  # Impact of ultra-low phase noise oscillators on system performance

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Introduction
To an electrical engineer, in an ideal word there would be no noise. But what is noise? What is electrical noise? Or more to the point of this paper: What is phase noise? As engineers, we know intuitively that low noise in a system is better than high noise. However, we must somehow quantify this noise in units and terms that we can all be in agreement with – and we will. We will also examine the difference in phase noise performance of commodity vs. low-cost, high-performance crystal oscillators. Understanding the cost performance trade-offs between oscillators is important to a system design. Many times we see two competitive systems separated widely in performance, but NOT in price. The oscillator phase noise characteristics will dominate the entire system performance and spending a few more dollars on the oscillator can turn a mediocre system into a superb system.

However, an engineer can easily over-specify the oscillator, and hence the key is to understand exactly how the oscillator phase noise (or jitter) limits the system performance. To help with this understanding, a tutorial on phase noise and jitter is in order.

Phase Noise and Jitter in Oscillators – A tutorial
In an oscillator, phase noise is the rapid random fluctuations in the phase component of the output signal. The equation of this signal is: Where:    A0 = nominal peak voltage
f0  = nominal fundamental frequency
t = time
Δø(t) = random deviation of phase from nominal – “phase noise”

Above, Δø(t) is the phase noise, but A0 will establish the signal-to-noise ratio. Figure 1 illustrates this. Figure 1: The signal to noise ratio is a function of A0

The Noise Floor
Noise signals are stochastic and, in a broad sense, noise can be characterized as any undesired signal that interferes with the main signal to be processed or generated. It can disturb any physical parameter such as voltage, current, phase, frequency (or time), etc. Therefore the idea is to maximize the signal and minimize the noise for a high signal-to-noise(S/N) ratio.

Noise power is quantified as Where
K is the Boltzmann’s contant = 1,38 x 10-23 (J/K)
T is the absolute temperature in °K
And Δf= B is the bandwidth in which the measurement is made, in hertz

In the absence of any signal, there is thermal noise floor. This floor level can be specified in a variety of units:  Watts, V2/Hz, √V/ Hz , dBm/Hz to name a few. For oscillators, it is convenient to use dBm/Hz to define noise density.

Before defining dBm/Hz we need to first define dBm. dBm refers to decibels above 1 milliwatt in a 50 Ohm system and is given by Thus from above equation, 1 milliWatt is equal to 0 dBm.
Equation 2 gives us the magnitude of thermal noise and substituting for K and T we get: Where B is the bandwidth of interest, for which we will use 1Hz to normalize the result. Using the equation of dBm (Equation 3) , and using the result from above we have: Setting the bandwidth B to 1Hz will give us the final result in dBm/Hz, and since the log(1) is zero, we have: The quantity of -174 dBm/Hz is the thermal noise power density of a 1Ω resistor at 290°K measured in a 1 Hz bandwidth.

If an oscillator has an output power of 1mW or 0 dBm, then: Where dBc is decibels relative to the carrier level. This result tells us the best obtainable noise floor for a 0 dBm oscillator is -174 dBc/Hz at 290 °K.

In general one can convert dBm to dBm/Hz with: and dBm/Hz to dBm with: For example: What is -50dBm in dBm/Hz in a 1 KHz bandwidth? Solution:
Power in dBm/Hz = -50-10log(1000) = -50-10(3) = -80 dBm/Hz

Noise Characteristics
Noise on a carrier can be separated into two categories; random and deterministic. Random noise spreads the carrier while deterministic noise generates sidebands on the carrier as illustrated in Figure 2. Adding the deterministic component to Equation 1, it now becomes, Where: md is the amplitude of the deterministic signal, which is phase modulating the carrier, and fd is the frequency of the deterministic signal. Figure 2: Noise on a carrier showing deterministic signal

Noise has infinite bandwidth, and hence the greater the bandwidth of the instrument being used to measure a carrier frequency with noise, the higher the noise it measures. For example, as you change the resolution bandwidth (equivalent to the physical bandwidth of the IF channel) on a spectrum analyzer, the noise magnitude changes. Hence, we must all agree on one measurement bandwidth to use when specifying spectral purity of an oscillator or signal source.

