Digital - : A Wideband 2.4GHz Delta-Sigma Fractional-N PLL with Calibrated

Chapter 3 : A Wideband 2.4GHz Delta-Sigma Fractional-N PLL with Calibrated

F. Digital

on the segmented quantizer’s design parameters to ensure certain statistical properties of the quantization noise and the running sum of the quantization noise. These proper-ties include the absence of spurious tones under application of non-linear distortion.

An example is presented that satisfies the conditions and is demonstrated via computer

simulation. The work borrows ideas from dc-free codes [23, 24] and dynamic element matching tree structured encoders [25, 26].

The paper consists of three main sections. Section II presents the principle of segmented quantization, as well as an example that illustrates the appearance of spuri-ous tones when the quantized sequence is subjected to non-linear distortion. Section III presents the sufficient conditions mentioned above. Section IV presents an exam-ple segmented quantizer that satisfies the sufficient conditions.

II.S

EGMENTED

Q

UANTIZATION A. Spectral Properties of Interest

As outlined above, fractional-N PLLs and delta-sigma DACs ultimately gener-ate analog waveforms. Each such waveform contains components corresponding to digitally generated quantization noise, s[n], and, in the case of fractional-N PLLs, its running sum,

[ ] ⁿ [ ]

t n s k

∑

^{. (53)}

Moreover, inevitable non-ideal analog circuit behavior generally causes non-linear dis-tortion. The distortion can be any non-linear function, but for almost all practical ap-plications can be represented by a memory-less, truncated power series. This gives rise to components in the output waveform corresponding to s^p[n] for p = 1, 2, 3, …, h_s, and t^p[n] for p = 1, 2, 3, …, h_t, where h_s and h_t are the highest significant orders of distortion for the given application applied on s[n] and t[n] respectively.

The proposed segmented quantizer architecture is shown in Figure 12a. Its in-put is a sequence of B-bit numbers, x₀[n], and its output is a sequence of B−K-bit

numbers, x_K[n], where n = 0, 1, 2, …, is the time index of the sequences. The seg-mented quantizer consists of K quantization blocks, each of which quantizes its input by one bit, so the segmented quantizer quantizes K bits overall.⁴

b) Sequence Generator LSB of

x_d[n]

B-d

B-d bit Adder

B-d-1

Discard LSB of x_d[n]+s_d[n]

B-K

x_K[n]

Quantization Block

B-1

x₁[n]

Quantization Block

x₀[n]

Quantization Block

B-K+1

x_K-1[n]

x_d[n]

s_d[n]

x_d+1[n]

c) Sequence Generator 2

1 2

x_d[n] x_d+1[n]

s_d[n]

o_d[n]

Figure 12: a) High-level block diagram of the segmented quantizer; b) quantization block details;

c) signal processing model

The high-level details of each quantization block are shown in Figure 12b and

4 Quantization blocks that quantize their input sequences by more than one bit could be used. However, it is straightforward to show that this is a trivial extension of the one bit-per-stage case.

the signal-processing model is shown in Figure 12c. Each quantization block gener-ates a quantization sequence, sd[n], with the property that xd[n]+sd[n] is an even num-ber for each n, where xd[n] is the quantization block’s input sequence. The quantiza-tion block adds s_d[n] to x_d[n] and discards the least significant bit (LSB) to implement the 1-bit quantization.⁵ Since xd[n]+sd[n] is an even number for each n, its LSB is zero, so discarding the LSB does not incur a truncation error. Hence, the quantization noise of the segmented quantizer is a weighted sum of the sd[n] sequences:

[ ] ^K 2^d _d[ ]

s n ⁻ s n

∑

^{. (56)}

So far, the only restriction on the sd[n] sequences is that xd[n]+sd[n] must be an even integer for each n and d. This leaves considerable flexibility in the design of the sd[n] sequences which is exploited in the remainder of the paper to achieve the desired quantization noise properties.

The versions of the segmented quantizer considered in this paper partially ex-ploit this flexibility to have first-order highpass shaped quantization noise, i.e., they are designed such that the running sum of each s_d[n] sequence,

[ ] ⁿ [ ]

d d

t n s k

∑

^{. (57)}

is bounded over all n and that the estimated power spectrum of sd[n] has a highpass spectral shape. It follows from (56) that the overall quantization noise, s[n], inherits

5 Without loss of generality, numbers within the SQ are taken to be integers with a two’s-compliment binary number representation.

the spectral shape of the s_d[n] sequences, and similarly that the running sum of the quantization noise,

[ ] ^K 2^d _d[ ]

t n ⁻ t n

∑

^{, (58)}

is bounded.

The restriction to first-order highpass shaped quantization noise still leaves flexibility in the design of the sd[n] sequences. This flexibility is exploited in the re-mainder of the paper to ensure that s^p[n] for p = 1, 2, …, h_s, and t^p[n] for p = 1, 2, …, h_t are free of spurious tones, where h_s and h_t are positive integers. By definition, if s^p[n] and t^p[n] contain spurious tones at a frequency ωn, then (54) and (55), respec-tively, are expected to be unbounded in probability at ω = ωn as L→∞. Therefore, to establish that there are no spurious tones in either s^p[n] or t^p[n], it is sufficient to show that (54) and (55) are bounded in probability for all |ω| ≤ π as L→∞. A spurious tone at ω = 0 is just a dc offset, so this case is excluded from consideration. Theorems 1 and 2 in the next section present sufficient conditions on the sd[n] sequences for (54) and (55) to be bounded in probability for every L ≥ 1 and 0 < |ω| ≤ π, thereby ensuring the absence of spurious tones in s^p[n] and t^p[n].

0, [ ] even

[ ], [ ] odd, [ 1] 0

[ ] 1, [ ] odd, [ 1] 1

1, [ ] odd, [ 1] 1

d d d

d d

x n

r n x n t n

s n x n t n

x n t n

 =

 = − =

= = − = −

− = − =



(59)

where r_d[n] is an independent random sequence that takes on the values 1 and –1 with equal probability. The results presented in [29] imply that neither s_d[n] nor t_d[n] con-tain spurious tones. Therefore, s[n] and t[n] inherit these properties provided the rd[n]

sequences for d = 0, …, K−1 are independent. This is demonstrated by the estimated power spectra shown in Figure 13 which correspond to a simulated segmented quan-tizer with K = 16, x₀[n] = 2457, and quantization blocks that implement (59).

