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A

Output spectrum for timing errors

A.1 Power spectrum of y(t) for random timing errors

The power spectrum of y(t) from eq. (7.10) will be calculated. The assumptions madeare that the DAC input z(m) is a ergodic stationary random process, and the timing errorsa stationary random process. Then y(t) is cyclostationary.

Initially, the mean of the empirical autocorrelation Ry(t, t + τ) is evaluated. Then wewill obtain the mean of the empirical power spectrum ESy( f ) with a Fourier trans-form of Ry(t, t + τ), from which the averaged probabilistic power spectrum 〈Sy〉( f ) isextracted since they are equal due to regularity [128]1 The probabilistic power spectrumSy( f ) matches that of 〈Sy〉( f ) due to stationarity. We start with the evaluation of theexpected empirical autocorrelation ERy(τ):

ERy(τ) =

E limD→∞

12D

∫ D

−D∑q

∑m

z(m+q)z(m)δ (t + τ − (q+m)Ts −µm+q)δ (t −mTs −µm)dt

= E limD→∞

12D

∫ D

−D∑m

z2(m)δ (t + τ −mTs −µm)δ (t −mTs −µm)dt

+E limD→∞

12D

∫ D

−D∑q=0

∑m

z(m+q)z(m)δ (t + τ − (q+m)Ts −µm+q)δ (t −mTs −µm)dt

(A.1)

1Regularity guarantees the existense of time average limits such as the empirical mean and autocorrelation.It is less strong than ergodicity.

199

200 Appendix A Output spectrum for timing errors

Let the first and the second parts of the sum of eq. (A.1) be T1 and T2, respectively.Recall the definition of the expectation of a function F(x) is EF(x)=

∫

σ f (σ)F(σ)dσwith f (σ) being the pdf of the random variable x. In our case f (σ) is the probability den-sity function of the time-jitter µm.

The term T1 is changed to

T1 = E limN→∞

1(2N +1)Ts

∫ (N+ 12 )Ts

−(N+ 12 )Ts

∑m

z(m)2δ (t + τ −mTs −µm)δ (t −mTs −µm)dt(A.2)

Since µm are in the neighborhood of mTs, we may rewrite (A.2) as

T1 = limN→∞

1(2N +1)Ts

N

∑m=−N

Rz(0)∫ ∞

−∞Eδ (t + τ −mTs −µm)δ (t −mTs −µm)dt

=1Ts

Rz(0)δ (τ)

(A.3)

The term T2 is written as

T2 = E 1Ts

limN→∞

12N +1

·∫ (N+ 1

2 )Ts

−(N+ 12 )Ts

∑q=0

∑m

z(m+q)z(m)δ (t + τ − (q+m)Ts −µm+q)δ (t −mTs −µm)dt

= E 1Ts

limN→∞

12N +1 ∑

q=0

N

∑m=−N

z(m+q)z(m)

∫ ∞

−∞δ (t + τ −qTs −mTs −µm+q)δ (t −mTs −µm)dt

(A.4)

and if we use

Im+q,m(τ) =∫ ∞

−∞δ (t + τ −qTs −mTs −µm+q)δ (t −mTs −µm)dt (A.5)

we transform T2 to

T2 =1Ts

limN→∞

12N +1 ∑

q=0

N

∑m=−N

Ez(m+q)z(m)EIm+q,m(τ) (A.6)

Next, we use the joint PDF cn−m(tn, tm) of the jitter to write kq(τ) = EIm+q,m(τ) as

kq(τ) =+∞∫∫∫

−∞δ (t + τ − (q+m)Ts −µm+q)δ (t −mTs −µm)cq(µm+q,µm)dµm+q dµm dt

(A.7)

A.1 Power spectrum of y(t) for random timing errors 201

and finally

kq(τ) =∫ ∞

−∞cq(t + τ −qTs, t)dt (A.8)

The time averaged probabilistic autocorrelation is obtained combining the equations(A.8) (A.4) and (A.6) into

ERy(τ) =1Ts

Rz(0)δ (τ)+1Ts

∑q=0

Rz(q)kq(τ) (A.9)

The function Rz(q) = Ez(m+q)z(m) represents the probabilistic autocorrelation of thestationary input signal z(m), and Rz(0) is its power.

