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.
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.
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.
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. M cke, M., et al, “Precision Digital Noise Source,” DesignCon 2007.
| FOR MORE INFORMATION | Click below | |
| Agilent Technologies | 81150A Pulse Function Arbitrary Noise Generator | Click here |
| dBm | WGN-1/200 White Noise Generator | Click here |
| Micronetics | CNG Series Carrier to Noise Generator | Click here |
| NoiseCom | J7000A High Crest Factor Noise Generator | Click here |