# An Algorithm to Generate Representations of  System Identification Errors

Authors: Wancheng Zhang; Patrick A. Naylor
Journal: Research Letters in Signal Processing
 (2008)

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## Abstract

An algorithm to generate representations of system identification (SI) errors, which enables systematic testing of the performance of system equalization techniques, is proposed. With this algorithm, the normalized projection misalignment (NPM) of the generated error representation can be chosen to suit the particular
characteristics of the application under test. Additionally, the generated error representation can represent all the
error vectors corresponding to different scaling factors in the estimates of the system impulse response (SIR), without influencing the signal-to-distortion ratio (SDR) of the equalized impulse response.


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## Body

## 1. Introduction

It is  commonly
known that the outcome of system identification (SI) usually includes some
estimation errors which can cause problems in the equalization of the system.
Suppose we have an estimateh^=[h^0h^1⋯h^L−1]T of some system impulse response (SIR) h=[h0h1⋯hL−1]T.
An equalization system g can be designed based on h^ to equalize the true SIR h.
Since h^ includes errors, we cannot obtain in general
an exact equalization system. The quality of g depends, to a large extent, on the robustness
of the equalization system design method to
the estimation errors induced
by the SI. Therefore, to evaluate equalization systems and their design methods
efficiently, we need to generate error vectors, which can be added to known h,
to represent the estimate h^.
Particularly, we need to generate error vectors of some desired levels, so that
the performance of equalization techniques can be tested with errors of
different magnitude. A well-recognized measure of the SI error level is the
normalized projection misalignment (NPM) [1]:(1)NPM=minβ∥h−βh^∥2∥h∥2.Another characteristic of SI is that the outcome of
many SI algorithms is determined only up to a multiplicative scaling factor
[2]. Estimates of
different scaling factors correspond to different error vectors. A natural
evaluation measure for system equalization quality is the signal-to-distortion
ratio (SDR) [3]. It
can be shown that for most of the equalization techniques, the SDR of the
equalized impulse response is independent of the scaling factor. That is to say
that, applying equalization systems designed based on estimates of different
scaling factors to the true SIR h will give the same equalization quality in the
sense of SDR.In this letter, we will first show the independence of
the SDR to scaling factors. Then, we will present an algorithm with which error
representation of desired NPM can be generated, so that the performance of
equalization techniques can be tested with SI errors at particular NPM. Since
the SDR is independent of the scaling ofh^,
we can use the error vector generated by the proposed algorithm to represent
all error vectors corresponding to different scaling factors in h^,
without influencing the evaluation result of SDR.


## 2. Problem Formulation

We formulate
the problem in Figure1. For illustration, we reduce this problem to
2 dimensions although extension to higher dimensionality is straightforward.
Consider the two vectors, the SIR h,
and its estimate (2)h^=αh^→,where h^→ is the unit vector in the direction of h^.
The estimate h^ is normally used as a basis for designing the
equalization system. The projection of h onto h^ is denoted as(3)h^p=βh^=βαh^→,where β=h^Th/(h^Th^).
Suppose that we have an equalization system(4)g=γg→such that(5)g*h^=γg→*αh^→=δ(n−τ)+Δ,where * denotes convolution, g→ is the unit vector in the direction of g, τ is an arbitrary delay, and Δ is included to represent any inversion error
in the case that g is not a perfect inverse of h^.
However, Δ vanishes when the inverse filter is perfect,
such as in MINT [4].
Then, the equalization system g is used to equalize the SIR h,
giving the result(6)g*h=g*(h^p−e),where e denotes the projection error vector
representing the difference between h and h^p.Figure 1
Illustration of the error representation.In fact, (6) can be rewritten as(7)g*h=βg*h^−βαtanθg*e→=βδ(n−τ)+β(Δ−αtanθg*e→),where e→ is the unit vector in the direction e.
The first term in (7) is the desired equalization result, and the second term
gives rise to distortions in the equalized impulse response. The SDR, that is,
the energy ratio between these two terms can be written as(8)SDR=∥βδ(n−τ)∥2∥β(Δ−αtanθg*e→)∥2.Using (4), (5), (8) together
gives(9)SDR=∥h^→*g→∥2∥Δ(∥h^→*g→∥)−tanθ(∥δ(t−τ)+Δ∥)g→*e→∥2.Above, we have shown the process of using an
equalization systemg designed based on h^,
of scaling factor α,
to equalize h.
It can be seen in (9) that in the resultant SDR of the equalized impulse
response g∗h,
the performance of the equalization system designed based on h^ is independent of the scaling factor α in h^,
subject to the assumption that Δ does not depend on α,
which is often the case. The equalization system designed based on h^p,
which corresponds to the same NPM as h^ of any α,
performs as well as that designed based on h^.
Therefore, we can represent the SI error vectors of same NPM, but corresponding
to different α,
by the projection error vector e in Figure 1. In Section 3, we will propose
an algorithm to generate e of desired NPM.


