# Theory and Practical Considerations for Measuring Phase Noise Better Than –165 dBc/Hz: Part I

#### Phase noise requirements have a major influence on the design of a communication system because they impact the local oscillator design and hence the cost of the radio. Since the number of wireless subscribers and thus, RF interference, continues to in...

Phase noise is a frequency domain measure of the short-term stability of an oscillator. It is measured and specified as noise power, relative to carrier power, in a one Hz bandwidth at a given frequency offset. Typically, for a communication system oscillator, the phase noise requirement is specified for several frequency offsets chosen according to system requirements. Poor in-band LO phase noise is seen as a distortion on the desired signal. In receivers, there are issues such as reciprocal mixing that set phase noise performance limits so that undesired signals do not mix with LO noise and fall in the IF band. Low out-of-band phase noise in transmitters ensures that spectral emission remains within specified limits. The out-of-band phase noise of a transmit LO is important since it gets amplified by the final transmit stages and reaches high enough levels that it can effectively jam other receivers operating in the vicinity. For a GSM system, the allowed handset transmitter noise, which can appear in the receive band of a co-located handset, is given in Table 1.

The phase noise specifications that can be derived for a transmit power level of +33 dBm for GSM and +30 dBm for DCS and PCS are shown in Table 2. Note that the GSM standard requires these specifications to be satisfied up to 2 MHz beyond the desired band of operation. Thus a little extra margin is needed.

In GSM handset transmitters, an offset phased-locked loop (PLL) architecture with IQ modulation within the transmit loop is the most common choice. The transmit PLL has a wide loop bandwidth and flicker noise is not a major concern in the design of the transmit VCO for this architecture. However, wideband noise is of concern. A bandpass filter at the output of the power amplifier (PA) can help to reduce this wideband noise to some extent at the cost of insertion loss. This would require a higher PA power to compensate for filter loss. For longer battery life, low noise performance is desirable for the transmit VCO so that post-PA filtering can be obviated.

Such issues continue to pose a major challenge for engineers to design very low phase noise oscillators. Thus, there is an obvious need to verify system performance and accurately interpret the results of design iterations.

To be able to set up accurate phase noise measurements one needs to understand the following:

- Various noise processes
- Distortion of an ideal sine-wave and modulation theory
- Bandpass filtered noise and noise modeling
- Measurement concepts and methods
- Special considerations at high frequency offsets
- Advantages and disadvantages of different techniques.

Noise Processes

Noise is a random process in time with unpredictable instantaneous levels. The noise from different sources has been characterized in terms of statistical properties and power spectral density. The different types of noise, thermal, flicker, shot, etc., are briefly summarized below.1,2 The actual mechanism for phase noise in oscillators is much more complicated than the simplified explanation given here.

Thermal noise is common in resistors and has a flat power spectral density (PSD) up to several hundred gigahertz. This PSD is independent of the resistor value or DC current, and is given by 4kT, normalized to 1 W. Thermal noise has a Gaussian probability density function (PDF).

Thermal noise is also seen in transistors and is bias dependent. Shot noise is associated with a DC current flowing across a semiconductor junction. The PDF is Gaussian and PSD is flat only up to a few gigahertz. The spectral density of shot noise is flat for wavelengths longer than the carrier transit time. Phase noise at large frequency offsets is predominantly due to this wideband noise, that is thermal and shot noise. At small frequency offsets, the main contributor to the VCO noise is flicker noise.

Flicker noise is associated with a direct current and is found in all active and some passive devices. It has a PSD that varies as 1/f. The flicker noise of a transistor is specified in terms of flicker corner frequency, which is a point on the frequency axis where flicker noise and white noise contribution are the same. The flicker corner for BJTs is of the order of a few kilohertz; it can exceed a few megahertz in the case of MOSFETs. A large device size helps to reduce the flicker corner by way of smoothing due to device capacitance, but that also reduces the useful frequency range of the transistor. The close-in phase noise of a VCO is caused by circuit nonlinearity that translates the flicker noise of active and passive devices to the carrier frequency. Certain oscillators use low noise Schottky diodes for signal limiting, while the amplifying device operates in the linear region. The performance of these circuits at low frequency offsets is limited by the flicker noise of Schottky diodes rather than of active devices.

