filter

Filter for vectors

Introduction

We consider a filter having the system function:

where $a_0=1,a_1,\ldots,a_N$, and $b_0,b_1,\ldots,b_N$ are real numbers. The function filter is implemented based on the transposed direct form II structure:

For $n=1,2,\ldots$, the difference equations for the above filter are given by

Usages

y = filter(b, a, x, wi)

[y, wf] = filter(b, a, x, wi)

• a and b are vectors of real numbers, containing filter coefficients $a_0,a_1,\ldots,a_N$​ and $b_0,b_1,\ldots,b_N$​​, respectively. The numbers a(1) and b(1) correspond to $a_0$ and $b_0$, respectively.
• If a and b have different lengths, the shorter vector will be packed with trailing zeros to match the length of the longer vector. As a result, $N$ is max(length(a), length(b)) - 1.
• If a(1) is not 1, each element of a and b are divided by a(1) before filtering. Hence, a(1) must be non-zero.
• Input x contains data to be filtered, and y contains the filtered data. The output y has the same size as x.
• wi contains values at the outputs of the delay blocks when the filtering begins.
• If wi is a vector, it should have length $N$​. If x is multi-dimensional, the same vector wi is applied repeatedly to different dimensions of x.
• If wi is multi-demensional, its size should be [N sz[1:(dim - 1) (dim + 1):ndims(x)]], where N is equal to max(length(a), length(b)) - 1, sz = size(x) and dim is the first nonsingleton dimension of x.
• If wi is empty, all $w_k$ will be initialized as zeros.
• If x is a vector, the filter is applied along the longer dimension. Otherwise, it is applied to vectors along the first nonsingleton dimension.
• wf contains values at the outputs of the delay blocks when the filtering ends. It has the size [N sz[1:(dim - 1) (dim + 1):ndims(x)]], where sz=size(x).

y = filter(b, a, x)

[y, wf] = filter(b, a, x)

• They give the same results as calling y = filter(b, a, x, wi) and [y, wf] = filter(b, a, x, wi) with wi being an array of zeros.

y = filter(b, a, x, wi, dim)

[y, wf]= filter(b, a, x, wi, dim)

• They give the same results as calling y = filter(b, a, x, wi) and [y, wf] = filter(b, a, x, wi) except that the filter is applied to vectors along the dim-th dimension.
• If wi is a vector, it should have length $N$, where $N$ is equal to max(length(a), length(b)) - 1. If x is multi-dimensional, the same vector wi is applied repeatedly to different dimensions of x.
• If wi is multi-demensional, its size should be [N sz[1:(dim - 1) (dim + 1):ndims(x)]], where N is equal to max(length(a), length(b)) - 1 and sz = size(x).
• If wi is empty, all $w_k$ will be initialized as zeros.
• dim should be a real positive integer scalar.