Local Oscillator Phase Noise and Its Effect on Receiver Performance All Superheterodyne Receivers Use One Or Phase Noise, and Can Only Be Reduced by in Figure 1

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The Communications Edgeª Tech-note Author: C. John Grebenkemper

Local Oscillator Phase Noise and its Effect on Receiver Performance All superheterodyne receivers use one or phase noise, and can only be reduced by in Figure 1. The input signal is assumed to more local oscillators to convert an input decreasing the phase noise of the oscillator. be an unmodulated carrier, and the local frequency to an intermediate frequency oscillator is phase-modulated by its phase before the signal is demodulated. In the ideal INTRODUCTORY THEORY noise. The output of the frequency converter receiver, these frequency conversions would A perfect oscillator would be described is at the sum frequency (or difference, not distort the input signal, and all informa- mathematically by a sinusoidal waveform, depending on the IF filter) of the unmodu- tion on the signal could be recovered. In a lated carrier and local oscillator frequencies. real-world receiver, both the mixer used for V = cos [ωot]. The phase noise which was present on the converting the signal’s frequency and the An actual oscillator will exhibit both an local oscillator has been transferred to the local oscillator will distort the signal and amplitude noise modulation, n(t), and a input signal and now appears as a phase limit the receiver’s ability to recover the modulation of the input carrier. This effect phase noise modulation, θn(t), modulation on a signal. Mixer degradations, can be extended to a modulated carrier, and such as undesired mixing products, can be V = [1 + n(t)] cos [ωot + θn(t)], results in the addition of an undesired phase minimized by proper design in the rest of noise modulation of the carrier to the where n(t) and θn(t) are random processes. A the receiver. The local oscillator degrada- good local oscillator will exhibit an ampli- desired signal modulation. This phase noise tions, which are principally random phase tude-noise modulation power that is much can result in additional noise at the output of the signal demodulator, depending on the variations known as phase noise, cannot be less than the phase-noise modulation power. type of modulation. decreased except by improving the perfor- Furthermore, receiver mixers are usually run mance of the oscillator. at a saturated input power, which will reduce PHASE-NOISE DEFINITIONS Low oscillator phase noise is a necessity for their sensitivity to local oscillator amplitude There are a number of ways to measure many receiving systems. The local oscillator variations. The net result is that amplitude oscillator phase noise. Table 1 lists some of phase noise will limit the ultimate signal-to- noise insignificantly contributes to degrada- the more common definitions, along with a noise ratio which can be achieved when lis- tions in the receiver performance due to the brief description on how each term is mea- tening to a frequency modulated (FM) or local oscillator. For this reason, the ampli- sured. The single-sideband (SSB) phase noise phase-modulated (PM) signal. The perfor- tude noise can usually be ignored. is the most common measure of oscillator mance of some types of amplitude modula- In its ideal form, the mixer in a receiver phase instability. It can be directly measured tion detectors may be degraded by the local multiplies the RF input by the LO input to on a spectrum analyzer, providing that the oscillator phase noise. When the receiver is produce the sum and difference of the two oscillator has low amplitude noise modula- used to monitor phase-shift keyed (PSK) or input frequencies. The mixer is usually fol- tion and the spectrum analyzer local oscilla- frequency-shift keyed (FSK) signals, the lowed by an IF filter to select the desired IF tors are lower in phase noise than the unit phase noise may limit the maximum bit output frequency. This process is illustrated under test. This latter condition is usually error rate which the system can achieve. In FM/FDM (frequency division multiplex) systems, phase noise will often limit the maximum noise power ratio of the receiving INPUT IF OUTPUT FILTER system. Phase noise can limit the maximum cos (ωt + φ) 1/2 cos [(ω + ωo)t + φ + θn(t)] angular resolution which can be achieved by an interferometric direction-finding receiver. Reciprocal mixing may cause the receiver noise floor to increase when strong signals LO are near the receiver’s tuned frequency; this cos [ωot + θn(t)] limits the ability to recover weak signals. All of these effects are due to local oscillator Figure 1. Effect of local oscillator phase noise on a frequency conversion.

