Understanding FFT Algorithm & Discrete-Time Convolution in MATLAB: Lab Experiment, Exercises of Electronic Circuits Analysis

Download Understanding FFT Algorithm & Discrete-Time Convolution in MATLAB: Lab Experiment and more Exercises Electronic Circuits Analysis in PDF only on Docsity! 9.8 Laboratory Experiment on Signal Processing Purpose: By performing this experiment, the student will receive a better understanding of the use and power of the FFT algorithm in evaluating the corresponding discrete (time) Fourier transforms, continuous-time Fourier transform, and discrete-time convolution. As the computational tool, we will use the MATLAB functions fft and ifft. Part 1. In this part we use the FFT algorithm, as implemented in MATLAB, to find the DFT of some discrete-time signals. In addition, we demonstrate the use of the IFFT in recovering original discrete-time signals. (a) Consider the discrete-time signal 
       ! #"%$& and find analytically its DTFT. (b) Use the MATLAB function X=fft(x,N) to find the DFT of the preceding signal for ' )( !* +  + ,  (   . Use the MATLAB function x=ifft(X,N) to recover the original discrete-time signal. Plot the DFTs and IDFTs, and comment on the results obtained. (c) Consider the signal -
./ 0 +  1%121% +   &3& #"%$& and repeat parts (a) and (b). (d) Consider the signal whose nonzero values are between 45 and 4 * , respectively defined by /6
  ( 7 (  8 +  #9 ;: , and repeat parts (a) and (b). Comment on the results obtained. Part 2. Formula (9.71), <>=@?A#BDCFEHG<DI+=@?KJ#B , can be used for an approximate evaluation of the continuous-time Fourier transform. In this formula, E G is the sampling interval used for sampling the continuous-time signal =MLNB into =  E G BPO -
 , and< I =@?KJ#B is the corresponding DFT. (a) Consider the continuous-time signals presented in Figures 3.22 and 3.23. Sample these signals with EHG  12 and find DFTs of the obtained discrete-time signals. Calculate and plot the corresponding magnitude spectra for the approximate Fourier transforms and compare them to the results obtained analytically. (b) Repeat part (a) with E G  1  . Part 3. Discrete-time signal convolution can be efficiently evaluated via the DTFT and its convolution property. The relation Q
R6-
TSVUW
 implies XY=M? JZB <P=8?KJ#B![P=M? JZB . Hence, discrete-time convolution via DFT can be evaluated as Q
\]&^V_-`Vab^V_-` = W
c B ^V_-` = U-
 Bd . Note that such an obtained signal Q
c is, in general, Docsity.com