Properties of dft

Gives the property of DFT

1. PROPERTIES OF DFT Linearity Periodicity Circular Time Shift Time Reversal Conjugation Circular frequency Shift Multiplication Circular Convolution Circular Correlation Parseval’s Theorem

2. LINEARITY Statement: The linearity property of DFT states that the DFT of a linear weighted combination of two or more signals is equal to similar linear weighted combination of the DFT of individual signals. Let DFT{x 1 (n)}=X 1 (K)&DFT{x 2 (n)}=X 2 (K) then, DFT{a 1 x 1 (n)+a 2 x 2 (n)}=a 1 X 1 (K)+a 2 X 2 (K) Where a 1 & a 2 are constants.

3. LINEARITY  

4. PERIODICITY Statement: If a sequence x(n) is periodic with periodicity of N samples then N-point DFT, X(K) is also periodic with a periodicity of N Samples Let x(n)=x( n+N ) for all n x(K)=x(K+N) for all K

5. PERIODICITY  

6. CIRCULAR TIME SHIFT  

7. CIRCULAR TIME SHIFT  

8. TIME REVERSAL Statement: The Time Reversal property of DFT says that reversing the N-Point Sequence in time is equivalent to reversing the DFT sequence. If DFT{x(n)}=X(K) then, DFT{x(N-n)}=X(N-K)

9. TIME REVERSAL Proof: DFT{x(N-n)}= Let m=N-n; n=N-m DFT{x(N-n)} = = W.K.T [Since k is an integer] = = [ ] = = DFT{x(N-n)} =X(N-K)  

10. CONJUGATION Statement: Let x(n) be a complex N-point discrete sequence x*(n) be its conjugate sequence If DFT{x(n)}=X(K) then, DFT{x*(n)}=X*(N-K)

11. CONJUGATION

12. CIRCULAR FREQUENCY SHIFT Statement: The Circular frequency shift property of DFT says that if a discrete time signal is multiplied by its DFT is circularly shifted by m units Let DFT{x(n)}=X(K) then, DFT{x(n). =x((K-m)) N  

13. CIRCULAR FREQUENCY SHIFT Proof: DFT{x(n) }= = DFT{x(n) }=X((K-m))N  

14. MULTIPLICATION Statement: The Multiplication Property of DFT says that the DFT of product of two discrete time sequences is equivalent to circular convolution of DFT’s of the individual sequences scaled by the factor of 1/N If DFT{x(n)}=X(K) then, DFT{x 1 (n) x 2 (n) =1/N{X 1 (K) Θ X 2 (K)}  

15. MULTIPLICATION  

16. MULTIPLICATION  

17. CIRCULAR CONVOLUTION  

18. CIRCULAR CONVOLUTION Proof: Let x 1 (n) & x 2 (n) be N-Point Sequences, Now by definition of DFT X 1 (K) = = [n=m] —(1) X 2 (K) = = [n=p] ---(2) Consider the Product of X 1 (K) X 2 (K). The inverse DFT of the product is given by, DFT -1 {X 1 (K) X 2 (K)}= -------(3) Substitute the value of equ 1 and 2 in equ 3 We get, DFT -1 {X 1 (K) X 2 (K)}=  

19. CIRCULAR CONVOLUTION  

20. CIRCULAR CONVOLUTION  

21. CIRCULAR CORRELATION Statement: The Circular Correlation of two sequence x(n) and y(n) is defined as r xy (m) = N Let DFT{x(n)}=X(K) and DFT{y(n)}=Y(K) then by Correlation property, X(K)= -----(1) Y(K)= [n=p] ---(2) Consider the product of X(K) Y * (K). The IDFT of the product is given by, DFT -1 {X(K) Y * (K)}= Let n=m = --------(3)  

22. CIRCULAR CORRELATION Substitute the value of equ 1 and 2 in equ 3 We get, = = ------(4) Consider the Summation in equ (4) Let n-m+p = qN and q is an integer = = = =N ------(5) Consider the summation in the equ (4)we get ((n-m)) N -------(6)  

23. CIRCULAR CORRELATION DFT -1 {X(K) Y * (K)}=1/N ((n-m)) N .N DFT -1 {X(K) Y * (K)}= ((n-m)) N = ((n-m)) N DFT -1 {X(K) Y * (K)}= r xy (m) X(K) Y * (K)=DFT{ r xy (m)} Thus proved  

24. PARSEVALS THEOREM Statement: Let DFT{x 1 (n)}=X 1 (K)& DFT{x 2 (n)}=X 2 (K) then by Parseval ’ theorem = Proof: Let and be N-point Sequences By the definition of DFT, X 1 (K)= ----(1) By the definition of IDFT, ---(2)