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Topic: sum of csc
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Posts: 1
From: china
Registered: 10/18/10
sum of csc
Posted: Oct 18, 2010 10:42 PM
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sum{csc[pi*(n+.5)/N]} with n

n,k,N are natural numbers

I doubt if it's concerned about the gamma function.

This question is from the problem X(k+1/2)=DFT[cot(pi*(n+.5)/N)*exp(-1j*pi*n/N)],where DFT denotes the discrete fourier transform.

while x(n) denotes a time sequence,X(k) denotes its DFT sequence and we have
X(k)=DFT[x(n)]=sum[x(n)*exp(-1j*2*pi*n*k/N)],(0<=n<=N-1),where N is the length of x(n),k is the sampling frequency,n is the sampling time.
=(1/N)*{sum[X(g)]+sum[X(g)*cot[pi*(k-g+1/2)/N]]},with 0<=g<=N-1.
and sum[X(g)*cot[pi*(k-g+1/2)/N]] denotes a circle convolution with X(k)and cot[pi*(k+1/2)/N].
I have known that
where T(k) is a sampling function, which is equal to
can be expressed by
with, 0<=g1<=k,k+1<=g2<=N-1,0<=k<=N-1
and there comes my question.
thanks for reading!

Message was edited by: stainburg

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