Industry has settled on a correlation bandwidth for phase noise measurements of 1Hz, known as the normalized frequency.There are few spectrum analyzers that have a 1 Hz resolution bandwidth. Such a spectrum analyzer is very expensive. In fact, the closer to the carrier you want to measure, the higher the instrument cost will be. A spectrum analyzer will specify how close to the carrier it can measure (known as the lowest resolution bandwidth possible); above this maximum frequency, one can normalize the reading to 1 Hz with the following: For example, figure as a given a point that is -40 dBc at an offset frequency of 10KHz from the carrier. In addition, the resolution bandwidth of the instrument is set to 1KHz. What is the phase noise at this point in dBc/Hz? Answer: Since the log(1000) = 3 we have: Therefore the phase noise at this point is -70dBc/Hz at 10KHz offset, or: The noise spectrum of a signal is symmetrical around the carrier frequency and therefore it is necessary to specify only one side. This one-sided spectrum is called a Single Side Band (SSB) spectrum. Hence, the spectral purity of a signal can be completely quantified by its single sideband (SSB) phase noise plot as shown in Figure 3. Figure 3: Typical SSB phase noise plot of oscillator vs. offset from carrier

This SSB plot has been assigned the script L{f} and is defined as one half the sum of both sidebands. L{f} has units of decibels below the carrier per Hertz (dBc/Hz) and is defined as where  Psideband (f0 + Δf, 1Hz) represents the signal power at a frequency offset of  away from the carrier with a measurement bandwidth of 1 Hz.

Below are three of the most popular ways in which phase noise is defined.

(1) The term most widely used to describe the characteristic randomness of frequency stability
(2) The short-term frequency instability of an oscillator in the frequency domain.
(3) The peak carrier signal to the noise at a specific offset off the carrier expressed in dB below the carrier in a 1-Hz bandwidth (dBc/Hz).

Jitter

So far all the discussion regarding noise has been presented in the frequency domain. An oscillator noise performance characterized in the time domain is known as jitter. Note that phase noise and jitter are two linked quantities associated with a noisy oscillator, and, in general, as the phase noise increases in the oscillator, so does the jitter.

Jitter is a variation in the zero-crossing times of a signal, or a variation in the period of the signal. Jitter is composed of two major components, one that is predictable and one that is random. The predictable component of jitter is call deterministic jitter. The random component of jitter is called random jitter. Random jitter comes from the random phase noise, while deterministic jitter comes from the deterministic noise.

Random Jitter (RJ)
Random jitter (RJ) is characterized by a Gaussian (Normal probability) distribution and assumed to be unbounded. As a result, it generally affects long-term device stability. Because pk-to-pk measurements take a long time to achieve statistical significance, random jitter is usually measured as an RMS (root mean square) value.

Why does jitter take on the characteristic of a Gaussian distribution function? The answer is the following: Random jitter is the result of accumulation of random processes including thermal noise, flicker noise, shot noise, etc. All of these noise sources contribute to the total jitter observed at the output of an oscillator. The Central Limit Theorem states that the sum of many independent random events (functions) converges to a Gaussian distribution, as depicted in Figure 5. Figure 5: Central Limit Theorem – the sum of independent random functions converges to a Gaussian distribution

Deterministic Jitter (DJ)
Deterministic jitter (DJ) has a non-Gaussian PDF and is characterized by
its bounded pk-to-pk amplitude. Deterministic jitter is expressed in units of time, pk-to-pk. The following are examples of deterministic jitter:

1)    Periodic jitter (PJ) or Sinusoidal – e.g., caused by power supply feedthrough
2)    Intersymbol interference (ISI) – e.g., from channel dispersion of filtering
3)    Duty cycle distortion (DCD) – e.g., from asymmetric rise/fall times
4)    Sub-harmonic(s) of the oscillator – e.g., from straight-multiplication oscillator designs
5)    Uncorrelated periodic jitter – e.g. from crosstalk by other signals
6)    Correlated periodic jitter

Total Jitter (TJ)
Total jitter (TJ) is the summation (convolution) of all independent jitter components.

