AN279 Silicon_Laboratories, AN279 Datasheet - Page 5

• Manufacturer Part Number: AN279
• Description: Estimating Period Jitter FROM Phase Noise
• Manufacturer: Silicon_Laboratories

A
The following calculations are based on the zero crossings derivation of Drakhlis (2001).
Let Δt represent the jitter accumulated in one period.
Where: T
We can drop the T
Per Parseval’s theorem:
Further:
The phase noise autocorrelation equals the cosine transform of the phase noise.
Δ
Since ϕ t ( ) is stationary:
<
<
Where R
Δt
<
<
<
Δ
Δ
<Δ
〈
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
Δt
ϕ
ϕ
( ) x
2
RMS
( ) x
P P E N D I X
( )
( ) x
t ( )
2
RMS
=
2
RMS
ϕ
t
t
2
t
t
1
1
1
1
〉
2
------
2
2
T
>
2
=
>
π
0
=
>
=
=
(
t
t
=
1
2
=
ϕ
ϕ
ϕ
⎛
⎝
ϕ
0
=
ϕ τ
4 S
------
2
2 S
2 S
T
( )>
[
( )>
( )
= first zero crossing.
= second zero crossing.
( )>
= nominal period.
( ) is the autocorrelation of Φ f ( ), and
∞
∫
0
t
<
∞
t
t
0
π
<
∫
0
t
∞
∞
∫
0
∫
0
2
2
1
2
⎞
⎠
S
ϕ
ϕ
ϕ
2
ϕ
( )
ϕ
( )
ϕ
–
〈
t
t2
f ( ) x
f ( ) 1
f ( ) x 2
f ( )
1
(
ϕ
=
=
=
ϕ
( )
2
2
(
( )
t
R
R
∞
> 2 <
∫
0
2
t
0
>
1
ϕ
ϕ
S
/2π factor for simplicity and add it back in later.
sin
–
–
)
=
(
( )
ϕ
[
t
τ
sin
–
cos
2
A —D
f ( )
<
sec
2
ϕ
–
ϕ
(
2
( )
cos
π
ϕ
t
t ( )
(
t
1
(
2
2
f
onds
( )> x <
π
)
τ
t
π
2
1
f
)df
)
(
>
τ
2
f
2
τ
)df
〉
π
)
)df
f
]
τ
)df
E R I V A T I O N
ϕ
( )>
t
2
+
<
ϕ
( )
t
2
2
τ
>
]
=
[
radians
T
Rev. 0.1
0
2
]
AN279
5