Conclusions

In practice, component mismatches could result in errors in the calibration technique. With mismatches in the component values of approximately 5%, the phase noise at the output of the PLL remains well behaved as shown in Figure 11, demon-strating the robustness of the calibration technique. It can be shown that component mismatches simply result in a replica of the CP and DAC charges appearing at the in-tegrator input. Since these charges have zero dc content due to the PLL loop, this re-sults in noise on the calibration signal, not misalignment. This demonstrates the vi-ability of this calibration technique for phase-noise canceling delta-sigma fractional-N PLL.

A

PPENDIX

A

Consider the continuous-time calibration loop model shown in Figure 8b. The impulse response, g(t), from the input of the filter to w(t) can be derived as

{

^{/ 1}

}

( ) ^m ( ) 1 ^{t RC}

int DAC

g t g u t e

C I

= ⋅ − − . (19)

The worst-case variation of (19) during a DAC pulse event occurs when the DAC gain is zero. Using (6) and (19), the deviation of w(t) over a duration of T_DAC during the n^th DAC and CP event can be expressed as

{

^{( ) / 1 / 1}

}

0

( ) ( _DAC) ^{m n} _CP[ ] ^{kT TDAC RC kT RC}

int DAC k

w nT w nT T g Q n e e

C I

− + −

=

− + =

∑

− ⁽²⁰⁾

Recalling from (3) that |QCP[n]| ≤ ICPTVCO, the right side of (20) can be bounded by

{

¹

}

¹

{

¹

}

1 ¹

1

1

( 1) /

/ / /

/ 0

/ /

1 1 1

1 1

1

DAC DAC

DAC

n T RC

T RC n kT RC T RC

CP VCO m CP VCO m

T RC

int DAC k int DAC

T RC

CP VCO m

T RC int DAC

I T g I T g e

e e e

C I C I e

I T g e

C I e

− − − − +

= −

−

−

− = − −

−

< −

−

∑

(21)

In order to ensure that w(t) does not change over the duration of a DAC pulse, it is necessary to keep (21) small. This leads to

1

1

/ /

1 1

TDAC RC CP VCO m

T RC int DAC

I T g e

C I e

−

−

− < ∆

− (22)

where ∆ is the specified level of gain calibration required for w(t).

A

PPENDIX

B

This appendix provides the claims necessary to prove the convergence of the calibration loop. Consider the signal-processing model shown in Figure 8c. With c[n]

given by (5), Claim 1 proves that the charge supplied by the CP and DAC are ergodic, i.e. C0 exists, and Claim 2 provides an intermediate claim to prove the stability of the calibration loop.

Definition: E_quant[n] is an integer-valued sequence representing the quantizer noise from the ∆Σ modulator such that

[ ] [ ] 2

quant

quant N

E n

e n
,

where 2^N is the quantizer step size of the ∆Σ modulator. This definition is provided to ensure congruency with the theorems outlined in [14].

Claim 1: Suppose that the correlation signals given by (5) are generated in conjunc-tion with a 2^nd order ∆Σ modulator designed to satisfy the conditions of Theorem 3 in [14] with a quantizer step size of 2^N. Then C₀ is given by

2 0

2 1

2

N

CP VCO N

C =I ⋅T ⋅ ⁻ − .

Proof: Consider the case where c[n]=sgn(e_quant[n−1])= sgn(E_quant[n−1]). Then from (3),

( )

( )

0 [ ] [ ] [ ] [ 1] sgn [ 1] [ 2]

2

CP VCO

CP CP N quant quant quant

C +C n =Q n c n⋅ = I T ⋅ E n− − E n− ⋅E n− .(23)

From (14), C₀ is defined as the sample average of (23). First consider the sam-ple average of |Equant[n]|. Theorem 1 in [14] proves that Equant[n] asymptotically ap-proaches a uniform random distribution as n→∞ given by

1 1

( ) 2 , ^N 2^N 1 2^N

P uU = ⁻ − ⁻ + ≤ ≤u ⁻ . (24)

And therefore,

lim E _quant[ ] 2^N 2

n E n E u ⁻

→∞  =   = . (25)

For any postive number m, define

[ ] 2^N 2

n quant

X = E n m+ − ⁻ , (26)

and Lemma 3 from [14] shows that E[X_n]→0 is sufficient to prove that the sample av-erage of |Equant[n]| converges in probability to 2^N-2, or in other words

1 2

1 2

lim [ ]

2

n m N

quant N

n k m

e k

n

+ − −

→∞ =

∑

= ^{. (27)}

Now consider the second term in (23). Theorem 2 in [14] also proves that the joint pmf of any two samples of the quantizer error converges in distribution to a jointly uniform random variable given by

