Vector operations

downsample(vec::InfiniteArrays, m::Int)

The resulting vector r satisfies r(k) = vec(mk)

upample(vec::InfiniteVector, m::Int)

The resulting vector r satisfies r(k) = vec(k/m), if k is multple of m, otherwise, r(k)=0

shift(vec::InfiniteArrays, m::Int)

The resulting vector r satisfies r(k) = vec(k+m)

shift!(vec::InfiniteArrays, m::Int)

In-place shifting of vector. (not always possible)

reverse(vec::BiInfiniteVector)

Time-reversel: vec(-k)

reverse!(vec::CompactInfiniteVector)

In-place time reversel: vec(-k)

alternating_flip(vec::BiInfiniteVector, pivot = 1)

From a given filter $h(i)$, compute a new filter satisfying the alternating flip relation, centered around the given pivot:

$g(k) = (-1)^k h(pivot-k)$

alternating(vec::InfiniteVector)

Compute a new 'alternating' filter satisfying $g(k) = (-1)^k h(k)`.$

evenpart(vec::InfiniteVector)

Return the even part of a sequence `s`, defined by `s_e[k] = s[2k]`.

oddpart(vec::InfiniteVector)

Return the odd part of a sequence s, defined by s_o[k] = s[2k+1]

inv(a::CompactInfiniteVector{T}, m::Int)

A solution of $[a*b]_{↓m}=δ$, given a.

adjoint(vec::BiInfiniteVector)

Returns the paraconjugate $\overline{vec(-k)}$`

transpose(vec::BiInfiniteVector)

Returns the time-reversed vector vec(-k)

ztransform(vec::InfiniteVector, z)

The Z transform of a sequence is a continuous function of z, defined by $S(z) = \sum_{k\in ℤ} s_k z^{-k}$

fouriertransform(s::BiInfiniteVector, ω)

The Fourier transform of a sequence is defined by $S(ω) = \sum_{k\in ℤ} s_k e^{-i ω k}$. It is like the Z transform with $z = e^{i ω}$. The Fourier transform is a 2π-periodic continuous function of ω.

moment(vec::InfiniteVector, j)

The j-th discrete moment of a sequence is defined as $\sum_{k\in ℤ} h_k k^j$.

leastsquares_inv(v::BiInfiniteVector, m::Int)

The inverse filter $f = v*[([v*v]_{↓m})^{-1}]_{↑m}$

returns by applying on b $v*[[f*b]_{↓m}]_{↑m}$ the least squares approximation of b in the range of v.

*(a::InfiniteVector, b::InfiniteVector)

The convolution is defined as $c(n) = \sum_{k\in ℤ} a(k)b(n-k)$

⊛(a::AbstractPeriodicInfiniteVector, b::AbstractPeriodicInfiniteVector)

The circular convoluation defined as $c(n) = \sum_{k=1}^N a(k)b(n-k)$ for $n\in ℤ$, where N is the period of a and b.

⋆(a::InfiniteVector, b::InfiniteVector)

The convolution is defined as $c(n) = \sum_{k\in ℤ} a(k)b(n-k)$