The goal of this analysis is to find an expression for the narrowband power spectrum
Pc(f)in terms of the baseband PSDP(f)of the random phase modulation. We can
Mathematical Preliminaries 91
f PC(f)
-BT BT
1/4 Pφ (f + BT) 1/4 Pφ (f - BT)
1/4 ( 1 - σφ2) 1/4 ( 1 - σφ2)
Figure 2.39 Narrowband power spectral density of a signal due to random phase modula- tion.
find the ESDSc(f)of this clock signal using the techniques of section 2.5, by first finding the time-autocorrelation function.
ac(;) =
Z NT=2
;NT=2c(t;)c(t+;)dt (2.230)
The expected value ofac(;)is the integral of the ensemble autocorrelation function.
Ac() =E[ac(;)] =Z NT=2
;NT=2Rc(t;)dt (2.231)
The narrow-band spectrum that we are looking for is the ESD ofc(t;), which is the Fourier transform ofAc().
Sc(f) =FfAc()g (2.232)
Explicit Expression forRc(t;) The above outlined analysis can be carried out by first finding an explicit expression for the ensemble autocorrelation functionRc(t;)
of the clock signal. By definition
Rc(t;) =E
[sin(2 BTt+(t;))]rect(t=NT)
[sin(2 BT(t+) +(t+;))]rect((t+)=NT)
(2.233) Now we can make make use of some trigonometric manipulations to separate terms into sum and difference frequencies. Recalling
sinAsinB= 12 cos(A;B);1
2 cos(A+B);
92 Chapter 2
then
Rc(t;) =E
1
2 cos (2 BT+(t+;);(t;))
;
1 2 cos(2(2BT)t+ 2 BT+(t+;) +(t;))
rect(t=NT)rect((t+)=NT);
(2.234)
and recalling also that
cos(A+B) = cosAcosB;sinAsinB;
then the fast varying center-frequency terms can be separated from the random phase- noise terms.
Rc(t;) =
1
2 cos(2 BT)E[cos ((t+;);(t;))]
;
1 2 sin(2 BT)E[sin((t+;);(t;))]
;
1 2 cos(2(2BT)t+ 2 BT)E[cos((t+;) +(t;))]
+ 12 sin(2(2BT)t+ 2 BT)E[sin((t+;) +(t;))]
rect(t=NT)rect((t+)=NT):
(2.235)
With one more application of the previous trig identity and
sin(A+B) = sinAcosB+ cosAsinB;
Mathematical Preliminaries 93
then terms involvingt, can be separated from terms involving, and we finally get the desired form of the auto-correlation function:
Rc(t;) =
1
2 cos(2 BT)E[cos((t+;);(t;))]
;
1 2 sin(2 BT)E[sin((t+;);(t;))]
;
1 2 cos(2(2BT)t)cos(2 BT)E[cos((t+;) +(t;))]
+ 12 sin(2(2BT)t)sin(2 BT)E[cos((t+;) +(t;))]
+ 12 sin(2(2BT)t)cos(2 BT)E[sin((t+;) +(t;))]
+ 12 cos(2(2BT)t)sin(2 BT)E[sin((t+;) +(t;))]
rect(t=NT)rect((t+)=NT):
(2.236) Since we are dealing with a narrow-band signal, then the baseband modulation, by definition, varies much slower than the tone. Therefore, the terms involving
E[cos ((t+;) +(t;))]
E[sin((t+;) +(t;))]
are expectations of a slowly varying signal, and these terms remain essentially constant over several periods of the double-frequency(2BT)signal. Hence, when (2.236) is integrated over time, the last four terms will vanish.