Total Jitter (TJ) = Random Jitter (RJ) + Deterministic Jitter (DJ)

Impact of phase noise/jitter on system
Phase noise or jitter of an oscillator has a direct impact on a system performance. In an RF communication system, high phase noise will affect communication distance, adjacent channel interference, Bit error rate to name a few.

For today’s advanced high-speed converters, a clean clock signal translates to more “effective number of bits,” or ENOB. The accuracy of an A/D is enabled by the purity of the clock being used and its inherent SNR. Hence a very low-jitter clock is essential to have good SNR.

In A/D converters, jitter limits the SNR by the following equation: Where:
t is the jitter in rms

Solving for the jitter term of the above equation we obtain: For example, suppose we have an input signal of 80 MHz and you require an SNR of 75 dB, then a clock with 470 fentoseconds is required. This assumes that jitter is the only limiting factor in the converter performance.

Commodity Clock vs. Ultra-low phase noise Clock

We will now compare the phase difference of two oscillators, one a commodity and the other a ultra-low phase noise. What dictates the title “ultra-low” to some oscillator could be a matter of “specsmanship.” To this author, the "ultra-low phase noise" designation should be given to an oscillator with a noise floor of -160dBc/Hz or lower, and lower than -130 dBc/Hz at 1K offset. This type of phase noise is easily achieved by many OCXO with SC-cut crystals at frequencies below 50 MHz. However, the comparison here is NOT for a reference type (OCXOs or TCXO for example) crystal oscillator, but rather for clock oscillators. Today, a commodity type 5x7mm +/-50 ppm stability clock can be purchased for less than \$2.00. What type of phase noise are you getting from this typical commodity clock?

Crystek’s oscillator family CCHD-950 (CLOCK) and CVHD-950 (VCXO) were designed as cost effective clean, low jitter Clocks and VCXOs. This family of oscillators uses discrete components to achieve “subpicosecond” jitter at a reasonable price. Figures 6 and 7 are actual SSB phase noise plots of a commodity Clock and the CCHD-950 at 100Mhz. Note that when comparing jitter specs from different oscillators, it is not sufficient to simply look at the quoted jitter of 1pS RMS, max. (12K to 20MHz). Both oscillators in Figures 6 and 7 will meet this spec, but clearly the CCHD-950 is a superior oscillator in terms of phase noise and wideband jitter. Figure 6: SSB phase noise plot of a commodity clock Figure 7: SSB phase noise plot of a true ultra-low phase noise oscillator (Crystek CCHD-950)

Oscillator Technology to Achieve Ultra-Low Phase Noise
A commodity oscillator is nothing more than an ASIC and a quartz crystal blank. In most cases it does not even have an internal bypass capacitor. The crystal blank is an AT-cut strip with Q of about 25K~45K. This low Q limits the close-in phase noise. The ASIC with all its transistors limits the floor noise to about -150 dBc/Hz. On the other hand, the true ultra-low phase noise oscillator uses a discrete high performance oscillator topology with a packaged crystal with a Q greater than 70K for excellent close-in phase noise. The discrete oscillator topology establishes the signal-to-noise ratio, and hence the floor is lower than -160 dBc/Hz. Therefore, superior performance is obtained with very high Q crystals and good a discrete topology. This lower phase noise does come with a price delta of approximately \$15. However, this a small price to pay (in most cases) considering the improvement gained.

References
Brannon, Brad, “Sampled Systems and the Effects of Clock Phase Noise and Jitter”, Analog Devices App. Note AN-756

Poore, Rick, “Phase Noise and Jitter”, Agilent EEs of EDA, May 2001

Vig, John R. "Quartz Crystal Resonators and Oscillators"

Footnote:

The Gaussian distribution is illustrated in Figure 4. Mathematically, this function is  Figure 4: Gaussian or Normal distribution curve

Properties of the Gaussian distribution follow: The Gaussian distribution is also commonly called the “Normal distribution” and often referred as a “bell-shaped curve”. Note that within +/- 1σ of the Gaussian distribution curve, 68.2% of the random events will occur and that 99.6% will occur within +/- 3σ.