10^-4 10^-3 10^-2 10^-1 10⁰

-80 -60 -40 -20 0

10^-4 10^-3 10^-2 10^-1 10⁰

-40 -20 0 20

[ ] t n

[ ] s n

Power Spectral Density

Normalized Frequency

Figure 13: Estimated power spectra of the quantization noise and its running sum for the SQ pre-sented in Section II.

However, if the quantization noise or its running sum is subjected to non-linear

distortion, spurious tones can be induced. For instance, Figure 14 shows the estimated power spectrum of t²[n] for the simulation example described above. Discrete spikes are evident in the plot, and it can be shown that the spikes grow without bound in pro-portion to the periodogram length. Therefore, the spikes represent spurious tones.

The presence of spurious tones implies that subjecting t[n] to second-order distortion is sufficient to induce spurious tones even though t[n] is known to be free of spurious tones.

10^-4 10^-3 10^-2 10^-1 10⁰ -40

-30 -20 -10 0 10 20

Estimated Power Spectra (dB)

Normalized Frequency

Figure 14: Estimated power spectra of the square of the running sum of the quantization noise for the SQ presented in Section II.

The spur generation mechanism can be understood by considering the first quantization block. Suppose the input to the segmented quantizer is an odd-valued constant and t₀[n−1] = 0 for some value of n. Then (59) implies that (s₀[n], s₀[n+1]) is either (−1, 1) or (1, −1) depending on the polarity of r₀[n]. It follows from (57) that

(t₀[n], t₀[n+1]) is either (−1, 0) or (1, 0), and, by induction, t₀[n] has the form {…, 0,

±1, 0, ±1, 0, ±1, 0, …}. Therefore, t02[n] has the form {…, 0, 1, 0, 1, 0, 1, 0, …}

which is periodic. A similar, but more involved analysis can be used to show that the t_d²[n] sequences for d > 0 also contain periodic components. These periodic compo-nents cause the spurious tones visible in Figure 3.

III.T

HEORY FOR

T

ONE

-F

REE

Q

UANTIZATION

S

EQUENCES

It is assumed throughout the remainder of the paper that the input to the quan-tizer, x0[n], is integer-valued and deterministic sequence for n = 0, 1, …, and that the segmented quantizer is designed such that the following properties are satisfied:

Property 1: xd+1[n] = (sd[n] + xd[n])/2 is integer-valued for n = 0, 1, …, and d = 0, 1,

…, K − 1.

Property 2: there exists a positive constant B such that |t_d[n]| < B, for n = 0, 1, 2, … .

Property 3: td[0] = 0, and

( )

[ ] [ 1], [ ], [ ]

d d d d

t n = f t n− r n o n (60)

where f is a deterministic, memoryless function, {r_d[n], d = 0, 1, …, K−1, n = 1, 2, …} is a set of independent identically distributed (iid) random variables, and

1, if [ ] is odd, [ ] [ ] mod 2

0, if [ ] is even,

d d

o n x n x n

x n

= =

 (61)

is called the parity sequence of the d^th quantization block.

Property 1 and the assumption that x₀[n] is integer-valued imply that s_d[n] is an even integer when xd[n] is even, and an odd integer otherwise. Therefore, (57) implies that td[n] is integer-valued, and Property 2 further implies that it is restricted to a finite set of values. Let T₁, T₂, …, T_N denote these values. Therefore, the function, f, in Property 3 takes on values restricted to the set {T1, T2, …, TN}.

It follows from Properties 1, 2, and 3 that x_d+1[n], s_d[n], and t_d[n], for d = 0, 1,

…, K−1, and n = 1, 2, …, depend only on the set of iid random variables {rd[n], d = 0, 1, …, K−1, n = 0, 1, 2, …} and the deterministic segmented quantizer input sequence, {x₀[n], n = 1, 2, …,}. Therefore, the sample description space of the underlying prob-ability space is the set of all possible values of the random variables {rd[n], d = 0, 1,

…, K−1, and n = 0, 1, 2, …}.

Equation (57) implies that

[ ]s n_d =t n_d[ ]−t n_d[ −1]. (62)

Therefore, it follows from Property 1 that

(

1 1 1

)

[ ] [ ] [ 1] [ ] / 2

d d d d

x n = t ₋ n −t ₋ n− +x ₋ n , (63)

for 1 ≤ d < K. Recursively substituting (63) into itself and applying (61) yields

( )

1 0

[ ] 1 [ ] 2 [ ] [ 1] mod 2

d k

d k k

o n x n ⁻ ⁻ t n t n

 

=  +

∑

− −  ^{. (64)}

Recursively substituting (60) into itself implies that for any integer n > 0,

( )

[ ] [ ], [ 1], , [1], [ ], [ 1], , [1]

d n d d d d d d

t n =g r n r n− … r o n o n− … o (65)

where gn is a deterministic, memoryless function. Similarly, for any pair of integers n2

> n1 > 0, recursively substituting (60) into itself m = n2 − n1 − 1 times implies that

( )

2 1 1 1 2 1 1 2

[ ] [ ], [ 1], [ 2], , [ ], [ 1], [ 2], , [ ]

d m d d d d d d d

t n =h t n r n + r n + … r n o n + o n + … o n (66) where h_m is a deterministic, memoryless function.

Repeatedly substituting (64) into (65) to eliminate the variables {od[n], …, o_d[1]} and then recursively substituting the result into itself to eliminate the variables {t_k[m], k = 0, …, d−1, m = 1, …, n} shows that t_d[n] is a random variable that de-pends only on x0[n] (which is deterministic), and the random variables {rk[m], k = 0, 1,

…, d, m = 1, 2, …, n}. This in conjunction with (64) implies that o_d[n] is a random variable that depends only on x0[n], and the random variables {rk[m], k = 0, 1, …, d−1, m = 1, 2, …, n}. In particular since the random sequence {o_d[n], n = 0, 1, 2, …} does not depend on the random sequence {r_d[n], n = 0, 1, 2, …} and since all the random variables {rk[m] d = 0, 1, …, K−1, n = 0, 1, 2, …} are statistically independent by Property 3, it follows that {o_d[n], n = 0, 1, 2, …} and {r_d[n], n = 0, 1, 2, …} are statis-tically independent random sequences. By similar reasoning, the random variable t_d[n] is statistically independent of the random variables {r_d[m], m=n+1, n+2, …}.