2 1 1

, ( , ) 2 ^N, 2^N 1 , 2^N

PU V u v = ⁻ − ⁻ + ≤u v≤ ⁻ , (28)

and therefore

( ) ^{[ ]}

lim E sgn _quant[ ] _quant[ 1] E sgn( ) 1

n E n E n u v

→∞  ⋅ − = ⋅ = . (29)

For any positive number m, let

( )

sgn [ ] [ 1] 1

n quant quant

X = E n m E+ n m+ − − , (30)

and Lemma 3 from [14] shows that E[Xn]→0 is sufficient to prove that the sample av-erage of sgn(E_quant[n])⋅Equant[n−1] converges in probability to 1. Therefore due to the linearity of the sample mean operator,

( )

{ }

²

1 1 2 1

lim [ ] sgn [ ] [ 1]

2

n m N

quant quant quant N

n k m

e k e k e k

n

+ − −

→∞ =

− ⋅ − = −

∑

^{, (31)}

which proves Claim 1 for c[n]=sgn(equant[n]), and the convergence is irrespective of the initial start index, m. Now consider the second correlation signal given by

( ) ( )

sgn [ ] sgn [ 1]

[ ] [ ]

2

quant quant

e n e n

c n − − s n

= + ,

where s[n] is by definition a zero-mean ergodic sequence. For this correlation signal, (23) can be expressed as

( )

{

( ) ( ) }

0

[ ] 1 [ ] [ 1] sgn [ ] [ 1]

2 2

sgn [ 1] [ ] 2 [ ] [ ] [ 1]

CP VCO

CP N quant quant quant quant

quant quant quant quant

I T

C C n E n E n E n E n

E n E n s n E n E n

+ = ⋅ ⋅ + − − ⋅ −

− − ⋅ + ⋅ ⋅ − −

. (32)

The first 4 terms have already been shown to converge to (31), and since s[n] is a zero-mean ergodic sequence uncorrelated from the quantization error, C0 is given by

2 0

2 1

2

N

CP VCO N

C =I ⋅T ⋅ ⁻ −

Therefore Claim 1 is proven for both correlation signals.

For the following stability claim, consider a system of non-linear difference equations defined by

1 2

[ ] [ ] [ ] [ ]

x n =u n −w n u n⋅ (33)

[ ] [ ] [ ]

w n =x n h n∗ (34)

where u₁[n] and u₂[n] are bounded inputs, and h[n] is a filter with the following char-acteristics

( 1)

[ ] 0 for 0, [1] 0, and [ ] [ 1] ⁿ

h n = n≤ h > h n −h n− =Ke^{− − α} (35)

where K and α are positive constants. Without loss of generality, the system is as-sumed to start at n = 0 such that u₁[n] = u₂[n] = 0 for n < 0.

Claim 2: Suppose that

[1] 2[ ] 1

h ⋅u n < (36)

Then the system of equations defined by (33) and (34) are BIBO stable if for any posi-tive integer m,

1 2

lim 1 ^{N m} [ ] 0

N n m

u n U N

+ −

→∞ =

= >

∑

⁽³⁷⁾

Proof: First, consider the product given by

(

²

)

1

1 [1] [ ]

n

l k

h u l

= +

∏

− ⁽³⁸⁾

It follows from (36) and the fact that |1 − x| ≤ e^−x that

(

2

)

^{[1] 12[ ]}

1

1 [1] [ ]

n

l k

n h u l

l k

h u l e^{− = +}

= +

− ≤ ∑

∏

⁽³⁹⁾

(37) implies that for any index m, and positive number ε, there exists an N_ε,m such that for all N > N_ε,m,

( )

1 2[ ]

N m

n m

u n U Nε

+ −

=

− <

∑

⁽⁴⁰⁾

For all m and ε, N_ε,m is finite. Therefore, for some ε < U, define

{ }

^, 0

' max _m

N N _ε m^∞

= = , (41)

and for n − k > N’, the right side of (39) is bounded by

( ) 1( ² ) ( )( )

[1] [ ]

[1] [1]

n

l k

h u l U

h n k U h n k U

e e ^{= +} e ^ε

 

 

−  − 

− − ∑  − − −

≤ (42)

Therefore, there exists positive numbers D, and β such that for all n and k,

(

2

)

^{( )}

1

1 [1] [ ]

n n k

l k

h u l De^{− − β}

= +

− ≤

∏

^{, (43)}

and therefore (38) is bounded by an exponentially decreasing function. The non-linear difference equations given by (33), (34) and (35) can be expressed as

( )

( ) ( )

( )

1 2

0

2

1 1 2

0 2

2 0

[ ] [ ] [ ] [ ] [ ]

