Signal Expansions, Filter Banks, and Subband Decomposition

Abstract

In this chapter, expansions of signals into orthogonal or biorthogonal functions, as well as into frames, are formulated as generalized transforms, where the expansion coefficients provide a discrete representation of a signal (globally seen as a generalized transform). The second topic is that of filter banks and subband decomposition, which implement generalized transforms and signal expansions with multirate components, such as decimators and interpolators. It will be shown that such an implementation is possible for generalized transforms that satisfy the condition of periodic shift invariance.

All these topics are preliminary to multiresolution analysis and wavelets, which will be developed in the next chapter. In fact, wavelets may be viewed both as generalized transforms and as signal expansions, and their practical implementation is obtained by filter banks.

Appendices

Appendix A: Proof of Theorem 14.1 on Projections

We first prove that the operator $\mathcal{P}_{b}$, defined by the kernel (14.57), is idempotent, that is, $\mathcal{P}_{b}^{2}=\mathcal{P}_{b}$. The kernel of $\mathcal{P}_{b}^{2}$ is given by

$$ \begin{array}{*{20}l} {h_{12} (t_3 ,t_1 )} & { = \int_I {{\rm d}t_2 h_b (t_3 ,t_2 )h_b (t_2 ,t_1 )} } \\ {} & { = {\rm d}(U)^2 \sum\limits_{p,p' \in P} {\int_I {{\rm d}t_2 \varphi _b (t_3 - p)\theta _b (t_2 - p)\varphi _b (t_2 - p')\theta _b (t_1 - p')} } } \\ {} & { = {\rm d}(U)\sum\limits_{p,p' \in P} {\varphi _b (t_3 - p)\left\{ {{\rm d}(U)\int_I {{\rm d}t_2 \,\theta _b (t_2 - p)\varphi _b (t_2 - p')} } \right\}\theta _b (t_1 - p')} } \\ {} & { = {\rm d}(U)\sum\limits_{p \in P} {\varphi _b (t_3 - p)\theta _b (t_1 - p) = h_b (t_3 ,t_1 )} ,} \\ \end{array} $$

where in {⋅} we have used condition (14.50b) with b=b′. Hence, $\mathcal{P}_{b}$ is a projector.

The sum of the kernels gives, after use of (14.21),

$$\sum_{b \in B} h_b(t_3,t_1)= \mathrm {d}(U)\sum_{b \in B} \sum_{p \in P}\varphi_b(t_3-p)\theta_b(t_1-p)=h_H(t_3,t_1),$$

which states the second of (14.58).

When the self-reciprocal condition holds, it is immediate to check that $h_{b}^{*}(t_{0}',t_{0})\!=h_{b}(t_{0},t_{0}')$, which states that the projectors $\mathcal{P}_{b}$ are Hermitian.

Appendix B: Alias-Free Condition in Subband Decomposition

In Sect. 7.7 we have developed the polyphase architecture of a filter, given by an I→J S/P converter, a polyphase filter on J, and a P/S converter (Fig. 14.37). The M×M polyphase matrix p _π (t)=[p _πji (t)] is obtained from the impulse response p(t ₀), t ₀∈I, of the filter as

$$p_{\pi j i}(t)=\frac{1}{L} p(t+\tau_j-\tau_i),\quad t\in J ,\tau_i,\tau _j\in A, $$

(14.164)

where we suppose that the PD is obtained with the same generator $A= \{\tau_{0},\ldots,\tau_{M-1}\}$ both at the S/P side and at the P/S side. We see from (14.164) that the polyphase matrix is redundant, since its j,i element depends only on the difference τ _j −τ _i .

