Properties of DFT | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

Properties of DFT 

1. Linearity

If h(n) = a h₁(n) + b h₂(n)

H (k) = a H₁(k) + b H₂(k)

2. Periodicity H(k) = H (k+N)

3. 

4. y(n) = x(n-n0)

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*(N-k)

h (-n) →H(-k) = H*(k) = H(N-k)

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 (N-k)

c. Imaginary component odd function

H_I (k) = -H_I (N-k)

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 Co-ordinates

      N=even

10.Parseval’s Relation

Proof: LHS      

11.Time Reversal of a sequence

Reversing the N-point seq in time is equivalent to reversing the DFT values.

DFT [x( N - n)] =

Let m=N-n

 m=1 to N = 0 to N-1

 = X(N-k)

12.Circular Time Shift of a sequence

DFT 

Put N+n-l = m

N to 2N-1-L is shifted to N ⇒ 0 to N-1-L

Therefore 0 to N-1 = (0 to N-1-L) to ( N-L to N-1)

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)

(m-fold 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(N-k) = 

N x(N-k) =  

 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(N-k) = 

= DFT of [Re[x (n)]]     LHS

Let y(n) =

Y(k) = 

Using central co-ordinate theorem

Y(0) =

Therefore