LINEAR MODEL

OF DIVIDER

Fig. 9. Formation of A.

OUTPUT IS 6(t)

-~nen the filter in Fig.

8

is a single-pole low-pass network with time constant 1/b, the mean square error can be obtained from Eq. 22 and is:

•

(31)

Then, for this case, A is obtained from Eq. 29 and is:

A= • (32)

But, from Eq. 30,

N

b

4jJ2

_•

(33)

Then

b + G₀

(34)

= ^•

.1'2b

Eq. 34 can be used in Eq. 24, 25 and 27 to obtain the desired statistical information for the PLL equivalent model

of the divider.

In order to demonstrate the effect of b, G₀ and

;?

^on

the performance of the divider, the graphs of Fig. 10, 11, 12, 13 and 14 are presented. Since the PLL represents the upper bound on

a:

and the linear model represents the lower bound, i t is necessary to include both of these curves for the data in the indicated figures. Measured data points for a divider connected as a divide by two are presented in Fig.

13 and 14. Similar performance would be expected for higher division ratios. However, the FLL model should provide a better representation of the divider for this condition

since the ratio of the divider bandwidth to loop filter band-width will be greater.

B. Nondeterministic Signals

1. Carrier plus narrow-band noise.

This signal type corresponds to the signal discussed in Section A-2 and defined by Eq. 23 and the paragraph follow-ing this equation when the bandwidth of the noise is much

greater than the loop bandwidth. Fig. 15 gives the method for generation of the r(t) under consideration in this part.

The discussion in Chapter IV, Section B, provides the neces-sary information to obtain the required statistical informa-tion for the wide-band noise case.

OdB

r=:..._::.=

--ldli

-2dB.

8 a

4 2 1 !_ 1

a a a 2a 4a

Pre-filter Time Constant ~

Fig. 10. Mean square error for Gaussian modulation vs.

pre-filter time constant. n=2,

p

^=450.

OdB

-ldB

-2dB

-3dB~~---~---._----~---_.---~---~---.-l.§. § ~ 2 1 1 1

a a a a a 2a 4a

Pre-filter Time Constant -~

Fig. 11. Mean square error for Gaussian modulation vs.

pre-filter time constant. n=2, ;:; =900.

OdB

O. OdB

..--~

Linear

-O.SdB

l-g2

t

---~==~==--~---~ter ~

-l.OdB

-1.5dB

-2.0dB

-2.5dB

~

Pre-filter = ~a

- 1

--

- 2 _a

-3.0dB

~---_.--~~--._---~----~--~---45° 90° 135° 180°

!>';odula tion Index ~

Fig. 14. Mean square error vs. modulation index for different pre-filter time constants. n=2. a corresponds to measured data points.

PO\'vER

DENSITY

2 V2P'

^{sin w}0 t

H( jw)

BAND CENTER

IS

Fig.

15.

Generation or carrier plus narrow-band noise.

r t

vllien the bandwidth or H(jw) is reduced to a range near the divider rilter bandwidth, the envelope amplitude varia-tion must be considered. This case is treated in Chapter IV, Section

c.

Parameters or interest concerning the output of the divider are:

(a) mean square error,

(b) cycle skipping characteristics,

The above parameters are computed usin£ the results of

Chapter IV and Eq. 24, 25 and 27 from the PLL model for the wide-band noise case.

When the envelope amplitude variation must be considered, two techniques are possible for obtaining the desired results.

The first of these techniques determines the cycle skipping

characteristics based on the amplitude variation o~ the out-put envelope, D(t), &s discussed in Chapter IV. This method provides good results when the bandwidth o~ H(Jw) is much less than the bandwidth o~ the divider loop ~ilter since this condition implies that D(t) and R(t) have nearly the same

amplitude variations. For this condition, the probability density ~unction o~ a sine wave plus narrow-band Gaussian noise is required in order to evaluate

Eq. 4-36.

