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

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

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

7. For real valued sequence

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

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

e. Phase function is odd function

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

8. For a complex valued sequence

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

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

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

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

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

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

9. Central Co-ordinates

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

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

10.Parseval’s Relation

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

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

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

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

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

11.Time Reversal of a sequence

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

Let m=N-n

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

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

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

12.Circular Time Shift of a sequence

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

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

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

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

Put N+n-l = m

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

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

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

13.Circular Frequency Shift

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

= DFT of [Re[x (n)]]     LHS

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

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

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

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

Using central co-ordinate theorem

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

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

The document Properties of DFT | Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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FAQs on Properties of DFT - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What is the purpose of the Discrete Fourier Transform (DFT)?
Ans. The purpose of the Discrete Fourier Transform (DFT) is to transform a discrete-time signal from the time domain to the frequency domain. It allows us to analyze the frequency components present in a given signal.
2. How is the DFT different from the Fourier Transform (FT)?
Ans. The DFT is a discrete version of the Fourier Transform (FT), which is a continuous transform. While the FT operates on continuous signals, the DFT operates on discrete signals, such as those obtained from digital recordings or sampled data.
3. What are the properties of the DFT?
Ans. The DFT has several properties, including linearity, time shifting, frequency shifting, time reversal, and conjugation. Linearity means that the DFT of a sum of signals is equal to the sum of the individual DFTs. Time shifting refers to the effect of delaying a signal in the time domain, while frequency shifting refers to shifting the frequency components in the frequency domain. Time reversal is the process of reversing the order of the samples in a signal, and conjugation involves taking the complex conjugate of each sample in the frequency domain.
4. How is the DFT computed?
Ans. The DFT is computed using the Fast Fourier Transform (FFT) algorithm, which efficiently calculates the DFT of a signal. The FFT breaks down the DFT into smaller sub-problems and recursively computes them. This algorithm significantly reduces the computational complexity, making it feasible to compute the DFT for real-world applications.
5. What are some applications of the DFT?
Ans. The DFT has numerous applications in various fields. It is commonly used in audio and image processing, telecommunications, radar systems, and spectral analysis. In audio processing, the DFT is used for tasks like audio compression, equalization, and effects processing. In image processing, the DFT is used for tasks like image compression, noise reduction, and image filtering. In telecommunications, the DFT is used for modulation and demodulation of signals. In radar systems, the DFT is used for target detection and range estimation. Lastly, in spectral analysis, the DFT is used to analyze the frequency content of a signal.
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