logging in or signing up properties of dft haichenna Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 1607 Category: Science & Tech.. License: All Rights Reserved Like it (6) Dislike it (0) Added: February 03, 2011 This Presentation is Public Favorites: 1 Presentation Description PROPERTIES OF DFT Comments Posting comment... By: jpjeya (10 month(s) ago) hello sir, i am professor in engg college. your presentation is very good in the DFT properties with its proof. i need a copy of yours. can you send it to my mail sir. my id : jpjeya@yahoo.co.in thank you Saving..... Post Reply Close Saving..... Edit Comment Close By: nilesh1991 (13 month(s) ago) nj Saving..... Post Reply Close Saving..... Edit Comment Close By: jram11 (13 month(s) ago) can i get the download link for these properties as i need them very urgently because they are very clear and easy to understand.Thank you. my email is sriram_11@yahoo.co.in.. Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript DIGITAL SIGNAL PROCESSING Properties of DFT: DIGITAL SIGNAL PROCESSING Properties of DFT PAMARTHY.CHENNARAO Associate professor PALADUGU PARVATHIDEVI COLLGE OF ENGINEERINGProperties of DFT: Properties of DFTSlide 3: LINEAR PROPERTY : Let x 1 (n), x 2 (n) are two finite duration sequences, with a equal duration of N samples and DFT[ x 1 (n) ] = X 1 (k), DFT[ x 2 (n) ] = X 2 (k), then according to linear property of DFT, DFT[ a x 1 (n) + b x 2 (n) ] = a X 1 (k) + b X 2 (k). Where a & b are arbitrary constants PROOF : From basic definition of DFT Replace x(n) by a x 1 (n) + b x 2 (n)Slide 4: PERIODIC PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to periodic property of DFT, X(N + k) = X(k) x(N + n) = x(n) PROOF: From basic definition of DFT Replace k by N + kSlide 5: TIME SHIFTING PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to time shifting property of DFT, DFT[ x(n – n 0 ) ] = It is also known as circular time shift property. PROOF: From basic definition of DFT Replace k by x(n) by x(n – n 0 )Slide 6: TIME REVERSAL PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to time reversal property of DFT, DFT[ x(N – n) ] = X(N – k). PROOF: From basic definition of DFT Replace x(n) by x(N – n)Slide 7: FREQUENCY SHIFTING PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to time shifting property of DFT, DFT[ x(n) ] = X(k – k 0 ). It is also known as modulation theorem . PROOF: From basic definition of DFT Replace x(n) by x(n)Slide 8: Replace x(n) by x*(n) CONJUGATE PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to conjugate property of DFT, DFT[ x*(n) ] = X*(N – k). DFT[ x*(N-n) ] = X*(N + k). PROOF: (a) From basic definition of DFT (b) From basic definition of DFT Replace x(n) by x*(N-n)Slide 9: PARSEVALLS THEOREM: Let x(n) be a finite duration sequence, with a duration of N samples and N-Point DFT[ x(n) ] = X(k), then Parsevalls theorem provides the relation between x(n) and its frequency domain X(k) as PROOF:Slide 10: CIRCULAR CONVOLUTION: Circular convolution of a first sequence x 1 (n) having N samples [0 n N – 1] and a second sequence x 2 (n) having N samples [0 n N – 1] can be defined as Where x(n) : Circular convoluted sequence, with a duration of N = N – 1 N : Duration of x 1 (n) or x 2 (n) or x(n), 0 n N – 1 Durations of circular convoluted sequence x(n), first sequence x 1 (n) and second sequence x 2 (n) are equal, therefore circular convolution is also known as periodic convolution. DFT supports circular convolution, due to equal durations. Procedure For Evaluating Circular Convolution: The following four steps were required to compute circular convolution 1. Folding : Fold x 2 (m) about k=0 and take periodic extension, to obtain x 2 (-m) 2. Shifting : Shift the folded sequence x 2 (-m) by n units left and/or right, to obtain x 2 (n-m) . 3. Multiplication : Multiply x 1 (m) and x 2 (n-m), to obtain the product sequence x 1 (m) . x 2 (n-m), 4. Summation : Sum all the values of product sequence at every instant, to obtainSlide 11: Derivation for Circular Convolution : Let x 1 (n) and x 2 (n) are two finite duration sequences with a equal duration of N samples, assume x(n) be the circular convoluted sequence with a duration of N samples x(n) = x 1 (n) x 2 (n), convolution in time domain leads to multiplication in frequency domain. i.e X(k) = X 1 (k) X 2 (k). IDFT of X(k) can be defined as Replace X(k) = X 1 (k) X 2 (k) Change the order of two sums You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
