Multidimensional Multirate Signal Processing

Chapter 2 Multidimensional Mult irate Signal Processing 2,1 Introduction This chapter extends the basic concepts of rnultirate signal proce~singto multidimensional multirate signal processing. Examples of multidimensional signals include images in two dimensions and video in three dimensions, Good reference texts for bwkground materid on. multidimensional signal promwing are Dudgeon and Mermresu[lfjj and Lim[26]. The important concept of sampling is related to the mathematics of lattices; see for example Cassels[41. The engineering analysis of sampling begari in the classic paper by Petersen and Middleton[38] and later extended to include decimation and expansion considerations by Mersereau and Speake[31] and Dubois[ltij. The multidimensional z-trclrrsform is carefully described by Viscito and Allebrzeh[55) The idea of the Smith form was first articulated by Smithpq, Many recent papers have dealt with applications of the Smith form to the multidimensional DFT (Guessoum and MersereaupO]), to the multidimensional DCT (Giindgzhan e t a1.[21]), and for the development of multirate CAD toolo (Evansf28I). Some of the concept8 dewloped in this chapter are also discussed in the text by Vaidyanathan[49]. Section 2.2 presents a framework for the study of multidimensional m l tira;te signd processing and it introduces two important representations far multidimensional signals. Section. 2.3 presents the multibirnensional building bloehs. Section 2.4 provides ways to interchange the mltidimensiorral building blocks.

A multidimnsional signal is a signal of more than one variable. This section systematically presents concepts that act as s kamework for our study

363

Chapter 2 Muftidimensionaf Muftirate Signal Processing

of the application of multirate s p t e m ~to multidimensional signals, These canceF>tsinclude sampling of a multidimensional signal, which involvw tbe mathematical concept of a sampling lattice, and introduce multidimensional sampled signals by way of multidimensional ;r-tramforms and multidirnensiclnal Fourier transforms.. Now, we will discuss the mathematics needed t o describe the s~mglingconsiderations of multidimensianal multirate. 2.3;.I Sampling lattices

Sampling a multidirnemional signal is more complicated than a one-dimensiorral signd because of the many ways t o choose the sampling geometry* Sampling points could be arranged on a rectangular grid in a straightforward manner, but many times there are more efZicient ways to sample mu1tidimensional signals, Some applications are mare suitalble for nonrectangular sampling than rectangular samplng. F"or example, the testing of a phared-array antenna rquires the measurement of the electric field on a plane in the near field of the antenna. Since phased-array antennm are typically designed with an hexagand arrangement of efennends, the resulting processing is mare wcurato if hexagonal sampling is used for the memurements. One reman to consider nonrectangular sampling is Lo minimize the number of points rtwded to charwterbe an nil-dimensional hypervolume. Moreover, if the functions of interest are bandlimited aver a circular region, then Figure 2.1 shows that significant savings are possible if hexagonal sampling is used instead of rwtangular sampling. In order to precisely describe both rectangular and nsnrectangulw sampling, we nwd a convenient way to describe an arbitrary sampling geometry. To do this activity9 we must appeal to the laxlguage af linear akgebra in order t o present the mathematical theory of sampling lattices. 2.2.1.1 Linear independence and sampling lattices

The idea of a vector in an integer-valued k-dimensional space is a generalization of vectors in a plane by ixl6eger-tralrred Cartesian coordinate. This leads to the foliowing definition. Definition 2.2.1.I. The set of a11 le-dirnensio~a1integer vectors wjfl be cdIed the fgndamental lattice and it will be denoted N.That is,

N = {r = [rl,rz,.. . ,rkjT

rI is an integer).

The set of all k-dimensional real vectors will be denoted 8. Mow, let us review a few definitions related to sums of vectors.

2.2 Multidimczngianal frame work

Signal Gtim-snsionaliQ

Figure 2.1. Percent swings: hexagon& versus rectangulw sampling.

Deftnition 2.2.1.2. A linear combination of N vectors {rl,. . .rN) E hi is an expression of the form N

where %, 1: = 1, . .. ,N , are integer8 and arc called cocfticients. The set of vectors {rl,. . .r ~ L4) aid t o be linearly independent if

Cniri = i= 2

* ni = 0 for all ii.

If the set of vectors {rl,. .. rN), is linearly independent, then the totality of vectors of the forr~

is called a N-dimensional lattice and it will be denoted by R. IB other words, the space R spaaned by the set of vectors {rl ,. . . ~ J )J is the space consisting of all h e a r combinations of the vectors, that is,

The set; 08 vectors (rl,...rN) Is a h s i s for R if dhey are linearly indepersdent and the g p x e is spanned by {rl,. ..rlv) is egud to R. Then, we say that R has dimension N. To better understand the definition of the lattice, consider the following illustration in two dimensions, Let r1

==

TI 1

T21

,

r2

"

T12

c22

and

n, ==

a1 7%

Then,

or equivalently., 72.1r1

+ 7L212 =

nlrll

t nzr12

nl Tzl + rb2T22

This can be rewritten as a matrk-vector product, i.e,

where the matrix R is cdallfed the sampli~gmatris, In general, let ri be the ith column of the matrix IR, that is

then the sampling miaLrix R is said to generate the lattice R. As such, the lattice R is dso given by

R=LAT(R) = { ~ E NmI= R n for n E N 1.

If R is the identity matrix, then each ri is a unit vector painting in the ith

direction, and the resulting lattice, 'R, is the fundamental lattice N. Let us present some examples of sampling lattices usirlg bfzk dots to represent the lattice points and white circles to represent points in N that are not in LAT(R). Example 2.2.1.1. Consider rectangular sampling defined by the sampling 2 0 rnatraX R = . It i s depicted in Figure 2 2 . 0 3

Figure 2.2. Lat tiee structure using rectangular sampling matrix.

Figure 2.3. Lattice structure using hexagand ;tlmplingmatrix,

Chapter 2 Multi&rnensr"o~a1Mulgirate Signal Processing

Figure 2.4, Latt;ice structure with quincunx smgfing matrix.

