The discrete fourier transform (dsp) 4
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The discrete fourier transform (DSP)
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The discrete fourier transform (dsp) 4
- 1. The Discrete Fourier Transform Mr. HIMANSHU DIWAKAR JRTGI 1 Mr. HIMANSHU DIWAKAR Assistant Professor JETGI
- 2. Mr. HIMANSHU DIWAKAR JRTGI 2
- 3. Sampling the Fourier Transform • Consider an aperiodic sequence with a Fourier transform • Assume that a sequence is obtained by sampling the DTFT • Since the DTFT is periodic resulting sequence is also periodic • We can also write it in terms of the z-transform • The sampling points are shown in figure • could be the DFS of a sequence • Write the corresponding sequence kN/2j kN/2 j eXeXkX ~ jDTFT eX]n[x kN/2j ez eXzXkX ~ kN/2 kX ~ 1N 0k knN/2j ekX ~ N 1 ]n[x~ Mr. HIMANSHU DIWAKAR JRTGI 3
- 4. Sampling the Fourier Transform Cont’d • The only assumption made on the sequence is that DTFT exist • Combine equation to get • Term in the parenthesis is • So we get m mjj emxeX 1N 0k knN/2j ekX ~ N 1 ]n[x~ kN/2j eXkX ~ mm 1N 0k mnkN/2j 1N 0k knN/2j m kmN/2j mnp~mxe N 1 mx eemx N 1 ]n[x~ r 1N 0k mnkN/2j rNmne N 1 mnp~ rr rNnxrNnnx]n[x~ Mr. HIMANSHU DIWAKAR JRTGI 4
- 5. Sampling the Fourier Transform Cont’d Mr. HIMANSHU DIWAKAR JRTGI 5
- 6. Sampling the Fourier Transform Cont’d • Samples of the DTFT of an aperiodic sequence • can be thought of as DFS coefficients • of a periodic sequence • obtained through summing periodic replicas of original sequence • If the original sequence • is of finite length • and we take sufficient number of samples of its DTFT • the original sequence can be recovered by • It is not necessary to know the DTFT at all frequencies • To recover the discrete-time sequence in time domain • Discrete Fourier Transform • Representing a finite length sequence by samples of DTFT else0 1Nn0nx~ nx Mr. HIMANSHU DIWAKAR JRTGI 6
- 7. The Discrete Fourier Transform • Consider a finite length sequence x[n] of length N • For given length-N sequence associate a periodic sequence • The DFS coefficients of the periodic sequence are samples of the DTFT of x[n] • Since x[n] is of length N there is no overlap between terms of x[n-rN] and we can write the periodic sequence as • To maintain duality between time and frequency • We choose one period of as the Fourier transform of x[n] 1Nn0ofoutside0nx r rNnxnx~ NkXNmodkXkX ~ kX ~ else0 1Nk0kX ~ kX NnxNmodnxnx~ Mr. HIMANSHU DIWAKAR JRTGI 7
- 8. The Discrete Fourier Transform Cont’d • The DFS pair • The equations involve only on period so we can write • The Discrete Fourier Transform • The DFT pair can also be written as 1N 0k knN/2j ekX ~ N 1 ]n[x~ 1N 0n knN/2j e]n[x~kX ~ else0 1Nk0e]n[x~ kX 1N 0n knN/2j else0 1Nk0ekX ~ N 1 ]n[x 1N 0k knN/2j 1N 0n knN/2j e]n[xkX 1N 0k knN/2j ekX N 1 ]n[x ]n[xkX DFT Mr. HIMANSHU DIWAKAR JRTGI 8
- 9. Example • The DFT of a rectangular pulse • x[n] is of length 5 • We can consider x[n] of any length greater than 5 • Let’s pick N=5 • Calculate the DFS of the periodic form of x[n] else0 ,...10,5,0k5 e1 e1 ekX ~ 5/k2j k2j 4 0n n5/k2j Mr. HIMANSHU DIWAKAR JRTGI 9
- 10. Example Cont’d • If we consider x[n] of length 10 • We get a different set of DFT coefficients • Still samples of the DTFT but in different places Mr. HIMANSHU DIWAKAR JRTGI 10
- 11. Properties of DFT • Linearity • Duality • Circular Shift of a Sequence kbXkaXnbxnax kXnx kXnx 21 DFT 21 2 DFT 2 1 DFT 1 mN/k2jDFT N DFT ekX1-Nn0mnx kXnx N DFT DFT kNxnX kXnx Mr. HIMANSHU DIWAKAR JRTGI 11
- 12. Example: Duality Mr. HIMANSHU DIWAKAR JRTGI 12
- 13. Symmetry Properties Mr. HIMANSHU DIWAKAR JRTGI 13
- 14. Circular Convolution • Circular convolution of two finite length sequences 1N 0m N213 mnxmxnx 1N 0m N123 mnxmxnx Mr. HIMANSHU DIWAKAR JRTGI 14
- 15. Example • Circular convolution of two rectangular pulses L=N=6 • DFT of each sequence • Multiplication of DFTs • And the inverse DFT else0 1Ln01 nxnx 21 else0 0kN ekXkX 1N 0n kn N 2 j 21 else0 0kN kXkXkX 2 213 else0 1Nn0N nx3 Mr. HIMANSHU DIWAKAR JRTGI 15
- 16. Example • We can augment zeros to each sequence L=2N=12 • The DFT of each sequence • Multiplication of DFTs N k2 j N Lk2 j 21 e1 e1 kXkX 2 N k2 j N Lk2 j 3 e1 e1 kX Mr. HIMANSHU DIWAKAR JRTGI 16
- 17. THANK YOU Mr. HIMANSHU DIWAKAR JRTGI 17