Fig. 14.37

Polyphase decomposition of a filter with impulse response p(t ₀), t ₀∈P

Now, we relate the polyphase matrix to the PD of the impulse response p(t ₀), given by

$$p_k(t)=\frac{1}{L} p(t+\tau_k),\quad t\in J,\tau_k\in A. $$

(14.165)

To this end, note that the differences τ _j −τ _i are points of I and therefore have the unique decomposition

$$\tau_j-\tau_i=t_{ji}+\tau_k=t_{ji}+\tau_{\alpha_{ji}},\quad t_{ji}\in J,\tau_{\alpha_{ji}}\in A, $$

(14.166)

where $\tau_{\alpha_{ji}}$ is a convenient point of the generator A. Hence, the j,i element of the polyphase matrix can be written in the form

$$p_{\pi,ji}(t) = \frac{1}{M}\, p(t+\tau_j-\tau_i)= \frac{1}{M} p(t+t_{ji}+\tau_{\alpha_{ji}})= \frac{1}{M} p_{\alpha_{ji}}(t+t_{ji}), $$

(14.167)

where $p_{\alpha_{ji}}(t+t_{ji})$ is the polyphase component of p(t ₀) of index α _ji shifted by −t _ji . In conclusion:

Proposition 14.6

For a filter with impulse response p(t ₀), t ₀∈I, the polyphase matrix p _π (t), t∈J, obtained with a cell A=[I/P) of cardinality M, has M ² elements defined by (14.164). Let p ^(k)(t)=p(t+τ _k ), t∈J,τ _k ∈A, be the PD of the impulse response p(t ₀), t ₀∈I. All the M ² elements of p _π (t) can be obtained from the M elements of the PD p ^(k)(t) according to (14.167).

As an example, consider a cell A={0,τ ₁,τ ₂}⊂I of cardinality M=3. Then, the polyphase matrix is

$$ \begin{array}{l} p_\pi\left( t \right) = \left[ {\begin{array}{*{20}c} {p_{\pi 00} \left( t \right)} & {p_{\pi 10} \left( t \right)} & {p_{\pi 20} \left( t \right)}\\ {p_{\pi 01} \left( t \right)} & {p_{\pi 11} \left( t \right)} & {p_{\pi 21} \left( t \right)}\\ {p_{\pi 02} \left( t \right)} & {p_{\pi 12} \left( t \right)} & {p_{\pi 22} \left( t \right)}\\ \end{array}} \right] \\ \quad \quad= \frac{1}{3}\left[ {\begin{array}{*{20}c} {p\left( t \right)} & {p\left( {t + \tau _1 } \right)} & {p\left( {t + \tau _1 } \right)}\\ {p\left( {t - \tau _1 } \right)} & {p\left( t \right)} & {p\left( {t + \tau _2- \tau _1 } \right)}\\ {p\left( {t - \tau _2 } \right)} & {p\left( {t - \tau _2+ \tau _1 } \right)} & {p\left( t \right)}\\ \end{array}} \right]. \\ \end{array} $$

The PD of the impulse response p(t ₀) has the 3 components p ₀(t)=p(t), p ₁(t)=p(t+τ ₁), p ₂(t)=p(t+τ ₂), and we can express the 9 elements of p _π (t) in terms of the 3 components p _k (t),k=0,1,2. In fact, p(t)=p ₀(t), p(t+τ ₁)=p ₁(t), p(t+τ ₂)=p ₂(t). For the other 6 elements, we note that −τ ₁, −τ ₂, τ ₂−τ ₁, −τ ₂+τ ₁ can be written in the form $t_{ij}+\tau_{\alpha_{ij}}$, then, e.g., $\tau_{2}-\tau_{1}=t_{21}+\tau_{\alpha_{21}}$, and then $p(t+\tau_{2}-\tau_{1})=p^{(\alpha_{21}}(t+t_{21})$. To be more specific, we continue the example in the 1D case, where A={0,T ₀,2T ₀}. Then −τ ₁=−T ₀=−3T ₀+2T ₀, so that p(t−τ ₁)=p ⁽²⁾(t−3T ₀), τ ₂−τ ₁=T ₀, so that p(t+τ ₂−τ ₁)=p ⁽¹⁾(t), etc. In conclusion,

$$\mathbf{p}_{\pi}(t)=\frac{1}{3}\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{}}p_0(t) & p_1(t) & p_2(t) \cr \noalign{\vskip 4pt \relax}p_2(t-T) & p_0(t) & p_1(t) \cr \noalign{\vskip 4pt \relax}p_1(t-T) & p_2(t-T) & p_0(t)\end{array}\right],\quad T=3T_0. $$

(14.168)

The symmetry of this matrix is referred to as pseudo-circulant. Evidently, p _π (t) is completely determined by the elements of its 0th row, which represent the PD of the impulse response of the filter.

