Receiver thermal noise is an ever-present source of noise in communications systems. This type of noise is modeled as Additive White Gaussian Noise (AWGN) and is often a primary consideration in communication link performance analysis. In many cases, however, receiver phase noise is often the limiting factor in determining system performance. For example, a crystal oscillator used in a mixer generates phase noise. It can also be caused by AWGN present at the input of a Phase Locked Loop (PLL) in a coherent receiver. Phase noise can cause several types of signal degradation that are usually very difficult to quantify analytically.

Ordinarily, a "pure" oscillator would produce a perfect sine wave, which could be described as:

In practice the signal always contains some noise and a noisy oscillator's signal can be represented as:

If you add in the carrier signal, the phase noise equation becomes:

where P (t) is the amplitude noise term and phy(t) is the phase noise term. For |P (t)| << Vo and |phy(t)| << 1 radian, equation 1 can be rewritten as:

The first term represents the desired RF signal and the last two terms represent the amplitude modulated RF signal due to the phase and amplitude noise, respectively. If we neglect the amplitude noise, an oscillator's output can be characterized by its Single Side Band (SSB) phase noise spectrum, which is also referred to as the phase noise mask.

Figure 1. The Phase Noise Mask

Phase noise, in practice, shows up as a continuous energy in the sidebands around the carrier frequency. When phase noise is significantly higher than amplitude noise, the spectrum around the carrier frequency is symmetric(1).

In operation, an individual oscillator has several components to its phase noise. Typical distortions resulting from oscillator phase noise are: Reciprocal Mixing and Phase Jitter. Reciprocal mixing is the process of corrupting the desired signal by conversion of the neighboring channel energy into the desired signal's bandwidth. Phase jitter is of particular interest for it has a direct impact on a typical fixed wireless multilevel QAM system.

It can be shown that the statistics of the stochastic process describing phase noise will approach Gaussian(2). Characterization of phase noise thus reduces to specifying a spectral shape and an overall total noise power. Phase noise testing characterizes the spectral purity of an oscillator by calculating the ratio of the desired energy being delivered by the oscillator at the specified output frequency to the amount of undesired energy being delivered at neighboring frequencies(3). The ratio is often expressed as a series of power measurements performed at various offset frequencies form the carrier. The units used to specify phase noise are dBc/Hz. It is a measure of noise power per unit bandwidth appearing at a given offset from the carrier frequency.

System performance is dependent on phase noise. Typical distortions caused by phase noise are phase jitter, spectral leakage and reciprocal mixing. Reciprocal mixing refers to the process of degradation to the desired signal caused by conversion of neighboring channel energy into the desired signal's bandwidth. This problem is particularly severe in multi-carrier systems where the carriers are closely spaced in frequency. The resulting spectrum is a consequence of simultaneously mixing both the desired signal with an interfering signal.

Figure 2. Phase Noise and Reciprocal Mixing(4)

The phase noise variance, expressed in radians squared, or integrated phase jitter is equal to:

where L-phy(fm) is the phase noise mask. Phase jitter shows up in the signal constellation as an angular rotation on a constellation point. A constellation is a two dimensional mapping of amplitude and phase for each symbol used in a multilevel QAM modulation.

The effects of phase noise in QAM systems

Figure 3a. An ideal 16QAM constellation

Figure 3b. A 16QAM constellation with phase jitter

Phase noise at a level acceptable for QPSK can cause problems when using higher-order modulation schemes. In figure 4, the received scatter diagrams for 16-QAM and 64-QAM are shown. (In these figures, Es/N0 is set to 100.0 dB, essentially eliminating the effects of AWGN. Note that for 16-QAM the constellation points are well within the decision regions, whereas, the 64-QAM case clearly shows that decision errors would be caused by only small noise excursions.)

Figure 4. Scatter Plots of 16-QAM and 64-QAM Systems

Symbol error rate results for QPSK, 16-QAM, 64-QAM and 256-QAM are shown overlaid in figure 5. Clearly, performance degradation increases with the alphabet size. As the decision regions become smaller, the dispersion in the scatter diagram becomes more critical to error rate performance.

Figure 5. Symbol Error Rates for Various Modulation Types

In addition to the total phase noise power, the shape of the phase noise spectrum can be critical. In cases where the oscillator bandwidth is much less than the symbol rate, the phase variations can be tracked and the effects of a slowly varying phase can be removed. This can be seen in the effects of phase noise on multi-carrier modulation.