Example 2.2..1 ..2. Consider .. hexagonal sampling defined by the sampling 1 1 . It is depicted in Figure 2.3, matrjac R .=: 2 -2 Example 2,2.i ,3. Consider quincunx sampling defined by t he sampliag 1 1 matrix R -It is depicted in Figure 2.4, -1 1

.

Far a given sampling matrix R, the corresponding Fourier domain sam- ~the lattice it generates is called the reciprocal pling matrix is ~ T R and lattice.

The m a t r k that generates a, la~tticeis not unique, AS we wiH gee later in this subsection, the follawing matrices generate the same lattice.

The theory underlying the nonuniqueness af these sampling litLtiGes is based on unimadular matrices. Thus, in order to dlkcus8 the nonuniqueness of sampling lattices, we must first briegy discuss unimoduiar matrices.

2.2 Multidimensional framew~rk

35

DeBxritian 2.2.1.3. An integer-valued matrix A is a unimortulizr matrix if

The following theorems provide properties of these mizlriees. Theorem 2.2.1.1. If A is an integer-valued unimoduiar matrix, then A-'

eksts and is an integer- vafued unimodular matrix.

Praof: Let A be an integer-valued unimodular matrix. Moreover, since det A f 0, then A-' e f i ~ t and s AA-' = I. Since det A det A"-' = 1, then = 1. Since A is unirnodular, then

where the (i,j ) elernelit of the adjugate(A1 is equd to the cofactor of the ( j ,i ) element of (A). Since A irs uriirnodufar, then A-' = f adjugate (A).

Since A is an integer-wlued matrix, then ad,jugatefG) ia an integer-valued matrix. Therefore, A-I is an integer-valued unimodular matrix. II With this background on integer-vaIued unimodular matrices, we are ready to discuss the isaue of nonurriqueness of sampling 1;jtetices. Theorem 2.2.1.2. Given a nonsingular maLrix M and an iateger-vdued

unimodular matrix V, then LATWV)

-- LATW),

Proof: If x E LAT(M), then there exists an n E N,such that x = Mn =

.

MVV-'n = MVm, where m = V-'n Since V is an integer-valued unimodular matrix, then V-I is also an integer-valued unimodular matrix by Theorem 2.2.1.1. Since V-' is integer-valued, then m = V-'n i s integervalued. Therefore, x E LAT(MV). Therefore, LAT(M) E LAT(MV). Obviously; LAT(MV) C LACP(M). Since N and MV generate the sarare la.ttice, then ZAT(MV) =: LAT(M). 1 FOF notation pufposes, let 3(M) =f det; M the absolute value of the determinant of the sampling matrix: ha. Theoxem 2.2.1.3. Given a nonsinguliar matrix R?1; and an integer-dued unirnodular matrix Y. Then, JfM) = J f M V ) = d(VM),

36

Chapttsr 2 Mul ti&mensiond MuItirate Signal Procesging

Prooh Sirice J(M) =I det M

1,

then

J(MV) = J(M)J(V) = J(V)J (M) = J (VM). Since V is an intqer-vrtlued unimodulax: matrix, then J(V) = 1. Herlce,

a

Therefore, J(M) is unique and independent af the chaiee of basis veetors. Moreover, J(M) can be interpreted g e o m e t r i d y a the k-dimensiond volume of the parallelepiped defined by M. Sometimes J(M) is called the sampling density. Now, consider the falltowing sampling matsices

R=

1

0

2 -4

and S =

1 1 2 --2

where J(R) = J(S) = 4. Sirice RE = S for unimodular matrix

then R and S generate the same sampling lattice. In this case, the sampling lattice is known aa an hexagonal saxnpEing laktice,

2.2,1,3 Unit c e b and fundamental p~dlelteplpeds Definition 2,3.1.4. Given an integer-vrtlued matrix R, a unit cell i ~ c f a d e s

one lattice point from &AT@) and J @ ) - l adjacent points in N that are not in &AT@).

If these unit cells are periodicdly replicated 0x1 LAT(R), then the entire spwe is tiled witfi no overlap. Thus, the unit cell is a footprint that characterizes tire samplirkg lattice. Definition 2.2.1,6, Given an irmteger-mlued matrix R, the fundawkentrzl parallelqipetf of 1Iattice L A T O , denobd FPD(a), is the unit cell fiat ineludes the origin mid i~ bounded by d Iat tice poini;s one positive unit away. Formdy, the f u a d a m e ~ t dpard1dep;tped is givm by

where E is the set crf aJf k-dimension& real wetors.

2.2 Multidimensional framework

31

Consider the fallowing example as an illustration of the concept of fun2 0 damental parallelepipeds. Assume the sampling matrix R = 0 3 Then, fundamental ppardlelepiped is given in Figure 2.5. By inspection, there are J(R) = 6 points in the fundomental parallelepiped. Theorem 2.2.3 -5%. Given an i n t e g e f i v d ~ e dunimod~larmatrix U and an

integer-vdued diagonitl rzlaitrk A, t f i e ~

Prosh By the definition of the fundamental psrauelepiped,

FPD(UA) = {y Let s = Ax. Then, x = A-'g.

EE y

= UAx for all x E

Hence,

Since iJ i s an integer-valued unirnodular matrix, then Uz is an integervalued vector if and only if a k an integer vector, Therehre,

FPD(UA) = U{y E 6' y = z for all A-'z E [ O , ~ ) ~ ) .

Chapter 2 Multidimc?nsionaiMuitirate Sigad Processing

38

But

2;

= AX. Eenee, FPD(f5A) = W(y E @ Y = Ax for all x E [O, I ) ~ )

or equivalently,

FPD(UA) = U FPDCA). 1

Sometimes im the literature, authors will refer to the Symmetric parallelepiped, denoted SPD(R). It is defined by

SPD(R) = {y E E y = Rx for all x E [-I., I.)~).

DefiniCisn 2.2.1.6. Let V1 and Vz be Ic x Ic integer matrices. LAT("V2) iu cdled ia sabladbice of LAT(ITl) if LATrJ2) C LATF1), that is, every point of LAT(V2)is d g a id p ~ j n of t L A T p lf .

Let V1and Vz be integer matrices, where LAT(V2)C LAT(VI). Then, for every m E H , there exist n E such that

or equivalently,

n =V ; ' V ~ ~ .