In the general multidimensional case, the symmetry of the polyphase matrix is stated by (14.165) and (14.166), and we referred to it as generalized pseudo-circulant. Proposition 14.6 represents a generalization of a theorem by Vaidyanathan and Mitra [13].

The conclusion is the following:

Proposition 14.7

The alias-free condition in a subband decomposition filter bank requires that the global polyphase matrix p _π (t)=(1/M)g _π ∗q _π (t) must be generalized pseudo-circulant. In this case the global Analysis–Synthesis is equivalent to a filter d(t ₀) whose polyphase components are given by the 0th row of p _π (t).

Hence, the filter d(t ₀) is obtained as the P/S conversion of the M elements of the 0th row of the matrix p _π (t) as (Fig. 14.38)

$$d(t+\tau_k)=p_k(t),\quad t\in J, k=0,1,\ldots,M-1, $$

(14.169)

and in the frequency domain, using (7.32)

$$D(f)=\frac{1}{M}\sum_{k=0}^{L-1}P_k(f)\mathrm{e}^{-\mathrm{i}2\pi f\tau_k}. $$

(14.170)

Fig. 14.38

Interpretation of the impulse response of the equivalent filter d(t ₀), t ₀∈I, in subband coding when the global matrix p _π (t) is pseudo-circulant

Once eliminated the aliasing according to the above statement, the perfect reconstruction imposes that d(t ₀)=δ _I (t ₀), which implies that p _k (t)=d(t+τ _k )=δ _I (t+τ _k ). In words: the first row of p _π (t) must contain pure shifts.

2.1 B.1 Alias-Free Condition in 1D Case

The alias-free condition requires the pseudo-circulant symmetry of global polyphase matrix

$$p_{\pi}(t)=\mathbf{g}_{\pi}*\frac{1}{M}\mathbf{q}_{\pi}(t)\quad\stackrel{\mathcal{F}}{ \hbox to 25pt{\rightarrowfill}}\quad \mathbf{P}_{\pi}(f)=\mathbf{G}_{\pi}(f)\frac{1}{M} \mathbf{Q}_{\pi}(f).$$

This symmetry can be formulated more specifically in the 1D case, where decomposition (14.166) becomes

$$\tau_j-\tau_i=(j-i)T_0=\begin{cases}(j-i)T_0, & j\geq0, \cr [M+(j-i)]T_0-MT_0, & j<i.\end{cases}$$

For instance, for L=4, this gives

$$\begin{array}{@{}rll}\mathbf{p}_{\pi}(t)&=&\displaystyle\frac{1}{4}\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}}p_0(t) & p_1(t) & p_2(t) & p_3(t) \cr \noalign{\vskip 4pt \relax}p_3(t-T) & p_0(t) & p_1(t) & p_2(t) \cr \noalign{\vskip 4pt \relax}p_2(t-T) & p_3(t-T) & p_0(t) & p_1(t) \cr \noalign{\vskip 4pt \relax}p_1(t-T) & p_2(t-T) & p_3(t-T) & p_0(t)\end{array}\right],\quad T=4T_0, \cr\noalign{\vspace*{6pt}}\mathbf{P}_{\pi}(f)&=&\displaystyle\frac{1}{4}\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}}P_0(f) & P_1(f) & P_2(f) & P_3(f) \cr \noalign{\vskip 4pt \relax}z^{-1}P_3(f) & P_0(f) & P_1(f) & P_2(f) \cr \noalign{\vskip 4pt \relax}z^{-1}P_2(f) & z^{-1}P_3(f) & P_0(f) & P_1(f) \cr \noalign{\vskip 4pt \relax}z^{-1}P_1(f) & z^{-1}P_2(f) & z^{-1}P_1(f) & P_0(f)\end{array}\right],\quad z=\mathrm {e}^{\mathrm {i}2\pi fT}.\end{array} $$

(14.171)

With this symmetry, the global subband system becomes a filter with frequency response (see (14.170))

$$D(f)=\frac{1}{4}\bigl[P_0(f)+z_0^{-1}P_1(f)+z_0^{-2}P_2(f)+z_0^{-3}P_3(f)\bigr],\quad z_0=\mathrm {e}^{\mathrm {i}2\pi fT_0}.$$

To eliminate the distortion, the further condition is D(f)=1. A solution may be obtained with $P_{k}(f)=z_{0}^{k}$.

