The Effects of Phase Noise on High-Order QAM Systems
When large, dense QAM constellations are employed at high-carrier frequencies, it is important that a systems designer understands the effects of phase noise, which can degrade a system’s performance.
By Douglas Barker
Modern digital communication systems now employ high-order quadrature amplitude modulation (QAM). The effect of ever-present additive white
Gaussian noise (AWGN) on QAM systems can be found in nearly any textbook that covers the subject of digital communications. Unfortunately, AWGN is not the only unwanted signal that degrades a system’s performance. Phase noise can be a major contributor to system performance degradation. Phase noise is introduced by the combined effect of the oscillators and synthesizers used in frequency translation of the modulated signal. It can also be caused by AWGN present at the input to the carrier recovery
phaselocked loop (PLL) in a coherent receiver. It’s important that a communication systems designer understands the effects of phase noise, particularly when large, dense QAM constellations are employed at high carrier frequencies. How can oscillators be specified to meet system requirements without over-specifying — which drives up the cost of the system — and how can a system be designed to minimize the degradations caused by phase noise?
Many references are available which describe
how to characterize and specify oscillator phase noise.
1
Unfortunately, the data is useless unless the systems designer can convert this information into a quantitative measure of performance degradation. This article addresses how to characterize the phase noise disturbance present at the symbol decision device in a coherent receiver, given measured phase noise data. Once the phase noise disturbance is known, the symbol error rate (SER) degradation can be calculated.
Receiver response
to phase noise
Oscillator phase noise is typically specified by discrete measurements of the oscillator output’s one-sided spectral density. Several techniques can be used to accurately measure the spectral density.
2
Table 1
is a typical specification from an oscillator manufacturer’s data sheet. In this case, the measurements represent discrete points at decade intervals in the power spectral density (PSD) of the oscillator output.
The measurement in dBc signifies dB below carrier level.
The root-mean-square (RMS) value of the phase noise can be computed by integrating the one-sided PSD over the offset frequency range of interest as in Equation (1):
3
| (1)
| 
|
Where
S(f)
is the one-sided PSD of the phase noise and
f
1
and
f
2
define the region over which to compute the phase noise. By performing this integration using the data from
Table 2.1
, a value of U
rms
= 3.6º results. It is assumed that phase noise is caused by a low-index incidental phase modulation. Equation (1) is not applicable if U
rms
> 1 rad
, therefore one must choose
f
1
and
f
2
carefully.
Modern QAM
systems use fully coherent detection by employing a PLL at the receiver. In most cases, the PLL is of the second order, meaning it has two integrators in its feedback path and is capable of tracking both phase and frequency offsets with zero-static phase error. The design criteria for the PLL is to minimize the loop’s phase-error variance, while tracking in response to noise at its input (while also maintaining acceptable acquisition performance). The damping factor and loop noise bandwidth are the
parameters of a second-order loop, which can be changed in order to meet the PLL’s design criteria. In general, by widening the loop bandwidth, faster acquisition time and wider capture range can be achieved at the expense of larger phase-error variance. The statistics of the phase noise caused by AWGN at the input to the PLL are taken as zero-mean Gaussian. This approximation is valid at signal-to-noise ratios (SNRs) of practical interest.
4
The variance of the phase noise can be found by using
the following equations:
4
| (2)
| 
|
where
| (3)
| 
|
is the loop noise bandwidth, v
n
is
the natural frequency, z is the damping factor,
R
b
is the information bit rate, and
E
b
/N
0
is the SNR per bit. Increasing the SNR, or decreasing
B
L
, results in lower phase-error variance.
Oscillator phase noise imparts another competing requirement in the design of the PLL. The PLL acts as a highpass filter to oscillator phase noise, suppressing phase noise close to the carrier. This highpass effect can be seen in the error response of a PLL. The high-pass cut-off frequency
of the error response is primarily set by the loop noise bandwidth, but is also affected by the damping factor. At frequencies close to the carrier, the PLL tracks with little error, but at frequencies beyond the loop’s noise bandwidth, the error is large and the incidental phase modulation on the carrier is left untracked. Suppression of oscillator phase noise can then be accomplished by increasing
B
L
at the expense of increasing the PLL’s phase error variance in response
to AWGN as implied by Equation (2).