Hence, (66) implies that t_d[n₂] conditioned on the random variables t_d[n₁], od[n1+1], od[n1+2], …, od[n2] is a function only of the statistically independent random variables rd[n1], rd[n1+1], …, rd[n2]. By definition, for i ≠ j the random variables {r_i[n₁], r_i[n₁+1], …, r_i[n₂]} are statistically independent of the random variables {rj[n1], rj[n1+1], …, rj[n2]}. Therefore, for i ≠ j the random variables ti[n2] and tj[n2] conditioned on t_i[n₁], t_j[n₁], o_i[n₁+1], o_i[n₁+2], …, o_i[n₂], o_j[n₁+1], o_j[n₁+2], …, o_j[n₂] are statistically independent. Consequently, for any positive real numbers p0, …, pK−1,

2 1 1 2

0 1

2 1 1 2

0 1

2 1 1 2

[ ] [ ], [ ]; 0, , 1, 1, ,

[ ] [ ], [ ]; 1, , ,

K p

j d d

K p

j d d

K p

j j j

E t n t n o n d K n n n

E t n t n o n d K n n n

E t n t n o n n n n

−

 = − = + 

 

 

 

=  = − = + 

 

=  = + 

∏

… …

…

(67)

where the second equality follows from (60) and the independence of the {rd[n], n = 1, 2, …,} sequences for d = 0, …, K − 1. This implies that the pmf of the random vari-able ti[n2] conditioned on ti[n1], oi[n1+1], oi[n1+2], …, oi[n2] is independent of any ad-ditional conditioning by tj[n1], oj[n1+1], oj[n1+2], …, oj[n2] for i ≠ j.

The statistical independence of o_d[n] and r_d[n] together with (60) imply that {td[n], n = 0, 1, …} is a discrete-valued Markov random sequence conditioned on the sequence {od[n], n = 0, 1, …}. Whenever xd[n] is odd the one-step state transition ma-trix for t_d[n] is given by

{

^d^{[ ]} ^j^{| [}^d ^1] ⁱ^{, [ ] 1}^d

}

_{N N}

P t n T t n T o n

  ×

= = − = = 

Ao . (68)

Similarly, whenever x_d[n] is even the one-step state transition matrix for t_d[n] is given by

{

^d^{[ ]} ^j^{| [}^d ^1] ⁱ^{, [ ] 0}^d

}

_{N N}

P t n T t n T o n

  ×

= = − = = 

Ae . (69)

The function f in Property 3 is independent of n and d, so neither matrix is a function of n and d.

Equation (62) implies that each possible value of s_d[n] is given by T_j− T_i for some pair of integers i and j, 1 ≤ i, j ≤ N, so

{

^d^{[ ]} ^j ⁱ ^d^[ ^1] ⁱ^{, [ ] 1}^d

} {

^d^{[ ]} ^j ^d^[ ^1] ⁱ^{, [ ] 1}^d

}

P s n = −T T t n− =T o n = =P t n =T t n− =T o n = . (70) Given that t_d[n] is restricted to N possible values, s_d[n] is restricted to N’ possible val-ues where N’ ≤ N². With identical reasoning to that used to proceed from (63) to (67), it follows that

2 0 1 1 1 1 2

0 1

2 1 1 2

[ ] [ ], , [ ], [ ]; 0, , 1, 1, ,

[ ] [ ], [ ]; 1, , .

K p

j K d

K p

j j j

E s n t n t n o n d K n n n

E s n t n o n n n n

−

= −

−

 = − = + 

 

 

 

=  = + 

∏

… … …

…

(71)

Given that {t_d[n], n = 0, 1, …} is a discrete-valued Markov random sequence condi-tioned on the sequence {od[n], n = 0, 1, …}, the conditional probability mass function (pmf) of td[n2] given td[n1] and od[n] is equal to the conditional pmf of td[n2] given t_d[n₁], t_d[n₁−1] and o_d[n]. Therefore, (62) implies that (71) is equivalent to

]

2 0 1 1 1 0 1 1 1

1 2 2 1 1 2

[ ] [ ], , [ ], [ ], , [ ], [ ];

0, , 1, 1, , [ ] [ ], [ ]; 1, ,

K p

j K K d

K p

j j j

E s n s n s n t n t n o n

d K n n n E s n t n o n n n n

−

− −

−





 

= − = + =  = + 

∏

… …

… … …

(72)

The following definitions are used by the theorems presented below. In analogy to the matrices Ao and Ae, let

{

^d^{[ ]} ^j^{| [}^d ^1] ⁱ^{, [ ] 1}^d

}

_{N N}_'

P s n S t n T o n

  ×

= = − = = 

So , (73)

and

{

^d^{[ ]} ^j^{| [}^d ^1] ⁱ^{, [ ] 0}^d

}

_{N N}_'

P s n S t n T o n

  ×

= = − = = 

Se , (74)

where {S_i, 1 ≤ i ≤ N’} is the set of all possible values of s_d[n]. Property 3 ensures that neither matrix is a function of n and d. It follows from (70) that each non-zero ele-ment of So or Se is equal to an element in Ao or Ae, respectively. For example, if Sk = T_j− T_i, then the element in the ith row and kth column of So is equal to the element in the ith row and jth column of Ao. In this fashion, once Ao and Ae are known, So and Se

can be deduced.

Let

( ) ( )

1 1

( ) ( )

, , and

p p

N N

T S

   

     

     

      

1 t s . (75)

Suppose a sequence of vectors, b[n] = [b₁[n], …, b_N[n]]^T converges to a constant vec-tor, b1, as n→∞. Then the convergence is said to be exponential if there exist

con-stants C ≥ 0 and 0 ≤ α < 1 such that

[ ] ⁿ

b ni − ≤b Cα (76)

for all 1 ≤ i ≤ N and n ≥ 0.

Theorem 1: Suppose that the state transition matrices Ae and Ao satisfy

e o o e

A A = A A , (77)

and there exists an integer h_t ≥ 1 such that for each positive integer p ≤ h_t

( ) ( )

lim ⁿ ^p _p , and lim ⁿ ^p _p

n b n b

→∞A t_e = 1 →∞A t_o = 1 (78)

where b_p is a constant and the convergence of both vectors is exponential. Then for every L ≥ 1,

[ p, ( )] ( )

E It L ω ≤C ω < ∞ (79)

for each 0 < |ω| ≤ π. Moreover, the bound C(ω), which is independent of L, is uniform in ω for all 0 < ε < |ω| ≤ π.