Appendix C: z-Domain Analysis of 1D Subband Decomposition

The frequency-domain analysis developed in Sect. 14.13 can be transferred to the z-domain using the rules outlined in Chap. 11 for the fundamental discrete-time components, where, for the complex variable, we use symbols z for low-rate signals with spacing T and z ₀ for high-rate signals with spacing T ₀. Since (in this chapter) T=MT ₀, the variables are related as $z=z^{M}_{0}$. We also recall that we find it convenient to define the z-transform as a Haar integral over ℤ(T) or ℤ(T ₀) and with the explicit presence of the time spacing, namely

$$X(z)= \sum_{n= -\infty}^{+\infty}T x(nT) z^{-n},\qquad Y(z_0)= \sum_{n= -\infty}^{+\infty}T_0 y(nT_0) z_0^{-n}$$

for low-rate and a high-rate signals, respectively. The frequency domain analysis is obtained as a particularization by¹²

$$z=\mathrm{e}^{\mathrm{i} 2 \pi fT},\qquad z_0=\mathrm{e}^{\mathrm{i} 2 \pi fT_0}.$$

At this point we invite the reader to reconsider in detail the second part of Chap. 11 from Sect. 11.5 on, in particular, the rules concerning the passage from the frequency domain to the z-domain. Here, we simply recall the relations in a ℤ(T ₀)→ℤ(T) decimator, with T=MT ₀, given by

$$Y(f)=\sum_{k=0}^{M-1}X(f-kF),\qquad Y\bigl(z_0^M\bigr)=\sum_{k=0}^{M-1}X\bigl(z_0 W_M^{-k}\bigr)\,\buildrel \varDelta \over= \,\mathop{\mathrm{dec}}_M X(z_0), $$

(14.172)

and the relations in a ℤ(T)→ℤ(T ₀) interpolator, given by

$$Y(f)=G(f)X(f),\qquad Y(z_0)=G(z_0)X\bigl(z_0^M\bigr). $$

(14.173)

So, caution must be paid with filters and interpolators, whose frequency relation is the same (but the difference comes from the context).

We also recall the relations of the P/S and S/P conversions, obtained with the cells (14.154). They are respectively (see (11.59) and (11.60))

$$\begin{array}{@{}rll}X(z_0)&=&\displaystyle\frac{1}{M} \sum_{i=0}^{M-1}z_0^{-i} X^{(i)} \bigl(z_0^M\bigr),\\[4mm]X^{(i)}\bigl(z_0^N\bigr)&=&\displaystyle\sum_{k=0}^{M-1} z_0^i W_M^{ki}X\bigl(z_0 W_M^{-k}\bigr)=\mathop{\mathrm{dec}}_M \bigl[z_0^i X(z_0)\bigr].\end{array} $$

(14.174)

Using the above rules, we are now ready to obtain the relation in the z-domain from the Fourier analysis. The relationships in the Analysis/Synthesis scheme, given by (14.156), become (considering that the Analysis consists of a decimator and the Synthesis of an interpolator)

$$\mathbf{C}\bigl(z_0^M\bigr)= \mathop{\mathrm{dec}}_M \bigl[\mathbf{Q}(z_0)S(z_0)\bigr],\qquad \tilde{S} (z_0) = \mathbf{G}(z_0) \mathbf{C}\bigl(z^M_0\bigr).$$

