, we refer to the NCO as a phase accumulator.

The difference equations for the phase accumulator are simply:

x = f*Ts + u(n-1) + vtune(n-1)*Knco; % cycles NCO phase u(n) = mod(x,1); % cycles NCO phase mod 1

A note on quantization: In a digital implementation, u must be quantized to a reasonable number of bits. For simplicity, we have not included quantization in this and subsequent models, but it can easily be added.

**Figure 2.1.**** Basic NCO**

**Figure 2.2.**** Phase Accumulator for DPLL. u = phase in cycles.**

### The Phase Detector

The phase detector is shown in figure 2.3. It performs a difference, then a rollover function when the phase crosses +/- ½ cycle (+/- π radians). The difference equations are:

pe= ref_phase(n-1) - u(n-1); % phase error pe= 2*(mod(pe+1/2,1) - 1/2); % wrap if phase crosses +/- 1/2 cycle phase_error(n) = pe;

The output of the phase detector has a range of +/-1 for an input phase difference of 1 cycle. Thus the phase detector gain is:

K_{p} = 2 in units of cycle^{-1}.

A plot of phase detector output vs. phase error is shown in Figure 2.4.

**Figure 2.3 Phase Detector**

**Figure 2.4 Phase Detector Characteristic**

** **

### The Loop Filter

For a 2^{nd} order loop, the loop filter consists of a proportional gain K_{L} summed with an integrator having gain K_{I}. It is called a Proportional + Integral or Lead-Lag filter. The difference equations are:

int(n) = KI*pe + int(n-1); % integrator vtune(n) = int(n) + KL*pe; % loop filter output

where pe is the phase error. KL and KI determine the damping and natural frequency of the PLL. We will show how to calculate KL and KI in Part 2.

**Figure 2.5 Loop Filter**

Putting all the components together, the DPLL time domain model is shown in Figure 2.6.

**Figure 2.6 DPLL Time Domain Model Block Diagram**

## 3. Example Cases of DPLL Time Domain model

The Matlab script for the time domain DPLL is listed in Appendix A. The input parameters for the examples are:

f_{s} 25 MHz

Reference frequency 8 MHz

Initial reference phase 0.7 cycles

NCO initial frequency error -100 ppm or -500 ppm (-800 or -4000 Hz)

K_{nco} 1/4096

N 30,000 or 100,000 samples

f_{n} 5 kHz or 400 Hz loop natural frequency. f_{n} = ω_{n}/(2π)

ζ (zeta) 1.0 loop damping coefficient

K_{L, }K_{I} calculate from f_{n} and ζ

The first example has initial frequency error less than loop natural frequency, and the second example has initial frequency error much greater than loop natural frequency. The formulas for K_{L} and K_{I} to obtain the desired natural frequency and damping will be covered in Part 2.

**Example 1.** Initial frequency error < Loop natural frequency. Loop natural frequency = 5 kHz and NCO initial frequency error = -800 Hz.

For this case, the loop behaves as a linear system. Note we have allowed Vtune to exceed [-1 1], rather than clipping at that level.

**Figure 3.1 Phase error. **

NCO Initial freq error = -800 Hz. Initial ref phase = 0.7 cycles. fn = 5 kHz, ζ = 1.0.

**Figure 3.2 Vtune. **

NCO Initial freq error = -800 Hz. Initial ref phase = 0.7 cycles. f_{n} = 5 kHz, ζ = 1.0.

** **

**Figure 3.3 Output Spectrum** (bin spacing = 25E6/2^14 = 1.53 kHz)

** **

** ****Example 2.** Initial frequency error >> Loop natural frequency. Loop natural frequency = 400 Hz and NCO initial frequency error = -4 kHz.

For this case, the phase detector repeatedly wraps due to the large initial frequency error. Thus the loop is highly non-linear during acquisition. Note the time scales of the plots are longer than for the previous example.

**Figure 3.4 Phase error **

NCO Initial freq error = -4 kHz. Initial ref phase = 0.7 cycles. f_{n} = 400 Hz, ζ = 1.0.