Since n and m are integer vectors and since V l n = Vzm, then v;'v~ must be an integer-valued matrix. Let L = V;'V~. Then, VIL = Vz. Since det Y rL = (det Vr)(det L) , then det V2 = (det V t ) (det L)

so that,

J(Vz) = J(V1) (det L) .

Hence, J(Vz) is an integer multiple of 3(V1). For example, if Vz =

2 0

4 0

0 4

, then V z is a sublattice of V1.AS such, J(Y2) = I6 0 2 and J(V1) = 4, then J(V2)iu, as expected, an integer multiple of J(V1), i.e. J(Vz) = 4J(V1). An important special case results when the lattice LAT(V1) coincides with the fundamental lattice N,i.e., LAT(V1)= LAT(I). In addition, p = represent8 the number of cells of FPD(V1) which can fit into FPD(V2). The lattice point in each one of these cells can m d V1 =

2.2 IMirltidI'mensr'ur~afframework

39

be thought of as a shift vector a, which if added t o each vector of LAT(V2) will generate an equivalence class of points called a coset. The union of all cosets is LAT(V1). Thus, the concept of a coset will provide a natural way to partition LAT(V1) into subsets, which is a necessary step for the generation of multidimensional polyghttse components. Befinition, 2.2.1.T. Let Vl and V2 be integer-vdued nzatrr'cm such that

Let a E LAT(Vl)fl FPD(V2). Define the coset C(V1,Vz,a) to be

If LAT(V1)= hi, then by convention, V1 is not explicitly identified, that is, the coset is simply

Given integer-valued matrices V1 and Vz such that LAT(V2)CLAT(V1), then denote the set of shifit vectors by

Similar to the convention for cosets, the convention for shift vectors when LAT(V1) = .A/' is not to explicitly identify V1, that is, the set of shift vectors is simpXy

4 0

and V1 = 0 4 then the cosets are uniquely defined by the following set of shift -vectors Returning to the example above, if TJZ =

Let us briefly review some elexnentary operatioas that be applied to integer matrices, These oper;;tlioxrs will be essential when we subsequently discuss the Smith form decomposition, a method for diagonalizing the sampling matrh.

Clxapeer 2 Multidimensional Muitirate Sign& Procrtsging

40

Etementary ruur (or column) operatima on integer matrices are important because they permit the patterning of integer matrices i&o simpler forms, such as triangular and diagonal forms. DeBnitian 2.1&.1,8. Any elementary row operatioxl on m integer-vdued m a t r k P i s d d e ~ e dto be any of the fo&owing:

Type-1: Interchaagtt two rows. Type-2: Multiply a row by a nonzerointeger constant c, Type-3: Add an i ~ t e p multiple r of a row to anather row, These aperatiotons can be represented by gremultiplying E) with an appropriaLLe square matrix called an. elementary m&rix, To illurJtrate these elementary operations, consider. the followhg examples, (By convellt;ion, the rows and columns are numbered starting with zero rather than one.) The first emmgle is a Type-l elemenlazy m a t r k that interchanges row 0 and raw 3, which h a the form

The second example is a Type-2 elementary matrix that multiplies elements in row 1 by c $ if:? which has the form

The third example is a Type-3 elementary matrix that replaces row 3 with row 3 fa * r w Q), which has the farm

+

Afl three types of elementary polynomial matrices w e integer-valued unimodular matrices.

41

2.2 Muft;idimensioxlal framework 2.lt.X.6

S d t h form decomposidfon

The Smith Form Decomposition p r o ~ d e sa method for dbgonalizing the sampling matrix, When matrices are diagonal, most one-dimensional results esn be edended automatically by performing operations in each dimension separrately. However, in the nondiagond cwe, these extensions are nontrivid and require complicated notations and m a t r k operations, Theorem 2.2.1.9, Every infeger-valued matrix corre~p ond i ~ Smith g form deeompo~itio~ aa

R ewl be expre~sedin its

where U and V are iateger-dued unimodular matn"ces md the Smitfr farm A is given by

where r is the rank of R and A;[ Xi+1,

i = 0,. .. ,T

- 2.

Proof: Assume the zeroth column of R contsbins a nonzero element, which may be broughL to the fQ,O) position by elementary operdions. This dement iar the ged of the zersth column. If the new (0,O) element does not divide all the elements in the zeroth row, then it may be replxed by the gcd of the elements of the zeroth rmv (the effect wilt be that it will contain fewer prime factors than belare). This process is repe&cd until ran elerneat in. the (0,O) position is obtained which divides every element of the zeroth row arid column, By elementary row and column operations, all the elements of the zeroth row and column, other than the (0, Of element, may be made zero, Denote the new submatrix formed by deleting the zeroth row and zeroth column by C. Suppose that the submatrix G contains an element C i j which is not divisible by ma. Add colu~lnj to column 0, Column O then consists of the elernents coo, el,j , . ,e,-1, j . Repeating the process, we replace by a proper divisor of itself using elementary operations. Then, we must finally reach the stage where the element in the (0,O) position divides every element of the matrix, and all other elements of the zeroth row and column are zero. The entire process is repeated with the submatrix obtained by deleting the zeroth row and column. Eventually a stage is reached when the m a t r k has the form

..

42

Chapter 2 Multidimensional MulGirate Signal Processling

where D = diag (A@, . . . , and &[&+li,i-- 0,. . . ,r - 2, But E must be the zero matrk, since otfierwiiie R woufd have rank larger than r. m

Note that although the two unimodular matrices U and '\r are not unique, the diagonal matrix II is uniquely determined by 33,.

Example 2,2.1.4. To illustritte the Smith form decomposition, consider the

sampling. If we divide the (1,0) element, which corresponds to hexa,~~onal 2, by the (0,O) element, 1, ure obtain

2== 2 (I)+ 0 * quotient remainder Therefore, if we apply a Type-3 row operation, which is defined by

to R, we will reduce the (1,O) element to zero. Therefore,

Dansform the (@,I)element t o zero by a Type-3 column operation, which 1, -1 is defined by . Then, we obtain 0 1

Finally, the (1,l)element is forced t o be positive by a Type-2 row operation, 1 Q which is defined by . The~l,we obtain Q -1

Thus,

2.2 Multidjmerr~ionaiframework

Le%E be the product of efementary raw operations, i.e.

