0.8 Filter banks and transmultiplexers  (Page 3/23)

A transmultiplexer is PR if and only if, for all $i,j\in \left\{0,,,1,,,...,,,L,-,1\right\}$ ,

Moreover, if the number of channels is equal to the downsampling factor (i.e., $L=\left|M\right|$ ), [link] and [link] are equivalent.

Consider a PR filter bank. Since an arbitrary signal is a linear superposition of impulses, it suffices to consider the input signal, $x\left(n\right)=\delta (n-n_1)$ , for arbitrary integer $n_1$ . Then (see [link] ) $d_i\left(n\right)=h_i(Mn-n_1)$ and therefore, ${\displaystyle y\left(n_2\right)=\sum _i\sum _n{g}_i(n_2-Mn)d_i\left(n\right)}$ . But by PR, $y\left(n_2\right)=\delta (n_2-n_1)$ . The filter bank PR property is precisely a statement of this fact:

Consider a PR transmultiplexer. Once again because of linear superposition, it suffices to cosnsider only the input signals $x_i\left(n\right)=\delta \left(n\right)\delta (i-j)$ for all $i$ and $j$ . Then, $d\left(n\right)=g_j\left(n\right)$ (see [link] ), and ${\displaystyle y_i\left(l\right)=\sum _n{h}_i\left(n\right)d(Ml-n)}$ . But by PR $y_i\left(l\right)=\delta \left(l\right)\delta (i-j)$ . The transmultiplexer PR property is precisely a statement of this fact:

Remark: Strictly speaking, in the superposition argument proving [link] , one has to consider the input signals $x_i\left(n\right)=\delta (n-n_1)\delta (i-j)$ for arbitrary $n_1$ . One readily verifies that for all $n_1$ [link] has to be satisfied.

The equivalence of [link] and [link] when $L=M$ is not obvious from the direct characterization. However, the transform domaincharacterization that we shall see shortly will make this connection obvious. For a PR filter, bank the $L$ channels should contain sufficient information to reconstruct the original signal (note the summation over $i$ in [link] ), while for a transmultiplexer, the constituent channels should satisfy biorthogonalityconstraints so that they can be reconstructed from the composite signal (note the biorthogonality conditions suggested by [link] ).

Matrix characterization of pr

The second viewpoint is linear-algebraic in that it considers all signals as vectors and all filtering operations as matrix-vector multiplications [link] . In [link] and [link] the signals $x\left(n\right)$ , $d_i\left(n\right)$ and $y\left(n\right)$ can be naturally associated with infinite vectors $\mathbf{x}$ , ${\mathbf{d}}_i$ and $\mathbf{y}$ respectively. For example, $\mathbf{x}=[\cdots ,x(-1),x(0),x(1),\cdots ]$ . Then the analysis filtering operation can be expressed as

where, for each $i$ , ${\mathbf{H}}_i$ is a matrix with entries appropriately drawn from filter $h_i$ . ${\mathbf{H}}_i$ is a block Toeplitz matrix (since its obtained by retaining every $M^{th}$ row of the Toeplitz matrix representing convolution by $h_i$ ) with every row containing $h_i$ in an index-reversed order. Then the synthesis filtering operation can be expressed as

where, for each $i$ , ${\mathbf{G}}_i$ is a matrix with entries appropriately drawn from filter $g_i$ . ${\mathbf{G}}_i$ is also a block Toeplitz matrix (since it is obtained by retaining every $M^{th}$ row of the Toeplitz matrix whose transpose represents convolution by $g_i$ ) with every row containing $g_i$ in its natural order. Define $\mathbf{d}$ to be the vector obtained by interlacing the entries of each of the vectors ${\mathbf{d}}_i$ : $\mathbf{d}=[\cdots ,d_0\left(0\right),d_1\left(0\right),\cdots ,d_{M-1}\left(0\right),d_0\left(1\right),d_1\left(1\right),\cdots ]$ . Also define the matrices $\mathbf{H}$ and $\mathbf{G}$ (in terms of ${\mathbf{H}}_i$ and ${\mathbf{G}}_i$ ) so that

$\mathbf{H}$ is obtained by interlacing the rows of ${\mathbf{H}}_i$ and $\mathbf{G}$ is obtained by interlacing the rows of ${\mathbf{G}}_i$ . For example, in the FIR case if the filters are all of length $N$ ,