AD9548SLASHPCBZ datasheet(107/112 Pages) AD

107 / 112 page

AD9548

Rev. 0 | Page 107 of 112

CALCULATING DIGITAL FILTER COEFFICIENTS

The digital loop filter coefficients (α, β, γ, and δ (see Figure 40))

relate to the time constants (T1, T2, and T3) associated with the

equivalent analog circuit for a third order loop filter (Figure 66).

FROM

CHARGE

PUMP

VCO

Figure 66. Third Order Analog Loop Filter

The design process begins by deciding on two design parameters

related to the second order loop filter shown in Figure 67: the

desired open-loop bandwidth (fP) and phase margin (θ).

FROM

CHARGE

PUMP

VCO

Figure 67. Second Order Analog Loop Filter

An analysis of the second order loop filter leads to its primary

time constant, T1. It can be shown that T1 is expressible in terms of

fP and θ as

)

cos(

)

sin(

−

where

An analysis of the third order loop filter leads to the definition of

another time constant, T3. It can be shown that T3 is expressible in

terms of the desired amount of additional attenuation introduced

by R3 and C3 at some specified frequency offset (fOFFSET) from the

PLL output frequency.

OFFSET

ATTEN

10 10

−

where

OFFSET

Note that ATTEN is the desired excess attenuation in decibels.

Furthermore, ATTEN and ωOFFSET should be chosen so that

3 ≤

With an expression for T1 and T3, it is possible to define an

adjusted open-loop bandwidth (fC) that is slightly less than fP. It

can be shown that ωC (fC expressed as a radian frequency) is

expressible in terms of T1, T3, and θ (phase margin) as

()

[]

⎥

⎦

⎤

⎢

⎣

⎡

−

)

tan(

)

tan(

It can also be shown that the adjusted open-loop bandwidth leads

to T2 (the secondary time constant of the second order loop filter)

expressed as

()

Calculation of the digital loop filter coefficients requires a scaling

constant, K (related to the system clock frequency, fS), and the PLL

feedback divide ratio, D.

125

578

517

where S, U, and V are the integer and fractional feedback divider

values that reside in the profile registers. Keep in mind that the

desired integer feedback divide ratio is one more than the stored

value of S (hence, the +1 term in the equation for D in this

equation). This leads to the digital filter coefficients given by

()

(

) ()

()

()2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

S T

−

S T

Calculation of the coefficient register values requires the

application of some special functions described as follows:

The if() function

y = if(test_statement, true_value, false_value)

where test_statement is a conditional expression (for example, x <

3), true_value is what y equals if the conditional expression is true,

and false_value is what y equals if the conditional expression is

false.

The round() function

y = round(x)

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