WJ Communications, Inc. ¥ 401 River Oaks Parkway ¥ San Jose, CA 95134-1918 ¥ Phone: 1-800-WJ1-4401 ¥ Fax: 408-577-6620 ¥ e-mail: [email protected] ¥ Web site: www.wj.com The Communications Edgeª Tech-note Author: C. John Grebenkemper

the limiting factor in the sensitivity of this methods can be used to measure phase noise FUNDAMENTAL RELATIONSHIPS measurement method. When this measure- with greater sensitivity and accuracy at the 1 S (f) = S f (f) ment is done using an analog spectrum ana- y 2 δ cost of considerably more complexity in fo lyzer, the result is usually two to three dB instrumentation. 1 S (f) = S (f) better than what the oscillator is actually δφ f2 δf achieving. The prime reasons for this error Incidental frequency modulation (IFM) is 1 ∞ often used to specify overall oscillator insta- 2 are averaging done in the spectrum analyzer £(f) = Sδf (f) if Sδφ(f’) df’<< 1 rad 2 ∫f after the log detector and the difference bility. For IFM to be well defined, it should ∞ sin2 (πfτ) sin2 (2πfτ) between the resolution bandwidth of the always be specified with a lower and upper δy2(τ) = 2 Sy (f’) - df o ( f )2 (2 f )2 analyzer and its noise bandwidth. Other frequency limit. For FM receivers, these lim- ∫ [ π τ π τ ]

fb βf = Sδf (f) df Symbol Units Definition √ ∫ fa £(f) dBc/Hz Single-sideband Phase Noise. This is the phase instability of the oscilla- f β = b S (f) df tor measured in the frequency domain. It is the most commonly used φ δφ √ ∫ fa measurement of phase noise. A spectrum analyzer can be used to mea- ∞ 2 sure it if the oscillator has no amplitude noise modulation and the phase if Sδφ(f) df<< 1 rad , then ∫ f noise of the spectrum analyzers oscillators are less than the measured a oscillator. The units of dBc/Hz refer to dB below the carrier measured in f 2 a 1-Hz bandwidth. βf = 2 b f £ (f) df √ ∫f a 2 Sδf(f) Hz /Hz Spectral Density of the Frequency Fluctuations. This is the power spectral density of a frequency discriminator’s output. It can be directly mea- f βφ = 2 b £ (f) df sured by connecting an audio spectrum analyzer to the output of a fre- √ ∫ f a quency discriminator whose input is the oscillator under measurement. FREQUENCY MULTIPLICATION RULE S (f) Radians2/Hz Spectral Density of the Phase Fluctuations. This is the power spectral δφ £Mfo(f) = £fo(f) + 20 log (M) density of a phase discriminator’s output. It can be directly measured by connecting an audio spectrum analyzer to the output of a phase demod- Table 2. Phase-noise relationships. ulator which has its input connected to the oscillator under test.

Sy(f) 1/Hz Spectral Density of the Fractional Frequency Fluctuations. This is S(f(f) divided by the oscillator frequency squared. The main advantage of this its are normally set to the lower and upper unit of measurement is that it is invariant under frequency multiplication limits of the video pass-band. For other and may therefore be used to judge the relative quality of oscillators at types of receivers the upper limit should be different frequencies. set equal to the IF bandwidth. If no upper σy(τ) Two-point Allan Variance. This is a time domain measure of oscillator instability. It can be directly measured using a frequency counter to repet- limit is set in an IFM specification, then its itively measure the oscillator frequency over a time period τ. The Allen magnitude tends to become very large. For variance is the expected value of the RMS change in frequency with phase-modulated signals, incidental phase each sample normalized by the oscillator frequency. modulation is preferred over IFM, since it βf Hz Incidental Frequency Modulation. This is a measure of the RMS frequency instability over a band of offset frequencies. It can be calculated by provides a better measure of overall oscilla- taking the square root of the spectral density of the frequency fluctua- tor instability for that type of signal. tions integrated from a lower frequency limit to an upper frequency limit. It can be directly measured by passing the output of a frequency discrim- All of these measures of phase noise can be inator, whose input is the oscillator under test, through a bandpass filter and determining the RMS frequency variation. related to each other by the appropriate mathematical formulas. Table 2 gives the βφ Radians Incidental Phase Modulation. This is a measure of the total RMS phase instability over a band of offset frequencies. It can be calculated by taking mathematical expressions that relate all of the square root of the spectral density of the phase fluctuations integrat- the phase noise definitions given in Table 1. ed from a lower frequency limit to an upper frequency limit. It can be directly measured by passing the output of a phase discriminator, whose Some of these formulas only apply under input is the oscillator under test, through a bandpass filter and determin- special conditions. SSB phase noise can only ing the RMS phase variation. be converted from the various spectral den- f Hz Offset Frequency. This is the frequency of the phase or frequency fluctu- sity measures if the power in the phase fluc- ations. When the oscillator is directly viewed on a spectrum analyzer, this becomes the offset from the carrier frequency. tuations at frequencies greater than the offset frequency is much less than 1 radian2. fo Hz Frequency of Carrier. This is the frequency of the oscillator which is being measured. The offset frequency at which this condition Table 1. Phase-noIse definitions. becomes valid can vary from tens of Hertz

