Subband Coder Signal Model

also show that if the FB is designed via the frequency dependent geometric mean decomposition (GMD), then the FB is optimal. Another advantage of the GMD FB is that the bit loading scheme is uniform. Therefore, it does not suffer from the bit loading granularity problem as in the traditional FB [70].

The theory of optimal filter banks is not only useful in data compression but also in digital com- munication [3, 18]. The notion of duality in the optimal DMT systems and biorthogonal subband coders has been reported in [53]. We also find GTD filter banks useful in wireless communication systems over slowly time-varying frequency selective channels with linear precoding and zero- forcing decision feedback equalizers. Our focus is on the quality of service (QoS) problem, namely, in minimizing the transmitted power subject to specified bit error rate and bit rate constraint. We will show that the optimal systems are related to the frequency dependent GTD of the channel response matrix.

The content of this chapter is mainly drawn from [133], and portions of it have been presented in [131, 132].

6.1 Outline

This chapter is organized as follows. Sec. 6.2 formulates the perfect reconstruction GTD FB opti- mization problems for subband coders. Sec. 6.3 and Sec. 6.4 provide mathematical derivations and solutions to the optimal orthonormal GTD FBs and the biorthogonal GTD FBs, respectively. Some discussions and performance comparisons of these FBs are presented in Sec. 6.5. Sec. 6.6 intro- duces the use of GTD filter banks in the context of wireless communications. Concluding remarks are made in Sec.6.7.

x(n)

M M

^

x(n) x(n)

x(n)^

z^{1 z0(n)} z

) (e^j

E R(e^j )

M M

z¹

z¹

z

) z

1(n z

)

2(n

M z M

M M

z ¹ z

)

2(n z

)

3(n z

Encoder Decoder

Figure 6.1: The biorthogonal GTD subband coders forM = 4.

allω.²

The signal x(n) first passes through a filter E(e^jω). Let z(n) = [z₀(n) z₁(n) · · · z_M−1(n)]^T denote the output ofE(e^jω). Before the quantizers, the signalz(n)passes through a frequency- dependent PLT stage, where the MINLAB structure [82] is used to ensure unity noise gain. Thekth quantizer inputvk(n)is the sum of the signalzk(n)and the filtered version of the quantized signal v0(n),v1(n), up tov_k−1(n). The filterPik(e^jω)is the estimation filter from thekth stream to theith stream. The decoder performs the inverse operations on the quantized data. The validity of the MINLAB structure assumptions must rely on the high-bit-rate assumption where we assume that the prediction based on the quantized data is not too much different from that on the unquantized data. Under this assumption, the signalv(n)before the quantizer is the filtered version ofx(n) passing through the filterL(e^jω)E(e^jω), whereL(e^jω)is the filter used to represent the frequency dependent PLT stage. In particular,L(e^jω)is a lower triangular matrix with unity on its diagonals for all frequencies.

Since x(n) is zero-mean and WSS, the quantizer inputsvi(n)’s are therefore zero-mean and jointly WSS with psd matrix

S_vv(e^jω) =L(e^jω)E(e^jω)S_xx(e^jω)E^†(e^jω)L^†(e^jω). (6.1)

Theith quantizer input signal variance is therefore

σ²_vi= Z 2π

0

[Svv(e^jω)]ii

dω

2π. (6.2)

2Note that if the vector process is obtained from blocking a scalar process, its power spectrum density matrix has pseudo- circulant structure [110, 107]. However, we will see that the theory developed in this section does not depend nor utilize this structure. Hence, the results of this section are not restricted to the blocked version of a scalar input process, but are also true for any WSS vector process with well-defined power spectrum density matrix.

To derive the coding gain expression, we model the quantizers with additive noise sourcesqi(n).

We assume these noise sources are jointly WSS, white, with zero mean and with variances of the form

σ²_qi =c2^−2biσ_v²_i (6.3)

wherebi is the number of bits assigned to theith quantizer. So the quantizer noise psdSqq(e^jω) is a constant diagonal matrix with diagonal elementsσ_q²_i. This is thehigh-bit-rateassumption, and is also used in the previous chapter. The average bit rateb = _M¹ PM−1

i=0 biis assumed to be fixed.

