The primary performance criterion for modern digital communications systems is the average bit error rate (BER) that the link can provide for a given signal-to-noise ratio. The achievable BER is a function of many parameters, including the sum of the phase noise contributions of the local oscillators (LO) used in the system.

As often is the case, the phase noise requirements of these LOs are overspecified—or worse yet, underspecified. This results in increased complexity and costs or, quite possibly, degraded BER performance.

**Figure 1** illustrates the basic components of a digital communications system. This figure indicates what is minimally required to transmit and receive digital signals. Data is used to phase modulate an intermediate frequency (IF) carrier which then is upconverted to the final transmit frequency Fc.

The modulated, upconverted signal is transmitted through a channel where additive white Gaussian noise (AWGN) is introduced. The signal then is incident to the receiver where an LO, usually a frequency synthesizer, downconverts the received signal to some IF before being directed to a demodulator.

Demodulation typically is realized by some form of matched filter detection, usually the correlator shown in **Figure 2**. Figure 2 reveals that the downconverted signal (R)^{ }is multiplied by a reference (S), and the resulting product is integrated over a symbol interval T. The output (Z) is sampled at rate T before entering a threshold detector of some form, at which point a decision is made regarding which symbol was received.

This example assumed that perfect phase and frequency synchronization existed in both the downconverter and the demodulator. Although our simplified block diagrams do not show it, the receiver usually has the capability to track the phase of the received signal to within a fraction of a degree. As a result, bit errors are due entirely to AWGN.

Even in the case of perfect tracking, there typically will exist a time-varying random phase error due to the phase noise of the synthesized LOs.

Theory

To develop a useful relationship between the phase noise of oscillators and the probability of bit error, we must analyze the demodulation process and the dynamics of noise-induced phase jitter in oscillators separately. We first will examine the basic concepts of phase noise dynamics and then derive an expression for the probability of bit error as a function of random phase of variance. Finally, we will find the phase noise variance of the synthesizer output spectrum and relate it to binary phase shift key (BPSK) BER performance.

Phase Noise Basics

Phase noise commonly is used to describe a range of phenomena that tends to perturb the phase of an ideal frequency source. **Figure 3a** illustrates an ideal signal in both the time and frequency domain. The ideal signal is depicted as a pure sinusoidal tone in the time domain and a discrete line spectra in the frequency domain. These attributes are the result of a constant rate of change of phase with respect to time, throughout all time.

Such a signal is not achievable because physical limitations exist that exert a time-varying perturbation on the phase of the signal source. Example effects include semiconductor impurities resulting in shot noise, environmental effects such as temperature and vibration, and random processes such as AWGN. These effects tend to introduce time-varying random disruptions in the instantaneous phase of the signal, which results in a departure from ideal signal characteristics as depicted in **Figure 3b**. Note the effects of random phase error on the signal’s spectrum. The presence of random phase changes tends to widen the spectrum.

Upon close examination of Figure 3b, we can identify the effects of phase perturbation on the power spectral density of the signal source. **Figure 4** illustrates a single-sideband (SSB) representation of the nonideal signal of Figure 3b. This is the phase noise profile of the signal source, and careful study of it can provide valuable insight into the origins of the LO phase noise.

Of the many causes of time-varying phase perturbations, the effects caused by random noise concern us the most. In turn, these effects can be treated as random processes themselves. If carrier phase perturbations are the result of a Gaussian random process, then the phase error itself is Gaussian distributed and can be modeled as such. This will simplify the mathematics involved and is a technique that lends itself well to the treatment of higher-order M-ary modulation schemes.

Referring to Figure 4, the SSB phase noise profile can be viewed as the result of the instantaneous addition of carrier power at frequency Fc and narrowband AWGN power at Fn to produce an instantaneous phase noise power contribution at Fc + Fn.^{1 }This contribution of energy at offset Fn is due to the instantaneous modulation of the carrier by the narrowband noise. The result of this process, in addition to unwanted energy at offset Fn, is an instantaneous shift of the carrier’s phase.

