Digital Receiver: Carrier Recovery

Summary: The phase-locked loop (PLL) is a critical component in coherent communications receivers that is responsible for locking on to the carrier of a received modulated signal. A PLL adjusts the phase of a numerically-controlled oscillator to match that of the received signal. You will simulate a carrier recovery sub-system in MATLAB and then implement the sub-system on the DSP.

Introduction

After gaining a theoretical understanding of the carrier recovery sub-system of a digital receiver, you will simulate the sub-system in MATLAB and implement it on the DSP. The sub-system described is specifically tailored to a non-modulated carrier. A complete implementation will require modifications to the design presented.

The phase-locked loop (PLL) is a critical component in coherent communications receivers that is responsible for locking on to the carrier of a received modulated signal. Ideally, the transmitted carrier frequency is known exactly and we need only to know its phase to demodulate correctly. However, due to imperfections at the transmitter, the actual carrier frequency may be slightly different from the expected frequency. For example, in the QPSK transmitter of Digital Transmitter: Introduction to Quadrature Phase-Shift Keying, if the digital carrier frequency is

π

2

 and the D/A is operating at 44.1 kHz, then the expected analog carrier frequency is f_c=

π 2

2π

 44.1=11.25kHz. If there is a slight change to the D/A sample rate (sayf_c=44.05kHz), then there will be a corresponding change in the actual analog carrier frequency (f_c=11.0125kHz).

This difference between the expected and actual carrier frequencies can be modeled as a time-varying phase. Provided that the frequency mismatch is small relative to the carrier frequency, the feedback control of an appropriately calibrated PLL can track this time-varying phase, thereby locking on to both the correct frequency and the correct phase.

Figure 1: PLL Block Diagram

Numerically controlled oscillator

In a complete coherent receiver implementation, carrier recovery is required since the receiver typically does not know the exact phase and frequency of the transmitted carrier. In an analog system this recovery is often implemented with a voltage-controlled oscillator (VCO) that allows for precise adjustment of the carrier frequency based on the output of a phase-detecting circuit.

In our digital application, this adjustment is performed with a numerically-controlled oscillator (NCO) (see Figure 1). A simple scheme for implementing an NCO is based on the following re-expression of the carrier sinusoid:

sin(ω_cn+θ_c)=sin(θ[n])(1)

where θ[n]=ω_cn+θ_c (ω_c and θ_c represent the carrier frequency and phase, respectively). Convince yourself that this time-varying phase term can be expressed asθ[n]=

n

∑

m=0

 ω_c+θ_c and then recursively as

θ[n]=θ[n−1]+ω_c(2)

The NCO can keep track of the phase, θ[n], and force a phase offset in the demodulating carrier by incorporating an extra term in this recursive update:

θ[n]=θ[n−1]+ω_c+d_pd[n](3)

where d_pd[n] is the amount of desired phase offset at time n. (What would d_pd[n] look like to generate a frequency offset?)

Phase detector

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ω_c is the believed digital carrier frequency. We can then represent the actual received carrier frequency as the expected carrier frequency with some offset, ω_c=ω_c+θ[n]. The NCO generates the demodulating sine and cosine with the expected digital frequency ω_c and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as ω_c=ω_c+θ[n]. Using the appropriate trigonometric identities 1, the in-phase and quadrature signals can be expressed as

z₀[n]=1/2(cos(θ[n]−θ[n])+cos(2ω_c+θ[n]+θ[n]))(4)

z_Q[n]=1/2(sin(θ[n]−θ[n])+sin(2ω_c+θ[n]+θ[n]))(5)

After applying a low-pass filter to remove the double frequency terms, we have

y₁[n]=1/2cos(θ[n]−θ[n])(6)

y_Q[n]=1/2sin(θ[n]−θ[n])(7)

Note that the quadrature signal, z_Q[n], is zero when the received carrier and internally generated waves are exactly matched in frequency and phase. When the phases are only slightly mismatched we can use the relation

∀θ,small:(sin(θ)≈θ)(8)

and let the current value of the quadrature channel approximate the phase difference: z_Q[n]≈θ[n]−θ[n]. With the exception of the sign error, this difference is essentially how much we need to offset our NCO frequency2. To make sure that the sign of the phase estimate is right, in this example the phase detector is simply negative one times the value of the quadrature signal. In a more advanced receiver, information from both the in-phase and quadrature branches is used to generate an estimate of the phase error.3

Loop filter

The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For this exercise you can use a one-tap IIR filter,

y[n]=βx[n]+αy[n−1](9)

To ensure unity gain at DC, we select β=1−α

It is suggested that you start by choosing α=0.6 and K=0.15 for the loop gain. Once you have a working system, investigate the effects of modifying these values.