Let E" be the product of elementary column operations, i.e,

since only one elementary column operation was performed. Therefore,

Then, the Smith form decomposition is given. by

where,

and

Theorem 2.2.1.6, Let and TC" be unimodular matrices and let A be a &agoad matrix. H the Srnith form decomposition of smpfing matrix R is

given b;y R

==

UAVl then

ProoE Since LAT(AV) = LAT(II) by Theorem 2.2.1.2, then FPD(AV) = FPD(A). Therefore, FPD(UAV) = FPD(UA). Since FPD(UA) = U FPD(A) by Theorem 2.2.1.4, then FPD(R) = U FPD(A). rn 2.2.2

Mul$idimensionill sampled sign&

Unfortunately, some important sampling structures can not be represented as a, lat;lice, Fbr example, consider an important sampling structure h r High Definition Television(HDTV) called line quincunz, where two samples are placed one vertically above the other in place of every sample in

44

Chapter 2 MuItidimerlsiorrd Multirate Sig~aXPraces~ing

the sampling grid, But, line quincunx can be represented as the union af twa shieed lattices using the rnuftidimensiond ~ t r a n ~ ~ f o r m VVe i , will firat define some underlying vector mathematics and, then we wiu present the multidimensional ;=-transform and the multidimensianal discrete 1Faurit;r transform, Then we will present two nnuftidimen~iodsign.& representations - the modulation representation. ~ n the d pollyphmt: representation. The theory of multirate and vavelet signal processing is considerably simplified by the use of these representations. 2,2.2, J

Vector

In order to gencrdize the definitions tihat we have grown accustomed to seeing in one-dimension, we wilX provide the definition of a vector raised t o a vector power and, subsequently, the definition of a vector ra&ed ta it m a t r k power. Given eomplttx-vdued vectorr = ..., 7 ~ - 1] and integer-valued vector s = [ so, .=. , s ~ - 1 T . Then, the vector r r&sed to the vector s power is a sealtar and it is deaned to be

T

Definition 2.2.2.1.

Then, building on this definition, we will define a vector raised to a matrix power.

.

Definition 2.2.2.2. Given a complex-valued vectorr = [ ro, . . , ~ j v - 1 ] a d an i~teger-vduedma&rix&= Id0, ..* L N " 4 ,where hi is the i tlx column of L, the^, the vector r r&~edto the matrix L pawer is a row v e e t ~ and r if;is defined to be

.

Definition 2.2.2.3. defined by

The b-dimensional z- transform of z(no,...,lab - 1 ) is

T

2.2 Mtlltidimeasiand framework

where, z = [zO,..., z ~ - i~~ a ]complex-valued ~ vector, n = (no,...,n k - l l T is aa integer-vdued vecCor, and

Let L be an integer-dued noasingufar m;ttrix, &en by Elefiaitjon 2.2.22, zL i~ given by where, Li irs $he ith colrrrnn of L, that is,

Theorem 2.2.2.1. Let L he an integer-valued matrix where Zi, z' = 0,. . . k1, are the columns of Z1, Then,

Proof: Using the Definition 2.2.2.2, vve can write

where, Li is the ith column of L. Substituting the definition of a vector raised to a vector power yields

where Lmli is the .nth comporlent of Lie a wctor power gives a scalar. Hence, k-l

/k-X

By definition, a vector raised

',

to

46

Chapter 2 Muitidimensiond Multirate Sip& Processing

Sirrce the product of terms with the same base equals the base to the sum of the exponents, the last equatkn becomes

I

rn

If the z-transform converges for all a;, of the form z, =; exp(jwm), = OO?. .. k - f then the n-transform car1 be represented as the sum of

harmonicdly refated sinusoids, i.e.

which is the multidimensional generalization of the discrete-lime Fourier transform, In order to quickly distinguish vectors in Fourier space, the vector Fourier variitble 2 i s denoted by an underlined omega rather than a Bold omega. Theorem 13.2.2.2, Let I; be an i~lteger-valuedmatrix. Then, exp ( j w j L

= exp ( j ~ ~ g ) .

Proof: Using the Definitiozl 2.2.2.2, we find that

where, Li is the ith column af L. Since the exponents of each term are simply inner products between g and a coXumn of L, their order can be interchmged, that is,

The multidimensiond discrete hur-ier transform is an e x x t Fourier regresentation for periodicasjly sampled arrays, Therefore, it takes the form of a periodically sampled Fourier trax~sftzlrm,As irz the one-dimensionai case,

the nnultidimensiond discrete Fourier transform can be interpreted ES a Fourier series representation for one period af a periodic sequence. 111 this formulatim, we wiH have to address t m types of geriodicities --orre due to the sampling lat;tice and one due to the s i p & (tbat is, defined on lattice points) to be Fourier trazuformed. Let V denote the sampling mat^, L e e , hexagonal, quincunx, rectangular, etc. Let N denote the periodicity matrix, which characterizes the periodicity of the lattice points on which the signd to be Rurier trmsformed is defined. Assume LAT(M) is a sublattice of LAT(V). Then, we define equivalence eIwses betvveen periodic replicw of the data by

[n] = { m E LAT(V) n - m e LAT(N) ). Therefore, if parailelogribms are drawn between the elements of LAT(PJ), then any tm vectors that occupy the same relati= pasition are in the same equivalence cfms, Many properties of the periodicity matrix, N,follow by a n d 0 0 from the corresponding facts far sampling matrices. For example, the density of the periodiciQ matrix is uniquely defined by , denoted J(N); but for a given periodic sequence the periodicity matrix N is not uxlique, since it, can be multiplied by any unirnodular miztriur and still describe the same periodic signd. In additioa, the columns of fiJ indicate the vectors along which it is periodically replicated, RefOniLion 2.2.2.4. A multidhcn.siond sequence a(n) is pesio&c FviGh pen'od N,that i ~for ? all n?r E JI/, z(n) = ~ ( nNr).

