CE 513: STATISTICAL METHODS

Transcription

1 /CE 68 CE 513: STATISTICAL METHODS IN CIVIL ENGINEERING Lecture: Introduction to Fourier transforms Dr. Budhaditya Hazra Room: N-37 Department of Civil Engineering 1

2 Fourier Analysis

3 Fourier Series

4 Fourier Series: Example

5 Fourier Series: Example a = ; a n = b n = 2 T T 2 T 2 x t sin 2nπt T dt = 2 nπ (1 nπ)

6 Fourier Series: Example

7 Amplitude & Phase Spectrum x t = c n e j2πnt T n= c n = 2 T T x t e j2πnt T dt

8 MATLAB: example clc; close all; clear all % Keep hitting enter button if you want to see the term by term approx. t=[:.1:1]; x=[ ]; x_tmp=zeros(size(t)); for n=1:2:39 x_tmp=x_tmp+4/pi*(1/n*sin(2*pi*n*t)); x=[x; x_tmp]; end figure, for i=1:2 drawnow plot(t, x(i,:)) %plot(t,x(i,:),t,x(7,:),t,x(2,:)); xlabel('\itt\rm (seconds)'); ylabel('\itx\rm(\itt\rm)') grid on pause end

9 FOURIER TRANSFORM Extension of Fourier analysis to non-periodic phenomena Discrete to continuous Skipping essential steps, in the limit T p Fourier transform Inverse Fourier transform

10 Examples 1) Dirac delta 2) Symmetric exponential

11 Examples 3) Sinusoid 4) Window function

12 Examples 5) Damped symmetrically oscillating function

13 6) Damped oscillating function

14 Windowing

15 Windowing

16 Windowing: Illustration The distortion due to the main lobe is sometimes called smearing, and the distortion caused by the side lobes is called leakage.

17 COMMON WINDOW FUNCTIONS Rectangular Window Smaller main lobe Hann Window Bigger main lobe

18 COMMON WINDOW FUNCTIONS Rectangular Window Bigger side lobe Hann Window Smaller side lobe

19 Discrete Fourier Transform Consider a sequence x(n ) at n =, 1, 2, 3, 4,.., N-1 points. The DFT is defined as : Note that this is still continuous in frequency

20 Discrete Fourier Transform Now let us evaluate this at frequencies:

21 FOURIER INTEGRAL VS DFT Fourier Integral Fourier transform of the sampled sequence

22 FOURIER INTEGRAL VS DFT

23 FOURIER INTEGRAL VS DFT

24 FOURIER INTEGRAL VS DFT

25 FFT ALGORITHM: GLIMPSES The DFT provides uniformly spaced samples of the Discrete-Time Fourier Transform (DTFT) DFT definition: X[ k] N 1 n x[ n] e 2nk j N x[ n] Requires N 2 complex multiplications & N(N-1) complex additions 1 N N 1 n X[ k] e 2nk j N

26 LET S TAKE AN EXAMPLE OF FFT: A SIMPLE MATLAB CODE Consider a signal: x = A sin 2πpt p = 1. Hz T p Set a sampling frequency fs = 1 T p. Hz A = 2, p =1 Hz What does Fourier integral give? X ( f ) = A 2 at p = 1 Hz

27 FFT MATLAB EXAMPLE-1 freq = 1; % Frequency of sinusoid. freq= 1/ Tp; Tp=1/freq; fs=1*freq; % Tp is the period % Define sample time and sample frequency. Take fs = 1/Tp ts=1/fs; T1= 1*Tp; % Time point where the data is truncated to t=:ts:t1-ts; x= 2*cos(2*pi*freq*t); L=length(x); FT = fft(x); % FFT command; gives Fourier amplitude f =fs* (:(L-1))/L; % Creating the X-axis or frequency axis fabs=abs(ft); % Single sided Fourier transform

28 Modulus of X(k) DATA TRUNCATED AT EXACTLY ONE PERIOD (1-POINT FFT) Since the exact number of period is taken for DFT, all the frequency components except p = 1. Hz are zero p = 1Hz N 2, fs 2 N 1, fs fs N Frequency (Hz)

29 Modulus of X(k) DATA TRUNCATED AT 5 PERIODS (5-POINT FFT) p = 1Hz N 2, fs 2 N 1, fs fs N Frequency (Hz)

30 LEAKAGE points 7 points Modulus of X(k) Frequency (Hz)

31 FFT MATLAB EXAMPLE-2 clc; clear all; close all A=2; p=1; Tp=1/p; fs=1/tp; No_of_periods = 2; %Keep changing this parameter T1=No_of_periods*Tp; t1=[:1/fs:t1-1/fs]; x1=a*cos(2*pi*p*t1); X1=fft(x1); N1=length(x1); f1=fs*(:n1-1)/n1; figure(1) subplot(1,2,1) plot(f1, abs(x1), 'b') xlabel('frequency (Hz)') ylabel('modulus of \itx\rm(\itk\rm)'); %axis([ 9.9 1]) subplot(1,2,2) stem(f1, abs(x1), 'b') xlabel('frequency (Hz)') ylabel('modulus of \itx\rm(\itk\rm)'); %axis([ 9.9 1]

32 8/28/217 EXTRA 32

33 UNIT IMPULSE FUNCTION Suppose a function d (t) has the form d 1 2, ( t), Then, I() = 1. t otherwise We are interested d (t) acting over shorter and shorter time intervals (i.e., ). See graph on right. Note that d (t) gets taller and narrower as. Thus for t, we have lim d ( t), and lim I( ) 1

34 DIRAC DELTA FUNCTION Thus for t, we have lim d ( t) and lim I( ) 1 The unit impulse function is defined to have the properties The unit impulse function is an example of a generalized function and is usually called the Dirac delta function., ( t) for t, and ( t) dt 1 In general, for a unit impulse at an arbitrary point t, ( t t) for t t, and ( t t ) dt 1

35 LAPLACE TRANSFORM OF The Laplace Transform of is defined by and thus, ) ( lim ) ( t t t d L t t L ) cosh( lim ) sinh( lim 2 lim 2 1 lim 2 lim 2 1 lim ) ( lim ) ( st st st s s st t s t s t t st t t st st e s s s e s s e e e s e e e s s e dt e dt t t d e t t L

36 LAPLACE TRANSFORM OF Thus the Laplace Transform of is L st t t ) e, t ( For Laplace Transform of at t =, take limit as follows: st L ( t) lim L d ( t t ) lim e 1 t For example, when t = 1, we have L{(t -1)} = e -1s.