3. Multisine excitations for SISO measurements
3.4 Algorithms to work with the phases
Crest Factor minimization
Phases in multisines were traditionally manipulated to keep their Crest-Factor low, i.e. to guarantee that:
|| min ) (
||
||
) (
||
)) ( (
| ) (
| )) max ( (
2
=
=
= ∞
t u
t u t u MSE
t t u
u
CR (3.49)
Crest-Factor minimized signal has the smallest amplitude range for the same power level, which means that such signal can be amplified (injecting more power into the system and improving the SNR of the measurement) without fearing that the excessively large amplitudes will hit the hidden non-linearities and introduce non-linear distortions into the linear FRF measurements [199-201]. Crest Factor is minimal for the binary signal yielding value of 1.
For any other signal this minimal value can be only approximated.
The required phases can be set algorithmically as so-called Schröder phases [214], [26-27], or by iterative minimizing algorithms: by suppressing the signal amplitudes by clipping [235], [189], or by minimizing ||u(t)||2p norms with increasing p (as (||u(t)||p)p → || u(t)||∞ for p →∞) [85]. Recently it is considered that the second algorithm is a winner, yielding on the average better (lower) values of the Crest Factor in the general case and for large number of harmonics. It should be also noted that whatever the algorithm, log-tone like multisines are more difficult to compress and have higher Crest Factor values, than the more or less uniformly spaced multisines.
L∞∞∞∞ multisines
The term was coined by [65] to denote the multisines with their Crest Factor minimized with the algorithm of [85].
Note: Manipulation of the phases seemingly destroys the randomness conditions imposed on the random multisines in (2.2.10-2.2.11), required in the proofs, so for a long time Crest Factor minimization was considered impossible in the non-linearly distorted FRF measurements. It turned out later (experimentally) that the situation isn’t hopeless and that the L∞ multisines seem to retain their random properties required for the BLA measurements. The phenomenon is possibly related to the richness in local minima of the Crest Factor minimization surface, where randomly started minimum search stops at still randomly distributed places. The phenomenon however resists any kind of formal analysis11.
Shaping amplitude density
Crest Factor minimization improves the SNR conditions of the measurement, but it changes drastically the amplitude density towards that of a binary signal (the lowest Crest Factor = 1).
Fig. 3.4.1 (left top) shows amplitude density of a Crest Factor optimized multisine. Such a design emphasizes strongly the extreme amplitudes; hence the linear approximation will be the best at the extreme amplitudes of the excitation. This unbalanced situation may be
11 Some experimental (not published) work of C. Evans, M. Solomou, J. Schoukens, and also my own.
undesirable for the general purpose excitation signal. An ideal signal would be a signal with e.g. a uniform (or normal) amplitude density. This requires an extended optimization method that would not only minimize the Crest Factor but would also impose the amplitude density with a specified power spectrum. The problem seems contradictory, because what improves Crest Factor, destroys good amplitude distribution and vice versa. The way out is the (heuristic) algorithm, which allows modifying the tails of the amplitude density function so that the user can balance between requirements regarding the distribution and the low Crest Factor [29*].
Algorithm 3.4.1: Shaping amplitude density - basic algorithm Consider a single period of the multisine u t =
∑
Mk=Uk k t+ k1 cos( 0 )
)
( ω ϕ , sampled at tk = kTs, with k = 0, 1, …, N-1 and ω0 = 2π/(NTs). Consider the desired amplitude density function: fd (u) with Fd(u)=
∫
vu=−∞fd(v)dv, and define Q as:) ( )
(l Fd1 Pl
Q = − , with Pl =21N +(l−1)N1 , l = 1, …, N. (3.4.2) One iteration of the algorithm consists of four steps:
(1) Consider the set U = {uL(tk), k = 0, 1, …, N-1} in the Lth iteration with:
∑
= += Mk k kL
L t U k t
u ( ) 1 cos( ω0 ϕ ); (3.4.3)
(2) Sort uL(tk) in the increasing order: (YL,τ) = sort(uL(tk)), with τ the time instances of the sorted points;
(3) Create a new multisine yL(tk) by replacing YL with Q, such that yL = sort-1(Q, τ);
(4) Calculate the spectrum of yL (with the DFT applied to the samples yL(tk)), retain the phases and restore the original amplitude Uk in this spectrum. The result is a new multisine:
∑
= ++ = Mk k + kL
L t U k t
u 1
1 0
1() cos( ω ϕ ). (3.4.4)
This process is repeated until a suitable convergence criterion is met.