+

Let XM represent one period of z(n). Tllea,

where V defines the unddying sarrrpling lattice, Moreover, since z(m) is periodic with period N, X(g) crzn also be written as

X(w) =

x

nEZr-3

But,

z (n)exp

- j g ~ n exp[] jwTv~r].

Therefore,

exp[- j w T v ~ r= ] 1,

which is equivalent to the condition,

where m is a vector of integers. Upon further examination of observe t bat uT = 2?rmT(VN)-I,

2,we

.n ."

or equivalently, Therefore,

w =2 n ( ~ ~ ) - ~ m . -

The matrix ~ ? I ( v N )serves - ~ as a Fourier domain sampling matrix. Substituting this equation inta the equation for Xfg) yields

or equivalently,

Let us further examine the inner product which occurs in the argument of the exponeatiaf,

Suppose the multidimenaiand sequence X(mf is periodic with period P, that is, X(m) = X(m Pq) for m, q E N.Also, let IF reprevent one period of X(m). Then, by analogy with the one-dimensional discrete Fourier ) the follovvlng form for some constarlt a, transform, assume that ~ ( ahas

+

Invoking the periodicity of X(m),that is, X(m) = X(m z(n) t o become

+ Pq),will cause

ar equivalently,

But,

Therefore,

2 z(n) = X(m)exp[jnT( 2 ? r ~ - ~ ) m ] . a mEZp

e ~ ~ [ j n ~ ( 2 ? r ~=-1~for) ~allqq] E N.

Since xl and q are integer-valued vectors, then or equivalently,

P = bIT. Therefore, X(m)is periodic with period NT,that is, X(m) = X(m + NTq). Hence,

+

Now let us determine the constant er: by substituting the equation for z(n) into the equation for X(m), Hence,

which is as expected, since J(N) =;I det N I is the number of samples in one period for LATIN), Therefare, the multidimensiond discrete Fourier transform pair are given by:

It should be noted that these equations reduce to the usuaf discrete Fourier trantzform. pair in the one-dimensional cme and t o the familitzr rectangular muldidimensional discrete Fourier transform when N is a diagonal matrk. As an illustration of this theoretical development, sometimes it is of interest to input data from an arbitrary lattice and output it an iz rectangular lattice, so that it could be conveniently displayed an a eompzoter display. Assume that V is degned by

For hexagonal input: a = 2, b = 1, c = 2, Moreover, for quincunx input: a == 2, b = 1, c = 1. In addition, for rectangular input: b == 0, Select ht periodicity madrk so that VN is a diagonal m a t r k so that the resulting Fourier analysis will1 be on a rectangular grid, Now, let ur~pick N to be

then this N m a t r k is a good choice for a periodicity mittrk. Therefore, the DFT becomes

where,

2.2 Muftidimensional framework This suggests the following algorithm: (I) Compute &HI-point FFTs, one for each row in the

51 7&1 direction.

(2) Apply. a. phauc shift to each paint of the resulting data.

Since we are mrking with a sampling grid with samples a t integer-valued locations, it is important that we perform the phase shift for integer multiples of But :nzml is real-valued. Therefore, we will need to quantize :nzml to integer values through the use of the round function. ( 3 ) Compute &&-point FFTs, one fbr each column in the 7 ~ 2direction.

R.

2.2.2.4

The S d t b foran and the DFT

First;, let us begin with the mul$idirnensionaI discrete Fourier trrtrrsfornn, that is, F(k) = f (n) exp

C

DE~M

Replace M with its Smith form. decornpositian, that is, M = TJAV. Then, using Theorem 2.2.1.2, LA'I"(UAV)=: LAT(UA). Novv, using elementary linear dgebraic operations, let us simplify the exyorrent of the exponentid

= exp [- j 2 ? r ( ( ~ h ~ ) - ' n ) ~ k = exp [-j2nkT(UhV)-In Then, the mzrltidirnensiond DF'T becomes

where

m = U-'n and pT= k T v - ' .

52

Chapter 2 Mult;idimensi'onalMuf tirate Signal Proeegsing

Therefore, the Smith form permits the use of a rectangulm DFT, when the initial data lies on. a nonrectangular sampling grid. The initial data must have pardlelepiped spatial support which becomes rectangular after being mapped by U-I. Moreover, the larger the values of the elements of U-I , the more the spatial suppart will be skewed. Thus, the aill;ar&hrrrfor the Smith form tversion of the multidimensiollal DFT is given by the following: (1) Shufee the input data samples by U-I. (2) Perform a septtrabk ~kultidimensionalDFT with len@'fis equal to the diagonal elements of A, (3) Shufee the output data samples by VT. Let us examine more carefu;tfy the mapping between the spwe of the initial data samples and the space defined by A. Assume that the data is defined on a quincunx grid with the hllowing SmitEl form decomposition.

Then, II characterizes the intermediate space for the Snrrith Farm decompasition and the mapping from M to A can be visualized by Figures 2.6 and 2,7. Si~iilarly,the inverse DFT is given by

where, det M

I=I det U

Defiinitrion 2.3.2.15. Let M be the sampfing matrix, Then, given a, multidimtm~ionalsequence ~ ( n )the , eompoxreats of the muZdidimensionaI modulat;ion representalion of the multidimensional n-transformof z f n) are defined

To more easily interpret this equation, assume z takes on the miue of .xp(jg). Then,

Figure 2.6, Ilnput data.

Figure 2.7. After shamng input data.

But z, = exp(jo,) for

m,

.- 0 , . .. ,k

- 1. Therefore,

Example 2.2.2.3. This example illustrates the idea of the multidimensional modulation representation of X ( z ) with respect to a rectangular 2 0 . In this case, I P Vwill I ~ contain four va.1sampling matrix; M = 0 2

X~?'(Z)

2.2.2.G

=

X(zo,zl)

for h a =

X(z0, -21)

for

X(-zo,zl)

for h z =

X(-al-xr)

far h3 =

hl = [ 0 1

2'

T

T

Polyphase represenkation

Definition 2.2.2.8. Given a multidimensional sequence s(n) and a nonsingular matrix R, then its Tgpe-I multidimensional polyphase components are defined as z.(n) = z (Rn a), where n E and a E N(R).