By Markov’s Inequality [30], this immediately leads to,

Corollary 1: Under the assumptions of Theorem 1, It Lp_, ( )ω is bounded in probability for all L ≥ 1 and for each ω satisfying 0 < |ω| ≤ π.

Proof of Theorem 1: The expectation of It Lp_, ( )ω can be expressed as

1 2

1 1 2

1 2

1 1

( )

1 2

, 0 0

1 1 1

( )

1 2

0 0 0

1 2

[ ( )] 1 [ ] [ ]

1 1

[ ] [ ] [ ]

L L

j n n

p p

t L n n

L L L

j n n

p p p

n n n

n n

E I E t n t n e

E t n E t n t n e

L L

J J

ω ⁻ ⁻ ⁻ ⁻

= =

− − −

− −

= = =

≠

 

=  

   

=  +  

∑ ∑

∑ ∑ ∑

. (80)

The notation above means that J1 and J2 are defined as the first and second terms, re-spectively, to the left of the symbol. Property 2 states that |td[n]| ≤ B, so it follows from (58) that t[n] ≤ B1 for some finite constant B1. Therefore, J1 ≤ B12p. The crux of the proof is showing that there exists a constant Ctp, positive constants D₁, D₂, and a constant 0 < α < 1 such that for n₁ ≠ n2

2 1 1

1 2 1 2

[ ] [ ] p

n n n

p p

E t n t n −Ct  ≤Dα ⁻ +Dα , (81)

The proof of (81), which is fairly lengthy, will be given later. Here (81) is used to complete the proof of the theorem. From (80), J₂ can be expressed as

( )

¹ ² ¹ ²

1 2 1 2

1 1 1 1

( ) ( )

2 1 2

0 0 0 0

2,1 2,2

1 1

[ ] [ ]

p p

L L L L

j n n j n n

p p

t t

n n n n

n n n n

J E t n t n C e C e

L L

J J

ω ω

− − − −

= = = =

≠ ≠

 

=  −  +

∑ ∑ ∑ ∑

(82)

From (81) it is seen that

( )

( ) ( )

1 2 1

1 2

1 2 1

1 1

2,1 1 2

0 0

1 1 1

1 2

0 0 0

1 2 1 2

1 1

2 2

1 1

L L

n n n

n n

n n

L L L

n n n

n n n

J D D

D D

D D D D

α α

− −

−

= =

≠

− − −

−

= = =

≤ +

≤ + − ≤ +

− −

∑ ∑

∑ ∑ ∑

⁽⁸³⁾

and the bound is independent of L. Similarly, J_2,2 can be bounded by

1 2

2,2

2 2

1 sin( / 2)

1 1

sin ( / 2)

p p

t L j n

t j L t

J C e L

C e C L

L L

L e L

ω ω

− −

−

≤ −

≤ − − = −

−

 

≤  + 

 

∑

. (84)

which is finite, independent of L, for each ω satisfying 0 < |ω| ≤ π; the bound is uni-form for all ω satisfying 0 < ε < |ω| ≤ π since sin(ω/2) > sin(ε/2). The result of the theorem then follows from (80) through (84).

To establish (81), it suffices to assume that n₂ > n₁. Using (58), E[t^p[n₂]t^p[n₁]]

can be expressed as

1 1

1 1 1 1

2 1 2 1

0 0 0 0 1 1

[ ] [ ] 2 ^p ^p _i[ ] _j[ ]

p p

K K K K

c c d d

p p

c d

c c d d i j

E t n t n ⁻ ⁻ ⁻ ⁻ ^{+ + + + +} E t n t n

= = = = = =

 

 =  

 

∑ ∑ ∑ ∑



∏ ∏

^{. (85)} It is seen that the above expression is a finite sum of terms of the form

( )

1 2 2 1

( , ) ^K _j^p^j[ ] [ ]^q_j^j

Q n n E ⁻ t n t n

 

=  



∏

^{, (86)}

where pj and qj are positive integers less than or equal to p. It thus suffices to establish a bound for Q(n₁, n₂) of the form

2 1 1

1 2 3 1 2

( , ) ⁿ ⁿ ⁿ

Q n n −C ≤Cα ⁻ +Cα . (87)

The right side of (86) is computed by conditional expectation as follows

1 2

1 1

1 2 1 1 2

0 0

( , )

[ ] ^j[ ] [ ], [ ], 0,1, , 1, 1, ,

K K

p q

i j d d

i j

Q n n

E ⁻ t n E ⁻ t n t n o n d K n n n

= =

  

 

= 

∏



∏

= ^… − = + ^… ^{. (88)}

Substituting (67) into the inner conditional expectation of (88) yields

( )

1 2 1 2 1 1 2

( , ) ^K ^p_j^j[ ] ^q_j^j[ ] [ ], [ ],_j _j 1, ,

Q n n E ⁻ t n E t n t n o n n n n

   

= 

∏

 = + ^…  ⁽⁸⁹⁾

Since {td[n], n = 0, 1, …} is a Markov process for any given parity se-quence,{od[n] = od,n, n = 0, 1, …} where o_{d n}_, ∈{0,1}, it follows from (68) and (69) that the m-step state transition matrix corresponding to t_d[n] from time n to time n + m can be written as

( )

, ,

[ , ] ^{n m} 1

d d k d k

k n

n m ⁺ o o

= +

 

∏

 ^o + ^e − 

A A A , (90)

where Ad[n, m] is an N×N matrix with elements of the form

{

^d^[ ^] ^j ^d^{[ ]} ⁱ^{, [}^d ^1] ^{d n}^, ¹^{, [}^d ^2] ^{d n}^, ²^{, , [}^d ^] ^{d n m}^,

}

P t n m+ =T t n =T o n+ =o ₊ o n+ =o ₊ … o n m+ =o ₊ .(91) Since o_d,n is either 1 or 0 for each n, (77) can be used to write (90) as

, 1

[ , ] ^y^m ^{m y}^m ^{m y}^m ^y^m, where ^{n m}

d m d k

k n

n m ⁻ ⁻ y ⁺ o

= +

= ^e ^o = ^o ^e =

∑

A A A A A . (92)

By definition, ym ≥ m/2 or m − ym ≥ m/2 depending on the given parity sequence. It follows from the exponential convergence of (78) that there exists positive numbers Cp,e and Cp,o and positive numbers αp,e and αp,o less than unity such that each element of

m ( )

y p

bp e −

A t 1 (93)

is less than Cp,eαp,e m/2 for ym ≥ m/2, and each element of

m ( )

m y p

− −

Ao t 1 (94)

is less than Cp,oαp,o m/2 for m − ym ≥ m/2.