[1] [ 1] [ ] [ 1 ] [ ] [ 1] 1 [1] [ 1]

[ ] [ 1 ] [ ] [ ].

n

k

n

k n

k

w n h n k u k w k u k

h u n h n k h n k u k w n h u n

h n k h n k w k u k

=

−

=

−

=

= − −

= − + − − − − + − − −

− − − − −

∑

∑

∑

(44)

Recursively substituting (44) into itself to eliminate w[k], k = 0, …, n − 1 yields

( )

( ) ( ) ( )

1 2

1 1

0 0

1

2 2 2

2 1

[ ] [1] [ 1] [ ] [ 1] [ ]

1 [1] [ ] [ ] [ ] [ 1] 1 [1] [ ] .

n n k

k l

k k k

m

m o m

w n h u n k h n l h n l u l

h u n m u n k h m h m h u n k o

− − −

= =

−

=

= =

 

=  − − + − − − − 

 

 

⋅ − − − − − − ⋅ − − + 

 

∑ ∑

∏ ∑ ∏

⁽⁴⁵⁾

The above equation consists of four separate terms. For a bounded input, there exists a B such that u1[n] and u2[n] are bounded by B in magnitude. From (43), the first term can be bounded by

( )

1 1

1 2

0 1 0

[1] [ 1] 1 [1] [ ] [1]

1 [1]

[1] 1 1

k

n n

k

k m k

n

h u n k h u n m h B De

e h B D h B D

e e

β

β

β β

− −

−

= = =

−

− −

− − − − <

− ⋅

= ⋅ <

− −

∑ ∏ ∑

(46)

Using (35) the second term can be expressed and bounded by

( ) ( )

( )

1 2

1 2

0 0 1

1 2 1

1 ( 1)

0 0 0 0

[ ] [ 1] [ ] ^k 1 [1] [ ]

n n k

k l m

n n k n n k

n l k k k l

k l k l

h n l h n l u l h u n m

B Ke ^α De ^β KBD e ^β e ^α e ^α

− − −

= = =

− − − − −

− − − − − − + −

= = = =

− − − − − −

≤ ⋅ = ⋅

∑ ∑ ∏

∑ ∑ ∑ ∑

. (47)

The expression on the right side of (47) can be further reduced resulting in

( )( )

( 1) ( )

1 ( 1)

( )

0

( )

1 1

1 1 1

1 1

n k n

n k k

k

e e e

B De Ke KBD

e e e

KBDe

e e

α α α β

β α

α α α β

α

α α β

− − + − − +

− − − +

− − − +

=

−

− − +

− −

⋅ ≤

− − −

≤ − −

∑

(48)

The third term from (45) can be expressed and bounded by

( )

¹

( )

1

1 2 2

0 2

2 1 ( 1) ( )

0 2

[1] [ 1] [ ] [ ] [ 1] 1 [1] [ ]

[1]

k

n k

k m o m

n k

m k m

k m

h u n k u n k h m h m h u n k o

h B Ke ^α De ^β

− −

= = =

− − − − −

= =

− − ⋅ − − − ⋅ − − +

≤ ⋅

∑ ∑ ∏

∑∑

(49)

The expression on the right side of (49) can be bounded by

( 1)( )

1 1

2 ( 1) 2 2( )

( )

0 2 0

2 2( ) ( )

( )

( )

2 2( )

( )

[1] [1] 1

1

1 1 1

[1] 1 1 1

1 1

[1] 1 1 1

n k n k

k m m k

k m k

n n

h B KD e e e h B KDe e e e

e

e e

h B KDe e e

e e e

h B KDe e e

e e e

β α β α α β β α β

α β

β α

α α β α β

α β β α

α α β α β

α β β α

− − −

− −

− − − − − −

= = = − −

− −

− − −

− − − −

− − −

− − − −

⋅ = −

−

 − − 

= −  − − − 

≤ −

− − −

∑ ∑ ∑

 

 

 

(50)

Finally, the fourth term in (45) can be expressed and bounded by

( )

( ) ( )

1 2

1 2

0 0

1

2 2

1 2

2 ( 1) ( 1) ( )

0 0 2

1 2

2 ( 1) ( )

0 0 2

[ ] [ 1] [ ] [ ]

[ ] [ 1] 1 [1] [ ]

n n k

k l

k k

m o m

n n k k

n l m k m

k l m

n n k k

k k l m

k l m

h n l h n l u l u n k

h m h m h u n k o

B e Ke De

KDB e e e e e

α α β

α β α α β α

− − −

= =

−

= =

− − −

− − − − − − −

= = =

− − −

− − + − −

= = =

− − − − ⋅ −

⋅ − − ⋅ − − +

≤ ⋅

= ⋅

∑ ∑

∑ ∏

∑ ∑ ∑

∑ ∑ ∑

(51)