By combination we find

$$ \begin{array}{l} \tilde S\left( {z_0 } \right) = {\rm G}\left( {z_0 } \right)\mathop {{\rm dec}}\limits_M \left[ {{\bf Q}\left( {z_0 } \right)V\left( {z_0 } \right)} \right] \\ \quad \quad= {\bf G}\left( {z_0 } \right)\sum\limits_{k = 0}^{M - 1} {{\bf Q}\left( {z_0 W_M^{ - k} } \right)S\left( {z_0 W_M^{ - \left( {M - 1} \right)} } \right)}\\ \end{array} $$

and, more explicitly,

$$ \begin{array}{l} \tilde S\left( {z_0 } \right) = {\rm G}\left( {z_0 } \right){\bf Q}\left( {z_0 } \right)S\left( {z_0 } \right) \\ \quad \quad+ {\bf G}\left( {z_0 } \right){\bf Q}\left( {z_0 W_M^{ - k} } \right)V\left( {z_0 W_M^{ - 1} } \right) \\ \quad \quad\vdots\\ \quad \quad+ {\bf G}\left( {z_0 } \right){\bf Q}\left( {z_0 W_M^{ - \left( {M - 1} \right)} } \right)V\left( {z_0 W_M^{ - \left( {M - 1} \right)} } \right). \\ \end{array} $$

The first line represents a filtered version of the original signal, and the rest represents aliasing. Hence, the perfect recovery condition is stated by

$$\mathbf{G} (z_0) \mathbf{Q}(z_0)=1 ,\qquad \mathbf{G}(z_0) \mathbf{Q}(z_0 W^{-k}_M)=0 ,\quad k=1,\ldots,M-1,$$

i.e.,

$$\fbox{$\mathbf{G}(z_{0}) \mathbf{Q}\bigl(z_{0} W^{-k}_{M}\bigr) =\delta_{k0}$.} $$

(14.175)

3.1 C.1 Polyphase Analysis

The general frequency domain relation (14.129) can be directly rewritten in the z-domain, since they are concerned with ordinary low-rate filters (hence we use the variable z). Thus,

$$\mathbf{C}(z)=\frac{1}{M} \mathbf{Q}_{\pi}(z) \mathbf{S}(z),\qquad \tilde{\mathbf{S}}(z)=\mathbf{G}_{\pi}(z)\mathbf{C}(z), $$

(14.176)

where S(z) is the PD of S(z ₀), and Q _π (z), G _π (z) are the polyphase matrices. Combination of (14.176) gives the global relation $\tilde{\mathbf{S}}(z)=\mathbf{P}_{\pi}(z)\mathbf{S}(z)$, where

$$\mathbf{P}_{\pi}(z)=\mathbf{G}_{\pi}(z)\frac{1}{M} \mathbf{Q}_{\pi}(z)$$

is the global polyphase matrix. Hence, the orthogonality condition becomes

$$\mathbf{P}_{\pi}(z)=\frac{1}{M}\mathbf{G}_\pi(z) \mathbf{Q}_\pi(z) = \mathbf{I}_M. $$

(14.177)

It remains to relate the polyphase vector S(z), with elements S ^(j)(z), to the z-transform S(z ₀) of the input signal s(t ₀). The extraction of the polyphase components (S/P conversion) in the z-domain is given by the second of (14.174). Then

$${S}^{(j)}\bigl(z^M_0\bigr) = \mathop{\mathrm{dec}}_M \bigl[ z^j_0 S(z_0)\bigr]$$

and, in matrix form,

$$\mathbf{S}\bigl(z_0^M\bigr)=\mathop{\mathrm{dec}}_M\mathbf{Q}_{\hbox{\footnotesize S/P}}(z_0)S(z_0).$$

Analogously, we can relate the elements of the polyphase matrices Q _π (z) and G _π (z) to the original subband filters Q(z ₀) and G(z ₀). Following (14.157), we find

$$Q_n^{(j)}(z_0^M)= \mathop{\mathrm{dec}}_M \bigl[z_0^j Q(z_0)\bigr],\qquad G_m^{(j)}\bigl(z_0^M\bigr)= \mathop{\mathrm{dec}}_M \bigl[z_0^j G(z_0)\bigr]. $$

(14.178)

As results from the above relationships, the approach in the z-domain may be viewed as a duplicate of the one carried out in the frequency domain (in the 1D case). Now, the z-domain approach can continue with the investigation on the alias-free and distortion-free conditions. But, now it is a simple exercise to transfer results from the frequency domain to the z-domain.