Biorthogonal GTD Filter Banks

QM−1 i=0 σ²_v

i=ÄR2π

0 detS_xx(e^jω)^{M1 dω}_2πäM

.The MSE of the direct quantization is

εdirect = c2^−b 1 M

M−1

X

k=0

Z 2π 0

[Sxx(e^jω)]kk

dω 2π

= c2^−b Z 2π

0

Tr(Sxx(e^jω)) M

dω 2π.

Substituting these in (6.6), the maximized coding gain can thus be calculated as

GC = R2π

0 1

MTr(Sxx(e^jω))dω R2π

0

Mp

detSxx(e^jω)dω

. (6.18)

Eq. (6.18) gives a nice closed-form expression for the optimal performance that orthonormal GTD FBs can have. For the traditional filter bank optimization problem discussed in the literature [106], it is well known that the coding performance can be further improved by relaxing the or- thonormal constraint. It is thus natural to ask how much performance improvement we can get if we relax the orthonormal precoder constraint in the GTD FBs. This will be addressed in the next section.

LDU decomposition ofE(e^jω)Sxx(e^jω)E^†(e^jω)isLe(e^jω)D(e^jω)L^†_e(e^jω),where D(e^jω)is the diagonal

matrix for all frequencies. ♦

Substituting the result of Corollary 6.4.2 in (6.5), we can establish a lower bound onφ:

φ ≥

M−1

Y

i=0

Z 2π 0

[L⁻¹_e ES_xxE^†L^−†_e ]_ii Z 2π

0

[R^†R]_iidω

2π (6.19)

≥

Z 2π 0

M−1

Y

i=0

[L⁻¹_e ESxxE^†L^−†_e ]ii[R^†R]ii2M¹ dω 2π

!^2M

(6.20)

≥

ÇZ 2π

0

detL⁻¹_e ES_xxE^†L^−†_e detR^†R_2M¹ dω 2π

å2M

(6.21)

=

ÇZ 2π

0

(detS_xx)^2M1 dω 2π

å2M

, (6.22)

where in (6.20) we have used H ¨older’s inequality for integrals (6.9 in [29]), in (6.21) we have used Hadamard inequality for positive definite matrices (7.8.1 in [31]), and in (6.22) we have used the fact thatdetL_e= 1anddetRE= detI= 1.

Now the question is, whether the bound (6.22) is achievable. The theorem below answers this question in the affirmative, and in particular, this bound can be achieved by a restricted case (where Ecan be decomposed as a paraunitary matrixU and a set of scalar filters) of biorthogonal GTD coder which is shown in Fig. 6.2.

Theorem 6.4.3 (The Cascade of Optimal Orthonormal GTD Filter Bank and a Set of Half-Whitening Filters is Optimal) The structure in Fig. 6.2, which is a restricted class of the biorthogonal GTD filter banks, achieves (6.22). In particular, (6.22) can be achieved by usingU(e^jω)and{λi(e^jω)}, whereU(e^jω)is the precoder solution to the optimal orthonormal GTD filter banks in Section 6.3, and{λi(e^jω)}is a set of half- whitening filters determined by the input psd. In particular, we can use the same filter for all subbands, i.e.,

λ_i(e^jω) =λ(e^jω). ♦

Proof: The proof is by construction. Consider the solution of the optimal orthonormal GTD filter banks in Sec. 6.3 which performs the frequency dependent GTD for the Cholesky factor of Sxx(e^jω)(we reproduce Eq.(6.14) here without some modifications on the notations):

S^†/2_xx =QD¹²ΦL^†_xP, (6.23)

where bothQandPare unitary matrices for all frequencies,Lxis a lower triangular matrix with 1on the diagonals. The middle diagonal matrixD(e^jω)is such that[D(e^jω)]_ii= ^Mp

detS_xx(e^jω)a_i, where the values in the set{a_i}satisfya_i>0andQM−1

i=0 a_i= 1.

Therefore,S_xx=P^†L_xDL^†_xP. Note thatPis exactly the precoder of the optimal orthonormal GTD coders described in Sec. 6.3. Now, consider the system structure in Fig. 6.2. LetΛ(e^jω) = diag(λ0(e^jω),· · ·, λ_M−1(e^jω))denote the frequency response of the scalar filters. If we takeU=P, the psd matrix ofy(n)can be expressed asSyy =ΛLxDL^†_xΛ^†. It can be proved thatΛLx =LyΛ whereLyis a lower triangular matrix with1on the diagonals and[Ly]ij =^λ_λ^j

i[Lx]ij. Therefore,Syy

can be expressed asSyy =LyΛDΛ^†L^†_y. We can take the PLT part to beL=L⁻¹_y , and the resulting psd matrix will beS_vv =ΛDΛ^†. Substituting these quantities in (6.5), we have

φ =

M−1

Y

i=0

Z 2π 0

[D]ii|λi|²dω 2π

Z 2π 0

|λi|⁻²dω

2π (6.24)

≥

M−1

Y

i=0

ÇZ 2π

0

([D]ii)^1/2dω 2π

å2

(6.25)

=

ÇZ 2π

0

(detSxx)^2M1 dω 2π

å2M

, (6.26)

where (6.25) is from the Cauchy-Schwartz inequality, and (6.26) is from the fact that[D(e^jω)]ii =

Mp

detSxx(e^jω)aiandQM−1

i=0 ai= 1. The equality in (6.25) can be satisfied by choosing λi(e^jω) = αi [D]ii(e^jω)^−1/4

= α_i a^1/4_i

detS_vv(e^jω)−_4M¹

, (6.27)

whereαi is any nonzero scalar multiplier. Thus,λi is thehalf-whitening filter in theith subband.