The receiver’s IF band-pass filters and matched filters will have a low-pass effect on oscillator phase noise. The combination of the PLL and matched filters will limit the oscillator phase noise present at the receiver’s decision device. The limits of integration in Equation (1) can now be set. The lower limit must be set just below the PLL’s noise bandwidth, while the upper limit must extend beyond the cut-off frequency of the receiver’s
matched filters. In addition to oscillator phase noise present at the symbol decision device, PLL phase noise must also be accounted for. This independent source of phase noise must be summed with the untracked oscillator phase noise to obtain the total disturbance which affects symbol decisions in the receiver.
The effect of the two sources of phase noise is best seen by use of a numerical example. For this example a 10-Mbaud system is evaluated. The system uses Nyquist filtering with 40% excess
bandwidth. The combined oscillator phase noise incident to the demodulator is given in
Table 1
.
Figure 1
shows the receiver’s overall frequency response to oscillator phase noise present at its input terminals. This is the combined effect of the receiver’s matched filter and PLL. The PLL’s loop bandwidth is equal to 10 kHz and the damping factor is 0.707. The PSD of the untracked phase noise is found by taking:
| (4)
| 
|
where
S
o
(f)
is the PSD of the oscillator phase noise and
S
u
(f)
is the PSD of the untracked oscillator phase noise, and
H(f)
is the receiver’s frequency response to phase noise. The RMS untracked oscillator phase noise can now be computed by substituting the result of
Equation (4) into Equation (1).
Table 2
gives the RMS oscillator phase noise and RMS untracked oscillator phase noise at the decision device as a function of the PLL’s loop bandwidth.
Figure 2
is a family of curves that shows the combined untracked oscillator phase noise and the PLL phase noise. When the PLL bandwidth is small, the untracked oscillator phase noise dominates, and when
B
L
is 0.1%
of the baud rate, the untracked oscillator phase noise is about 3.3º (the PLL phase noise has little effect on the total). As the loop bandwidth is made larger, the PLL phase noise starts to dominate at the lower SNRs. The data from
Figure 2
can be used to determine the PLL noise bandwidth which minimizes phase noise at a specified SNR.
Calculation of symbol-error probability
Now that the amount of phase noise present at the
receiver’s decision device has been quantified, the degradation in SER, and hence system performance for
M
-ary QAM constellations can be determined. No other signal perturbations are considered in the calculations. Three assumptions are made concerning the statistics of the phase noise. The first assumption concerns the statistics of the phase noise resulting from AWGN present at the input to the receiver’s PLL. It can be shown that the resulting stochastic process has a Gaussian probability density
function (pdf) at SNRs of practical interest.
4
The second assumption is that oscillator phase noise is Gaussian as well.
5
The argument is that many physical sources of the oscillator phase noise exist. For example, an individual oscillator has several components to its phase noise spectrum (which include white noise, phase-flicker noise, flat-frequency noise, and frequency-flicker noise).
4
Multiple oscillators may be used in the communication system. Due to the central limit
theorem, the composite probability density function of the resulting stochastic process describing phase noise will approach Gaussian.
6
The literature lacks any solid proof that this assumption is true, but it seems to be used frequently, possibly because it makes the problem mathematically tractable. The assumption is partially verified by measurements taken on actual communication systems. The third assumption states that the carrier phase offset caused by phase noise is constant over the
duration of a baud interval. For high-speed QAM, this assumption is valid since both the baud interval and the index of modulation causing the phase noise are small.
Given the three assumptions previously mentioned, the pdf of the phase noise stochastic process can be found. Since the oscillator phase noise and PLL phase noise are independent zero-mean Gaussian processes, the sum of these processes is also a Gaussian process with zero mean and a variance equal to the sum of the variances of the independent
processes. If the variance of the PLL phase noise is
and the variance of the untracked oscillator phase noise is
, then the pdf of the phase noise which affects symbol decisions in the receiver is:
| (5)
| 
|
where the variance of the process is given by:
| (6)
| 
|
The probability of symbol error for
i
is given by:
7
| (7)
| 
|
where
is the probability of symbol error, given a fixed carrier phase offset U. This can be easily
evaluated for any square QAM constellation. The requirement that the constellation be square greatly simplifies the computations and allows the same algorithm to be used on different size constellations, due to the fact that the decision regions are horizontal and vertical as shown in
Figure 3
.