Algorithm 3.4.2: Amplitude density with controlling the crest factor with a don’t care zone
If fd has long tails, the resulting multisine would have a very large Crest Factor. For this reason the amplitude domain must be partitioning into Dcrest and Df.
Assuming symmetric distributions:
U
] , ]] ,
]−∞− ∞
= a a
Dcrest , and Df =[−a,a], (3.4.5)
where [-a, a] is the support of the desired amplitude density with −a=Fd−1(0),a=Fd−1(1).
Consequently after imposing Fd, all samples of the signal are concentrated in Df, however after imposing Uk some of the samples xL+1(tk) will be smeared outside Df, into Dcrest, increasing the crest factor over the value of a.
The amplitude domain will be now partitioned in three parts by adding a don’t care region.
The amplitude distribution is only imposed in the given amplitude interval Df and left free
outside this interval:
U
] , ]] ,
]−∞− ∞
= a a
Dcrest , Ddc =[−a,Fd−1(ε)[
U
]Fd−1(1−ε),a], and 5. 0 0 )], 1 ( ), (
[ 1 1 − ≤ ≤
= d− ε d− ε ε
f F F
D . (3.4.6)
To reduce the crest factor the samples belonging to Dcrest are clipped to the borders a or –a before a new iteration cycle is started.
Note: In practice Fd is imposed on Nε = [(1-2ε)N]ceil points belonging to Df and distributed following the Alg.
3.4.1 as: Q(l)=Fd−1(Pl),
N N Pl (l /2)
2
1+ − ε
= , l = 1, …, Nε. (3.4.7)
During the design of the desired density function fd it is necessary to take care that no conflicting constraints are imposed. The power of the multisine is imposed by its amplitude spectrum by
∑
|Uk|2 , on the other hand also the amplitude density function sets the power via its second moment∫
∞∞
−
du u f
u2 d( ) . In case of desired densities of the infinite support, they must be truncated to a suitable finite interval (e.g. ±α σ) to get an acceptable Crest Factor. The second order moment is restored by selecting a proper scaling factor S for fd:
− ≤ ≤
= elsewhere
a u a S u u f
fd
0 / ) ) (
( (3.4.8)
with S the area under the truncated density.
Algorithm 3.4.3: Amplitude density – optimizing the Crest Factor
The algorithm resembles the Alg. 3.4.2., only the clipping algorithm is improved. Instead of clipping all the samples in Dcrest to the borders a and –a, a varying clipping level CL≥Fd−1(ε) is chosen using the algorithm of [235] to compress the signal. All samples with amplitude larger than CL are clipped towards this level. That way even better Crest Factor can be achieved.
In the presentation of the algorithm the uniform frequency grid was used, however in theory the algorithm could work with any frequency grid. Experience shows that the irregular grids are much more difficult to obtain a fast convergence. Although no formal proof of the convergence is available, the tests indicate a good convergence of the algorithms in case of uniform grids. However theoretical handling of the convergence question seems hopeless, considering:
1. A particular solution is a fixed point (by construction). More solutions are possible for dense grids and finite error approximation.
2. The operator of Alg. 3.4.1 is a non-linear, discontinuous, and non-expansive operator.
3. The multisine in the algorithm is confined to a non-convex set (a shell of a fixed MSE value, as the amplitude sorting and similarly the imposing of the pdf and the amplitude spectrum is ||.||∞, ||.||2 invariant).
Fig. 3.4.1 Multisine signal containing 204 linearly spaced harmonics and 8192 data points: (left top) amplitude spectrum of the Crest Factor optimized multisine, (left bottom) the same multisine with amplitude density shaped to be uniform. Log tone multisine with 204 harmonics shaped to the Gaussian amplitude density (top).