MATLAB Simulation

Simulate the PLL system shown in Figure 1 using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.

Use Equation 3 to implement your NCO in MATLAB. However, to ensure that the phase term does not grow to infinity, you should use addition modulo 2π in the phase update relation. This can be done by setting θ[n]=θ[n]−2π whenever θ[n]>2π.

Figure 2 illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case the transmitted carrier frequency was set to ω_c=

π

2

 +

π

1024

 to simulate a frequency offset.

Figure 2: Output of PLL sub-system for BPSK modulated carrier.

Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by

π

2

 . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadrature component to zero (and the in-phase component to 1/2). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?

DSP Implementation

As you begin to implement your PLL on the DSP, it is highly recommended that you implement and test your NCO block first before completing the rest of your phase-locked loop.

Sine-table interpolation

Your NCO must be able to produce a sinusoid with continuously variable frequency. Computing values of sin(θ[n]) on the fly would require a prohibitive amount of computation and program complexity; a look-up table is a better alternative.

Suppose a sine table stores N samples from one cycle of the waveform: ∀k,k={0,…,N−1}:(sin(

2πk

N

 )). Sine waves with discrete frequencies ω=

2π

N

 p are easily obtained by outputting every p^th value in the table (and using circular addressing). The continuously variable frequency of your NCO will require non-integer increments, however. This raises two issues: First, what sort of interpolation should be used to get the in-between sine samples, and second, how to maintain a non-integer pointer into the sine table.

You may simplify the interpolation problem by using "lower-neighbor" interpolation, i.e., by using the integer part of your pointer. Note that the full-precision, non-integer pointer must be maintained in memory so that the fractional part is allowed to accumulate and carry over into the integer part; otherwise, your phase will not be accurate over long periods. For a long enough sine table, this approximation will adjust the NCO frequency with sufficient precision.4

Maintaining a non-integer pointer is more difficult. In earlier exercises, you have used the auxiliary registers (ARx) to manage pointers with integer increments. The auxiliary registers are not suited for the non-integer pointers needed in this exercise, however, so another method is required. One possibility is to perform addition in the accumulator with a modified decimal point. For example, with N=256, you need eight bits to represent the integer portion of your pointer. Interpret the low 16 bits of the accumulator to have a decimal point seven bits up from the bottom; this leaves nine bits to store the integer part above the decimal point. To increment the pointer by one step, add a 15-bit value to the low part of the accumulator, then zero the top bit to ensure that the value in the accumulator is greater than or equal to zero and less than 256.5 To use the integer part of this pointer, shift the accumulator contents seven bits to the right, add the starting address of the sine table, and store the low part into an ARx register. The auxiliary register now points to the correct sample in the sine table.

As an example, for a nominal carrier frequency ω=

π

8

 and sine table length N=256, the nominal step size is an integer p=

π

8

 N

1

2π

 =16. Interpret the 16-bit pointer as having nine bits for the integer part, followed by a decimal point and seven bits for the fractional part. The corresponding literal (integer) value added to the accumulator would be 16×2⁷=2048.6

Extensions

You may want to refer to Proakis and Blahut. These references may help you think about the following questions:

• How does the noise affect the described carrier recovery method?
• What should the phase-detector look like for a BPSK modulated carrier? (Hint: You would need to consider both the in-phase and quadrature channels.)
• How does α affect the bandwidth of the loop filter?
• How do the loop gain and the bandwidth of the loop filter affect the PLL's ability to lock on to a carrier frequency mismatch?

FOOTNOTES

1. cos(A)<times></times>cos(B)=1/2(cos(A−B)+cos(A+B)) and cos(A)<times></times>sin(B)=1/2(sin(B−A)+sin(A+B)).
2. If θ[n]−θ[n]>0 then θ[n] is too large and we want to decrease our NCO phase.
3. What should the relationship between the I and Q branches be for a digital QPSK signal?
4. Of course, nearest-neighbor interpolation could be implemented with a small amount of extra code.
5. How is this similar to the addition modulo 2π discussed in the MATLAB Simulation?
6. If this value were 2049, what would be the output frequency of the NCO?

REFERENCES

1. J.G. Proakis. (1995). Digital Communications. (3rd edition). McGraw-Hill.
2. R. Blahut. (1990). Digital Transmission of Information. Addison-Wesley.

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