Commutativity of D-Dimensional Decimation and Expansion Matrices, And Application to Rational...

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Commutativity of D-Dimensional decimation and expansion matrices, and application to Rational Decimation Systems

Dept. of ECE, SIT, Tumkur 1

CHAPTER1

INTRODUCTION

Multi-Dimensional Multi Rate Systems have a lot of applications in the field of image

and video processing. The key building blocks in these systems are the decimation matrix M

and the expansion matrix L. These areDxDnon-singular integer matrices, whereD denotes

the number of dimensions.

Most of the one dimensional results can be applied directly (i.e. by performing

operations in each dimension separately) to the multi-dimensional (MD) case, when these

decimation and expansion matrices are diagonal. However, for the non-diagonal case the

extensions are non-trivial and they need extra set of rules and notations to be specified. One

such peculiar case wherein the one dimensional results cannot directly be extended into the

MD form is the commutativity of the decimation and the expansion matrices.

The concept of commutativity refers to the ability to interchange the positions

between two entities; in the case at hand it is the ability to interchange the Mand Lmatrices.

In the 1D case Mand Lwere mere integers and hence the commutativity could be established

with a simple condition, wherein the two integers had to be relatively prime i.e. coprime.

Figure 1 interprets the commutativity concept clearly.

Figure 1: Interchange of a 1D decimator and expander.

Theorem 1: An M-fold decimator and an L-fold expander commute if and only if

ML=LMand Mand Lare coprime.

Generally speaking it is essential to show that the right coprimeness and left

coprimeness are different for the matrix case, but it can be showed that these two are

equivalent for the case at hand i.e. when ML=LM.

One of the major applications of commutativity in such cases is the option availed to

implement polyphase decomposition via the interchangability that is hence availed after the


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commutativity is exploited. Figure 2 shows a rational decimation system that allows us to

vary the sampling rate of a signal by a factor of M/L. Such factors are of the rational nature

as evident, hence the rational decimation systems. The implementation of this paper aims at

the very same concept, polyphase implementation of rational decimation systems.

Figure 2: Rational decimation system.

The filter H(z)which operates at a rate L times the rate of the input signal, is used to

suppress image components generated by the L-fold expander and to avoid aliasing due to theM-fold decimator. The major fault or the problem that arises in such an implementation is the

load on the filter and also the redundancy that is created due to the fact there lies a decimator

after the filter. The expander that is present prior to the filter increases the samples and hence

feeds it to the filter, thus increasing the load on it. This results in the requirement of higher

order filters. Also, the filtering process results in a redundancy after the complete

implementation is observed, as the decimator reduces the number of samples. This indirectly

mocks at the performance of the filter. In order to get a better implementation of such rational

decimation systems the concept of polyphase implementation is used. The polyphase

decomposition of the filter H(z) followed by a successive implementation of the same

concept via the exploitation of the commutative nature of the decimator and expander results

in a more efficient implementation. Such an implementation is referred here as the Rational

Polyphase Implementation (RPI), wherein the filter operates at a rate 1/M times the input

rate, so the efficiency is LM times that of the direct implementation. As mentioned earlier,

the RPI technique can be applied only if the decimation and interpolation factors Mand L are

co-prime.

The same concept can be applied to MD decimation systems with rational decimation

ratio but under certain conditions as discussed. The RPI of the MD systems will be hence

derived in the latter part of the report, such that it can be applied to any MD system wherein

M, Lcommute and also ML=LM.


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CHAPTER2

Prerequisites

Capital and lowercase boldfaced letters denote matrices and vectors, respectively. The

notations ATand A-1denote the transpose and the inverse of A, respectively. The row and

column indices typically begin from zero. WithDdenoting the number of dimensions,

n=[ no nl nD-1]Tis the time domain index of MD discrete signals. The symboldenotes

the set of allD x 1integers, so that n.The complex vector z=[zoz1... zD-1]Tis the variable

of the z-transform of MD signals. The z-transform of x(n), where it converges, is given by

X(z) = n x(n)z-n. The notation zkis a scalar.

2.1 Decimation and Expansion

The M-fold decimated version of x(n) is defined as y(n) = x(Mn), where the

decimation matrix M is a non-singular integer matrix. On the other hand, the L-fold

expanded version of x(n)is defined as,

y(n) = x(L-1

n), nLAT(M);

0 , otherwise.

In the above equation, LAT(V) (the lattice generated by V) denotes the set of all

vectors of the form Vm, m. The corresponding z-domain relation of expansion is

Y(z)=X(zL). The notation zLis a Dx1 vector.

2.2 Polyphase decomposition

Polyphase decomposition is a technique that was initially employed for the

purpose of getting a computationally efficient implementation which would result in

the optimal usage of the elements used in a typical rational decimation system.

The polyphase components of x(n) with respect to a given Mare defined as

ei(n)=x(Mn + ki), (Type 1)

ri(n)=x(Mn

ki), (Type 2)


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where ki(M) and (M) is the set of all integer vectors of the form Mx,

0xi


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a. R0is a non-singular crm (M, L).b. If Ris another non-singular crm (M, L), then R =R0Sfor some integer matrix

S.

The common left multiple (clm) and the least common left multiple (lclm) are defined

on similar lines.

Singular crm is of less importance because given any non-singular crm (M, L), we

can always post multiply it by a singular matrix to get a singular crm ( M, L). Also, if either

M or L is singular, all crm (M, L)s are singular and it is meaningless to discuss the

lcrm(M,L). For these various reasons, by definition we restrict the lcrm to be non-singular

and to be defined only for non-singular Mand L.