+

Now, let us investigate the multidimensional z-transform of the Type-I muEtidimexlsioxla1 polypfime components, that is,

or equivalently,

Far notatioriaf purposes, let

then, X ( z j becomes

55

2.2 MuXtidim ension& framework

where zea corresponds to shifts of the multidimensional signal by a vector amount a, Novv consider the Eollawing example, which illustrates the idea of a Type-I multidimensional polyphase decomposition of the multidimensional filter Xfs) . Example 2.2.2.2. What is the Type-I multidimensional polyphase decom-

position of the multidimensiorkal filter X (1;) with respect to a rectangular 2 0 ? In this case, there are J(R) = ldet R I = sampling mabtrk R = a 2 4 values of the shift vectors are given by [0, O I T , [I,O I T , (0,1 Let s = [za, rl] T and a = [(a), ,(a),lT then, zma = z~"'~z;(a'z. ThusI the quantity ia given by for a = 10, 0jT

( z,z' ;' r,'z;'

=

for a = [l,OIT

zcaz;'

for a = 10,

z

for a = (I,ljT,

I'

Since for arbitrary R = Hence, the quantity

llT

for a = [o,o

zo

=Cl

Yo

o1

x i R ) (sR)is

] ~

,zR is given by zR = ( ~ , 2 ~ zz:' Y , z? ) . defined by

So, the polypkitse decomposition of Xfz;) is giiven by

X(z) =

+

xiR) (zi,2:) + z;lxLR) -1

z1

x,

(R)

2

(Zi, zf )

2

(Z~,L~)+Z~=Z;~X~~~(Z~Z,I~).

Definition 2.2.2.7. Given a multidimensional vector z(n)and a nonsingu1ar matrix Et, its Type-11nzultidimewionad palyphme components are given by z,(n) = g(Rn - a), where a E hi(R) and n E N.

Chapter 2 A/lulfidirneusioud Mul tirate Signal Processi~g

56

Now, led us inwstigate the multidinrensionat ~~transfornr of the Type-I1 multidimensionalt polyphasse components, that is,

then where za advances the multidimensional signal by a vector amount a. T ~ u s , the set of dl integer xctora can be partitioned into J(R) equivalence classes using either Type-I or Type-I1 muttidimensiona! polypbitse decomposition. Moreover, the success of these decompositions rests on the following theorem. Theorem 2.2.2.3. (Division Theorem. for Integer Vectors) Let R be a k x k nonsingular integer matrix, let; p and n he m integer veetors, and let a be s sbr'ft vector of R, the^, w can uniquely express p w

Praof: Write g

+EM

P'PP+P~;

where, pp and pr, are unique vectors with

Since R is an integer matr2, then pr; ia an integer vector, Moreover, since p and E)L are integer vectors, then p~ is an integer vector. Moreover, since p~ E FPD(R), it follows that p~ E M(R). By letting p~ = a and p ~ = Rn, we obtain

p=Rn+a.

The remainder, a, can be expressed

I

%s

a ZE p mod R or simply ((a)) =:

2-3 Multidimensional building blocks

Figure 2.8. Mult idimensiond ezcpantler,

2.3

Mdtfdimemional building blocks

Thits section defines and analyzes two types of multidimensional building blocks used in multidimensional rnultirate signal processing. One type of building block deals with changes ia the sampling rate of the input signal, and the other type of building; block deals with changes in filter length, Analogous t o the one-dimensional c a e , there are two types of multidirnensionaf building blocks, which change the sampling rate of the input, signal ---- decimators to reduce it and expanders t s increwe it. But in the multidimensional cme, deeirnators iznd expanders aRect not only the sampling rate but also the geometry of the sampling lattice, Then, a multidimensional building block is discussed which cha~rgesthe length of a given iilter - multidimeni~ionalcomb filters to increase its Imgtfr. fn this way, a multidimensional comb filter version, of a given filter is analogous to ~nultidimen~io~al expander acting on an input signal. 2.3.1

Multidlmemiond expanders

Definition 2.3.1.1. Let x(n) be a k-dimmsiond signd m d let L be an integer-vdued matrix. Then, $file muftidimensional procesg of L-fold expruldillg mtqps a signat on I\( to another signal that;is nonzero only rct points on $he subIattice LATP), The output of the R-dimensional expander is related to the input ~ i g n dz(n) by

where XI is a nonsingular k x k integer matrix. Since LAT(L) denotes all vectors of the form Lm, where m E N9then the condition L-'n E N is equivalent to n 6 LAT(L). The matrix L is knowxk the expansion matrix. The multidimensional expander is depicted pictorially in Figure 2.8, Now, led us anitlyze the multidimensianal expander using the definition of the multidimensional 2-transform,

Chapter 2 MuliCidimensional Muidirate Signal P r o c e s ~ i ~ g

Figure 2.9, Cascade of multidimensional expmders.

Then, applying the definition of the nrultidimension.Ellexpander will require that Y ( z ) is zero for the lattice points defined by L-'n $! N.So, let n. = Lm.Then, Y(z) = y(~m)e-Lm.

By the definition of the rnultidimensiond expander, z(m) = y(L@

m E hl* . Henee,

for d l

~(rn)z-~~,

Y(z)= EN

or equivalently, using Theorem 2.2.2.1, the equation becomes

Hence,

Y ( s ) = x(zL).

In order to investigate the behavior of the multidimensionizl expander in the Fourier domdn, replace rt with exp(jg) to yield

Utilizing Theorem 2.2.2.2 this equation becomes

By suppressing the exponential and, as such, changing the notation so that Y (2) denotes Y (exp(jo)), then

Now consider the carrcade of two expanders fcrl and L2,which can be depicted grapkicdly by Figure 2.9. Substituting ][I F= LILz into the equation yields for Y (x)

Y(o)= x ( ( L , L ~ ) ~ s ) ,

or simply,

Y ( o )=

x(~TLT~),

2.3 Multidimensional building Mocks

Figure 2.11. Smith form cascade of expanders.

which can be represented by Figure 2.10. This cwcade of tvvo multidimensional expanders can be verified by redizing that

~(= o~(LTo) )

and

Therefore,

S ( 0 ) = x(L:*).