Hence,20,21

M ( 2RA) (

^R2A)

p(r) =../2P Io p exp -A- 2P •

(35)

where A is the signal-to-noise ratio as de~ined in Eq. 25;

except, in this case, the noise bandwidth o~ the ~ilter given in Fig. 15 must be considered. P is the signal power and R is the envelope amplitude o~ r(t).

In order to obtain the cycle skipping rate ~rom Eq.

35,

a threshold ~or D(t) must be established. The exact value or this threshold point is not of extreme importance when A>> 1 since the slope of the function de~ined in Eq.

35

is not very great ~or small

R;

i.e., A is the dominating parameter.

\'J"hen the bandwidth of H ( jw) is of the same order of mag-nitude as the loop ~ilter bandwidth, the correlation between R(t) and D(t) must be considered. Then from Eq. 4-36 and the above consideration, the probability of cycle skipping is:

p ^S ^:

1\ ..

^L ⁽^{1 -} ^{(RDDn (}

J.))

⁽³⁶⁾

and

(37)

but

(38)

where _(RDR is the correlation between R(t) and D(t) and

fin

^is

the value of R(t) that would make D(t) = Dn if R(t) remained at ~ :for a long period o:f time. Then from Eq.

35, 36, 37

and

38

and the assumption that rR00, (A) =

1/2,.

I ^Rn

P ₈ =

t

^lR,,.

p (R ) dR (39)

where

p(R)

is defined in Eq.

35.

For most conditions, the threshold,

Rn,

will be such that

(40)

and A> 1. With this assumption, Eq.

35

can be simplified and used in Eq. 39 to obtain:

-A

n -.. ~ .fO RnA

e

•a ^~^1a '-"'oo

".~

v 2P

~nen the bandwidth of the signal pre-filter, H(jw),

(41)

is approximately the same or greater than the divider band-width, Eq. 41 will not provide accurate results. This is because fast variations in ~(t) must be considered and the divider will not be able to follow all of these fast varia-tions in phase, thus resulting in cycle skipping due to phase variations.

For wide bandwidths of H(jw) such that Eq. 41 is no longer valid, the cycle skipping characteristics of the PLL are applicable as an upper bound.

The curve of Fig.

16

is a graph of the cycle skipping rate vs. ratio of pre-filter bandwidth to divider bandwidth for different input-noise power densities. This graph demon-strates the relationship between equations for determining the cycle skippinG behavior.

The probability density function for 6(t) and the mean square error for the wide-band case is readily obtained from the PLL model and the linear model and is given by Eq. 22, 24 and

26.

For the medium bandwidth case, the ~~ to

PM

con-version factor must be considered. This is given by

Eq.

4-28.

The term involving the ratio

D/R

is obtained from

Eq.

4-27 and results in

R(A-B) ⁼

^A+B^A-B.

I

^e

^-a(t-A)

-oo

(42)

t

^Cycle

where A-B is assumed to be constant.

Since the input to the pre-filter is a sine wave plus white noise, an approximation to the power ratio required for evaluation of Eq. 4-28 can be obtained by considering the difference in noise power between R and D. This assumption requires that the signal-to-noise ratio of r(t) is large.

Then,

(43)

where ~1 is the noise bandwidth of the filter H1(jw) and ~2 is the noise bandwidth of the composite filter defined by Hl(jw) and the divider loop filter. This approximation is obtained by obtaining a series expansion for

~f~~

and taking only the first two terms of the expansion.

Then using Eq. 43 in Eq. 4-28, one can obtain the

approximate mean square error for the medium bandwidth case.

The curves of Fig. 17 give the Eeasured and computed mean square error for the divider with the ratio of pre-filter noise bandwidth to divider noise bandwidth as the independent variable.

An exact mathematical representation for the probability density function, p(6), for the medium bandwidth case is not possible since the assumption made in the analysis was that

the variations in phase were such that no nonlinear effect on

E[~2J

In document Follow this and additional works at: (Page 88-100)

8