LAB # 5 HANDOUT. »»» The N-point DFT is converted into two DFTs each of N/2 points. N = -W N Then, the following formulas must be used. = k=0,...

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1 EEE4 Lab Handout. FAST FOURIER TRANSFORM LAB # 5 HANDOUT Data Sequence A = x, x, x, x3, x4, x5, x6, x7»»» The N-point DFT is converted into two DFTs each of N/ points. x, x, x4, x6 x, x3, x5, x7»»» N =e -jπ/n x, x x, x N = N/ (+N/) N = - N Then, the following formulas must be used. =,,,3 ( ) ( ) ( ) = =,...,7 + N =,,,3 = ( ) = ( ) ( ) ( ) ( ) ( ) + N / 3 + N / 4 =, =, =, =, ( ) = x + N / 4 x4 ( ) = x + N / 4 x6 3( ) = x + N / 4x5 4 ( ) = x3 + N / 4x7 ()= x + x 4 ()= x + x 6 3 ()= x + x 5 4 ()= x 3 + x 7 ()= x - x 4 ()= x x 6 3 ()= x x 5 4 ()= x 3 x 7 ( ) = ( ) + ( ) ( ) = 3 ( ) + 4 ( ) () = () j () () = 3 () j 4 () ( ) = ( ) ( ) ( ) = 3 ( ) 4 ( ) ( 3) = () + j () () 3 = () + j () 3 4 By using fft and ifft functions in matlab: In command window: >> x=[ ] x = >> fft(x) ans = Columns through i i i.44-.i >> ifft(ans) ans =.....

2 EEE4 Lab Handout. THE BUTTERFLY () () = () + () () () = () - () FFT Butterflies () () = () + () () (3) = () - () The butterfly that shows all of the connections is shown below: x () () () x 4 () () () x () () () x 6 () (3) (3) x 3 () () (4) x 5 3 () () (5) x 3 4 () () (6) x 7 4 () (3) 3 (7)

3 EEE4 Lab Handout 3. SAMPLING OF A SINUSOIDAL SIGNAL In this program we will investigate the sampling of a continuous-time sinusoidal signal x a (t) at various sampling rate. Since MATLAB cannot strictly generate a continuoustime signal, we will generate a sequence {x a (nt H )} from x a (t) by sampling it at a very high rate T H such that the samples are very close to each other. A plot of x a (nt H ) using the plot command will then loo lie a continuous-time signal. As a result, x[n] is generated by periodically sampling a (t) at uniform time interval T. x[n]= a (n,t) T=sampling interval F s =sampling rate, sampling frequency= /T M-file: % Illustration of the Sampling Process in the Time-Domain clf; % Clear current figure clear all; % Clear all variables % Generation of the continuous-time signal t = :.5 : ; f = 3; xa = cos(*pi*f*t); subplot(,,) plot(t,xa);grid xlabel('time, msec'); ylabel(''); title('continuous-time signal x_{a}(t)'); axis([ -..]); subplot(,,) % Sampling in time-domain T =.; n = : T : ; xs = cos(*pi*f*n); = : length(n)-; stem(,xs,'r');grid xlabel('time index n'); ylabel(''); title('discrete-time signal x[n]'); axis([ length(n)- -..]) 3

4 EEE4 Lab Handout Continuous-time signal x a (t) Time, msec Discrete-time signal x[n] Time index n 4. ALIASING EFFECT IN THE TIME-DOMAIN In this program we will generate a continuous-time equivalent y a (t) of the discretetime signal x[n] generated in previous program to investigate the relation between the frequency of the sinusoidal signal x a (t) and sampling period. To generate the reconstructed signal y a (t) from x[n], we pass x[n] through an ideal lowpass filter. The higher frequency analog signal can be reconstructed in lower freq. form after sampling. Formula of Reconstruction: n= [ Fs( t nts ] a ( t) = x( n).sinc ) To prevent aliasing f s >f. The colors of the plots: Some other representations: b represents blue o represents circle g represents green * represents star r represents red. c represents cyan. m represents magenta. y represents yellow represents blac 4

5 EEE4 Lab Handout M-file: % Illustrating of Aliasing Effect in the Time-Domain clf; clear all; T =.; f = 3; n = (:T:)'; xs = cos(*pi*f*n); %linspace(x,x,n) generates N points between x and x. %for N<, linspace returns x. t = linspace(-.5,.5,5)'; %reconstruction formula is shown below. ya = sinc((/t)*t(:,ones(size(n))) - (/T)*n(:,ones(size(t)))')*xs; plot(n,xs,'o',t,ya);grid; xlabel('time index n'); ylabel(''); title('reconstructed continuous-time signal y_{a}(t)'); axis([ -..]) Reconstructed continuous-time signal y a (t) Time index n 5

6 EEE4 Lab Handout 5. EFFECT OF SAMPLING IN THE FREQUENCY-DOMAIN In order to convert a continuous-time signal x a (t) into an equivalent discrete-time signal x[n], the former must be band-limited in the frequency-domain. To illustrate the effect of sampling in the frequency-domain we choose an exponentially decaying continuous-time signal with a CTFT that is approximately band-limited. M-file: % Illustrating of the Aliasing Effect % In the Frequency-Domain clf; clear all; % Generation of the continuous-time signal t = :.5 : ; xa = *t.*exp(-t); subplot(,,) plot(t,xa);grid % grid is used for grid lines xlabel('time, msec'); ylabel(''); title('continuous-time signal x_{a}(t)'); subplot(,,) wa = : /5 : ; % H=freqs(B,A,) returns the complex frequency response % vector H of the filter B/A; the frequency response is % evaluated at the points specified in vector. Ha = freqs(,[ ],wa); plot(wa/(*pi),abs(ha));grid xlabel('frequency, KHz'); ylabel(''); title(' _{a}(j\omega) '); axis([ 5/pi ]); subplot(,,3) % Sampling of the signal T = ; n = : T : ; xs = *n.*exp(-n); = : length(n)-; stem(,xs);grid; xlabel('time index n'); ylabel(''); title('discrete-time signal x[n]'); subplot(,,4) wd = : pi/55 : pi; 6

7 EEE4 Lab Handout % H=freqz(B,A,) returns the frequency response at % frequencies designated in vector. Hd = freqz(xs,,wd); plot(wd/(t*pi),t*abs(hd));grid; xlabel('frequency, KHz'); ylabel(''); title(' (e^{j\omega}) '); axis([ /T ]);. Continuous-time signal x a (t) a (jω) Time, msec.5.5 Frequency, KHz. Discrete-time signal x[n] (e jω ) Time index n.5 Frequency, KHz 7