Vu) = X ( L f ( L T w ) ) = X((LIL~)~A) = X(LTr??).

So, for example, if L is repfaced by itis Smith form decomposition, that is, -L=: ULnlLVLt then we can graphically depict it in Figure 2.11..

L

Definition 2.3.2.1. Let z(n) bc a k-dimensional signal and let M be a k x k integer-vdued matrix. Tlxe multidimensianal M-fold decimalor sitmples input z (n) by mapping points on the sublat tice LAT(M) to .A/' according $0

yln) = s(Mn) and dises~rdingsamples ofaln) not on LATfM).M irj. ca,Ifed the decimatioa matrix.

The multidimensionaf decimator is depicted pictoridly in Figure 2.12. Let us analyze the muXticiimenaiorra1 decirnator using the definition of the

Figure 2.12. Multidlixeewianal deeinn8tor.

multidimensional z-transform, i. e.

Substituting the definition of the muXtidimensiona1decimator., we obtain

Note that zfn) is not zero for noninteger multiples of Mn, So, define an intermediate mapping of points that i s zero f'or aoninteger multiples of Mn, that is, z(n), where M-'n EN

ZI(~) =

0,

otherwise

.

So, y (xr) = 4 M n ) = zl(Mn). Hence,

Let m = Mn.Then, by the definition of zl(m),if M-'n $N,then sl(m)= 0. Therefare, Y ( z )= al(rn)a-M-'my

C

m€N

ar equivalently,

Y(.) =

C zl(m) (8M-'

)--me

=EN

Next, we need t o express XI (z) in terms of X(e). By the definition of sl(n), we can write the foliowing:

2.3 Multidimensional building blocks

61

where CM(n)is a scalar-valued sampling function associated with the sampling matrix M, that is,

G M ( ~=)

1, whenever M-'n E

0, otherwise

.

Since CM(n)is periodic in LAT(M) with spatial variable n, i.e.

+

C M ( ~=) C M ( ~Mm), it can be expressed as a complex Fourier series

Applying the definition of the multidimensional rtransform to zl(n)yields

Substituting the equation for GM(n) yields

or equivalently,

Then, performing a rnultidimensiond z-transform yields

Since Y ( B ) = X I (zM-I), then

Y (z) becomes

If z-" equals e x y ( - j g n ) , then sM-' is equivalent to e ~ ~ ( - j ~ - ~ ~ ) . Therehre,

Chapter 2 Multl"dimensiondt1Multirate Sign& Proces~ing

Figure 2.13. Cwcade af mtlltidimensiond dedmstars.

Figure 2.14. Interpreting the cmcade of multidimertsiond decinzators.

Utilizing Theorem 2.2.2.2, Y(exrpba]) becomes

By suppressing the exponential and, za such, changing the notation so that Y (o) denotes Y (expb*]), then

Now consider the emcade of two decirnators MI and M2, which can be depicted grqhically in Figure 2.13. Substituting M = M5M2i ~ t the o definition of the multidimensional ddecimator yields

which can be repre~entedin Figure 2.14. This cascade of multidimensional deeimators can be veriGeb by realizing that

Therefore,

y(n) = z(M1M2n).

So, for examplie, if M is r e p k e d by its Smith hrrn decomposition, that is, M = UMAMVnn, then we can graphicdly depict it as Figure 2.15.

Let ucr brieAy examine unimodular deeimators with decimation matrix Y. Consider multidimensional decimator,

2.3 Multidimensio11d building block8

Figure 2.15, Smith form cascade of decimatms.

Figure 2.16, Unimodular decimator,

Since V is a, unirnodular, then J(V) = 1. Therefore,

Therefore, there is no diasing and, as such, unimodvtar decimation can be viewed as just a rearrangement of samples. This inte~pretationof unirnodular decimation is depicted in Figure 2.16.

Definition 2.3.3.1. Given the impulse response h(n) o f a mull-idimensioaal filter, one can build a muttidimensional comb filter g(n) by mapping the filter h(n) on to another one that is nonzero only at points on the subfat tice LATCL), i.e.

As mentioned earlier, &-In E N is equivalent to n E LAT(L). The impulse response of a multidimensional comb filter can be represented by

where 6,,L, is the multidimensional Kronecker delta function. Taking the multihensional x-transform of gfai), we obtain

Chapter 2 Malt;idimensiond Maltirate Signal Proce~~ing

64

or equivalently,

If LAT(L) is not rectangular, that is, L is not a diagonal malrix, then the components of ~ ( z ' ) are no longer separable. To illustrate F

this point, let L =

I0

%I

90

YI

, then ~ ( a l =) H(z,'~r?,

z,Z1zp) and

H(z) = H(ro,zl). For example, we will consider both rectangular Sampling and hexagonal sampling, For rectangular sampf ng, let

then ~ ( s =~~ () r i z yz:z,:) = H(zi z:) and H ( z ) = B(ro,zl). This indicates the separability of components of ~ ( zfor~ a )rectangular sampling. Far hexagond sampling, let

) and H(z) = H ( z & zl). , then ~ ( z ' ) = x ( z ~ z ~ , z ~ z=; ~H(z~z:,z~z;~) T h k indiea-tes the lack of separability of components of the m-tlllidimensiortal comb filter H(zL)for non-rectangular sampling.

2.4

Interchanging fsdlding blocks

This section defines w q s to interchange multidimensional Building blockis. First, the conditions for irtterchanging muldidimensiond dceimators and expanders Tor sampling rate conversion are presented. Secondly, we consider the multidimensional noble identities, an approach for interchanging multidimensional building blocks with multidimensiond decimation and interpolatiox~filters. 2,4.1.

Irrderchangimg deciurralors and expanders

Multidimensional sarnplirxg rate conversion is irnportaxxt for many signal processing applications, because many times it is necessary to interfact: irnagc or video data between systems which use different sampling lai;t;ices. Examples include the conversion Between European and American television systems and the conversion between high definition television (HDTV)

2.4 I~Cercha~ging building blocks

Figure 2.17. Decimatoz preceding expander.