to tens of kiloHertz, depending on the quali- The phase noise of the local oscillator will fb 2 2 Kf(f) Kd(f) Sv(f) df ty of the oscillator. The Allan variance can generate a constant level of noise at the out- S fa = ∫ be directly computed from the fractional fre- put of the FM receiver. If the RF input to N fb 2 quency fluctuations. However, the reverse is the receiver is sufficiently strong, this source Kd(f) Sδf(f) df ∫ fa not true unless an assumption is made about of noise will dominate and therefore limit This latter formula is more difficult to evalu- the power law slope of the spectral density of the maximum signal-to-noise ratio that the ate than the simpler formula, which does the fractional frequency fluctuations. The receiver can achieve. The power in the noise not include the effects of preemphasis and frequency multiplication rule relates the is found from, increase in the SSB phase noise to multipli- deemphasis. However, for most FM trans- cation integer, M. If an oscillator is multi- 2 fb mission systems, the simpler formula will Pn = Kd Sδφ(f) df plied in frequency by a factor of ten in an ∫ fa provide an answer which is within a few dB ideal multiplier, the oscillator’s SSB phase Taking the ratio of these two powers yields a of the correct result. Since local oscillator phase noise performance can vary by this noise will increase by 20 dB. Similarly, if the signal-to-noise ratio of, oscillator’s frequency is divided by ten in an much, it is usually sufficiently accurate to 2 fb use the simpler formula. ideal frequency divider, its SSB phase noise Kd Sv(f) df S fa will decrease by 20 dB. = ∫ N f EXAMPLE b S (f) df f δφ LIMIT OF FM SIGNAL-TO-NOISE ∫ a What is the local oscillator limited signal-to- RATIO which simplifies to, noise ratio for an FM signal which has a 5- kHz RMS frequency deviation and a video The phase noise of a local oscillator will 2 fb Kd Sv(f) df modulation bandwidth of 300 Hz to 3 kHz? limit the maximum signal-to-noise ratio that S fa = ∫ The local oscillator SSB phase noise is a con- can be achieved with an FM receiver. The 2 N β oscillator phase noise is transferred to the f stant -70 dBc/Hz from 100 Hz to 10 kHz. carrier to which the receiver is tuned and is The local oscillator limited signal-to-noise SOLUTION then demodulated by the FM discriminator. ratio is equal to the power in the frequency The phase noise results in a constant noise deviation of the signal divided by the inci- The solution requires us to find the modu- power output from the discriminator. If the dental frequency modulation squared. lating power in the signal and the incidental phase noise has a power spectral density of, frequency modulation of the local oscillator. If the FM transmission system uses preem- S (f), the output of the discriminator due The signal has a 5-kHz RMS frequency δφ phasis and deemphasis, then the modulator to the phase noise is f 2 S (f). Figure 2 illus- deviation. The square of this is the modulat- δφ and demodulator gain constants change with trates a simplified block diagram of an FM ing power contained in the signal. frequency. Under this condition, the local receiver. The bandpass filter on the output 2 7 2 Ps (5 kHz) = 2.5 × 10 Hz limits the video bandwidth to that required oscillator limited sign al-to-noise ratio to pass the signal. The output signal-to-noise becomes, Since this oscillator exhibits a low phase ratio is the power in the signal divided by the power in the noise. The power in the signal can be found by,

lim 1 T 2 2 BANDPASS P = K K v2 (t) dt, INPUT LIMITER OUTPUT s T→∞ T o d f DISCRIMINATOR FILTER ∫ fa << f << fb where Kf is the modulator gain constant, Kd is the demodulator gain constant, and v(t) is the instantaneous modulating voltage. If we take the single-sided power spectral density LOCAL OSCILLIATOR of v(t), which is Sv(f), then this equation becomes,