The next step is to use eq. (A.9) and with a Fourier transformation to obtain thepower spectrum of the process y(t). The difficulty is posed by the transformation ofkq(τ), defined as Kq( f ). Therefore, we define the double Fourier integral of the jitterMk−l( fk, fl) for k = l

Mk−l( fk, fl) =∫ ∞

−∞

∫ ∞

−∞fk−l(µk,µl)e− j2 pi( fkµk+ fl µl) (A.10)

Observe that Mk−l( fk, fl) is related to the characteristic function

Ck−l( fk, fl) = Ee− j2π( fkµk+ fl µl)Because the timing error process is assumed stationary the characteristic function

Cm,n( f ,− f ) depends only on the difference k− l = q, hence

C0( f ) = 1 (A.11)

|Cq( f )| ≤ 1 (A.12)

Then it is easy to show that for q = 0

Kq( f ) = e− j2π f qTsMq( f ,− f ) = e− j2π f qTs = Ee− j2π f (µm+q+µm) = e− j2π f qTsCq( f ,− f )(A.13)

The Fourier transformation of eq. (A.4) with the use of eq. (A.10) gives

Sy( f ) =1Ts

∑q

Rz(q)Cq( f ,− f )e− j2πq f Ts (A.14)

Eq. (A.14) gives us the power spectrum of the impulse position modulated waveformthat is subject to stationary timing uncertainties with for general statistical properties andcorrelation.

The analysis when z(m) is deterministic is very similar. In place of the probabilisticautocorrelation function Rz(q) of the stationary signal z(m) the empirical autocorrelationRz(q) is used. This follows directly from eq. (A.2) and (A.4) where the factors T1 and T2

are calculated. Indeed, if z(m) is not a random process, the expectation in eq. does notapply to the factors z2(m) and z(m+q)(z(m), which subsequently are combined with thediscrete average operant 〈 〉 = 1

2N+1 ∑Nm=−N to form Rz(0) and Rz(q), respectively.

202

A.2 Spectrum of y(t) for deterministic timing errors

The general magnitude spectrum will be calculated in this section. The following twoproperties of the Bessel function of the first kind are used for the calculations:

eM sinθ = ∑k

Jk(M)e jkθ (A.15)

J−k(x) = (−1)kJk(x) (A.16)

(A.17)

The time modulated signal z(t−µ(t)) is calculated first. Using the definition of the Besselfunction, it is found that

z(t −µ(t)) =

(−1)P

∑p=1

∑q

ApJq(ωpM)e j(ωp+qωµ )t

= (−1)P

∑p=1

∑q

Bp,q(ωpM)cos((ωp +qωµ)t)

(A.18)

where Bp,q(ωpM) = ApJq(ωpM).Next we calculate in the same way the factor

1Ts

∑m

e jmωs(t−µ(t)) =(−1)

Ts∑m

∑r

Jr(mωsM)e j(mωs+rωµ )t (A.19)

The combination of eq. (A.18) and (A.19) gives

y(t) =1Ts

P

∑p=1

∑q,m,r

cos((ωp +qωµ)t)Bp,q(ωpM)Jr(mωsM)e j(mωs+rωµ )t (A.20)

where we have used Γp,q,r(ωpM,mωsM) = Bp,q(ωpM)Jr(mωsM).Applying a Fourier transformation in y(t) leads to

Y ( f ) =1Ts

P

∑p=1

∑q,m,r

Γp,q,r(ωpM,mωsM)2

[

δ(

f −m fs − r fµ ± ( fp +q f µ))]

(A.21)

We define fA(p,q) = fp +q f µ and fB(m,r) = m fs + r f µ , and we evaluate the magnitudespectrum |Y ( f )|

|Y ( f )| = 1Ts

P

∑p=1

∑q,m,r

|Γp,q,r(ωpM,mωsM)|2

[δ ( f − fB(m,r)± fA(p,q))] (A.22)

Appendix A Output spectrum for timing errors

B

Literature data

203

204 Appendix B Literature data