The Colorful World of Noise
by Tom Lecklider, Senior Technical Editor
Although much telecom testing involves
additive white Gaussian noise (AWGN), white noise does not need
to be Gaussian, nor is Gaussian noise necessarily white. White
noise is defined by two characteristics: It has a zero mean
value, and its autocorrelation is represented by a delta
function. In other words, successive values are completely
uncorrelated with previous values.
In the frequency domain, such a time-domain
function has a constant power spectral density. This means that
the spectrum of an ideal white noise source has constant power
per cycle regardless of frequency.
Practical white noise sources are flat within
some small deviation across
a defined frequency band. For example, the Model WGN-1/200 White
Noise Generator from dBm is specified as producing -87-dBm/Hz
noise density with 0.5-dB flatness from 1 MHz
to 200 MHz.
By definition, successive values of a truly
random variable cannot be predetermined. Nevertheless, all of
the values that occur within an arbitrarily large set of
observations determine a distribution. The Gaussian or normal
distribution is perhaps the most common and defined by the
probability density function (PDF)
where μ = the mean
σ2 = the variance
σ = the standard deviation
When σ = 1.0 and μ = 0.0, the definition simplifies to the standard form of the normal distribution
This equation describes the familiar
bell-shaped curve shown in Figure 1. The probability
density at the mean is 0.3989, at 1σ larger or smaller = 0.2420, 2σ = 0.0540, and 3σ = 0.004432.
Because of the square term in the exponent, the probability
density falls off very quickly above 3σ so that at 5σ away from the
mean N(x) = 0.000001487.
Figure 1. Normal Distribution PDF and CDF
A large series of observed values can conform
to a Gaussian distribution but occur in time in a deterministic,
highly ordered manner. The signal would not have a flat spectrum
and could not be used for noise testing. It would have a
Gaussian distribution but would not be white—successive values
would not be statistically independent.
Gaussian white noise has the benefit of a
well-understood and compact mathematical description. Even if
the actual distribution is not quite Gaussian, the Normal
distribution often is assumed to apply because it simplifies
further analysis.
The integral of the PDF is the cumulative
probability density function (CDF), also shown in Figure 1. It
indicates the probability that a value is to the left of any
arbitrary point. For example, the probability that a sample
within a Gaussian distribution has a value less than +1.0 is
about 0.84. Obviously, the probability that a value lies between
-5 and +5 is close to 1.0
Values far out on the tails of the
distribution are very large compared to the standard deviation.
The crest factor is a measure of the ratio of peak to rms values
and a good indication of how well a generator preserves these
infrequent events.
Bob Muro, product manager at NoiseCom,
commented that a crest factor corresponding to at least 7σ or as high as 18
dB is needed to correctly emulate rare data events for stringent
bit error rate (BER) testing. The relationship between the crest
factor and the probability level is shown in Figure 2 and
follows directly from definitions of the PDF and CDF.
Figure 2. Normal Distribution Crest Factor vs. Probability
Because CDF(-x) is the probability that a
value is less than -x and 1 - CDF(x) the probability that a
value is greater than x, 1 - [CDF(x) - CDF(-x)] equals the
probability that a value lies outside the interval from -x to x.
Figure 2 results from this equation.
The error function, erf(x), is related to the
CDF, but has an output range from -1 to +1. It is based on an
integral similar to the Normal distribution function taking into
account the sign of x. However, because of the usual scaling
used in the definition
It follows that
The crest factor has been defined as the
ratio of peak to rms values, but for a distribution, the
standard deviation is equivalent to the rms value. So, the crest
factor for a Normal distribution is numerically equal to the
value of the variable x, which is a multiple of σ. In dB, the
crest factor = 20 log (|peak|/rms) or for a Normal distribution
20 log (|x|).
From equation 2, a value 7σ (16.9 dB) from
the mean has a probability density of just 9.1347 x 10-12.
A more meaningful number is the probability that a value will
lie outside the range from -7σ to +7σ, which is about
2.57 x 10-12. Nevertheless, for a noise
source to be useful in testing the very low BER of
communications receivers, it must reliably produce these rare
events.
