MODULE 7

Question 1

DISCRETE FOURIER TRANSFORM

Answer 1

• The Discrete Fourier Transform (DFT) refers to a signal processing operation that converts n-sample, periodic discrete-time signal into discrete frequency signal.

Question 2

MATHEMATICAL FORMULA

Answer 2

X(k)=∑_(n=0)^(n-1)▒〖x(n) 〖 e〗^(-jΩ_k n) 〗
WHERE,
Ω_k=2π/N k

Question 3

TWIDDLE FACTOR

Answer 3

〖 e〗^(-jΩ_k )=W_n

Question 4

W_4

Answer 4

-j

Question 5

W_8

Answer 5

(√2-j√2)/2

Question 6

2. CIRCULAR SHIFT

Answer 6

〖x(n-n_0 )〗_(mod N)↔〖 e〗^(-jΩ_k n_0 ) X(k)

Question 7

3. FREQUENCY SHIFT

Answer 7

〖 e〗^(-jΩ_k0 n) x(n)↔X(k-k_0 )

Question 8

4. CIRCUILAR FOLDING

Answer 8

〖x(-n)〗(mod N)↔〖X(-k)〗(mod N)

Question 9

5. SYMMETRY

Answer 9

X(k)=X*(N-K)

Question 10

6. CIRCULAR CONVOLUTION

Answer 10

x_1 (n)⊛x(n)↔X_1 (k)X(k)

Question 11

7. MULTIPLICATION PROPERTY

Answer 11

〖x_1 (n)∙x〗_2 (n)↔1/N (X_1 (k)⊛X(k))

Question 12

8. FREQUENCY INTERPOLATION

Answer 12

Interpolation in f.d ↔ replication in f.d