Based on the above definitions the below two lemmas can be defined and proved.

Lemma1: Let Rbe a non-singular crm (M, L), i.e., R = MP = LQ. Then, Ris an lcrm(M, L)

if Pand Q are right co-prime.

Proof:

1. If Pand Qare not right co-prime, then there exists an Xwhich is not unimodular suchthat P = P'Xand Q=Q'X. Therefore, we have

R = MP'X = LQ'X

R' R'

Clearly, R = R'X, and R'is a crm of M andL. Suppose Ris an lcrm of Mand

L, then R' = RSaccording to the definition of lcrm. Then, R = R'X = RSX, which

implies both Xand Smust be unimodular and thus leads to contradiction. Hence we

conclude that if Pand Qare not right co-prime, Ris not an lcrm of Mand L.

2. Next, suppose Pand Qare right co-prime, we have to prove that Ris an lcrm of Mand L. Let R' be any other non-singular crm of M and L, i.e., R' = MP' = LQ'.

Clearly, P' is non-singular, because P and Q are right co-prime, there exist integer

matrices Aand Bsuch that AP + BQ = I(generalized Bezout theorem). Replacing Q

with Q'P'-1

P, we can rewrite this as

AP'P'-1

P + BQ'P'-1

P = I

Post-multiplying both sides by P-1P', we get

AP' + BQ' = P-1

P'

SSo, P'=PSand hence R'=RS. From the definition of lcrm, Ris indeed an lcrm(M,L).


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The left multiple (lm), common left multiple (clm), and least common left

multiple (lclm) can be defined and also defined similarly.

Lemma2: When ML = LM, any lcrm(M,L) and any gcrd(M,L) can be related as

lcrm(M,L).U.gcrd(M, L) = ML for some unimodular U. Also, when ML = LM, the

following four statements are equivalent:

i. MLis an lcrm(M, L).ii. Mand Lare right co-prime.iii. Mand Lare left co-prime.iv. MLis an lclm(M, L).

For the 1D case, this nicely reduces to lcm(M,L) = MLif and only if Mand Lare

co-prime, which is a well-known fact.


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CHAPTER3

1D Rational decimation systems

A simple rational decimation system in the 1D frame can be easily explained and

understood as well. The objective behind understanding this to get a clear picture about the

successive redrawing of the polyphase implementations of a 1D rational decimation system

yielding an efficient system eventually.

Figure 4: 1DRational decimation system

Consider an up sampling factor of 2 and down sampling factor of 3 (so that the

concept of commutativity can be exploited and explained simultaneously). The two types of

polyphase implementations can be expressed as shown in figure 5, wherein (a) represents the

type 1 implementation and (b) represents type 2.

Figure 5: Types of polyphase implementations of the 1D rational decimation system at hand.

Successive redrawing of the polyphase implementations results in an efficient system

eventually. Figure 6 hierarchically explains the successive implementation that is being

discussed as a remedy to overcome the efficiency problem.


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CHAPTER4

MD Rational decimation systems

Starting from the basics of commutativity and the conditions related to the application

of this concept to a rational decimation system, a series of implementations of the polyphase

decomposition can be illustrated in a similar manner as shown in chapter 3. The only

difference is the extension of the concepts to the MD case.

The polyphase implementation of an MD decimation system without performing

successive redrawing is shown in figure 8.

Figure 7: Polyphase implementation of an MD decimation system

Figure 8: Successive redrawing of polyphase implementations of a MD rational decimation system.


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In figure 8, the equivalent subparts indicate the following,

(a)Type 1 polyphase implementation of the MD decimation system. Suppose it ispossible to replace every ki in (M) with Mki1 + Lki2,where kil and ki2 are some

integer vectors.

(b)With the help of noble identities and the flexibility availed in the positioning of thesampler relative to the summation, this subpart can be realised.

(c)Exploiting the concept of commutativity, the up and down samplers can beinterchanged under the specified conditions. Hence this subpart.

(d)Employing the type 2 polyphase implementation, arrival at this sub part from theprevious is justified.

In summary, Figure 8(d) is equivalent to Figure 7, but each arithmetic operation is now

performed at its lowest rate, i.e., l/J(M) times the input rate.

Thus we can see that the following two issues should be considered for the above

technique to work:

1. The decimators and the expanders should commute.2. Every k, in(M) should be expressible in the form of ki= Mki1+ Lki2, where ki1and

ki2are some integer vectors.

The conditions for commutativity have been given earlier. So, it only remains to consider the

second issue. Using the generalized Bezout theorem, we know that when Mand L are left

coprime, there exist integer matrices Pand Qsuch that I = MP + LQwhere Iis the identity

matrix. Then, every integer vector k can be expressed as k = MPk + LQk = Mkl + Lk2.

Using Lemma 2, we can conclude that the above mentioned MD RPI technique is feasible if

and only if, (i) ML = LM, and (ii) M and L are co-prime.


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REFERENCES

1.

Tsuhan Chen and P. P. Vaidyanathan, Commutativity of D-dimensional decimationand expansion matrices, and application to rational decimation systems, IEEE, 1992.

2. Tsuhan Chen and P. P. Vaidyanathan, The Role of Integer Matrices inMultidimensional Multirate Systems, IEEE transactions on signal processing, vol 41.

no.3, march 1993.

3. Tsuhan Chen and P. P. Vaidyanathan, least common right/left multiples of integermatrices and applications to multidimensional multirate systems, IEEE, 1992.

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