How well a noise generator’s output conforms
to the Gaussian distribution is termed its Gaussinity.
Typically, generators are limited in some way, compromising
their Gaussinity especially at high multiples of σ.
For example, an electrodynamic shaker used in
vibration testing is constrained in its maximum force and
displacement. This means that beyond some limit the high values
corresponding to infrequent events far out on the Gaussian tails
will be clipped. They will occur but not with the correct
amplitude. If the controller driving the shaker does not require
greater than, say, 5σ events, then values farther out on the Gaussian tails
will not occur at all.
Digital noise generators take many forms, but
all have a finite number of bits to describe the output signal.
This means that there is a minimum resolution to the system and
that high σ values beyond some limit cannot be accurately
represented. Of course, with modern devices, the limit can be
quite large, and many manufacturers have included digital AWGN
generators in their communications test instruments.
Noise Colors and Shapes
Naturally occurring sounds made by wind or
waterfalls have less power at higher frequencies although the
human ear perceives these sounds as being equally loud at all
frequencies. A white noise source that is filtered to have a
-3-dB/octave amplitude vs. frequency slope has a power spectral
density proportional to 1/f and is called pink noise. Pink-noise
filters typically are used to simulate the kind of background
noise spectrum found in nature.
Red or Browian noise falls off at a
-6-dB/octave rate, and its energy density is proportional to 1/f2.
Conversely, blue and purple noise have increasing slopes of +3
and +6 dB/octave, respectively. Grey noise is used in
psychoacoustic testing, and its energy density looks like a
bathtub curve, decreasing at low frequencies and increasing at high frequencies.
Most of these colored noise sources are used
in the audio frequency range and based on Gaussian white noise
that has been appropriately filtered. In mechanical vibration
testing, Gaussian white noise often is used to simultaneously
subject a DUT to all frequencies rather than one at a time as in
swept sine vibration testing.
Depending on the application, large forces
may be under-represented. In these cases, the distribution
Kurtosis can be adjusted to raise the tails and narrow the
central peak.1 Rather than a value of 3.0 that
corresponds to a Gaussian distribution, Kurtosis = 5.0 or larger
will ensure that higher force events occur more often. Changing
the actual distribution shape cannot easily be done in a
traditional analog noise generator. Instead, it generally is
achieved through digital techniques.
Analog Noise Sources
Analog noise sources are based on passive
components that naturally provide near-Gaussian white noise
signals. NoiseCom’s Bob Muro explained, "Precision manufacturing
techniques used in today’s RF and microwave diodes have
decreased the number of noise-generator defects and improved
statistical performance. In addition, better circuit-board
materials and amplifiers help to maintain the original Gaussian
distribution."
Micronetics, NoiseCom, and dBm are among a
small group of companies specializing in analog noise
generation. The Micronetics Carrier to Noise Generators (CNG)
Series has built-in calibration routines that use an internal
power meter. Several amplifier gain stages are required to boost
the noise to levels suitable for testing over a wide dynamic
range. The auto-cal routine updates the gain calibration tables
as required. In addition, a separate routine calibrates the
power meter against a stable internal reference source.
Two advantages of analog sources are the
direct generation of noise in the required frequency band and
with high amplitude resolution. Typically, a digital generator
cannot directly provide greater than 14-bit or 16-bit resolution
at frequencies approaching 1 GHz. Many digital generators
instead produce a noise signal at baseband and upconvert to the
RF frequencies required. According to the Micronetics CNG
datasheet, this approach typically compromises the
distribution’s Gaussinity because of mixer product spurs and
local oscillator bleed-through.
Mr. Muro of NoiseCom said that the basic AWGN
source has been extended to provide exact signal-to-noise (SNR)
test signals. Precision SNR generators measure receiver
performance including dynamic range and robustness. A further
variant, the noise power ratio (NPR) instrument, is used in the
measurement of adjacent channel power ratio (ACPR), receiver
selectivity, and amplifier linearity.