2 2 fb Ps = Kd Kf Sv(f) df ∫ fa Figure 2. Block diagram of an FM receiver.

modulation power, the incidental frequency shifted in integer multiples of a minimum predict the exact degradation in the bit error modulation may be found from, phase step. For bi-phase shift keying, the rate for a non-specific case. However, a rule phase shift is integer multiples of 180°; for of thumb can be used to predict the system f β = 2 b f2 £(f) df quad-phase shift keying, it is multiples of performance. The rule states that for bit f f √ ∫ a 90°; and for eight-phase shift keying, it is error rates greater than 10-6, the system per- = 2 3000f2 10-7 df = 42 Hz multiples of 45° Other forms of digital formance can be maintained to within a few √ ∫ 300 transmission are used, but these usually dB of theoretical bit error rates for that The power in the FM demodulator output involve both amplitude and phase-shift key- modulation type if the incidental phase is the square of the incidental frequency ing of the carrier. Local oscillator phase noise modulation of the local oscillator is less than modulation. will effect the bit error rate performance of one-tenth of the minimum phase step of the

2 a phase-shift keyed digital transmission sys- phase-shift keyed carrier. The incidental P = β = 1800 Hz2 N f tem. A transmission error will occur any phase modulation should be computed from The ratio of these two numbers yields the time the local oscillator phase, due to its the natural frequency of the carrier recovery local oscillator limited signal-to-noise ratio. noise, becomes sufficiently large that the phase-lock loop to one-half of the IF band- S = 41 dB digital phase detection makes an incorrect width. For instance, in a QPSK system, the N decision as to the transmission phase. For incidental phase modulation should be less instance, a QPSK transmission system will than 9 degrees RMS to meet this rule. INCIDENTAL PHASE make a transmission error if the instanta- MODULATION neous oscillator phase is offset by more than EXAMPLE The local oscillator phase noise can limit the 45° since the phase detector will determine A receiver has a local oscillator SSB phase signal-to-noise ratio of a phase-modulated that baud to be in the incorrect quadrant. noise given in the table below. What is the signal to which the receiver is tuned. A sim- Digital transmission systems with smaller incidental phase modulation of the local plified block diagram of such a receiver is phase multiples are more sensitive to degra- oscillator integrated from 100 Hz to 1 shown in Figure 3. In this case, the limiting dation due to local oscillator phase noise. MHz? signal-to-noise ratio is determined by the The bit error rate degradation due to local power in the phase modulation divided by f(f) oscillator phase noise can only be deter- the incidental phase modulation squared, 100 Hz -70 dBc/Hz mined if the probability distribution of the 1 kHz -70 dBc/Hz 2 fb local oscillator phase is known. This cannot Kp Sv(f) df S ∫ fa be determined uniquely from the measure- 10 kHz -70 dBc/Hz = 2 N βφ ment of the phase noise without using a 100 kHz -90 dBc/Hz detailed model of the oscillator. 1 MHz -120 dBc/Hz where, K is the phase modulator gain con- p Furthermore, if the oscillator is within a stant. phase-locked loop, the probability distribu- SOLUTION Phase modulation is usually used to transmit tion of the phase will be modified by the If we assume that the integrated power in digital signals rather than analog signals. For parameters of the phase-locked loop. For the phase modulation is much less than 1 digital signals, the phase of the carrier is these reasons, it is not practical to attempt to radian2, then we can evaluate the approxi- mate integral of the SSB phase noise, given in Table 2, to determine the oscillator inci- BANDPASS OUTPUT dental phase modulation. If the approxima- PHASE FILTER SIGNAL DEMODULATOR fa << f << fb tion is true, then the resulting answer will be much less than 1 radian. The numerical evaluation of this integral yields, 1 MHz βφ = 2 £(f) df LOCAL 100 Hz OSCILLIATOR √ ∫ = 0.062 radians

