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Frequency Domain Analysis of Jitter Amplification in
Clock Channels
Fangyi Rao Sammy Hindi
Agilent Technologies, Inc. Juniper Networks
Santa Clara, CA Sunnyvale, CA
[email protected] [email protected]
Abstract--Clock channel jitter amplification factor in terms of where 0 is the fundamental frequency of the clock signal, 0 a
transfer function or S-parameters is derived. Amplification is constant phase offset, and the phase modulation that
shown to arise from smaller attenuation in jitter lower sideband represents jitter. When is small Eq. 1 can be linearized.
than in the fundamental. Amplification scaling with loss is
obtained analytically.
A
v in (t ) [exp( j 0 t j 0 ) j (t ) exp( j 0 t j 0 ) (2)
Index Terms--jitter, amplification, loss. 2
exp( j 0 t j 0 ) j (t ) exp( j 0 t j 0 )]
I. INTRODUCTION
High speed interconnect performance is increasingly Consider a sinusoidal jitter at frequency .
influenced by jitter as data rate advances. The amount of jitter
is modulated by channel dispersion as signals propagate in the (t ) ( ) exp( j t ) ( ) * exp( j t ) (3)
system. It is observed in both measurements and simulations
that jitter can be amplified by a lossy channel even when the Substitution of Eq. 3 into Eq. 2 yields
channel is linear, passive and noiseless [1]-[4]. The mechanism
A
of jitter amplification is discussed in terms of channel v in {exp( j 0 t j 0 )
impulse/step response in [2]-[4]. In particular, duty-cycle- 2
distortion (DCD) and random jitter (RJ) amplifications in clock j ( ) exp[ j ( 0 )t j 0 ]
signals are shown to scale uniquely with channel loss [2], (4)
indicating that loss is responsible for the effect. j ( ) * exp[ j ( 0 )t j 0 ]
In this paper jitter amplification in clock channels is exp( j 0 t j 0 )
analyzed analytically in frequency domain. The advantage of j ( ) exp[ j ( 0 )t j 0 ]
using clock signals is that the periodicity of the 1010 clock
pattern eliminates the inter-symbol-interference (ISI) jitter so j ( ) * exp[ j ( 0 )t j 0 ]}
jitter at the channel output is entirely induced by input jitter. A
phase modulation (PM) approach is employed to derive the Assume the signal transfer function of the channel is H(). The
jitter transfer function and amplification factors in terms of output signal, vout, is given by
signal transfer function or channel S-parameters for sinusoidal
A
jitter (SJ), DCD and RJ. Results show that jitter amplification is vout (t ) {H (0 ) exp( j0t j 0 )
the consequence of smaller attenuation in the jitter lower 2
sideband (LSB) than in the fundamental. The scaling of DCD jH ( 0 ) ( ) exp[ j ( 0 )t j 0 ]
and RJ amplifications with channel loss is explained by using
jH ( 0 ) ( )* exp[ j ( 0 )t j 0 ]
an approximate loss model. It is shown that jitter is amplified
by lossy channels at any frequency below Nyquist and the H (0 ) exp( j0t j 0 ) (5)
effect grows exponentially with jitter frequency and data rate. jH ( 0 ) ( ) exp[ j ( 0 )t j 0 ]
The theory is verified by simulation results.
jH ( 0 ) ( )* exp[ j ( 0 )t j 0 ]}
II. JITTER TRANSFER FUNCTION AND AMPLIFICATION A
In lossy channels high order harmonics are heavily H (0 ) exp( j0t j 0 )[1 j (t )]
2
attenuated and the 1010 clock pattern can be approximated by a
A
sinusoidal wave with frequency at one half of the data rate. H (0 ) exp( j0t j 0 )[1 j (t )]
Jitter in the input clock signal, vin, can be represented by phase 2
modulation as
where + and - are defined as
vin (t ) A cos[ 0 t 0 (t )] (1)