The director of the Micronetics CSE Division,
Patrick Robbins, added, "The primary application for our noise
generators is SNR vs. BER testing. Sometimes, SNR is replaced
with Eb/No or bit energy vs. noise spectral density. In this
case, noise generators mimic the actual noise channel that
occurs in digital radio applications such as satellite
communications, point-to-point radio, and CDMA markets.
"The noise in these applications needs to be
as close as possible to Gaussian, and broadband testing requires
that the noise be broadband as well. Micronetics makes a noise
generator that adds a precise amount of noise to a clean
digitally modulated signal connected to the instrument. This
provides a very accurate SNR." He continued, "You also can enter
the duty cycle for burst-mode operation, and the generator will
add the correct amount of noise synchronized to the signal
burst."
The CNG Series of noise generators from dBm
is based on a resistive termination rather than a diode. dBm
claims that the use of thermal noise rather than the shot noise
developed by diodes avoids amplitude distortion errors.
These instruments monitor power on both the
input signal channel and the noise channel to ensure a precise
ratio is maintained. Similar to the Micronetics CNG, operation
has been streamlined, requiring only single-button selections to
set up various configurations. Further, the unit automatically
compensates for bit rate, signal bandwidth, duty cycle, and
power level, maintaining the carrier/noise (C/N) ratio
previously selected.
The WGN Series of AWGN generators also is
available from dBm, with models covering applications requiring
a passband as high as 3,600 to 4,200 MHz. A
temperature-stabilized and accurate attenuator in the noise path
supports 0.016-dB resolution. You can add a precise amount of
noise to a signal through the built-in and calibrated low
amplitude and phase ripple combiner. An optional attenuator in
the signal path provides greater flexibility by allowing you to
separately set the signal level.
Unlike WGN instruments, the CNG continuously
monitors both the signal and noise powers to ensure that their
ratio is constant. The WGN only ensures that the amount of noise
added to your signal accurately matches the level you selected.
Digital Noise Sources
A paper presented at DesignCon 2007 described
an interesting approach to digital noise generation.2
As the authors point out, any arbitrary waveform generator (Arb)
can produce an AWGN signal. On the other hand, whiteness depends
on sample independence, and these kinds of signals actually are
pseudorandom, not truly random. The data pattern eventually will
repeat although the time to do so ranges from hours to days.
Because of the limited memory in most Arbs, only a small σ value can
be developed, and the signal will have poor Gaussinity.
The signal-generation method described in the
paper recognizes that a completely Gaussian distribution cannot
be generated because there is a finite but very small
probability that a huge value will occur. Instead, the smallest
probability that must be supported is determined, and the design
progresses from there. Although a Gaussian distribution is used
as an example, the method is suitable for almost any
distribution.
A memory stores a range of values that
describes the distribution. A random-address generator then
reads these values. The cumulative distribution function is
calculated and inverted to derive the required relationship
between data value and memory location as shown in Figure 3.
Figure 3. Digital Noise Source Data Compression Scheme
Courtesy of Agilent Technologies
For a Gaussian distribution, almost all of
the memory will be filled with values within σ or 2σ of the mean.
There will be far fewer values stored corresponding to higher
σ values, reducing to only one value at the highest supported σ—the value
with the lowest probability.
Because the data points have been weighted as
described, when they are read with a random address generator,
the output will conform to the correct probability distribution.
Without further refinement, a massive memory is required to
cater for the large dynamic range associated with a 7σ generator. The
paper explains how taking advantage of symmetry and using data
compression the necessary dynamic range can be accommodated in a
reasonably small memory.
Especially at high frequencies, the output
DAC determines quantization in the output signal. This is the
case regardless of the larger number of bits that might be
supported by the digital circuitry controlling the DAC. Because
of this, there is little reason to account for greater
resolution in the distribution memory.
The authors determined that at least three
more bits of address space should be provided beyond the DAC
resolution. So, for example, 17 bits are required for a 14-bit
output. The increased address space is required to overcome
linearization errors caused by the system’s data-compression
scheme.
