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: where θn= ω c n+ θ c θ n ω c n θ c ( ω c ω c and θ c θ c represent the carrier frequency and phase, respectively). Convince yourself that this time-varying phase term can be expressed as θn=∑m=0n ω c + θ c θ n m 0 n ω c θ c and then recursively as The NCO can keep track of the phase, θn θ n , and force a phase offset in the demodulating carrier by incorporating an extra term in this recursive update: where d pd n d pd n is the amount of desired phase offset at time n n. (What would d pd n d pd n look like to generate a frequency offset?)

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ω c 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 ω c ˜ ω c θ ˜ n . The NCO generates the demodulating sine and cosine with the expected digital frequency ω c 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 ω c ^ ω c θ ^ n . Using the appropriate trigonometric identities 1, the in-phase and quadrature signals can be expressed as After applying a low-pass filter to remove the double frequency terms, we have Note that the quadrature signal, z Q n 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 and let the current value of the quadrature channel approximate the phase difference: z Q n≈ θ ˜ n- θ ^ n 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

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

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

Your NCO must be able to produce a sinusoid with continuously variable frequency. Computing values of sinθn 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 N samples from one cycle of the waveform: ∀k,k=0…N-1:sin2πkN k k 0 … N 1 2 k N . Sine waves with discrete frequencies ω=2πNp ω 2 N p are easily obtained by outputting every p th 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 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=π8 ω 8 and sine table length N=256 N 256 , the nominal step size is an integer p=π8N12π=16 p 8 N 1 2 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 16 2 7 2048 .6

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?