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Properties of DFT
1. Linearity
If h(n) = a h_{1}(n) + b h_{2}(n)
H (k) = a H_{1}(k) + b H_{2}(k)
2. Periodicity H(k) = H (k+N)
3.
4. y(n) = x(nn0)
Y (k) = X (k) e
5. y (n) = h (n) * x (n)
Y (k) = H (k) X (k)
6. y (n) = h(n) x(n)
7. For real valued sequence
a. Complex conjugate symmetry
h (n)→H(k) = H*(Nk)
h (n) →H(k) = H*(k) = H(Nk)
i. Produces symmetric real frequency components and anti symmetric
imaginary frequency components about the N/2 DFT
i. Only frequency components from 0 to N/2 need to be computed in order to define the output completely.
b. Real Component is even function
H_{R} (k) = H_{R} (Nk)
c. Imaginary component odd function
H_{I} (k) = H_{I} (Nk)
d. Magnitude function is even function
e. Phase function is odd function
f. If h(n) = h(n)
H (k) is purely real
g. If h(n) = h(n)
H (k) is purely imaginary
8. For a complex valued sequence
Similarly DFT [x*(n)] = X*(k)
9. Central Coordinates
N=even
10.Parseval’s Relation
Proof: LHS
11.Time Reversal of a sequence
Reversing the Npoint seq in time is equivalent to reversing the DFT values.
DFT [x( N  n)] =
Let m=Nn
m=1 to N = 0 to N1
= X(Nk)
12.Circular Time Shift of a sequence
DFT
Put N+nl = m
N to 2N1L is shifted to N ⇒ 0 to N1L
Therefore 0 to N1 = (0 to N1L) to ( NL to N1)
Therefore
= X(k) e RHS
13.Circular Frequency Shift
x(n)e ⇔ X (k l ) _{N}
_{DFT }
= NX (k  l ) _{N} RHS
14. x(n) ⇔ X(k)
{x(n), x(n), x(n)…….x(n)} ⇔ M X(k/m)
(mfold replication)
x (n/m) ⇔ { X (k ), X (k ),......X (k )} (M fold replication)
2, 3, 2, 1 → 8, j2, 0, j2
Zero interpolated by M
{2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1} → {24, 0, 0, j6, 0, 0, 0, 0, 0, j6, 0, 0}
15. Duality
x(n)⇔X(k) 0 ≤ K ≤ N  1
(n) =
x(Nk) =
N x(Nk) =
DFT [ X(n) ] LHS proved
16. Re[x(n)]
x _{ep}(n) = Even part of periodic sequence =
x _{op} (n) = op Odd part of periodic sequence =
Proof: X(k) =
X(Nk) =
= DFT of [Re[x (n)]] LHS
Let y(n) =
Y(k) =
Using central coordinate theorem
Y(0) =
Therefore
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