Assignment on Electronics

Added on - 21 Apr 2020

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Table of ContentsPART A.......................................................................................................................................................1i.Radix-2 fft Algorithms........................................................................................................................1ii.FFT Calculate using Matlab Output....................................................................................................21.Decimation-in-time FFT Twiddle Factures and Flow Graph...............................................................21.1Signal Flow Graph.......................................................................................................................22.Block Diagram Representation in Butterfly Method............................................................................43.Text Output..........................................................................................................................................54.Matlab Code........................................................................................................................................5PART B.......................................................................................................................................................6i.Matlab Function...................................................................................................................................6ii.Input and Output Stages in Twiddle Factor.........................................................................................8iii.FFT Output Implementation..........................................................................................................111.Block Diagram of FFT.......................................................................................................................122.Output................................................................................................................................................123.Matlab Code......................................................................................................................................12References.................................................................................................................................................13
PART Ai.Radix-2 fft AlgorithmsThe radix N=2 point DFT is used by the divide and conquer approach. The N-point datais two N/2 point data sequencef1(n) andf2(n). The values are even and odd number samples ofx(n), respectively, as follows:f1(n) =x(n)f2(n) =x(2n+1) n=0,1,,,,,,N/2-1f1(n) andf2(n) isx(n) by a factor of 2, and the resulting FFT algorithm is called as adecimation-in-time algorithm. Now, the N-point DFT can be expressed in terms of the DFT's, of thedecimated sequences, as follows:X(k)=n=0N1x(n)WN,knK=0,1,,, N-1=n=0N1x(n)WN,kn+n=0N1x(n)WN,kn=n=0N1x(2m)WN,2kn+n=0N1x(2m+1)WN,k(2m+1)But,WN2= WN/2. With this substitution, the equation can be expressed as mentioned below:X(k)=n=0N21f1(m)WN,kn+WN,Kn=0N/21f2(m)WN/2,km,=F1+WN,kF2(k) k=0,1,,,, N-1. (PHAM, M. S)F1(k) andF2(k) are theN/2-point DFTs of the sequences off1(m) andf2(m).Since,F1(k) andF2(k) are periodic with periodN/2, we haveF1(k+N/2)= F1(k) andF2(k+N/2)=F2(k).In addition, the factorWNk+N/2= -WNkX(k)=F1(k)+WN,kF2(k)k=0,1,,,,,N/2-1k=0,1,,,,,N/2-11
ii.FFT Calculate using Matlab OutputXX1WNKY -1Y1Decimation in timeX1= x+WNK=YY1= x+WNK=Y>>Din = (1.99*rand(1,16)-1) + j*(1.99*rand(1,16)-1);(Amidror, 2013).i=0,15plot(1000*t(1:25),x(0,16))>>res_analysis(fft(Din)*Scale,FFTOutput);title(‘matlab function fft output’) (Marković and Brodersen, 2012).1.Decimation-in-time FFT Twiddle Factures and Flow GraphKthDIT twiddle factor angle =[[k2p/N]]bit-revThe N-point DIT FFTT haslog2(N) stages and numbered p=1,2,...log(N)These stages contain N/2 butterflies.Not counting the -1 operation, the pthstage is N/2 twiddle factors and is numbered as,k=0,1,2,,,,N/2.1.1Signal Flow GraphThe DIT FFT uses the signal flow graph in the complex number(W).The 16 point, radix 2 of signal graph is using to discrete-in-time, inW0¿W16¿.Symmetry property:WNk+N/2= -WNkPeriodicity property:WNk+N/¿¿=WNk2
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