If we chooseαi =a^1/4_i , for all the subband we can use the same half-whitening filter that has the frequency response

λi(e^jω) =λ(e^jω) = detSxx(e^jω)⁻4M¹ .

Therefore we just need to design one scalar filterλ(e^jω)for all the subbands.

To summarize, we have constructed a biorthogonal GTD filter bank in the structure of Fig. 6.2 that achieves exactly (6.22), and is thus optimal for the problem (6.8).

)

0(

! e^j !0¹(e^j )

)

0(n y

)

1(

!1 _e^j

1

)

1(

! e^j

x (n)

) (e^j U

^

) (e^j U

x (n)

)

1(n y

)

2(n y

)

1(

!2 _e^j

)

1(

!3 _e^j

)

2(

! e^j

)

3(

! _e^j )

3(n y

Middle frequency ! dependent PLT Part

R_(e^j ₎ )

(e^j E

Figure 6.2: A restricted case of the biorthogonal GTD subband coders forM = 4.

Theorem 6.4.3 proves that the bound (6.22) is achievable. The proof also suggests a design method for the optimal biorthogonal GTD filter banks. By working out the optimal bit loading formula (6.4) of this design scheme, we can find that the bit loading is exactly the same as (6.17).

That is,the optimal biorthogonal GTD FBs have the same bit loading scheme as the corresponding optimal orthonormal GTD FBs.This design method is summarized below.

Design of optimal biorthogonal GTD filter banks:

1. Perform the design procedure for optimal orthonormal GTD filter banks as described in Sec.

6.3 to obtained the precoderU(e^jω)and the estimatorsLx(e^jω).

2. Design the half-whitening filter according to (6.27). The precoder is chosen as E(e^jω) = Λ(e^jω)U(e^jω). The estimators need to be recomputed according to Corollary 6.4.2.

3. Design the optimal bit loading scheme as (6.17).

For the case when the optimal orthonormal GTD filter banks are designed as GMD filter banks, the diagonal matrixD(e^jω)can be written asD(e^jω) = ^Mp

detSxx(e^jω)I. Therefore, for alliwe have[D(e^jω)]ii = ^Mp

detSxx(e^jω). This makes the optimal bit loading formula (6.4) correspond to uniform bit loading, which does not have the granularity problem if the average bit budget is an integer!

Theorem 6.4.3 not only shows the lower bound (6.22) can be achieved but also provides more insight to this problem. The optimal solution to (6.8) must have φ equal to (6.22), and thus it must satisfy all the equalities from (6.19) to (6.22). These conditions are necessary conditions of the

optimal systems, and will be examined in the following. First, in (6.20) for the H ¨older’s inequality

M−1

Y

i=0

Z 2π 0

[L⁻¹_e ES_xxE^†L⁻¹_e ]_iidω 2π ≥

Z 2π 0

M−1

Y

i=0

[L⁻¹_e ES_xxE^†L⁻¹_e ]_iidω 2π

!^M

to have equality, we require the spectrum equalizing property (6.13) to be satisfied. This gives us another necessary condition for the optimal biorthogonal GTD filter banks:

Corollary 6.4.4 Spectrum equalization of Subband Signals is Necessary for Optimal Biorthogonal GTD Filter Banks: The subband signals of the optimal biorthogonal GTD filter banks have the spectrum equalizing

property (6.13). ♦

Second, in (6.21) for the Hadamard inequality

M−1

Y

i=0

[R^†R]ii ≥det(R^†R)

to have equality,R^†Rneeds to be diagonal. This shows that the matrixR(e^jω)must be decompos- able as a paraunitary matrix multiplying with a diagonal matrix. Therefore, the optimal solution to (6.7) will definitely have the structure as in Fig. 6.2. This is stated as the following corollary:

Corollary 6.4.5 (Optimal Biorthogonal GTD FBs Necessarily Has a Decomposable Structure) The optimal biorthogonal GTD filter banks have the structure as shown in Fig. 6.2. ♦ To summarize from Theorem 6.4.1, Corollary 6.4.4, and Corollary 6.4.5, the optimal biorthogonal GTD filter bank also hastotal decorrelationandspectrum equalizationas the two necessary conditions.

Furthermore, the optimal systems can be decomposed as the structure in Fig. 6.2. Surprisingly, all of these results have a parallel fashion to the theory of traditional biorthogonal filter banks devel- oped in [109] and [70]: Theorem 2.3 in [70] suggesting that total decorrelation is necessary, Lemma 2.6 in [70] suggesting that spectral majorization is necessary, and Theorem 2.7 in [70] suggesting that the optimal biorthogonal filter banks have the decomposable structure as shown in Fig. 3 of [70].