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.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?