# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 2 Preliminaries
### Filter Banks and Frames  
The general structure of a non-uniform analysis FB is presented in Figure 1 (e.g., [17]). It is a collection of \(K+1\) analysis filters \(H_{k}(z)\), where \(H_{k}(z)\) is the \(z\)-transform of the impulse response \(h_{k}[n]\) of the filter, and downsampling factors \(d_{k}\), \(k\in\{0\ldots K\}\), that divides a signal \(x\) into a set of \(K+1\) sub-band components \(y_{k}\), where  
\[y_{k}[n]\,=\downarrow_{d_{k}}\left\{h_{k}*x\right\}[n]\quad. \tag{1}\]  
The special case where all downsampling factors are identical, i.e., \(d_{k}=D\,\forall\,k\in\{0\ldots K\}\), is referred to as a uniform FB.  
By analogy, a synthesis FB is a collection of \(K+1\) upsampling factors \(d_{k}\) and synthesis filters \(G_{k}(z)\) (see Figure 2) that recombines the sub-band components \(y_{k}\) into an output signal \(\tilde{x}\) according to  
\[\tilde{x}[n]\,=\,2\Re\left(\sum_{k=0}^{K}\left(g_{k}*\uparrow_{d_{k}}\left\{ y_{k}\right\}\right)[n]\right)\,, \tag{2}\]  
where \(\Re\), denoting the real part, and the factor of 2 are a consequence of considering the positive frequency range only.  
A synthesis FB can be generalized to a _synthesis system_ (shown in Figure 3), which is a linear operator \(\mathcal{S}\) that takes as an input sub-band components \(y_{k}\) and yields an output sequence \(\tilde{x}\). For the synthesis operation, we use the notation \(\widetilde{\mathcal{S}}(\cdot,(G_{k},d_{k})_{k})\), where \((G_{k},d_{k})_{k}\) is the synthesis FB. An analysis  
Figure 1: General structure of a non-uniform analysis filter bank (FB) \((H_{k},d_{k})_{k}\) with \(H_{k}\) being the \(z\)-transform of the impulse response \(h_{k}[n]\) of the filter, also denoted as \(\mathcal{A}(\cdot,(H_{k},d_{k})_{k})\).  
FB is _invertible_ or _allows for perfect reconstruction_ if there exists a synthesis system \(\mathcal{S}\) that recovers \(x\) from the sub-band components \(y_{k}\)_without error_, i.e., \(\tilde{x}=x\) for all \(x\in\ell_{2}(\mathbb{Z})\). In other terms, the analysis-synthesis system \(((H_{k},d_{k})_{k},\mathcal{S})\) has the _perfect reconstruction property_. In practice, the implementation of that operation might introduce errors of the order of numerical precision.  
We use the mathematical theory of frames in order to analyze and design perfect reconstruction FBs (e.g., [20; 21; 22]). A _frame_ over the space of finite energy signals \(\ell_{2}(\mathbb{Z})\) is a set of functions spanning the space in a stable fashion. Consider a signal \(x\) and an analysis FB \((H_{k},d_{k})_{k}\) yielding \(y_{k}\). Then, an FB constitutes a frame if and only if \(0<A\leq B<\infty\) exist such that  
\[A\|x\|^{2}\leq\sum_{k}\|y_{k}\|^{2}\leq B\|x\|^{2},\forall x\in\ell^{2}( \mathbb{Z}) \tag{3}\]  
where \(A\) and \(B\) are called the lower and upper frame bounds of the system, respectively. The existence of \(A\) and \(B\) guarantees the invertibility of the FB. Several numerical properties of an FB can be derived from the frame bounds. In particular, the ratio \(\sqrt{B/A}\) corresponds to the _condition number_[49] of the FB, i.e., it determines the stability and reconstruction error of the system. Furthermore, the ratio \(B/A\) characterizes the overall frequency response of the FB. A ratio \(B/A=1\), for instance, means a perfectly flat frequency response. This is often desired in signal processing because, in that particular case, the analysis and synthesis FB are the same. Specifically, the synthesis filters are obtained by time-reversing the analysis filters, i.e., \(G_{k}(z)=\overline{H}_{k}(z)\).  
The frame bounds \(A\) and \(B\) correspond to the infinimum and supremum, respectively, of the eigenvalues of the operator \(\widetilde{\mathcal{S}}(\mathcal{A}(\cdot,(H_{k},d_{k})_{k}),(H_{k},d_{k})_{k})\) associated with the system \((H_{k},d_{k})_{k}\). In practice, these eigenvalues can be computed using iterative methods (see Sections 3.2 and 3.3).