Simulation experiments

0 P[𝑥 = 𝑎] 𝑎 𝑝 𝑝 𝑝 𝑝 1 −𝑝 −𝑝 −𝑎 +𝑎 −𝑎 +𝑎

Figure 5.5: Designing the p.m.f. of a discrete symmetric quinary sequence unity and match𝔐 ,𝔐 and 𝔐 with those of a Gaussian sequence, giving:

𝔐 ≜ Eq𝑢 y= 𝑎 𝑝 + 𝑎 𝑝 = 1 (5.6a)

𝔐 ≜ Eq𝑢 y= 𝑎 𝑝 + 𝑎 𝑝 = 3!! = 3 (5.6b)

𝔐 ≜ Eq𝑢 y= 𝑎 𝑝 + 𝑎 𝑝 = 5!! = 15 (5.6c)

𝔐 ≜ Eq𝑢 y= 𝑎 𝑝 + 𝑎 𝑝 = 7!! = 105. (5.6d)

Solving the simultaneous equations gives:

𝑎 = 5 −√10, 𝑎 = 5 +√10,

𝑝 = 7 + 2√10 𝑝 = 7 − 2√10 , (5.7)

so that the sequence levels (to four significant figures) are ±1.356, each with prob- ability 0.2221 and ±2.857, each with probability 0.01126; and lastly the zero level with a probability of0.5333. Thep.m.f. of this tuned sequence is plotted in Fig.5.6.

0 P[𝑥 = 𝑎] 𝑎 0.0113 0.0113 0.0222 0.0222 / −1.36 +1.36 −2.86 +2.86

Figure 5.6: The p.m.f. of a symmetric quinary sequence tuned for Gaussianity

5.3

Simulation experiments

Two sets of simulation experiments on systems with a Wiener structure were con- ducted to confirm the theory developed in Section 5.2. In both experiments, meas-

5.3. Simulation experiments urements were taken at steady-state.

5.3.1 Experiment 1: Ternary sequences

In this experiment, ternary input sequences are compared with Gaussian input se- quences, both having a period 𝑁 = 1024. The ternary sequence experiments were performed for several different values of𝑝 —the probability that the sequence is at zero (note that 𝑝 = (1 − 𝑝)); including 𝑝 = 0 (a binary sequence), 𝑝 = / (a uniform ternary sequence) and𝑝 = / (from (5.3), the value for which𝔇should be minimised). 𝑝 serves as an independent variable to vary the higher order moment of the ternary input in this set of experiments. The outputs were noise-free and were measured for 𝑀 = 1024 independent realisations of each input. The 1024 in- put and output auto- and cross-power spectra were averaged, with anon-parametric

estimate of the Bla then being obtained according to (4.5) with 𝑃 = 1. The aver- aging is needed even though the system is noise-free because the contribution of the nonlinear distortion terms is different for each realisation as noted in Section 3.1. The linear part of the Wiener system was arbitrarily chosen as a digital first-order low-pass filter with time constant of 4 sampling intervals, i.e. with a transfer func- tion given by 𝐺(z) =z/ z−𝑒 . Two different static nonlinearities were used, as described in the next two subsections.

Although the full analysis could be performed with non-parametric estim- ates, in order to obtain a more robust estimate of the Df 𝔇, parametric estimates of the Blawere obtained using ELiS, an iterative weighted least squares estimator in the Frequency Domain System Identification Toolbox (Fdident) (Kollár,1994). The variances of the complex frequency response estimates were supplied to ELiS

for weighting purposes. The cost function used internally was 𝐶 = ∑ 𝑊 |𝑒 | where _{|𝑒 | = ⏐⏐⏐ ̂𝑌[𝑘] − 𝑌 [𝑘]⏐⏐⏐} are the squared errors between modelled output spec- trum𝑌̂and actual output spectrum𝑌in terms of frequency line number𝑘, and 𝑊 are the weighting factors proportional to the reciprocal of the variance at each 𝑘. For the Gaussian input sequences, a parametric model of order 1/1 (i.e. one zero, one pole) gave satisfactory fitting, but for the ternary input sequences, a model of order 8/8 was needed because the Bla then contains higher order dynamics than those of the underlying linear system; this is due to nonlinear effects (see example 4.1 on p. 42 of Enqvist,2005). For the purposes of finding theDf 𝔇, the high order of the parametric model with a ternary input sequence is not of concern.