**Figure 3.5 Vtune**

NCO Initial freq error = -4 kHz. Initial ref phase = 0.7 cycles. f_{n} = 400 Hz, ζ = 1.0.

### Appendix A. Time domain model of DPLL with fn = 5 kHz

% pll_time2.m 6/3/16 nr % Digital PLL model in time using difference equations. % fn = 5 kHz NCO initital freq error = -100ppm*8 MHz = -800 Hz N= 30000; % number of samples fref = 8e6; % Hz freq of ref signal fs= 25e6; % Hz sample rate Ts = 1/fs; % s sample time n= 0:N-1; % time index t= n*Ts*1000; % ms init_phase = 0.7; % cycles initial phase of reference signal ref_phase = fref*n*Ts + init_phase; % cycles phase of reference signal ref_phase = mod(ref_phase,1); % cycles phase mod 1 Knco= 1/4096; % NCO gain constant KI= .0032; % looop filter integrator gain KL= 5.1; % loop filter linear (proportional) gain fnco = fref*(1-100e-6); % Hz NCO initial frequency u(1) = 0; int(1)= 0; phase_error(1) = -init_phase; vtune(1) = -init_phase*KL; % compute difference equations for n= 2:N; % NCO x = fnco*Ts + u(n-1) + vtune(n-1)*Knco; % cycles NCO phase u(n) = mod(x,1); % cycles NCO phase mod 1 s = sin(2*pi*u(n-1)); % NCO sine output y(n)= round(2^15*s)/2^15; % quantized sine output % Phase Detector pe= ref_phase(n-1) - u(n-1); % phase error pe= 2*(mod(pe+1/2,1) - 1/2); % wrap if phase crosses +/- 1/2 cycle phase_error(n) = pe; % Loop Filter int(n) = KI*pe + int(n-1); % integrator vtune(n) = int(n) + KL*pe; % loop filter output end plot(t,phase_error),grid axis([0 0.5 -1 1]) xlabel('t (ms)'),ylabel('phase error'),figure plot(t,vtune),grid axis([0 0.5 -3.5 1]) xlabel('t (ms)'),ylabel('vtune'),figure psd(y(11000:end),2^14,fs/1e6) axis([7.95 8.05 -80 40]),xlabel('MHz')

### Appendix B. Phase of a sampled sine wave

For a continuous sine wave y = cos(2πft), the phase is 2πft radians. For a *sampled* sine wave, we make the following definitions:

N= 50; % number of samples fs= 25e6; % Hz sample frequency Ts= 1/fs; % s sample time f= 1e6; % Hz sinewave frequency

Noting that 2πf*t = 2πf*nTs, the phase is:

n= 0:N-1; % sample index phi= 2*pi*f*n*Ts; % radians phase

To make the phase wrap at phi = 2π, we modify this expression using the modulus function:

phi= 2*pi*mod(f*n*Ts,1);

The phase and its cosine are plotted here:

Top: phase phi of sampled cosine wave (radians).

Bottom: cos(phi)

### References

1. Gardner, Floyd M., __Phaselock____ Techniques, 3 ^{rd} Ed.__, Wiley-Interscience, 2005, Chapter 4.

2. Rice, Michael, __Digital Communications, a Discrete-Time Approach__, Pearson Prentice Hall, 2009, Appendix C.

6/3/2016 Neil Robertson

**Next post by Neil Robertson:**

Digital PLL's -- Part 2

## Comments:

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In my experience of QAM receivers, it is typically possible for the clock recovery loop to lock without any frequency acquisition aids. This is because the frequency error is not so large compared to the loop bandwidth. On the other hand, the carrier loop typically requires an acquisition aid, such as sweeping.

One interesting point made by Gardner in section 8.3 of his book "If the loop filter contains a perfect integrator, pull-in will be accomplished no matter how large the initial frequency error. (This statement neglects clipping limits; the loop clearly cannot pull in a signal that requires excessive control voltage to the VCO)."

This applies to the digital PLL's we are discussing. As a practical matter though, it can take a loop too long to acquire.

Thanks. I hope you like part 2 of the article.

regards, Neil

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