properties of dft haichenna Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 1607 Category: Science & Tech.. License: All Rights Reserved Like it (6) Dislike it (0) Added: February 03, 2011 This Presentation is Public Favorites: 1 Presentation Description PROPERTIES OF DFT Comments Posting comment... By: jpjeya (10 month(s) ago) hello sir, i am professor in engg college. your presentation is very good in the DFT properties with its proof. i need a copy of yours. can you send it to my mail sir. my id : jpjeya@yahoo.co.in thank you Saving..... Post Reply Close Saving..... Edit Comment Close By: nilesh1991 (13 month(s) ago) nj Saving..... Post Reply Close Saving..... Edit Comment Close By: jram11 (13 month(s) ago) can i get the download link for these properties as i need them very urgently because they are very clear and easy to understand.Thank you. my email is sriram_11@yahoo.co.in.. Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript DIGITAL SIGNAL PROCESSING Properties of DFT: DIGITAL SIGNAL PROCESSING Properties of DFT PAMARTHY.CHENNARAO Associate professor PALADUGU PARVATHIDEVI COLLGE OF ENGINEERINGProperties of DFT: Properties of DFTSlide 3: LINEAR PROPERTY : Let x 1 (n), x 2 (n) are two finite duration sequences, with a equal duration of N samples and DFT[ x 1 (n) ] = X 1 (k), DFT[ x 2 (n) ] = X 2 (k), then according to linear property of DFT, DFT[ a x 1 (n) + b x 2 (n) ] = a X 1 (k) + b X 2 (k). Where a & b are arbitrary constants PROOF : From basic definition of DFT Replace x(n) by a x 1 (n) + b x 2 (n)Slide 4: PERIODIC PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to periodic property of DFT, X(N + k) = X(k) x(N + n) = x(n) PROOF: From basic definition of DFT Replace k by N + kSlide 5: TIME SHIFTING PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to time shifting property of DFT, DFT[ x(n – n 0 ) ] = It is also known as circular time shift property. PROOF: From basic definition of DFT Replace k by x(n) by x(n – n 0 )Slide 6: TIME REVERSAL PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to time reversal property of DFT, DFT[ x(N – n) ] = X(N – k). PROOF: From basic definition of DFT Replace x(n) by x(N – n)Slide 7: FREQUENCY SHIFTING PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to time shifting property of DFT, DFT[ x(n) ] = X(k – k 0 ). It is also known as modulation theorem . PROOF: From basic definition of DFT Replace x(n) by x(n)Slide 8: Replace x(n) by x*(n) CONJUGATE PROPERTY : Let x(n) be a finite duration sequence, with a duration of N samples and DFT[ x(n) ] = X(k), then according to conjugate property of DFT, DFT[ x*(n) ] = X*(N – k). DFT[ x*(N-n) ] = X*(N + k). PROOF: (a) From basic definition of DFT (b) From basic definition of DFT Replace x(n) by x*(N-n)Slide 9: PARSEVALLS THEOREM: Let x(n) be a finite duration sequence, with a duration of N samples and N-Point DFT[ x(n) ] = X(k), then Parsevalls theorem provides the relation between x(n) and its frequency domain X(k) as PROOF:Slide 10: CIRCULAR CONVOLUTION: Circular convolution of a first sequence x 1 (n) having N samples [0 n N – 1] and a second sequence x 2 (n) having N samples [0 n N – 1] can be defined as Where x(n) : Circular convoluted sequence, with a duration of N = N – 1 N : Duration of x 1 (n) or x 2 (n) or x(n), 0 n N – 1 Durations of circular convoluted sequence x(n), first sequence x 1 (n) and second sequence x 2 (n) are equal, therefore circular convolution is also known as periodic convolution. DFT supports circular convolution, due to equal durations. Procedure For Evaluating Circular Convolution: The following four steps were required to compute circular convolution 1. Folding : Fold x 2 (m) about k=0 and take periodic extension, to obtain x 2 (-m) 2. Shifting : Shift the folded sequence x 2 (-m) by n units left and/or right, to obtain x 2 (n-m) . 3. Multiplication : Multiply x 1 (m) and x 2 (n-m), to obtain the product sequence x 1 (m) . x 2 (n-m), 4. Summation : Sum all the values of product sequence at every instant, to obtainSlide 11: Derivation for Circular Convolution : Let x 1 (n) and x 2 (n) are two finite duration sequences with a equal duration of N samples, assume x(n) be the circular convoluted sequence with a duration of N samples x(n) = x 1 (n) x 2 (n), convolution in time domain leads to multiplication in frequency domain. i.e X(k) = X 1 (k) X 2 (k). IDFT of X(k) can be defined as Replace X(k) = X 1 (k) X 2 (k) Change the order of two sums