The matrices A^{m y}_o⁻ ^m and A_e^y^m are stochastic matrices, so A^{m y}_ο⁻ ^m1 1= , A 1 1_e^y^m = and

(

^{( )}

)

^{( )}

m m m m

m y y p m y y p

p p

b b

− − = − −

o e o e

A A t 1 A A t 1, (95)

(

^{( )}

)

^{( )}

m m m m

y m y p y m y p

p p

b b

− − = − −

e o e o

A A t 1 A A t 1. (96)

Since the elements of the vectors in (93) and (94) are exponentially bounded, the same must be true for the vectors in (95) and (96). From (92) it follows that the right side of either (95) or (96) is equal to

[ , ] ( )^p

d n m −bp

A t 1 . (97)

Therefore, in general each element of (97) has a magnitude less than Cα^m/2 where C=max{C_g,e,C_g,o} and α=max{αg,e, αg,o}, which implies that

[ ] | [ ], [ ] , , 1, ,

d d d d n j p

E t n m t n o n + + =j o ₊ j= … m→b (98) as m → ∞ uniformly in n where the convergence is also exponential. This result is

independent of the given deterministic sequence {o_d,n, n = 0, 1, …}, so it implies that

[ ] | [ ], [ ], 1, ,

d d d p

E t n m t n o n + + j j= … m→b (99)

almost surely as m → ∞ uniformly in n where the convergence is also exponential.

Thus, the inner conditional expectation in (89) converges exponentially to

b as n₂ − n1 → ∞ with probability one so that

1 1

1 2 1

0 0

( , ) ⁱ[ ]

K K

q i

j i

Q n n ⁻ b E ⁻ t n

= =

 

→  

 

∏ ∏

^{. (100)}

More precisely, the exponential convergence of (100) implies that for every n2 > n1

2 1

2 1 1 2

[ ] | [ ], [ ], 1, , ( )

q n n

j j j q j

E t n t n o n n n= + … n −b ≤C q α ⁻ . (101) with probability one where C(qj) is a constant that depends on qj. For every n2 > n1

2 1 2 1

1 1

1 2 1

0 0

1 2 1 1 2

0 1

1 0

( , ) [ ]

[ ] [ ] | [ ], [ ], 1, ,

( )

i j

j j

K K

q i

j i

K p q

j j j j q

K p n n n n

j j

Q n n b E t n

E t n E t n t n o n n n n b

C q B α Cα

− −

= =

−

− − −

 

−  

 

   

≤   = + − 

≤

∏ ∏

∏

…

. (102)

where B is given from Property 2. By similar reasoning, it can be established that

1 1

1 2

0 0

E ^K ^q_j^j[ ] ^K _q_j ⁿ

j j

t n b Cα

− −

= =

 − ≤

 



∏



∏

⁽¹⁰³⁾

Hence, the above two bounds imply there exist positive constants C₁ and C₂ such that for all n2 > n1

2 1 1

1 1

1 2

0 0

1 1 1 1 1 1

1 2 1 1

0 0 0 0 0 0

1 2

( , )

( , ) [ ] [ ]

i j

i i

j j i j

K K

p q

i j

K K K K K K

p p

i q i q p q

i j i j i j

n n n

Q n n b b

Q n n E t n b E t n b b b

Cα Cα

− −

= =

− − − − − −

= = = = = =

−

   

≤ −   +   −

   

≤ +

∏ ∏

∏ ∏ ∏ ∏ ∏ ∏

^{. (104)}

Consequently, there exists a constant C₃ such that

2 1 1

1 2 3 1 2

( , ) ⁿ ⁿ ⁿ

Q n n −C ≤Cα ⁻ +Cα (105)

which is of the required form.

Theorem 2: Suppose that the state transition matrices Ae and Ao satisfy

e o o e

A A = A A , (106)

and there exists an integer h_s ≥ 1 such that for each positive integer p ≤ h_s, the se-quence transition matrices Se and So satisfy

( ) ( ) ( ) ( )

lim ⁿ ^p lim ⁿ ^p lim ⁿ ^p lim ⁿ ^p _p ,

n n n n c

→∞A S s_e _e = →∞A S s_e _o = →∞A S s_{o e} = →∞A S s_{o o} = 1 (107) where cp is a constant and the convergence of all vectors are exponential. Then for

every L ≥ 1,

[ s Lp, ( )] ( )

E I ω ≤D ω < ∞ (108)

for each 0 < |ω| ≤ π. Moreover, the bound D(ω), which is independent of L, is uniform in ω for all 0 < ε < |ω| ≤ π.

By Markov’s Inequality, this immediately leads to,

Corollary 2: Under the assumptions of Theorem 2, Is Lp_, ( )ω is bounded in probability for all L ≥ 1 and for each ω satisfying 0 < |ω| ≤ π.

Proof of Theorem 2: The proof is similar to that of Theorem 1, so only the non-trivial differences with respect to the proof of Theorem 1 are presented.

Similarly to the proof of Theorem 1, it is necessary to show that

[ ]| [ ], [ ], 1, ,

d d d p

E s n m t n o n + + j j= … m→c (109) almost surely as m → ∞ uniformly in n where the convergence is also exponential.

With this result and sd[n], cp, and (72) playing the roles of td[n], bp, and (67) in the proof of Theorem 1, respectively, the proof of Theorem 2 is almost identical that of Theorem 1. Therefore, it is sufficient to prove (109).

Since the random variables td[n−1] and od[n] are statistically independent, for any given parity sequence,{o_d[n] = o_d,n, n = 0, 1, …} where o_{d n}_, ∈{0,1}, it follows from (73), (74), and (91) that

( )

, 1 , 1

[ , 1] [ , ] 1

d n m+ = d n m  ^ood n m_{+ +} + ^e −od n m_{+ +} 

S A S S , (110)

where Sd[n, m+1] is an N×N’ matrix with elements of the form

{

^d^[ ^1] ^j ^d^{[ ]} ⁱ^{, [}^d ^1] ^{d n}^, ¹^{, , [}^d ^1] ^{d n m}^, ¹

}

P s n m+ + =S t n =T o n+ =o ₊ … o n m+ + =o _{+ +} , (111) where i is the row index and j is the column index. By similar reasoning to that used

in the proof of Theorem 1, (106) and (107) together imply that there exists a positive number Dand a positive number β less than unity such that each element of the vector

[ , 1] ( )^p

d n m+ −cp

S s 1 , (112)

has a magnitude less than D⋅β^m/2. Thus, (112) implies that

[ ]| [ ], [ ] , , 1, ,

d d d d n j p

E s n m t n o n + + =j o ₊ j= … m→c (113) as m → ∞ uniformly in n where the convergence is also exponential. This result is

independent of the given deterministic sequence {o_d,n, n = 0, 1, …}, so it implies that (109) holds almost surely as m → ∞ uniformly in n where the convergence is also ex-ponential.

