The basic ideas is to simply
reorder the
DFT computation to expose the redundancies in the
DFT , and
exploit these to reduce computation!
Three conditions must be satisfied to make
this map serve our purposes
Each map must be one-to-one from
$0$ to
$N-1$ , because we want to do the
same computation, just in a different
order.
The map must be cleverly chosen so that computation is
reduced
The map should be chosen to make the short-length
transforms be
DFTs . (Not essential, since fast algorithms for
short-length
DFT -like computations could be developed, but it
makes our work easier.)
Conditions for one-to-oneness of general index map
Case i
${N}_{1}$ ,
${N}_{2}$ relatively prime (greatest common denominator
$1$ ) i.e.
$\gcd ({N}_{1}, {N}_{2})=1$
${K}_{1}{K}_{4}\mod N=0$ exclusive or
${K}_{2}{K}_{3}\mod N=0$ Common Factor Algorithm (CFA). Then
$$X(k)={\mathrm{DFT}}_{Ni}(\text{twiddle factors}{\mathrm{DFT}}_{Nj}(x({n}_{1}, {n}_{2})))$$
${K}_{1}{K}_{4}\mod N$and${K}_{2}{K}_{3}\mod N=0$ Prime Factor Algorithm (PFA).
$$X(k)={\mathrm{DFT}}_{Ni}({\mathrm{DFT}}_{Nj})$$No twiddle factors!
A PFA exists only and always for relatively prime
${N}_{1}()$ ,
${N}_{2}()$
Conditions for short-length transforms to be dfts
${K}_{1}{K}_{3}\mod N={N}_{2}$ and
${K}_{2}{K}_{4}\mod N={N}_{1}$
Convenient choice giving a PFA
${K}_{1}={N}_{2}$ ,
${K}_{2}={N}_{1}$ ,
${K}_{3}={N}_{2}{N}_{2}^{-1}\mod {N}_{1}\mod {N}_{1}$ ,
${K}_{4}={N}_{1}{N}_{1}^{-1}\mod {N}_{2}\mod {N}_{2}$ where
${N}_{1}^{-1}\mod {N}_{2}$ is an integer such that
${N}_{1}{N}_{1}^{-1}\mod =1$
radix-2 ,
radix-4 eliminate all multiplies in short-length
DFTs, but have twiddle factors: PFA eliminates all twiddlefactors, but ends up with multiplies in short-length
DFTs .
Surprisingly, total operation counts end up being very similarfor similar lengths.
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
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Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
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Damian
Is there any normative that regulates the use of silver nanoparticles?