Figure 3. Block diagram of a phase modulation receiver. βφ = 3.6° RMS

The answer is indeed much less than 1 radi- noise. The phase noise on each carrier will The apparent noise floor of the receiver is an, which means that the initial assumption dominate until the noise power is reduced to the sum of these two powers. To compute is true. The resulting answer indicates that the noise floor of the receiver. Otherwise, the this sum, the powers must be converted to this local oscillator could be used in a receiv- noise floor is flat across the IF passband. The absolute power, summed, and then converted ing system for an 8-PSK modulation which central carrier is the strongest, and therefore back to dBm. If it is necessary to compute has a minimum phase shift of 45°. exhibits the strongest phase-noise compo- the apparent noise floor at different frequen- nent. The weaker carrier on the left has a cies, this process can be repeated at the desired RECIPROCAL MIXING smaller range over which its phase noise offset frequencies. The net effect is that the Reciprocal mixing will cause the receiving dominates. The weaker carrier on the right is receiver’s apparent noise floor decreases as system to lose sensitivity when there is a nearly masked by the phase noise from the the receiver is tuned away from the carrier strong signal near the frequency to which strong carrier. If the receiver was tuned to until it reaches the underlying noise floor the receiver is tuned. This effect is due to the this carrier, it would achieve a much worse generated by the receiver’s front end. phase noise of the local oscillator modulating signal-to-noise ratio performance than the carrier of the strong signal. The carrier is would be predicted from the receiver’s noise EXAMPLE spread in frequency by the phase noise mod- figure. This poorer performance is due exclu- A receiver with a 15 dB noise figure is tuned ulation, which results in a power spectral sively to the local oscillator phase noise. to a carrier with a -20 dBm power level. density that is proportional to the local oscil- What is the equivalent receiver noise figure The increase in the noise floor of the receiver lator’s SSB phase noise. When the receiver is 1 MHz from the carrier when £(1 MHz) = can be computed using the following tuned to a frequency near the strong carrier, -120 dBc/Hz? the power density in the strong carrier’s methodology: The receiver noise floor in a noise sidebands may exceed the noise floor one-Hertz bandwidth is the sum of the SOLUTION receiver’s noise figure, F, in dB and -174 of the receiver. If it does exceed the noise The noise flow due to the receiver’s front dBm/Hz, floor, then the receiver sensitivity is limited end is, by reciprocal mixing. Pn = F - 174 (dBm/Hz) Pn = F - 174 = -159 dBm/Hz This effect is illustrated by Figure 4. The The noise generated in the receiver from a At a 1 MHz offset frequency, the noise power receiver is tuned in the frequency range of nearby carrier is the sum of the carrier due to the local oscillator phase noise is, three carriers. The strongest carrier is in the power, Pc, in dBm and the SSB phase noise center, with a weaker carrier on each side. of the local oscillator at an offset frequency Po = Po + £(1 MHz) = -140 dBm/Hz The local oscillator has an SSB phase noise equal to the difference between the carrier which decreases with increasing offset fre- The sum of these two powers at this offset frequency and the frequency to which the quency. The three carriers will appear at the from the carrier is an apparent noise floor of receiver is tuned. IF output, and each carrier will have been -140 dBm/Hz. The equivalent noise figure modulated by the local oscillator phase Po =Pc + £(f) (dBm/Hz) of the receiver is the difference between the apparent noise floor and -174 dBm/Hz,

Feq = 34 dB The receiver noise figure is increased 19 dB RECEIVER when it is tuned 1 MHz away from the -20 dBm carrier. INPUT SIGNALS IF OUTPUT LOCAL OSCILLATOR SPURIOUS SIGNALS Besides exhibiting phase noise, local oscillators may also be phase or amplitude modu-

LOCAL OSCILLATOR lated by discrete frequencies. These oscillator modulations may produce a different effect Figure 4. Reciprocal mixing model. than phase noise on the receiver perfor-