IV. A S

EGMENTED

Q

UANTIZER THAT

S

ATISFIES

T

HEOREMS

1

AND

2

0 3 4 0 1 4 0 1 4 0 3 4 0 0

3 16 0 3 4 0 1 16 0 5 8 0 3 8 0

, and .

0 1 2 0 1 2 0 1 8 0 3 4 0 1 8

1 16 0 3 4 0 3 16 0 3 8 0 5 8 0

0 1 4 0 3 4 0 0 0 3 4 0 1 4

   

   

=  = 

   

   

o e

A A (115)

From (62) all possible s_d[n] values are {−4, −3, −2, −1, 0, 1, 2, 3, 4}, and further de-fine

( )^p = −( 4)^p ( 3)− ^p ( 2)− ^p ( 1)− ^p 0 1^p 2^p 3^p 4^p^T

s . (116)

Applying (70) yields

0 0 0 0 0 3 4 0 1 4 0

0 0 0 3 16 0 3 4 0 1 16 0

, and

0 0 0 1 2 0 1 2 0 0 0

0 1 16 0 3 4 0 3 16 0 0 0

0 1 4 0 3 4 0 0 0 0 0

0 0 0 0 1 4 0 3 4 0 0

0 0 0 0 5 8 0 3 8 0 0

. 0 0 1 8 0 3 4 0 1 8 0 0

0 0 3 8 0 5 8 0 0 0 0

0 0 3 4 0 1 4 0 0 0 0

 

 

= 

 

 

 

 

= 

 

 

(117)

Multiplying the matrices in either order yields

0 9 16 0 7 16 0

9 64 0 3 4 0 7 64

0 1 2 0 1 2 0

7 64 0 3 4 0 9 64

0 7 16 0 9 16 0

 

 

= = 

 

 

e o o e

A A A A (118)

so the matrices commute. Direct computation reveals that the eigenvectors of both Ae

and Ao are linearly independent, and therefore Ae and Ao are diagonalizable [31].

Specifically, Aⁿ_e =V Λ V_e ⁿ_e _e⁻¹, where

1 0 0 0 0 1 1 1 0 0

0 1/ 4 0 0 0 0 0 0 1 1

, ,

0 0 0 0 0 1 0 1/ 3 0 0

0 0 0 1 0 0 0 0 1 1

0 0 0 0 1/ 4 1 1 1 0 0

1/ 8 0 3/ 4 0 1/ 8

1/ 2 0 0 0 1/ 2

and 3/ 8 0 3/ 4 0 3/ 8 ,

0 1/ 2 0 1/ 2 0

n n

−

   

   − 

   

= = −

   

   − 

   

 

 − 

 

= −

 

 − 

 

e e

Λ V

(119)

and Aⁿ_o =V Λ V_o _oⁿ _o⁻¹, where

( )

1 0 0 0 0 1 1 1 1 1

0 1 0 0 0 1 1 1/ 2 1/ 2 0

, 1 1 0 0 1/ 3 ,

0 0 1/ 4 0 0

1 1 1/ 2 1/ 2 0

0 0 0 1/ 4 0

1 1 1 1 1

0 0 0 0 0

1/16 1/ 4 3/ 8 1/ 4 1/16 1/16 1/ 4 3/ 8 1/ 4 1/16

and 1/ 4 1/ 2 0 1/ 2 1/ 4

1/ 4 1/ 2 0 1/ 2 1/ 4

3/ 8 0 3/ 4 0

n n

−

 −   − 

  − − − 

   

= −  =− − − 

− −

= − −

− −

−

o o

Λ V

V .

3/ 8

 

 

 

(120)

By inspection of (119), Λ_eⁿ converges to

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

 

 

= 

 

 

Λe . (121)

The vector given by V Λ V t is equal to b_e _eⁿ_,1 _e⁻^{1 ( )}^p _p1, where b_p = 0, 1 and 0 for p = 1, 2 and 3 respectively, which is of the form required by Theorem 1. To show exponential convergence, consider

( )

( ) 1 ( )

1 ( )

n p n p

n p

b ⁻

−

− = −

≤ −

e e e e e

e e e e

A t 1 A V Λ V t

A V Λ V t , (122)

where || || is the l2 norm, and p = 1, 2 or 3. Evaluating ||t^(p)|| for p = 3, and ||Aen − VeΛΛΛΛe,1Ve-1|| yields 130 and ^{2 1/ 4}

( )

ⁿ respectively therefore the right side of (122) is equal to

( )

260 1/ 4 ⁿ (123)

and therefore each element of the vector given by A tⁿ_e ^{( )}^p −b_p1 converges exponen-tially to zero.

By inspection of (120), Λ_oⁿ does not converge, however it is sufficient to show that the vector V Λ V t_o ⁿ_o _o⁻^{1 ( )}^p converges. Consider Aⁿ_o =V Λ V_o _oⁿ_,1 _o⁻¹+V Λ V where _o ⁿ_o_,2 _o⁻¹

,1 ,2

0 0 0 0 0 ( 1) 0 0 0 0

0 1 0 0 0 0 0 0 0 0

, and ,

0 0 ( 1/ 4) 0 0 0 0 0 0 0

0 0 0 1/ 4 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

n n n

 − 

 

 

= − =

 

   

o o

Λ Λ (124)

Multiplying V Λ V by t_o ⁿ_o_,2 _o⁻¹ ^(p) for p = 1, 2 or 3 results in a vector with all zero ele-ments for all n≥1. Therefore, for all n≥1 and p = 1, 2 or 3, Α tⁿ_o ^{( )}^p =V Λ V t . By _o _oⁿ_,1 _o⁻^{1 ( )}^p inspection, Λ converges to _oⁿ_,1

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0

 

 

= 

 

 

Λo (125)

The vector given by V Λ V t is equal to b_o _o_,3 _o⁻^{1 ( )}^p p1, where bp = 0, 1 and 0 for p = 1, 2 and 3 respectively. Replacing A V Λ , and ⁿ_e, _e, _e_,1 V_e⁻¹ in (122) with A V Λ , and ⁿ_o, _o, _o_,3

−1

Vo respectively shows that ||Aont^(p) − bp1|| converges exponentially to bp1. Therefore the state transition matrices given by (115) satisfy the conditions of Theorem 1 for h_t = 3.