Solving the right side of (51) yields

( ) ( )

( )( ) ( )

( )( )

( 1) 2( ) ( 1)( )

2 1 ( 1)

( )

0

2 2( ) 1

( ) ( 1)( )

( )

0

2 2( ) 1 1

( ) ( )

( )

0 0

1 1

1 1

1 1 1

1 1

n k k

n k k

k

n k k

k

n n

k k

k k

e e e

KDB e e e

e e

KDB e

e e

e e

KDB e

e e e

e e

α β α β α

α β α

α β α

β α β α β α

α β α

β α β α β α β α β α

α β α

− − − − − − −

− − − +

− − −

=

− −

− + − − −

− − −

=

− − −

− + − − + + −

− − −

= =

− −

⋅ − −

≤ ⋅ −

− −

⋅  

= − −  − 

∑

∑

∑ ∑

( )( )

( )( )( )( )

2 2( ) ( ) 2

( ) 2

( )

2 2( )

( ) ( ) 2

1 1

1 1

1 1

1 1 1 1

n n

KDB e e e

e e

e e

KDB e

e e e e

β α β α β

β α β

α β α

β α

α β α β α β

− − + −

− + −

− − −

−

− − − − + −



⋅ − −

= − − − −

≤ ⋅

− − − −

(52)

which proves the BIBO stability of (33) and (34).

R

EFERENCES

1 . I. Galton, “Delta-Sigma Fractional-N Phase-Locked Loops”, in Phase-locking in High Performance Systems: from Devices to Architectures, Wiley-Interscience, 2003.

2 . S. Pamarti, L. Jansson and I. Galton. “A Wideband 2.4GHz Delta-sigma Frac-tional-N PLL with 1Mb/s In-Loop Modulation,” IEEE Journal of Solid-State Circuits, vol. 39, no. 1, Jan. 2004.

3 . E. Temporiti, G. Albasini, I. Bietti, R. Castello, and M. Colombo. “A 700kHz bandwidth Sigma-Delta Fractional Synthesizer with Spurs Compensation and Linearization Techniques for WCDMA Applications,” IEEE Journal of Solid-State Circuits, vol. 39, no. 9, Sept. 2004.

4 . S. E. Meninger and M. H. Perrott. “A Dual Band 1.8GHz/900MHz 750kb/s GMSK Transmitter Utilizing a Hybrid PFD/DAC Structure for Reduced Broad-band Phase Noise,” 2005 Symposium on VLSI Circuits, June, 2005.

5 . F. L. Martin. “A Wideband 1.3GHz PLL for Transmit Remodulation Suppres-sion,” IEEE International Solid-State Circuits Conference, vol. 44, Feb. 2001.

6 . S. Pamarti and I. Galton. “Phase-Noise Cancellation Design Tradeoffs in Delta-Sigma Fractional-N PLLs,” IEEE Transactions on Circuits and Systems II : Analog and Digital Signal Processing, vol. 50, no. 11, Nov. 2003.

7 . M. Gupta and B-S Song. “A 1.8GHz Spur-Cancelled Fractional-N Frequency Synthesizer with LMS-based DAC Gain Calibration,” Digest of the Interna-tional Solid-State Circuits Conference, pp. 478-479, Feb. 2006.

8 . R. M. Gray and T. G. Stockham Jr. “Dithered Quantizers,” IEEE Transactions on Information Theory, vol. 39, no. 3, May, 1993.

9 . B. De Muer and M. Steyaert. “A CMOS Monolithic DS-Controlled Fractional-N Frequency Synthesizer for DCS-1800,” IEEE Journal of Solid-State Circuits, vol. 39, no. 7, July 2002.

10 . E. Fogleman and I. Galton. “A Digital Common-Mode Rejection Technique for Differential Analog-to-Digital Conversion,” IEEE Transactions on Circuits and Systems II:Analog and Digital Signal Processing, vol. 48, no. 3, Mar. 2001.

11 . I. Galton. “One-Bit Dithering in Delta-Sigma Modulator-Based D/A Conver-sion,” Proceedings of the International Symposium on Circuits and Systems, 1993, May, 1993.

12 . M. Gupta. Private communication.

13 . M. H. Perrott, M. D. Trott and C. Sodini. “A Modeling Approach for Σ∆ Frac-tional-N Frequency Synthesizers,” IEEE Journal of Solid-State Circuits, vol. 37, no. 8, Aug. 2002.

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Chapter 2 : A Digital Quantizer with Shaped

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