The reconstruction error vector is e(n) = bx(n)−x(n). Based on unity noise gain property in the MINLAB structure [82], the error vectore(n)can be regarded as the output of the synthesis matrixR(e^jω)in response to the quantization errorq(n). Thus, the psd matrix of the errore(n)is S_ee(e^jω) =R(e^jω)S_qq(e^jω)R^†(e^jω)The average mean square error of the coderε_coderis³

ε_coder = 1

ME[e^†(n)e(n)] = 1

MTr(E[e(n)e(n)^†])

= 1

M Z 2π

0

Tr(S_ee)dω 2π = 1

M Z 2π

0

Tr(RS_qqR^†)dω 2π

= 1

M Z 2π

0

Tr(R^†RSqq)dω 2π

= 1

M

M−1

X

i=0

σ_q²_i Z 2π

0

[R^†R]ii

dω 2π Using (6.3) andσ_v²_i =R2π

0 [LESxxE^†L^†]iidω

2π, we get σ_q²

i =c2^−2bi Z 2π

0

[LES_xxE^†L^†]_iidω 2π Substituting into the preceding equation, this yields

εcoder = 1 M

M−1

X

i=0

c2^−2bi Z 2π

0

[LESxxE^†L^†]ii

dω 2π ×

Z 2π 0

[R^†R]ii

dω 2π

≥ c2^−2b

M−1

Y

i=0

Z 2π 0

[LES_xxE^†L^†]_iidω 2π×

Z 2π 0

[R^†R]_iidω 2π

!_M¹ ,

3For simplicity, from now on we drop the argument “e^jω” when there is no confusion. For example, bySeewe mean the psd matrixSee(e^jω).

where we have used the AM-GM inequality [29]. This becomes an equality if the terms in the summation are identical for all i, and can be accomplished by choosing theb_i according to the optimum bit loading formula⁴similar to that in [109], i.e.,

bi=b+ 0.5 log₂σ²_viK_i²−0.5X

i

log₂σ²_viK_i²/M, (6.4)

whereK_i²=R2π

0 [R^†R]iidω

2π.Let us define φ=

M−1

Y

i=0

Z 2π 0

[LES_xxE^†L^†]_iidω 2π

Z 2π 0

[R^†R]_iidω

2π. (6.5)

Therefore, the average MSE of the coder under optimal bit allocation becomes

coder=c2^−2bφ^1/M.

The coding gain of a coder is defined by comparing the average mean square valueεcoderof the reconstruction errorx(n)−x(n)ˆ with the mean square valueεdirectof the direct quantization error (roundoff quantizer) with the same bit rateb. An expression for the coding gainGCcan be written as

GC= εdirect

εcoder

. (6.6)

Thus, maximizing the coding gain is equivalent to minimizingφby choosing{E(e^jω),R(e^jω),L(e^jω)}

subject to some constraints. In this section, we consider problems similar to what was considered in [106] and [109] – a theoretical performance bound of the infinite order perfect reconstruction filter banks. We consider two classes of subband coders. The first class is when the precoderE(e^jω)in Fig. 6.1 is restricted to be paraunitary, i.e.,E(e^jω)is unitary for allω. We call such filter banks the orthonormal GTD filter banks, since the columns ofE(e^jω)are orthonormal for every frequency. In

4The optimum bit loading formula for conventional perfect reconstruction filter banks usually has the granularity prob- lem, i.e., the number of bits in the formula needs to be rounded off to the nearest integer when used in practice. This results in some performance loss. However, it will be seen later that this problem does not exist in the optimal orthonormal and biorthogonal GTD FBs where uniform bit loading can be applied.

this case, the optimization problem can be written as

minE,R,L φ

s.t. (a)R(e^jω)E(e^jω) =I

(b)E(e^jω)is paraunitary. (6.7)

The second class is when the paraunitary precoder constraint is absent; we call such filter banks the biorthogonal GTD filter banks, since the matrixE(e^jω)andR(e^jω)are biorthogonal pairs. In this case, the optimization problem is

minE,R,L φ

s.t. R(e^jω)E(e^jω) =I. (6.8)