A Simulation Environment

A basic end-to-end QPSK communication system simulation is shown in figure 6. The Microwave Office environment (MWO)(5) allows the construction of communication link models from basic building blocks contained in a variety of communication sub-system libraries. We will describe some of these presently. This type of block-diagram representation has been employed in various CAD tools for many years. The MWO environment provides a virtual laboratory in which candidate communication system designs may be assembled and tested without the time and expense of hardware prototyping.

Figure 6. Screen Image of a Basic End-to-End QPSK Communication Link Simulation

In the diagram of figure 6, a pseudo-random bit stream is generated with a "digital source" model and modulated onto a carrier with QPSK modulation. As with many RF system simulators, complex baseband representation can be used to reduce required sampling rates in the time-domain sampled-data simulation. In our system model, Additive White Gaussian Noise (AWGN) perturbs the signal during transmission and has the effect of introducing errors in the receiver output. The receiver, in this case, consists of a linear matched-filter demodulator with the appropriate following detection circuitry that is responsible for regenerating the original transmitted bits. An error counter model regenerates the psuedo-random bit stream at the receiving end for the purposes of error counting. Of particular interest to us is the model labeled "1/f Noise". This is fairly flexible model for the generation of phase noise possessing various types of spectral characteristics. The model assumes that phase noise power spectral density characteristics are piecewise-linear on a log-log plot, where the independent and dependent axes represent logarithm of frequency and logarithm of the spectral density, respectively.

This representation has been shown to be sufficiently flexible to represent phase noise generated in most frequency sources. In figure 6, the phase noise generated with this model is applied to an input port associated with the demodulator block. MWO demodulator models provide auxiliary ports that allow various perturbations on phase, timing and AGC estimates to be applied. As we will see, this capability is very useful in assessing the degradations due to imperfect estimates of these quantities.

The time-domain phase noise waveform generated from the 1/f model is shown in the top plot of figure 7. For simulation of digital communication signals, it is convenient to parameterize time/frequency domain quantities, which is the normalization employed here. In this figure, the 1/f characteristic can be seen in the power spectrum plot, which is plotted on a log/linear axis. In this specific example, the spectrum is flat from 0.0001. to 0.01 cycles per symbol and then declines at a rate of 20dB per decade until the edge of the digital sampling bandwidth is reached. Therefore, this spectrum shape specification is based on two piece-wise linear segments. The sampling rate employed in this simulation is 8 samples per digital symbol, therefore, the digital sampling bandwidth is (-4.0, 4.0) cycles/symbol. Because of the log/linear scaling in the digital spectrum estimate, the high-frequency portion of the total noise spectrum is most obvious to the eye and does, indeed, possess a 1/f decline.

Figure 7. Time-Domain Waveform and Spectral Density Estimate of Phase Noise

The results of a simulation run with the phase noise model are shown in figure 8. In this case, total phase noise power is held fixed at a value of --45.0dB, while received Es/N0 is varied to sweep out a symbol error rate curve. MWO allows the use of "virtual instruments", such as the scatter diagram shown in the right of the figure. In the figure, the scatter diagram has been taken at an Es/N0 value of 8.0 dB. The demodulator model has the capability to output ideal and phase-noise corrupted versions of the I/Q output statistics. These are overlaid on the same set of axes. Note that the phase noise is not significantly degrading the signal constellation over and above the dispersion due to AWGN on the channel. Symbol error rate curves are shown for both cases in the left of the figure. The 'noisy' case is plotted in solid squares and corresponds to the performance with phase noise added. Note that both cases are nearly indistinguishable from each other and from the ideal reference curve.

Figure 8. Symbol Error Rate Results for QPSK with Phase Noise

Orthogonal Frequency-Division Multiplexing (OFDM) is in common use in wireless broadband systems today. This technique employs closely-spaced carriers, which are placed at orthogonal frequencies so they do not interfere with one another. An Inverse Fast Fourier Transform (IFFT) is normally employed to synthesize the tones efficiently. A good modern treatment of OFDM systems is the recent book by Van Nee and Prasad(6). An OFDM encoder/modulator is shown in figure 9. Here, bits are mapped into QAM modulating coefficients which enter an IFFT serially. Guard segments, called cyclic extensions, are added to allow for gradual turn-on and turn-off of the signal. The receiver consists of an FFT block, followed by circuitry for the detection of the serial QAM channels that are output from the FFT. OFDM is particularly sensitive to reciprocal mixing effects, since the channels are spaced at only the inverse of the FFT sample period. In multicarrier systems, the interference between adjacent carriers is referred to as InterCarrier Interference (ICI).