The actual address generator has a large
number of bits, such as 42, to provide a long, nonrepeating
sequence even at high data rates. Data compression results in
replacing the upper 2k-1 bits of a generated address
with k bits of an actual memory address. For example, with k =
5, the memory space is divided into 32 sections with widths
weighted as 1/2, 1/4,
1/8…1/65,536 of the
total memory. Figure 3 shows the memory space divided into eight
sections corresponding to k = 3.
Rather than a 31-bit address, the compression
scheme provides a 5-bit address to identify the appropriate
section of the memory. The remaining lower-order bits of the
42-bit word address data in that section.
Although it’s true that a limited number of
DAC output values will occur again and again, a different time
history will precede each occurrence. Within one complete cycle
of a 42-bit pseudorandom address generator—4.4 x 1012
addresses—the value with the lowest probability of occurrence
will be output once. It is stored in the 32nd section
of memory for k = 5. Those values stored in the 31st
section will be output twice, those in the 30th
section four times, and so on. This demonstrates how the data
compression varies depending on the probability associated with
the data value.
The noise signal is repeatable and
deterministic with a digital system based on a pseudorandom
address generator. These characteristics are especially
important for communications testing at very low BERs where a
small deviation in Gaussinity between two test generators would
give very different results.
Against digital noise generation is the
limited performance of very high-speed DACs. The approach
outlined in reference 2 appears to have advantages and is
representative of current work in digital noise generation.
Summary
Three factors can give you a good idea of the
best noise generator for your application: crest factor, power
output, and frequency range. A digital generator can have a high
crest factor, but its resolution will suffer at high frequencies
because of the output DAC limitations. On the other hand, a
digital generator could provide a very good AWGN signal with
high crest factor and high resolution up to a few hundred
megahertz. For RF and microwave applications, the noise signal
must be upconverted.
For example, the Agilent Model 81150A Pulse
Function Arbitrary Noise Generator provides AWGN with selectable
crest factors of 3.1, 4.8, 6.0, and 7.0 with a signal repetition
rate of 26 days. The widest noise bandwidth available is 120 MHz
generated as a baseband signal. This instrument combines a noise
source with a function generator capable of AM, FM, PM, FSK, and
PWM modulation and bursts of pulses and standard as well as
arbitrary waveforms.
An analog instrument can directly generate
AWGN in the required frequency band. However, the signal is not
deterministic so tests cannot be repeated under exactly the same
conditions. Statistically, they may be equivalent, but the time
order of noise peaks always will be different from one test to
the next. This can be important when trying to track down
seemingly intermittent performance problems. If the noise
pattern can be exactly repeated, it often helps to pinpoint the
cause of the error.
White noise is specified by its power
density, such as dBm’s Model WGN-1/200 White Noise Generator
with -87-dBm/Hz noise density from 1 MHz to 200 MHz. Of course,
you can amplify an AWGN signal, but the output then will be
distorted by the amplifier’s own noise characteristics as well
as any gain errors.
A comment in the Micronetics CNG datasheet
provides insight regarding amplifiers. "[The instrument’s
carrier path loss] …is caused by…the coupler…, the combiner…,
the attenuator…, and the impedance transformer…. Generally, if
the loss does not pose a problem, the [zero carrier path loss]
option should not be ordered. Despite the high quality amplifier
used, it is better not to have any unnecessary active devices in
the test signal path."
References
1. Lecklider, T., "Trends in Vibration Test,"
EE-Evaluation Engineering, January 2006, pp. 42-46.
2. Mücke, M., et al, "Precision Digital Noise Source,"
DesignCon 2007.
| FOR MORE INFORMATION |
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| Agilent Technologies |
81150A Pulse Function Arbitrary Noise Generator |
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| dBm |
WGN-1/200 White Noise Generator |
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| Micronetics |
CNG Series Carrier to Noise Generator |
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| NoiseCom |
J7000A High Crest Factor Noise Generator |
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