Using the decomposition in (119) and (120) and the sequence transition matri-ces given by (117), it can be shown by direct computation that

( ), ( ), ( )

n p n p n p

o e o o e e

A S s A S s A S s and A S sⁿ_e _o ^{( )}^p converges to c_p1, where c_p = 0, 1.5, 0, 6, and 0 for p = 1, 2, 3, 4 and 5 respectively. Furthermore, the convergence of each vector at index n can be bounded using (122), replacing ||t^(p)|| alternately with ||Ses^(p)|| and

||Sos^(p)||, which implies that the convergence of A S sⁿ_{o e} ^{( )}^p ,A S sⁿ_{o o} ^{( )}^p ,A S sⁿ_e _e ^{( )}^p and

( )

n p

e o

A S s are exponential. Therefore, the matrices Ae, Ao, Se, and So given in (115) and (117) also satisfy the conditions of Theorem 2 for hs = 5.

10^-3 10^-2 10^-1 10⁰ -60

-40 -20 0 20

10^-3 10^-2 10^-1 10⁰

-60 -40 -20 0 20

10^-3 10^-2 10^-1 10⁰

-20 0 20 40

10^-3 10^-2 10^-1 10⁰

-20 0 20 40

10^-3 10^-2 10^-1 10⁰

-20 -10 0 10 20

10^-3 10^-2 10^-1 10⁰

-20 -10 0 10 20

10^-3 10^-2 10^-1 10⁰

-20 -10 0 10 20

10^-3 10^-2 10^-1 10⁰

-20 -10 0 10 20

( )

P SQ3

( )

P SQ5

∆ΣM

( )

P5 ∆ΣM

∆ΣM

( )

P3 ∆ΣM

Estimated Power Spectra (dB)

Normalized Frequency Normalized Frequency

Estimated Power Spectra (dB)

Figure 16: Estimated power spectra of a) the quantization noise sequences, and b) the running sums of the quantization noise sequences of the first-order ∆Σ modulator and the segmented quantizer presented in Section IV before and after application of non-linear distortion.

Simulation results for the segmented quantizer presented above and the ∆Σ modulator with K =16 and a constant input ofx n₀[ ] 2048= are shown in Figure 16.

The quantization noise, as well as its running sum for both the segmented quantizer and the ∆Σ modulator are subjected to the following distortion polynomials,

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

3 2

5 4 3 2

[ ] 0.15 [ ] 0.32 [ ] 0.99 [ ] 0.23

[ ] 0.32 [ ] 0.27 [ ] 0.64 [ ] 0.12 [ ] 1.03 [ ] 0.13.

P t n t n t n t n

P s n s n s n s n s n s n

= + + −

= − − + + + (126)

Figure 16 shows the estimated power spectra of the quantization noise and in-tegrated quantization noise before and after application of the distortion polynomials.

The estimated power spectra of the sequences, t^p[n] or s^p[n], are taken to be the aver-age of the periodograms of the M windowed sequences, t^p[n − kL]w[n − kL] and s^p[n − kL]w[n − kL], for k = 1, 2, …, M, where w[n] is a Hanning window of length L. As expected from the theoretical results presented above, no spurious tones are apparent in the figures for the segmented quantizer before or after application of the distortion polynomials. In contrast, spikes, which imply the presence of spurious tones, are evi-dent in the estimated power spectra of the quantization noise from the ∆Σ modulator after application of the distortion polynomials.

A

CKNOWLEDGEMENTS

The authors would like to acknowledge Sudhakar Pamarti, and Jared Welz for helpful discussions relating to this work. The text of Chapter 2, in full, has been sub-mitted for review to the IEEE Transactions on Signal Processing. The dissertation au-thor was the primary investigator and auau-thor. Ian Galton supervised the research

which forms the basis of this paper. Andrea Panigada contributed to the theorem statements, and Elias Masry contributed to the structure of the proofs.

R

EFERENCES

15 . R. Schreier, G. C. Temes, Understanding Delta-Sigma Data Converters, Wiley-IEEE Press, 2004.

16 . B. Razavi, Phase-locking in high-performance systems: from devices to architec-tures, Wiley-Interscience, 2003.

17 . I. Galton, “Delta-sigma data conversion in wireless transceivers,” IEEE Transac-tions on Microwave Theory and Techniques, vol 50, no. 1, pp. 302-315, January, 2002.

18 . T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, Second Edition, Cambridge University Press, 2003.

19 . S. Pamarti, J. Welz and I. Galton. “Statistics of the Quantization Noise in One-Bit Dithered Single-Quantizer Digital Delta-Sigma Modulators,” submitted to IEEE Transactions on Circuits and Systems I.

20 . W. Chou and R. M. Gray. “Dithering and Its Effects on Sigma-Delta and Multi-stage Sigma Delta Modulation,” IEEE Transactions on Information Theory, vol.

37, no. 3, May 1991.

21 . S. Pamarti, L. Jansson and I. Galton. “A Wideband 2.4GHz DS fractional-N PLL with 1Mb/s in-loop modulation,” IEEE Journal of Solid State Circuits, vol. 39, no. 1, Jan. 2004.

22 . B. De Muer, M. Steyaert. “A CMOS Monolithic ∆Σ-Controlled Fractional-N Frequency Synthesizer for DCS-1800,” IEEE Journal of Solid-State Circuits, vol. 37, no. 7, July 2002.

23 . B. H. Marcus and P. H. Siegel. “On Codes with Spectral Nulls at Rational Sub-multiples of the Symbol Frequency.” IEEE Transactions on Information The-ory, vol. IT-33, No. 4, July 1987.

24 . G. L. Pierobon. “Codes for Zero Spectral Density at Zero Frequency,” IEEE Transactions on Information Theory, vol. IT-30, no. 2, Mar. 1984.