Approximations for Small Angles To continue the analysis it is helpful at this point to make some approximations assuming that the phase modulation is small. This is a valid assumption, because any clock signal that has a large cycle-to-cycle phase jitter
(t;)is of no use to us, so there is no need to analyze it. Instead we will be considering a clock signal with small phase deviations. Recalling the series expansions for sine and cosine around zero
sin=;3!3 +5!5 ; cos= 1;2!2 +4!4 ;;
and ignoring any terms of3rdorder or greater, then
E[sin((t+;);(t;))]'E[(t+;)];E[(t;)]; (2.237)
94 Chapter 2
and for a zero-mean phase-noise process
E[(t+;)];E[(t;)] = 0;0 = 0: (2.238)
Using the small-angle approximation for the cosine function
E[cos((t+;);(t;))]'1;1 2
h
E[2(t+;)] + E[2(;)]
;2E[(t+;)(t;)]i:
(2.239) If the base-band phase-noise process is assumed to by wide-sense stationary with a variance of2and an auto-correlation functionR(), then
E[cos((t+;);(t;))]'1;2+R(): (2.240)
Energy Spectral Density Now the expected value of the time-autocorrelation function
Ac() can be found by integrating Rc(t;). Define an effective ensemble auto- correlation functionRcc()as,
c
Rc() = 12 cos(24 BT)1;2+R(): (2.241)
Integrating the horrendous expression in (2.236) reduces to
Ac() =Z NT=2
;NT=2Rc(t;)dt
=Rcc()Z NT=2
;NT=2rect(t=NT)rect((t+)=NT)dt
=Rcc()[rect(=NT)rect(=NT)]
(2.242)
Taking the Fourier transform ofAc()will finally give us the energy spectral density function of the narrowband signal in terms of the baseband ESD. In anticipation of the final result we’ll define a power spectral densityPc(f)as the Fourier transform of
c
Rc().
Pc(f)4=FnRcc()o (2.243a)
Pc(f) = 14(1;2)(fBT) + 14P(f BT); (2.243b)
Mathematical Preliminaries 95
whereP(f)is the PSD of the baseband phase noise. Therefore,
Sc(f) =FnRcc()oFfrect(=NT)rect(=NT)g (2.244a)
Sc(f) =Pc(f)[NTsinc(fNT)]2; (2.244b)
and using the now familiar approximation of the sinc2 function with an impulse of equal area we obtain
Sc(f) =Pc(f)NT(f) (2.244c)
Sc(f) =NTPc(f); (2.244d)
dividing the ESD by the time intervalNT the power spectral density of the phase- modulated signal is as anticipated
Pc(f) = 14(1;2)(fBT) + 14P(f BT): (2.245)
Determining the Phase Noise Variance fromPc(f) It was assumed that the time domain signal corresponding to the energy spectrum was a constant amplitude tone with small-signal phase modulation. The expression for the narrowband PSD was expressed in terms of the PSD of the baseband phase-noise as in (2.245). The result is that the PSD consists of a pure tone plus the baseband noise PSD shifted toBT. This is illustrated in Fig. 2.39. The phase-noise variance can be found by taking the ratio of the tone power and the sideband power. From (2.245) the tone power is
Ptone= 14Z;11(1;2)(fBT)df = 12(1;2); (2.246)
and the sideband power is
Psb= 14Z;11 P(fBT)df (2.247a)
= 12Z;11 P(f)df (2.247b)
= 22: (2.247c)
Therefore the ratio of the two powers is simply related to the phase-noise variance by
Psb
Ptone = 1;22: (2.248)
96 Chapter 2
Alternatively, we can express the noise variance in terms of the power ratio
2= 1 +PsbPsb=P=Ptonetone
: (2.249)
Returning to the example of edge-detected NRZ data passing through an ideal bandpass filter of selectivityQ, the above power ratio, which is the same as the energy ratio, is just equal to1=Q. The noise variance of the recovered clock signal is therefore,
2= 1 1 +Q: (2.250)
For an arbitrary filter we use the equivalent selectivity, so that the general result is
= 1
p1 +Qeq radians (2.251)
This result, however, assumes that all of the sideband energy is converted to phase- noise, and there is no contribution to the envelope deviation. Therefore, (2.251) gives an upper-bound on the phase-noise obtained by filtering random edge-detected data.
In the following section we will show how a nonlinear phase filter distributes the noise power between amplitude and phase modulation. Before, ending this section, however, we will give some simulation results that verify the theory.