BEE604 Digital Signal Processing

Transcription

1 BEE64 Digital Signal Processing Copiled by, Mrs.S.Sherine Assistant Professor Departent of EEE BIHER.

2 COTETS Sapling Discrete Tie Fourier Transfor Properties of DTFT Discrete Fourier Transfor Inverse Discrete Fourier Transfor FAST FOURIER TRASFORMS

3 Sapling Continuous signals are digitized using digital coputers hen we saple, we calculate the value of the continuous signal at discrete points How fast do we saple hat is the value of each point Quantization deterines the value of each saples value

4 Sapling Periodic Functions - ote that wb = Bandwidth, thus if then aliasing occurs (signal overlaps) -To avoid aliasing -According sapling theory: To hear usic up to KHz a CD should saple at the rate of 44. KHz

5 Discrete Tie Fourier Transfor In likely we only have access to finite aount of data sequences (after sapling) Recall for continuous tie Fourier transfor, when the signal is sapled: Assuing Discrete-Tie Fourier Transfor (DTFT):

6 Discrete Tie Fourier Transfor Discrete-Tie Fourier Transfor (DTFT): A few points DTFT is periodic in frequency with period of p X[n] is a discrete signal DTFT allows us to find the spectru of the discrete signal as viewed fro a window

7 Eaple of Convolution Convolution e can write [n] (a periodic function) as an infinite su of the function o [n] (a non-periodic function) shifted units at a tie This will result Thus

8 Finding DTFT For periodic signals Starting with o[n] DTFT of o[n]

9 DT. Fourier is in radian and Transfors it is between and p in each discrete tie interval. This is different fro w where it was between IF and + IF 3. ote that X() is periodic

10 Properties of DTFT Reeber: For tie scaling note that > Signal spreading

11 Discrete Fourier Transfor e often do not have an infinite aount of data which is required by DTFT For eaple in a coputer we cannot calculate uncountable infinite (continuu) of frequencies as required by DTFT Thus, we use DTF to look at finite segent of data e only observe the data through a window In this case the o[n] is just a sapled data between n=, n=- ( points)

12 Discrete Fourier Transfor It turns out that DFT can be defined as ote that in this case the points are spaced pi; thus the resolution of the saples of the frequency spectru is pi. e can think of DFT as one period of discrete Fourier series

13 A short hand notation reeber:

14 Inverse of DFT e can obtain the inverse of DFT ote that

15 Eaple of DFT Find X[k] e know k=,.., 7; =8

16 Eaple of DFT Polar plot for Tie shift Property of DFT

17 Eaple of DFT Suation for X[k] Using the shift property!

18 Eaple of IDFT Reeber:

19 Fast Fourier Transfor Algoriths Consider DTFT Basic idea is to split the su into subsequences of length and continue all the way down until you have subsequences of length Log(8)

20 Radi- FFT Algoriths - Two point FFT e assue =^ This is called Radi- FFT Algoriths Let s take a siple eaple where only two points are given n=, n=; = y Butterfly FFT Advantage: Less coputationally intensive:.log() y

21 General FFT Algorith First break [n] into even and odd Let n= for even and n=+ for odd Even and odd parts are both DFT of a point sequence Break up the size subsequent in half by letting The first subsequence here is the ter [], [4], The second subsequent is [], [6], ) sin( ) cos( ]) [ ( ] [ j k k k k k j e p p p

22 Eaple ) sin( ) cos( ]) [ ( ] [ ] [ j k k k k k j e k X p p p Let s take a siple eaple where only two points are given n=, n=; = [] [] [] [] []) ( [] ] [ [] [] []) ( [] ] [.... k X k X Sae result

23 FFT Algoriths - Four point FFT First find even and odd parts and then cobine the: The general for:

24 FFT Algoriths - 8 point FFT