Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Document Description: Properties of DFT for Electronics and Communication Engineering (ECE) 2022 is part of Signals and Systems preparation. The notes and questions for Properties of DFT have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Properties of DFT covers topics like and Properties of DFT Example, for Electronics and Communication Engineering (ECE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Properties of DFT.

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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.  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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

Y (k) = X (k) e  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

5. y (n) = h (n) * x (n) 

Y (k) = H (k) X (k)

6. y (n) = h(n) x(n)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

7. For real valued sequence

  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  

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

HR (k) = HR (N-k)

c. Imaginary component odd function

HI (k) = -HI (N-k)

d. Magnitude function is even function

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

e. Phase function is odd function

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

f. If h(n) = h(-n)

H (k) is purely real

g. If h(n) = -h(-n)

H (k) is purely imaginary     Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

8. For a complex valued sequence

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Similarly DFT [x*(-n)] = X*(k)

9. Central Co-ordinates

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)     Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)     Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) N=even

10.Parseval’s Relation

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Proof: LHS       Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

11.Time Reversal of a sequence

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  

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

DFT [x( N - n)] = Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Let m=N-n

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) m=1 to N = 0 to N-1

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) = X(N-k)

12.Circular Time Shift of a sequence

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

DFT  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Put N+n-l = m

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

= X(k) e Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) RHS

13.Circular Frequency Shift

x(n)e  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) ⇔ X (k l ) N

DFT Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) = 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) =   Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  

x(N-k) =  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

N x(N-k) =   Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) DFT [ X(n) ]    LHS proved

16. Re[x(n)]  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

x ep(n) =  Even part of periodic sequence = Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

x op (n) = op Odd part of periodic sequence = Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Proof: X(k) =  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

X(N-k) =  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

= DFT of [Re[x (n)]]     LHS

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Let y(n) = Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Y(k) =  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Using central co-ordinate theorem

Y(0) = Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Therefore  Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The document Properties of DFT Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
All you need of Electronics and Communication Engineering (ECE) at this link: Electronics and Communication Engineering (ECE)
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