Figure 9. Diagram of OFDM Modulator

In single carrier systems, reciprocal mixing is not usually a severe problem, since adjacent channels are normally separated by a guard band. In multi-carrier systems, phase noise causes a phase offset common to all channels and also ICI between nearby channels. To provide for accurate tracking and correction of phase variations, most OFDM systems transmit a pilot signal that is embedded within the set of carriers generated with the OFDM multiplexer. One or several carriers may be devoted to carry this pilot signal information. We consider the Hiperlan2 system, in which 4 of 48 sub-carriers are devoted to the transmission of pilot tones. A simplified receiver for the Hiperlan2 system is diagrammed in figure 10.

Figure 10. Simplified Hiperlan2 Receiver

In this configuration, a symbol disassembler removes the 4 equally spaced pilot channels from the QAM channel stream and applies them to the input of a Hiperlan2 Phase Estimator model. This model performs a complex cross-correlation between the received pilot coefficients and a reference copy of the pilot coefficient sequence. This procedure allows for accurate tracking of phase variations that are slow compared to an OFDM symbol interval of 250 microseconds. When phase variations are fast compared to the symbol interval, pilot tracking cannot remove them. The result of demodulation of the Hiperlan2 signal, with and without phase correction is shown in figure 11. One OFDM symbol, consisting of 48 I/Q outputs is shown on the plot. In all scatter diagrams that follow, no channel noise is present. The phase noise generator used in figure 11 has the same specifications as in the previous single-carrier experiments.

Figure 11. Scatter Diagram of Raw and Corrected Hiperlan2 Receiver Output

Note that there is substantial rotation in the scatter diagram before correction is applied. The derotation according to the pilot estimates provides nearly perfect phase alignment, however. Figure 12 shows the results of the same receiver/tracking arrangement, but with a different phase noise spectrum. In this case, the 3dB bandwidth is equal to the symbol rate, producing a great deal of phase fluctuation within the same symbol.

Figure 12. Scatter Diagram of Raw and Corrected Hiperlan2 Receiver Output -- Wideband Phase Noise Case

OFDM systems are very sensitive to this type of high-frequency phase variation. In this case, phase noise obliterates the orthogonality of the carriers, rendering the signal unusable. In contrast, ordinary 16-QAM does not suffer significantly more for this wideband noise than for the narrowband case. (Figure 13 illustrates the scatter plot output of the single-carrier 16-QAM receiver for the same noise conditions as figure 12. Note that the degradation is actually slightly less than for the narrowband case.)

Figure 13. Scatter Diagram for Single-Carrier 16-QAM - Wideband Phase Noise Case.

The design environment described here enables very accurate evaluation of non-ideal hardware effects on end-to-end communication link performance. We have demonstrated particular examples of single-carrier and multi-carrier communication architectures where oscillator phase noise characteristics can have a profound effect on system performance. As indicated, many other analyses are possible with this tool and more will be described in the future.

References

(1) Bhasin, K.B., Downey, A.N., Ponchak, G.E., Romanofsky, R.R., Anzic, G. and Connolly, D.J.: "Monolithic Microwave Integrated Circuits --Interconnections and Packaging Considerations," prepared for the Fourth Annual International Electronics Packaging Conference sponsored by the International Electronics Packaging Society, Baltimore, MD, October 29-31, 1984, NASA TM- 83774. p. 84.

(2) Barker, Douglas, "The Effects of Phase Noise on High-Order QAM Systems," Communication Systems Design, October 1999.

(3) Vectron International, "Phase noise", Application Note, p. 2.

(4) Bateman, Andrew, "Radio Design Course Transmitter and Receiver Architectures," p. 5.

(5) Microwave Office User's Guide; Applied Wave Research Corporation, 1960 Grand Avenue, El Segundo, California 2001.