mance, since they are not generated by a tive to a different bandwidth, the amplitude sary to specify the spurious signals to a lower random process. The sources of these dis- of the random phase noise component will level at some frequencies. This is particularly crete frequencies within the receiver are change, whereas the discrete spurious signal true of digital demodulators, which usually numerous. The power line frequency will component will remain constant in amplitude. contain phase-lock loops that can false-lock often modulate the local oscillator. If the to a spurious modulation. Any local oscillator which is generated using local oscillator is generated using frequency a phase-locked loop will always have some Outside the video passband, the spurious synthesis techniques, then reference frequen- spurious signals present in its output. The signals should not degrade the receiver’s spu- cies used in the synthesizer will generate spu- amplitude and frequency of these spurious rious-free dynamic range. This condition is rious signals. Other oscillators and digital modulations may vary as the local oscillator guaranteed if the spurious signals at offset frequency dividers in the receiver can gener- is tuned. Poor layout of the phase-locked frequencies greater than the narrowest IF ate frequencies which modulate the local loop oscillator circuitry may increase the bandwidth are further below the carrier than oscillator. If the receiver uses a switching amplitude and number of these spurious sig- the spurious-free dynamic range specifica- power supply, the switching frequency may nals. However, even under ideal conditions tion. A receiver specified this way will not modulate the local oscillator. While there are some of the spurious signals will always be have any spurious responses from the local other potential sources of modulation, those present. It is therefore necessary to define an oscillator, which occur at power levels less mentioned above are the most common ones. acceptable level of oscillator spurious modu- than the input power required to generate Figure 5 shows a plot of SSB phase noise for lations. intermodulation spurious responses. a local oscillator which is generated using an A spurious specification can be broken down However, when the power level is sufficiently indirect frequency synthesizer. The peaks in into two regions of interest: inside the video high, the receiver will have spurious respons- the noise spectrum are generated by the dis- passband and outside the video passband. es that are due to the local oscillator. These crete frequency modulation of the local Any spurious signals present with a modula- spurious responses may be detected as if they oscillator. Visible in this spectrum are spuri- tion rate which is in the video passband of were real signals. ous components due to the 60-Hz line fre- the receiver output must not degrade the A more stringent specification would require quency, the 30-kHz power supply switching incidental frequency modulation or inciden- that the spurious signals not degrade the rec- frequency, and the 250-kHz reference fre- tal phase-modulation performance required iprocal mixing performance of the receiver. quency of the phase-lock loop synthesizer. of the receiver. If this condition is not met, This condition will guarantee that the The spurious signals are given in units of the receiver will not meet its desired local receiver will never detect any of the spurious dBc rather than the dBc/Hz of SSB phase oscillator limited signal-to-noise ratio. With signals as a real signal. This condition will be noise. If the SSB phase noise is plotted rela- some types of demodulators it may be neces- met if no spurious signals can be observed in the SSB phase noise when it is measured with a resolution bandwidth equal to the 0 narrowest IF bandwidth used in the receiver. HUM SWITCHING REFERENCE In effect, the local oscillator SSB phase-noise MODULATION POWER SUPPLY HARMONICS 60 120 180 30 60 90 250 500 1000 power in this bandwidth exceeds the spurious-signal power. A specification of this type -50 can be very difficult to meet.

CONCLUSIONS

(dBc/Hz) The local oscillator phase noise will effect

PHASE NOISE, £(f) -100 SINGLE SIDEBAND the overall performance that can be achieved in a receiving system. Great care should be exercised in determining the desired receiver performance. Once these requirements have -150 been determined, they can be translated into 10 Hz 100 Hz 1 kHz 10 kHz 100 kHz 1MHz OFFSET FREQUENCY, f a required level of local oscillator performance. Conversely, if the local oscillator perfor- Figure 5. Local oscillator spurious signals. mance is already known, it can be translated

into a set of system level performance data. cle. The interested reader is referred to the 4. Dieter Scherer, “Learn About Low-Noise references for more information in this area. Design,” Part I, Microwaves, April 1979. Overly stringent specifications for the local oscillator performance should be avoided. REFERENCES 5. Dieter Scherer, “Learn About Low-Noise Improving the oscillator performance of a Design,” Part II, Microwaves, May 1979. given design is usually very expensive, both 1. “Time and Frequency: Theory and in engineering and production times. Fundamentals,” Byron E. Blair, Editor, 6. Michael C. Fischer, “Analyze Noise Generally, a local oscillator used in an FM NBS Monograph 140, US GPO, 1974. Spectra With Tailored Test Gear,” receiver does not have to be nearly as low in 2. Floyd M. Gardner, “Phase-lock Microwaves, July 1979. phase noise as one used in a PSK receiver. Techniques,” ,John Wiley & Sons, 1979. 7. “Understanding and Measuring Phase Finally, no data regarding the measurement 3. Peyton Z. Peebles, “Communication Noise in the Frequency Domain,” of phase noise has been presented in this arti- System Principles,” Addison-Wesley, 1976. Hewlett-Packard Applications Note 207.

WJ Communications, Inc. ¥ 401 River Oaks Parkway ¥ San Jose, CA 95134-1918 ¥ Phone: 1-800-WJ1-4401 ¥ Fax: 408-577-6620 ¥ e-mail: [email protected] ¥ Web site: www.wj.com