## 3. Error Vector Generation

The most
commonly used scaling independent evaluation measure of SI errors is NPM
[1] which corresponds,
in terms of Figure 1, only to the angle θ between h and h^.
It can be written as(10)NPM=(∥e∥∥h∥)2=sin2θ.Now we build the link betweenh and e so that, for a given h,
we can generate SI errors of specific NPM. The error vector e can be decomposed into two components, of
which one is parallel to h and the other is normal to h.
The magnitude of the parallel component ep is ∥ep∥=sin2θ∥h∥.
For the component ev normal to h,
we can write(11)hTev=0,(12)∥ev∥=∥h∥sinθcosθ,and substituting (10) into (12)
gives(13)∥ev∥=∥h∥NPM(1−NPM).It can be seen that the
direction of ev is constrained by (11) and its length is
determined from (13).The ensuing procedure is first to generate a random
vector orthogonal toh,
and then adjust it to the desired length. The error vector can be generated
following the steps below.(1)
Generate an i.i.d. random vector written asy=[y(0)y(1)⋯y(L−1)]T,
the elements of which are uniformly distributed on (−0.5,0.5),
where L is the dimension of h.(2)
Test if the vectory is parallel to h.
If yes, go to step (1); if not, go to step (3).(3)
Apply Gram-Schmidt orthogonalization [5] to h and the random vector to obtain a new vector y⊥ which is orthogonal to h.(4)
Adjust the length ofy⊥ according to (13) to obtain ev.(5)
Generateep=−sin2θh.(6)
Sumev and ep to obtain e.The projection error vectors of same NPM can be in
different directions in theL-dimension space. Since the vector is randomly
generated in step (1), the generality of the direction of e is not lost.


## 4. Experiment Results

The following
results show some illustrative examples of the use of our algorithm in the
context of SI and equalization of room impulse responses (RIRs). Firstly, we
compare the algorithm proposed above with existing methods. Secondly, the SDR
results are shown when using MINT and least-square
(LS) [4] inverse
systems to perform equalization of the RIRs with different scaling factors in h^.Two existing error generation methods are (i)
proportional errors [6], in which error samples amplitudes are proportional to h(n),(14)ξ(n)=ϵ(n)×h(n),with ϵ(n) being white Gaussian noise (WGN) of variance
corresponding to the desired NPM, and (ii) additive errors such that WGN is
added to h.
These error generation methods will be compared to our approach.Figure2(a) shows the NPM results of the generated
errors for an RIR of L=1600 generated with image method [7] for 20 experimental runs.
The desired NPMs are −20 and −40dB. The results show that with the proposed
algorithm, the errors of the desired NPM can be exactly generated. The
proportional errors are less accurate and less consistent over different runs.
Although the white Gaussian process is more accurate and consistent than the
proportional errors, it does not match the desired NPM exactly. In Figure 2(b),
we used a random channel of L=32,
and it can be seen that for the short-impulse response, both proportional
errors and white Gaussian process are not accurate and not consistent.NPMs of generated errors with
different algorithms.
(a)
RIR,L=1600(b)
random channel,L=32Table1 shows the SDR results with the equalization
systems designed with MINT and LS. The 2-channel RIRs used in the MINT experiments are from the image method
with reverberation time (T60) 0.2 s (L=1600) and sampling frequency 8kHz.
The RIR used in the LS experiments is one of the above two channels and it is
truncated to L=512.
The SDR results of (a) group in Table 1 are obtained with MINT and errors of
−20dB NPM, and the 3 values of SDR correspond to 3 random scaling factors α in h^.
The (b) group is obtained with −40dB NPM. It can be seen that the SDR results
obtained using our approach are independent of scaling of h^,
as desired. The NPM in the LS experiments is −20dB. The (c) group is obtained
with inverse filter of length Li=512,
and the (d) group is obtained with Li=1024.
For LS, the Δ in (5) cannot be zero since LS inverse filters
are generally not exact, and different Li give different Δ.
However, the SDR results still show independence of scaling of h^.Table 1
SDR of
different scaling factors. (a)−20dB, (b) −40dB, (c) Li=512,
(d) Li=1024.
MINT(a)α−0.43−1.670.13SDR4.484.484.48(b)α1.191.18−0.04SDR23.6223.6223.62LS(c)α−0.190.73−0.59SDR2.382.382.38(d)α0.111.070.60SDR2.742.742.74


## 5. Conclusions

We have
presented an algorithm for the generation of SI errors. We have shown by
experiments that the generated error representations can meet the desired NPM
exactly. We have also shown that the generated error representation can
represent all the error vectors corresponding to
different scaling factors in the system estimates, without influencing the SDR
of the equalized impulse response. Therefore, we conclude that the proposed
error generation algorithm facilitates reliable and repeatable testing of
system equalization methods.



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*Source: 529291-2008-11-16.xml*