5.3. Simulation experiments

Cubic nonlinearity

The first nonlinearity was a pure cubic, i.e. f(𝑥) = 𝑥 , for which it is possible to obtain a theoretical expression for 𝔇 using (3.48). From (5.2b), the deviation of the 4th _{moment of the ternary sequence with an arbitrary value of𝑝 from that of a} Gaussian sequence is given by:

𝛿 = 3!! −𝑎 𝑝 = 3 − 𝑎 (1 − 𝑝 ). (5.8)

For the ternary sequence to have a power of unity, 𝑎 𝑝 = 1, so that 𝑎 = = . Substituting in (5.8),

𝛿 = 3 − 1

1 − 𝑝 . (5.9)

By writing (3.48) and (3.26) in terms of𝛿 , the theoretical expression for 𝔇is given by:

𝔇_theory= 3 − 1 1 − 𝑝

∑ 𝑔 (𝑘)

9 ∑ 𝑔 (𝑘) . (5.10)

The results are shown in Fig. 5.7. For the cubic nonlinearity, the theoretical Df 𝔇_theory is shown as a solid line and the simulation results are shown as circles; it can be seen that there is excellent agreement between the two.

Ideal saturation nonlinearity

The second nonlinearity was an ideal saturation characteristic with limits (i.e. clip- ping levels) set at ±0.75 and a slope of 1 between these limits. Only simulation results are shown in Fig. 5.7 for this nonlinearity, for which there is currently no theory for finding 𝔇. From these, it can be seen that the minimum value of 𝔇 is obtained with a lower value of 𝑝 than for the cubic nonlinearity. The difference is not unexpected, because there is no reason to expect that the contributions across multiple higher order terms will be minimised at the same value. It is worth noting that despite this, a lower value of 𝔇 is obtained for this nonlinearity when 𝑝 =

2_{⁄3than for the more widely used values of 𝑝 = 0(binary input) and} 1_⁄3(uniform

ternary input).

5.3.2 Experiment 2: Multilevel sequences

In this experiment, Gaussian, binary, ternary, quaternary and quinary input se- quences were used; all had a period 𝑁 = 2048. The linearity was a 3-tap FIR filter with transfer function given by 𝐺(z) = (1 + 0.6z +0.1z ), with the rel- atively short impulse response magnifying the levels of discrepancy (Section 3.5.2)

5.3. Simulation experiments Simulation: x3 Theory: x3 Simulation: Saturation Optimised ternary Uniform ternary Uniform binary Discrepancy factor

Symmetric discrete ternary sequence

zero level probability p′

10−6

10−4

10−2

100

0 0.2 0.4 0.6 0.8

Figure 5.7: Discrepancy Factors of random ternary sequences with various zero-level probabilities

and the actual choice of the filter coefficients here was arbitrary. For the ternary, quaternary and quinary sequences, two versions were created, one with a uniform distribution of the levels and the other with levels and probabilities optimised for

Gaussianity—using Equations (5.3) (ternary), (5.5) (quaternary) or (5.7) (quinary). The outputs were noise-free and were measured for 5000 independent realisations of each input. Similar to the method used in Section 5.3.1, the 5000 input and output auto- and cross-spectra were averaged (because the contribution of the non- linear terms is different for each realisation) with a non-parametric estimate of the Bla then being obtained according to (4.5). The nonlinearity was a static poly- nomial function f(𝑥) = 𝑥 + 𝑥 + 𝑥 + 𝑥; the higher degree than in Experiment 1

was chosen to emphasise performance differences between sequences with different numbers of levels and between sequences with uniform probability distributions and those optimised for Gaussianity. TheMatlabprogram code used in this simulation experiment is included in AppendixB.4.

5.3. Simulation experiments 0 200 400 600 800 1000 20 30 40 50 60 Gaussian Binary (Uniform) Ternary (Uniform) Quinary (Uniform) Ternary (Optimised) Quaternary (Optimised) B LA Magnitude (dB) Frequency line

Gaussian and quinary(opt.) Quaternary (opt.) Ternar y (opt.) Quinar y (uni.) Ternar y (uni.) Binar y (uni.)

Figure 5.8: Illustration of the difference between Bla’s estimated from various multilevel uniform and optimised sequences

quency line for most of these sequences in Fig. 5.8. The top band shows the magnitude estimated using the Gaussian sequences, while the next two show the estimates using the optimised quaternary and ternary sequences. The other three bands show the estimates obtained using quinary, ternary and binary uniformly distributed sequences. The estimates using the quaternary uniformly distributed sequences are not shown, because they come between those for the quinary and ternary uniform bands, which are already relatively close to each other.

The results for the optimised quinary sequences are also not shown in Fig.5.8, because the Blaestimated from them is indistinguishable from the Gaussian Bla. This is as expected, since from Equations (5.6a) to (5.6d), it is possible to match moments 𝔐 up to𝑚 = 8, which is higher than7, the highest degree contained in the polynomial nonlinearity. It can be seen that increasing the number of levels of the input sequences decreases the difference between the estimated Bla and that estimated using a Gaussian sequence and that the difference is considerably smaller using an optimised sequence than it is using a uniformly distributed sequence.