Figure 3
shows that the probability that
i
is in error is:
| (8)
| 
|
where:
x
u
= upper decision boundary in
x
direction
x
l
= lower decision boundary in
x
direction
y
u
= upper decision boundary in
y
direction
y
l
= lower decision boundary in
y
direction
n
x
= noise perturbation
x
direction
n
y
= noise perturbation
y
direction
This equation states that the probability of
i
being in error is simply the probability that noise will cause the received constellation point to move beyond the decision boundaries in either the horizontal or vertical direction. Note that:
| (9)
| 
|
and
that:
| (10)
| 
|
which is the integral over the Gaussian pdf outside the decision region in the positive
x
direction.
8
The other terms of the expression for symbol-error probability are taken in the same manner. It should be noted that
n
x
is the magnitude of the noise, with variance s
2
in the
horizontal direction, and likewise for
n
y
. The previous expression for symbol-error probability can be easily modified to include the effect of a carrier phase offset. If a constant phase offset is imparted (assumption 3) on a constellation point, the (
x
i
,
y
i
) pair is moved from its correct position which affects the probability of symbol error (see
Figure 3
). The effect is different for each and
every symbol in the constellation. The new position (
xx
i
,
yy
i
) of the constellation point in Cartesian coordinates can be found from the correct (
x
i
,
y
i
) position using the phase offset U as follows:
| (11)
| 
|
| (12)
| 
|
where the
tan
-1
function is taken over all four quadrants. Substituting (
xx
i
,
yy
i
) from Equations (11) and (12) into Equation (8) for (
x
i
,
y
i
) yields the probability of error conditioned on a constant phase offset U, or
. This
expression can be substituted into Equation (7) which gives the probability of
i
being in error. Assuming all
M
symbols in the
M
-ary QAM constellation are equally probable, then the total probability of error is:
| (13)
| 
|
It should be noted that:
| (14)
| 
|
where
E
s
is the average energy per symbol and
E
si
is the energy of
i
. This can be converted to SNR per bit (
E
b
/N
0
) using Equation (15).
| (15)
| 
|
System performance
in terms of SER can be evaluated numerically using a tool such as Matlab.
Figure 4
and
Figure 5
show the results for two high-order QAM systems. For each QAM constellation, a family of curves is graphed, plotting probability of symbol error versus
E
b
/N
0
for four different values of RMS phase error and for 0º phase error. The only other perturbation considered in the
analysis is AWGN. Note that the
y
-axis is probability of symbol error and not BER, and that the abscissa is SNR per bit. This may cause some confusion when comparing the curves to those found in other literature. To convert SER to BER requires knowledge of the bit-to-symbol mapping.
Figures 4
and
Figure 5
show that high-order QAM systems are sensitive to phase noise. The specifications of local oscillators and
synthesizers are critical to system performance. Since the cost of these components can be a significant portion of the system’s total cost, it is prudent to correctly specify these devices. Inferior specifications can lead to undesired system performance degradation while over-specifying can drive up system cost.
Douglas W. Barker is a member of technical staff with Stanford Telecommunications, Inc. He holds a BSEE from Pennsylvania State University, and an MS and a PE degree in
electrical engineering from the George Washington University. He can be reached at
doug.barker@stelhq.com
.
| References
|
- Spilker, J.J.,Digital Communications by Satellite, Prentice-Hall, Inc., Upper Saddle River, NY, 1977.
- Vectron
Crystal Oscillators 1994 Handbook and Catalog, Vectron Laboratories, Inc., 166 Glover Avenue, PO Box 5160, Norwalk, CT 06856-5160.
- Howald, R.L.,“Analyzing Phase Power Spectral Density For Noise Power,” Microwaves & RF, June 1994.
- Gardner, F.M.,
Phaselock Techniques
, John Wiley & Sons, New York, NY, 1979.
- Howald, R.L., “Isolating Sources of Phase Errors,” Microwaves &
RF, May 1994.
- Papoulis, A.,
Probability Random Variables, and Stochastic Processes
, McGraw-Hill, Inc.,New York, NY, 1984,.
- Howald, R., “The Communications Performance of Single-Carrier and Multi-Carrier Quadrature Amplitude Modulation in RF Carrier Phase Noise,” Ph.D. dissertation, Drexel University, December 1997.
- Proakis, J.G.,
Digital Communications
, Third Edition, McGraw-Hill, Inc., New York, NY, 1995.
|
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