Little Known Characteristics of Phase Noise

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AN-741 APPLICATION NOTE
Little Known Characteristics of Phase Noise
by Paul Smith
INTRODUCTION There is a wealth of information available on the topic of phase noise, its characteristics1, how it can be measured2, and how it affects system performance3. It is well known that phase noise in oscillators and clocks becomes one of the limiting degradations in modern radio systems. However, most of the traditional analyses concentrate on degradations to sine wave signals in single carrier radio systems. The effects of phase noise on multicarrier receivers, wideband systems, or digital radios are very rarely discussed. This application note will address some of the rarely discussed issues related to phase noise in sampled data systems. It will focus primarily on multicarrier radios, wideband signals, and undersampled radio architectures. PHASE JITTER IN SAMPLED DATA SYSTEMS The easiest way to calculate the SNR degradations incurred by phase noise in a sampled data system is to convert phase noise to phase jitter. This is most easily accomplished by recognizing that a time delay is the same as a phase delay at a given frequency. Extending this concept and writing it in terms of noise power yields Equation 1.
In addition, using the differentiation theorem of the Fourier transform, which states that the Fourier transform of a derivative is just the Fourier transform of the original function multiplied by iω , as shown below,
1
2 2 4π 2fO σt
(4a)
SNR sig =
, for a single carrier system.
(4b)
Many modern radio systems don’t use narrow-band carriers. Modulated data often occupies a fairly wide spectrum. In order to determine how click jitter effects the SNR for such systems, it is convenient to assume the data has zero mean and a flat spectrum uniformly distributed between fL and fH , fL < fH as shown in Figure 2. When squared and integrated over its bandwidth, the total signal power σ 2 out is obtained.
{ E {ν'
E ν out (t ) = power in ν out (t ) =
2 2 out
} (t )} = power in ν'
A2 2 2 ω2 OA out (t ) = 2
REV. 0
AN-741
Using Equation 3,
SNRsig si
A2 1 1 1 = 2 22 2 = 2 2 σt ω O A σt ω O 2
+∞ 2 ∫ |ν(t )| dt = +∞ 1 +∞ 2 2 ∫ |g (ω )| dω = ∫ |g (f )| df 2π – ∞ –∞
{ E {ν'
E ν out (t )
2 2 out
} A (ω (t )} =
nA 2 = 2
2
where |g (f )|2 is the power spectral density in Watt / Hz .
where σ 2 t is in (rms seconds)
{
}
2
From this it is seen that the noise power is a function of the jitter power and the power in the signal derivative. The SNR of a signal sampled with a jittery clock is defined as
2 2 σ2 θ = ω clk σt , where σ θ = phase noise in rms radians σt = phase jitter in rms seconds ω clk = clock frequency in radians /sec
νOUT
∆ν TIME ∆τ
Figure 1. Traditional sampled data SNR analyses use Figure 1 as an aid to determine how noise on a clock generates an error in the sampled data. From this it is seen that ∆ν( t ) = ∆t × ν'out ( t )
σout fH – fH
FREQ
This is the standard SNR equation for a single sine wave sampled by a clock with jitter and can be found in many publications5. Intuitively what is happening is that higher frequency signals have larger slew rates. This results in larger voltage changes as the sample time changes. It should be remembered that quantization noise and thermal noise must also be added to this to obtain the total noise out of a data converter. Extending this to a multicarrier signal is a simple matter. Using the same procedure as before with out defined as a summation of n equal amplitude sine waves,
SNR sig =
=
=
1 E ν out t σ
2 t 2 out
{ E {ν'
2
( )} (t )}
(3)
For example, in a single sine wave,
SNR clk (dB ) = –10 log
( )
σ2 θ
sin ω Ot
–∞
2 1
2 + ω2 2 + ...ω n
)
2
SNR sig
nA 2 1 = 2 2 2 22 = σ t A ω 1 + ω 2 + ...ω 2 n
ℑ ν' (t ) = i ωℑ ν(t )
[ ]
[ ]
(
)
2 1 n n = 2 2 2 2 2 2 σ t ω 1 + ω 2 + ...ω n 4 π σ t ∑ f i2
power in signal power in noise σ2 out σ
2 err
(1)
That is, for a given jitter error, a higher frequency signal will have more phase error. The term σ θ is the total integrated phase noise of the clock4 and defines the clock SNR by