If we consider the long-term average of the sum of many of these instantaneous phase noise contributions, we will arrive at the phase noise profile of Figure 4. As a result, Figure 4 provides the average SSB energy density due to the long-term AWGN- induced phase noise contributions from Fc to Fc + F offset.

If we consider the long-term contribution of phase noise energy density as the result of AWGN, then the associated long-term phase error also will be Gaussian. Consequently, the long-term effect of superimposed Gaussian noise at frequency Fn is to produce carrier phase error at rate Fn and varianceProbability of Bit Error for BPSK With Static Phase Error

The general expression relating the probability of bit error (P_{b})_{ }to the existence of static phase error has been developed in Reference 2 and is given by:

where N_{o} is the thermal noise floor, E_{b} is the amount of energy in an information bit, and Q is the error function. **Figure 5** is a plot of P_{b} as a function of E_{b}/N_{o }for several values of static phase error.

The average value or mean of a random processes x is the integrated product of the instantaneous value of the random process (x) and the probability density function (pdf) of the process f(x)

Now suppose that the phase error between the receiver LO and the carrier no longer is static but varies randomly throughout time. Also suppose that this phase error has a pdf of q (f ) and the instantaneous value of a bit error as a function of this phase error is P_{b}(f ). We then would expect the average value of the probability of bit error as a function of random phase error to be

The addition of AWGN to a carrier will result in instantaneous modulation of the carrier which gives rise to phase noise sidebands (frequency domain) and carrier phase jitter (time domain).

Since the origin of carrier phase noise then can be viewed as the result of AWGN, the phase noise, or phase jitter, also will have a Gaussian distribution. The probability distribution of phase jitter of varianceWe now can rewrite our expression of the probability of bit error:

Figures

5 and **6** compare ideal BER performance to nonideal BER performance when a static phase error (Figure 5) exists and when the phase error varies randomly (Figure 6). Note how the existence of a random phase error has a greater effect on P_{b} than does the case of static phase error.

For example, Figure 5 indicates that 10° of static phase error results in a BER of 10^{-5 }for an E_{b}/N_{o} of 9.7 dB which is about 0.2 dB worse than the 9.5 dB ideally required. On the other hand, 10° of random phase error results in a further increase of 0.4 dB, which represents a 0.6-dB degradation of required E_{b}/N_{o }due to phase noise alone.

You would expect an increase of required E_{b}/N_{o} for random phase error over the static phase error. Static phase error could introduce a fixed decrease between signaling points in the BPSK signaling constellation. This distance and the corresponding phase error between signal points remain fixed for all detection instances.

. It is this tendency of large phase error that accounts for the degraded BER performance in the presence of random phase error.

Summary

Phase noise contributions to degraded communications link performance are an important consideration in modern digital communications systems. The analysis performed for this effort revealed that as little as 10° rms of random phase error results in » 0.6-dB increase of required E_{b}/N_{o }for a desired BER. This is impressive if you consider that the decision threshold for BPSK nominally is 90° . The degradation for higher order M-ary modulation schemes is even more dramatic as decision regions can be separated by as little as 11.25° .

References

1. Howald, R., “Understand the Mathematics of Phase Noise,” *Microwaves and* *RF*, December 1993.

2. Besse, J., “The Effects of Phase Noise on BPSK Bit Error Rate Performance in* *Wireless Communications” Proceedings, *RF Design*, 97.

About the Author

John Besse is vice president and co-founder of In-Phase Technologies Special Products Division. He also serves as an adjunct at the City University of New York, College of Staten Island. Mr. Besse holds a B.S.E.E. from the State University of New York and an M.S.E.E. from Polytechnic University.

*In-Phase Technologies, 4 Hytek Corporate Center, Route 526, Clarksbury, NJ 08510, (609) 259-8555, e-mail: [email protected]..*

**Copyright 2000 Nelson Publishing Inc.**

April 2000