Flicker noise is also a major concern in direct down-conversion receivers where the modulated signal is directly translated from RF to DC. The close-in phase noise of the LO driving the down-converter can be suppressed by choosing a sufficiently large PLL bandwidth. However, down-conversion has to be done through a frequency mixer whose performance can be limited by the flicker noise of the diodes (in passive mixers) and transistors (in Gilbert cell mixers). Noise in bias circuits can modulate the VCO signal by FM through supply pushing and by AM through amplitude dependence on supply voltage.

Low Noise Design and Measurements

Several principles pertaining to very low noise measurements are directly applicable in the design of low noise VCOs, that is most VCOs have high isolation amplifiers at the output to minimize load pulling. The noise figure of this kind of circuit can add to the phase noise at large frequency offset measurements. In order to achieve a high Q with available components, it is desirable to have the resonator coupling be as loose as possible. That comes at the cost of increased resonator insertion loss, however, which means that the active devices need to provide a higher gain. If a closed loop analysis is done on such a design, it can be seen that the signal level at the resonator output, and hence the absolute noise level, is pushed way down closer to the thermal noise floor. A subsequent amplification will only reduce the signal-to-noise ratio (SNR), so there is a direct trade-off in the optimization of phase noise at low offset versus higher offsets. Most IC designs attack the noise problems by looking at the percentage contribution of individual components, which provides good information as to where the noise is coming from but does not give any design insight unless the circuit is partitioned into different blocks and the performance of each one is optimized from an RF perspective. RF measurements follow the same basic principles.

Distortion of an ideal sine wave

The output of a practical oscillator differs from the desired ideal sinusoid in the sense that the actual VCO output is frequency and amplitude modulated, in addition to having undesired harmonics. Harmonic noise can easily be filtered out by conventional bandpass filtering, but the close-in noise, which is often termed as phase noise, must be minimized by design.

The output of an oscillator can be mathematically expressed as

where

Ac = peak voltage of the unmodulated carrier

m = amplitude modulation index, with 0 < m < 1

ωm = angular frequency of modulating signal

ωc = angular frequency of carrier signal

Δφ(t) = phase fluctuation and represents a combined contribution of various noise sources that result in phase noise and harmonics. For a simplified analysis, this term is often considered to represent the result of a single tone modulation

Frequency modulation and phase modulation

Frequency modulation (FM) and phase modulation (PM) are two different forms of ‘angle’ modulation. Phase modulation varies the instantaneous phase of the carrier. Since frequency is the rate of change of phase, PM can also be seen as changing instantaneous frequency that is FM. Mathematically, both these forms of angle modulation can be represented by

Δφ(t) = βcos(ωmt) gives the instantaneous phase fluctuation

where the maximum frequency deviation, Vf = AmKvco and Δf(t) = –fmβsin(ωmt), gives the instantaneous frequency fluctuation.

The PSD of phase and frequency fluctuations can be calculated as

Since the PSD of a sinusoid is the same as that of its phase-shifted version, it follows that

Irrespective of the noise modulation process inside the VCO circuit, the phase noise at its output can be measured by frequency or phase demodulation. It might seem that the processes responsible for phase noise in a complex nonlinear VCO circuit are as simple as phase and frequency modulation, but that is not true. Due to nonlinear operation, there are complex factors like AM-PM conversion and frequency translation of noise, etc. Equation 5 is useful in converting the results from frequency to phase domain and vice-versa.3,4

Sideband Level Relative to Carrier

By the very definition of phase noise, the matter of concern is the sideband power relative to signal power. It is important to review the two basic modulations, which may result in the undesired sideband power in a VCO. When β<<1, using the identities

the narrowband nature of FM modulation can be verified. For higher values of β, Equation 2 can be expanded in terms of Bessel functions and the signal occupies a much larger bandwidth in the frequency domain.

It can be seen that the power spectrum of narrowband FM looks similar to that of AM, except that the modulation indices are different by definition. For narrowband angle modulation, the level of each sideband is

below carrier. For AM, it is

below carrier, where

In FM, with an increasing modulation index, the total power in the modulated signal remains constant while the occupied bandwidth increases. The power in sidebands is generated at the expense of carrier power.