Figure 2.18. Substituting Smith form dewmpositrioxls.

signals and conventional tefevision. signals, Thus, sampling rate conver~ion will require the cascade of a multidimensional I;-foid exprlnder and a multidimensional.M-fold decirnator, separated by a filler. Under what conditions can the decimdor and the expander be interchanged? We will see that we will need to a s s u m that (a) M and L commute, that is, L M = ML, and (b) M and L are coprime. We wiH see that through the utilization sf the Smith h r m that coprimeness in muftidimensions can be xhieved in each dimension independently: Conr~iderthe configuration in whicEl the decinzator precedes the exy a d e r , which is depicted in Figure 2.17, Let us assume the following Smith farm decompositions for M and L:

M == iJMAMVlyr and XI = fJLALVL where UM,VM,UL, VL are unimodular matrices and AM,AL are diagonal matrices. Therefore, the resulting configuraLion is depicted in Figure 2J.8, or equivalently in Figure 2.19, Without loss of generdity, we can ilssurvze that Hence, we obtain the configuration depicted in Figure 2.20, Since AM and

Figure 2.19. Substituting cmcade definitions.

Chapter 2 Multidimensional Multirate S i g ~ a Procesgiag l

Figure 2.20. Simplifying the cascade.

Figure 2.2 1, Interchan[;;inglambda matrices,

AL are diagonal matrices, then interchanging AM and .ALcan be? achiemd in each dimension individually, provided that the associated decimation and expansion ratios in each dimension are relatively prime. In this way, the Smith form decomposition. has llelped us transform a multidimensional problcrn into a series of one-dimensional problems. Therehre, we obtain the conliguration depicted in Figure 2.21, Since unirnodular deeimators me equivalent to inverse uxxirnodrrlsr expanders (as depicted in Figure 2-22), then for a ~ i yunirnariulw matrh T, Figure 2.23 results. Now choose a unimadulatr m a t r k UA such that

or equivalently,

UA = &UMh;'.

Choose a matrix VB such. that

or equivalently,

Figure 2.22. Unimodular deeimatars and unimodular expanders,

2.4 In terehangirjlg building block8

Figure 2.23, Rmulting figure.

Substituting Ui' into the last equation yields Since AM and AL are relatively prime in each. dimension, we can interchange A 2 and AL. Therefore,

vB= U

~ A ~ A ~ ~ U ~ L - ~ M .

Since VM = VL,then VLvkl ; :1. Therefore,

VB = u ~ A ~ v ~ v & ' A ~ ' u G ' L - ~ M , or equivalently,

vB= L M - I L - ~ M .

If L and M are assumed to commute, then L-'M = ML-'. Therefore,

VB = 1. Hence, we ltave shown that the nrultidimensionaf L-fold expar~derand the multidimensional M-fold decimator can commute provided (a)3L ztr~dM commute and (b) L artd M are coprime, Exampla 2.4,li.T. Let us consider the following example. Assume a multidimensional decirnator is defined by the decimation matrix D is given

and a multidimensional expander is defined by the expan~iorlm a t r k E is

Clan these multidimensiond Building blocbs be inderchlzngedl First, we will find the Smith form decomposition of rnaLtrices D a ~ l dE, that is,

Chapter 2 MaltidimensionaJ Malgirate Sign& Proee~sing

Figure 2.24. Multidimeasionial;expander followed by after,

Figure 2-25. Multidimensionaf dedmator preeded by filter.

Since the diagonal entries in Ax> and are relatively prime, then r f r , and ljlE me coprime, which in turn implies that 33 and E are coprime. Secondly; we need to check to see whether D and E commute, that is,

and

Since DE = ED, then I)m d E commute, Since D and E are coprime and since D and E commute, then the multidimensional decimator (defined by the decimation matrix D) and the multidimensional expander (defined by the expansion matrix E)can be interchanged.

In moat izppkcations involving multidimensional interpolation, an interpolation filter foUows an expander as in Figure 2.24. Similarly, in many app1icatiorra involving multidimensional becimators, a decimation filter precedes the decimator as in Figure 2.25. If H ( z ) and K ( B )are multidimensional comb filters, then can the multidimensional building blocks be interchanged?

Figure 2.26. MuILidimensional, expmder with comb filter.

Figure 2.27. Filter preceding mnltidimensiond expmder .

Consider the configuration depicted in Figure 2.26. Writing the elerrlentd equ;ztions, vve obtain V ( z )= x(sL)

But, this equation could dm be interpreted as Figure 2.21, where and

T ( z ) =: G ( z ) X ( e ) .

These two equivalent block diagrams present a systematic approwh for interchanging multidimensional filters with multidimensional expanders, and together they will be referred to cw the Nable identity for multidimensional expanders,

Consider the configuration depicted in Figure 2.28, Writing the! elemental equations, V ( e )= c(sM)X(Z)

Chapter 2 Mull;r.dimensiond Muftirate Sign& Proces~ing

Figure 2.28. Comb fitter preceding malt idirnensional decimator.

Figure 2.29. Mufdidimensiond decimator preceding filter.

Hence, using the multidimensional z-transform identities msociated with x raised to a matrix power, we obtain the following

But this equatioxt. could be interpreted Y (z)

a r ~Figure

2.29, where

=: G(z-)T(z)

These two equivalent block diagrams present, a systematic approxh for interchanging multidimensianal filters with multidimensional decimatorts, and together they will be referred to as the Noble identity far multidimensional decimadors, 2.5

Problem

1, Consider the following sampling matrix:

Sketch the lattice LAT(T). Clearly indicate the fundmental parallelepiped FPD(T) and highlight the J(T) points in N which belong to FPDIT).

fat

n

72

Ghapter 2 Multidimensio~~d Multirate S i p & Processing 7, Let us generalhe the notion cvf multidimensional expanders and dlecimatarg. Let N be a nonsingular mat^. In the definition of the multidimensional expander, replace JV'with LAT(N), where LAT(L) C LATIN), Similarly, in the definition of ttte mullidirnensional decimator, replace A/' with LAT (N), where LAT(M) C LAT(N). Interpret the resulting operators and provide examples to illustrate the use of eaeh of them,