Dr. Navneet Yadav
X(k+N)=X(k) for all k
Formula for DFT
X(k+N)=X(k)
x(n+N)=x(n) for all n
Formula for IDFT
If two finite duration sequence x1(n) & x2(n) are linearly
combined as
The DFT of x3(n)
is
The shifted version of x(n) is
Same Way
DFT{x(n)}=x(k)
Consider 1st part
1
N-m+n=l
n=l+m-N
if n=0 then l=N-m
if n=m-1 then l=N-m+m-1=N-1
2
Consider 2nd part
N-m+n=l
n=l+m-N
if n=m then l=N
if n=N-1 then l=N-m+N-1=2N-m-1
3
Add 1st and 2nd part of equation 2 & 3
N-n=m
N-M=n
if n=0 then m=N
if n=N-1 then m=1
Similarly
If two finite duration sequence x1(n) & x2(n) are finite duration
Sequence both of length N with DFTs X1(K)& X2(K)
Now we find x3(n) for which the DFT is X3(K) Where
Similarly
N
Let
n-m-l=PN
l=n-m-PN
-P=1
l=n-m+N=((n-m))N
n-m-l=PN
m=n-l-PN
m=n-l+N
if l=0 m=n+N
if l=N-1m=n+1
Arrange by circular shift
m=n+1n+1-n-1=0
similarly m=n+Nn+N-n-1=N-1
N