In AM, the carrier power remains as is and increasing the modulation index increases the total power.

PSD and PDF of filtered noise

For white noise with a Gaussian distribution and small variance, there is a high probability that the noise samples will be close to a mean value. Let the PSD of white Gaussian noise be defined as

Then for ideal white noise, the total power is given as

Figure 1 shows the noise in the time domain and the PDF of this random process is shown in Figure 2. For a zero mean random process, the variance is numerically equal to the power

From Equations 7 and 8, the variance of an ideally ‘white’ Gaussian noise turns out to be very high. This means that there is always a finite non-zero probability that an instantaneous noise sample may have a very high amplitude. This argument is also supported by the fact that the area under a function is equal to the central ordinate of its Fourier transform.5 More insight can be gained by generating and analyzing random numbers using a math tool. Such high noise impulses as may theoretically seem possible are not seen in practical situations because all practical noise sources and measurement circuits are band-limited. Moreover, the PSD of thermal noise is known to get smaller at very high frequencies, although that happens much beyond the bandwidth of noise sources and measurement systems.

If noise is low pass filtered with a cutoff frequency of fbHz, then, assuming a rectangular filter transfer function

From Equations 7 and 9 it can be seen that filtering reduces the noise variance. Therefore, in practice, two consecutive noise samples cannot differ by a very large magnitude. The PDF may remain unchanged but the PSD is shaped by the filter transfer function. Even analog circuit simulators working in discrete time domain and infinite bandwidth cannot be supported, if aliasing effects are to be avoided. Hence, the noise samples under consideration are ‘always’ low pass filtered and only the cutoff frequency varies according to different situations.

Modeling VCO Noise as a Single Tone Modulation Sideband

Consider a very narrow band rectangular filtering window

centered at frequency fm and having a bandwidth b Hz. Then the filtered noise

Y(f) = N(f)H(f)

where

N(f) = frequency domain representation of noise

Including negative frequencies for completeness of analysis, from Equations 6 and 10,

As the filter bandwidth is made infinitesimally small

Introducing Equation 12 in Equation 11,

Equation 13 resembles the PSD of a sinusoid with frequency fm, and amplitude scaling factor η.

For all practical purposes, in the frequency domain, the flat averaged PSD of noise can be thought of as the power in each 1 Hz band due to a large number of sinusoids spaced by 1 Hz. This result is very important in VCO noise analyses. The effective FM and PM noise at the output of the VCO can be modeled by an input noise source at its frequency tuning port, while the VCO itself is considered noiseless. Using the discussion of FM and PM given before, it is possible to modulate using FM and demodulate using PM and vice-versa. The relative level of the modulation sideband in the original signal can still be determined using relation Equation 5.

Phase Noise Measurements Using Direct Spectrum Analysis

This is the most common method for measuring phase noise using a spectrum analyzer. It works well for low frequency offsets where noise levels are much higher than the instrument noise floor. The measurement is simply the difference between the carrier power and average noise power at the offset frequency of interest in a 1 Hz bandwidth. This needs to be compensated by numerous factors like the resolution bandwidth (RBW), the ratio of necessary bandwidth (NBW) to RBW and the effects of log amplification before averaging. An actual noise power measurement requires that the average of the squared noise voltage be calculated. In practical analyzers, envelope detection is used that is followed by a logarithmic amplifier before getting averaged by a video filter. Also, the envelope of the Gaussian noise has a Rayleigh distribution, which has a null at the origin. Log processing of noise values close to zero generates sharp negative peaks that result in under-estimating the noise power by approximately 2.5 dB.6 The resulting data can be represented in various forms such as log plots or spot noise using software programs. Averaging is important for repeatable measurements. For time-critical measurements, most of the processing is done in the digital domain using FFT. Waveform capturing is a time-consuming process. To reduce the test time, averaging FFT magnitude at a number of points around the offset frequency of interest helps reduce the variation in measurement and gives an effect equivalent to averaging the FFT of an ensemble of waveform captures. Some spectrum analyzers use this method in reading out the ‘noise marker’ function. The number of points should be low enough such that actual well-averaged noise in that region does not vary too much. Too many points may also result in the inclusion of an undesired spur near the frequency point of interest. To further double the sample size, points around the offset frequency on both sides of the carrier can be included if β<<1 such that phase noise is symmetrical around the carrier, which is the case considering the offset frequencies of interest in most communication sources. The user should refer to the instrument manual to analyze measurement uncertainty due to different settings of the spectrum analyzer such as:

- RBW switching during the measurement
- Input attenuator switching during the measurement
- Frequency response flatness
- Display scale fidelity
- Reference level calibration accuracy

Whenever possible, relative measurements should be chosen over absolute measurements. Depending on instrument specifications, certain steps can be taken to minimize the experimental error.

Limitations of Direct Spectrum Measurements

As the frequency offset increases, the measurement error tends to increase because the oscillator noise floor typically is monotonically reduced with increasing frequency offset, as is evident from the well-known Leeson’s model.3,4,7,8

The instrument noise floor is fairly flat. At large measurement offsets, the device under test (DUT) noise floor can be too close to the instrument’s noise floor for a low noise DUT.

The proximity of the DUT noise to the measurement system noise floor introduces a measurement error that makes the DUT noise appear higher than actual, as shown in Figure 3. The resulting measurement error is plotted in Figure 4.

It is possible to measure the system noise floor and determine how close it is to the DUT noise floor, and the resulting error can be calculated. Some instruments, like the Agilent PSA, use a technique called DANL cancellation, where the instrument noise floor is measured and normalized to the DUT carrier power before the actual measurement. The instrument noise power is then subtracted from the measured phase noise at each frequency offset and a corrected phase noise trace is plotted.

This method can easily measure noise levels down to the instrument’s own noise floor and a few decibels below that. However, this calculation may result in large errors if the DUT noise is too far below the instrument noise. Note that conventional measurements were recommended with an instrument noise floor at least 10 dB below the DUT noise level. This type of cancellation method can accommodate DUT noise levels approximately 15 dB below that limit. Figure 5 shows how measurement inaccuracies and finite averaging can result in large errors for DUT noise levels more than 5 dB below the instrument noise floor. In this graph, it is assumed that instrument noise normalized to DUT carrier power is –153 dBc/Hz and trace error using conventional measurement is ±0.1 dB.

In direct spectrum analysis, one actually measures the net power in a given noise sideband (in a 1 Hz bandwidth), regardless of the type of noise modulation contributing to that power. This method measures the overall spectral purity but cannot distinguish between the AM and PM noise. Most VCOs have active devices in saturation. These VCOs have much reduced AM noise levels at the cost of increased harmonic levels and increased phase noise due to frequency translation. In this case, the measurement can be called a ‘phase noise’ measurement.

For those designs that prevent saturation of the active device by using AGC techniques, the phase noise is low and the amplitude noise can be comparable. In such a case, direct spectrum analysis is not really the ‘phase noise’ measurement. Amplitude noise is included as well. Separate measurement of AM and PM noise may be desirable, at least in the VCO design phase, even if the final system just specifies a spectral mask requirement.

Improving Direct Spectrum Analysis

DUT noise amplification is an obvious choice to bring it several decibels above the instrument noise floor. However, if the carrier power of DUT is already high to begin with, further amplification may result in saturation of the amplifier or of the front end of the measuring instrument, which should be avoided.

The amplifier also adds its own noise to the DUT noise floor but noise floor degradation (in dB) is not the same as its noise figure. Definition of the noise figure as SNRi/SNRo holds good only for Ni = –174 dBm, which applies well in the case of radio receivers, where the received signal is so low that the SNR is measured against kT.

The noise contribution of the amplifier is given as

The total noise floor is given as

From Equation 15, it is obvious that

The measurement error can be calculated as

error_dB =

where

G = amplifier gain

k = 1.38 × 10–23 Joules/K

T = 16.85 + 273K

F = noise factor

Figure 6 shows the error introduced by the amplifier noise figure for various input noise levels (NDUT).