Direct Lookup: Ve = θe

The piecewise-linear sawtooth PD outputs the result of ( θr - θo θr-θo )mod-2π. In other words, the sawtooth PD outputs exactly θe = tan-1 θe=tan-1 (q/i). Of course, such a device must be easily implemented in hardware for an analog PD or must be compuatationally efficient for a software PD. As tan-1 tan-1 (x) is not, a suitable approximation is often sought. The first solution might be to utilize a lookup table for the argument x mapping it to tan-1 tan-1. As accuracy is improved via a larger lookup table, more memory is required and lookup time increased.

Figure 6: A sawtooth Phase Detector.

Sawtooth PD

Traditional: Ve = q

Another solution involves replacing tan-1 tan-1 (q/i) with the mathematically equivalent sin-1 (qi2+q2 ) sin-1(qi2+q2 ) . Assuming the PLL is locked and the phase error θeθe reasonably small, sin-1 (qi2+q2 ) sin-1(qi2+q2 ) can be approximated by the argument qi2+q2 qi2+q2 . Since i2+q2 i2+q2 is simply the sqrt of the input power, it can be removed using Automatic Gain Control (AGC) for a flat-fading channel or considered part of the fixed value KdKd for a non-fading channel. In this case the error voltage becomes Ve =q ≈ Kd θeVe=q≈Kdθe. This corresponds to the bottom "leg" of Figure 3.

Figure 7: The "traditional" mixer-type phase detector.

Traditional PD

Of course, cosine shifted by π/2 is just a sinusoid and the mixer produces both DC and double-frequency terms. Therefore, this figure can be more accurately represented as

Figure 8: The more common representation of the traditional phase detector.

Traditional PD (Common Representation)

In a PLL, the double frequency terms will likely be removed by the loop filter and are of little concern.

Costas BPSK: Ve = iq

For the pure sinusoidal input considered up to this point, zero phase error is synomonous with a single point in the I-Q plane, located on the positive I-axis. For a Binary Phase-Shifted Signal (BPSK), the signal with zero phase error actually exists at two points within the I-Q plane, as the data stream varies between a positive and negative I-axis value. This results in θeθe being defined as zero in two locations on the I-Q plane. The approximation Ve =q Ve=q indeed gives these two zeros, but for the left-half plane has a negative slope! We must modify the approximation by taking into account the sign of I. In a first attempt, we may multiply by the value i directly: Ve =iq Ve=iq.

Figure 9: The "S-curve" of a Costas Phase Detector.

Costas PD Response

Modified Costas BPSK: Ve = Sign[i]q

The simplicity of the Costas solution has resulted in widespread use. However, allowing the magnitude of i to affect the approximation of Ve has limited the linear range of the phase error estimate to about (-45,+45) degrees. With the added complexity of a hard-limiter, the I-channel data can be made to affect only the sign, resulting in a much improved phase error estimate: Ve =Sign[i]q Ve=Sign[i]q. The linear range of the phase error estimate now approaches (-90,+90) degrees as seen in Figure 10.

Figure 10: The "S-curve" of a Modified Costas Phase Detector.

Modified Costas PD Response

The resulting S-curve is nearly equivalent to the ideal sawtooth.

Linearity and the Effect of Noise

Such a PD is considered to be a true-linear PD only if the phases are all "unwrapped" with respect to some absolute location in time. That is, a true-linear PD must have access to θr θr values of the form (eg.) 576.123 π and the PLL must output θo θo values of the form (eg.) 243.163 π. The (unfortunate?) truth is that in a system for which the reference signal and the PLL are operated on different clocks, there is no way to define and track such an absolute time reference. The (fortunate) truth is that outside of perhaps a non-critical phase-ambiguity and some modeling details, an absolute time reference is not needed. Knowing θr θr and θo θo modulo-2π is both practical and sufficient.

When the input is noisy, the effect is for the PD to occassionally map the true phase error across the discontinuity of the defining S-curve. Large-magnitude phase errors (which sit closer to the discontinuity) are affected more so than small ones. Therefore, as the SNR falls, the first effect is a softening of the sawtooth's "teeth." The sawtooth becomes more sinusoidal. As the SNR continues to fall, eventually even small-magnitude phase errors are affected and the apparent slope (defining the PD gain constant KdKd near θdθd=0) decreases in value.