25 . I. Galton, "Spectral shaping of circuit errors in digital-to-analog converters,"

IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Proc-essing, vol. 44, no. 10, pp. 808-817, Nov., 1997.

26 . 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. 38, no. 3, Mar. 2001.

27. A. V. Oppenheim, R. W. Schafer, J. R. Buck, “Discrete-Time Signal Processing,”

Prentice-Hall, Inc, Second Ed., 1999.

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30 . A. Papoulis and S. Unnikrishna Pillai. Probability, Random Variables and Sto-chastic Processes, McGraw-Hill, 2002.

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Chapter 3 : A Wideband 2.4GHz Delta-Sigma

tional-N PLL for operation in the 2.4GHz ISM band. The technique enables the frac-tional-N PLL to achieve the required phase noise for Bluetooth with a bandwidth of 730kHz and a reference of 12MHz without incurring either increased phase noise or power consumption that results from the design of wide bandwidth PLLs. The wide bandwidth makes it possible to integrate a passive 2^nd order loop filter on chip [33], which does not suffer from the increased noise inherent to an integrated active loop filter [34]. Moreover, the wide bandwidth also reduces the sensitivity of the voltage-controlled oscillator (VCO) to pulling [35], and attenuates the 1/f² and 1/f³ phase noise from the VCO within the PLL loop bandwidth.

The presented calibration technique differs from current techniques in that the existing PLL loop is modified so that minimal extra circuitry is required for calibra-tion. In addition, the calibration circuitry is an analog, continuous-time technique, and avoids many of the problems of existing techniques, such as reference clock feedthrough from sampling the loop filter voltage, and the low calibration loop band-width required to filter quantization noise added during the calibration process [36].

The calibration technique demonstrated in this paper has a wide bandwidth and thus fast settling time. The resulting improvement in the quality of phase noise cancella-tion allows for enhancements to be made, such as reducing the reference frequency, which lowers the power consumption of the digital logic at the cost of increased phase noise. A dynamic bias technique allows the charge pump to operate with significantly high output current to meet circuit noise requirements without dissipating constant power in the bias network.

2nd-Order Digital ∆Σ Modulator N+y[n]

R C₂ C₁

VCO

1 1

z z

−

− −

16 16

Pseudo-Random Generator

PFD and Charge

Pump

5 2-b Binary-Weighted

I-DAC 16 1-b

Unit-Weighted

I-DAC

y[n]

10 16

Segmented Mismatch Shaping

DAC Encoder

MSB LSB

e_Q[n]

c[n]

Channel Select

Integrated Circuit

12MHz Digital Logic e_int[n]

gm V_cal

C_int C_int

19 18

8 sgn(e n_int[ ])

Figure 17: High-level functional diagram of the implemented PLL

A high-level block diagram of the implemented fractional-N PLL is shown in Figure 17. It differs from a typical phase noise canceling fractional-N PLL in that the charge pump, digital-to-analog converter (DAC), loop filter and VCO have been modified to implement the calibration technique. The additional circuitry required to

The core of a phase-noise canceling PLL is shown in Figure 18a [37, 38, 39].

It consists of a Phase/Frequency Detector (PFD), charge pump (CP), voltage-controlled oscillator (VCO), divider, DAC, and digital logic required to generate the divider input, y[n], as well as the DAC input. The operation of a delta-sigma frac-tional-N PLL is discussed at some length in [40], but will be summarized in order to present the salient points necessary to understand the operation of the calibration tech-nique. The divider output, V_div(t), is a two level signal, where the n^th and (n+1)^th rising edges are separated by N + y[n] VCO cycles. The PFD compares the rising edges of V_div(t) with the rising edges of a reference signal, V_ref(t), and then generates control signals for the CP, resulting in a pulse of current, icp(t), which deposits charge propor-tional to the time difference between Vdiv(t) and Vref(t) onto the loop filter. This serves to either increase, or decrease the loop filter voltage, V_ctl(t), and hence increase or de-crease the frequency of the VCO. In this way, the PLL attempts to lock the phase of the divider output with the reference signal.

Ideal Matching DAC Gain Mismatch

ref( ) V t

div( ) V t

cp( ) i t

dac( ) i t

ctl( ) V t

ref( ) V t

div( ) V t

cp( ) i t

dac( ) i t

ctl( ) V t 2nd-Order

Digital ∆Σ Modulator N+y[n]

1 1

z z

−

− −

PFD and Charge

Pump

ref( ) V t

div( ) V t

[ ] x n

[ ] y n

Q[ ]

e n e n_int[ ] Digital Requantizer Current

DAC

ctl( ) ( ) V t

i tcp

dac( )

i t R

C₂ C₁

K_vco VCO

Figure 18: Phase Noise Canceling PLL; a) Block Diagram; b) Timing Diagram

If y[n] is a constant, then the VCO frequency is N + y[n] times the reference frequency. The VCO can be locked to a fractional multiple of the reference frequency by maintaining a fractional average value on y[n]. This is done by quantizing a

frac-tional number, x[n] to an integer y[n] with a delta-sigma modulator [41] such that the quantization noise, eQ[n] is high-pass spectrally shaped. Since it is only possible for the divider edge to occur after an integer multiple of the VCO period, instantaneously the divider output will never be phase-locked to the reference. At the output of the CP, this results in current pulses always adding or subtracting charge from the loop filter and, on average, the net charge added to the loop filter is zero. This CP charge depos-ited each reference period can be well modeled by the following expression [37]

[ ] ⁿ [ ] [ ]

cp CP VCO Q CP VCO int

Q n I T ⁻ e k I T e n

∑

= ^{, (127)}

where T_VCO is the period of the VCO output under steady-state conditions, I_CP is the magnitude of the charge pump current, and e_int[n] is the integrated quantization noise.

Since the noise introduced by eint[n] is generated by the digital logic, the phase noise cancellation technique uses this information to subtract the error charge given by (127).

The DAC converts e_int[n] into current pulses which have a charge nominally equal in magnitude and opposite in sign to the CP charge. Thus with ideal cancella-tion, the net voltage change on V_ctl(t) over a period of the reference clock is zero, and is shown in Figure 18b. The DAC pulses can be mismatched from the CP pulses due to static current mismatch or pulse width mismatches [42]. This results in a residual charge error remaining on the loop filter, thereby limiting the achievable phase noise cancellation, also shown in Figure 18b.

The DAC circuitry is often simply a scaled copy of the CP circuitry in order to

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