3777901 dpit exercises

8/13/2019 3777901 Dpit Exercises

1/18

DATA PROCESSING AND

INVERSE THEORY:

PRACTICAL EXERCISES REPORT

Rafael Fernando Daz Gaztelu

3777901


2/18

Geophysical Data Processing: Exercise 1

1) Fourier transform of sine and cosine

a) Calculate analytically the Fourier transform of f(t)=cos(2M t)

f(t)=cos(2M t)=

1

2 (e2j Mt

+e2j Mt

)

FT(f(t))=F()=

+

e2j t 1

2(e2j M t+e2j M t) dt=1

2

+

(e2j (M)t+e2j(+M)t)dt =

=1

2[(M)(+0)]

b) What is the Fourier transform of f(t)=sin (2 M t)

f(t)=sin (2 M

t)=1

2j (e

2j Mte2j M t

)

FT(f(t))=F()=

+

e2j t 1

2j(e2j Mte2j Mt) dt= 1

2j

+

(e2j(M)te2j(+M)t)dt

j

2[(+M)(0)]

c) Use matlab to compute the Discrete Fourier Transform of f(t)=cos(2M t) andf(t)=sin (2 M t) and plot the amplitude and phase spectra; use M=5Hz

Why is the computed result different from the analytical result?

t=[0:0.001:4.95]

frax=[0:0.101:500];

nu=5;

f=cos(2*pi*nu*t);

g=fft(f)

subplot(311)

plot(tf);

subplot(312)

plot(tg)

axis([!1 # !1000 2000])

subplot(313)

plot(fraxg)axis([!100 #00 !1000 2000])


3/18

2) Time and frequency domain.in! the ten "i#en functions in the time domain to their

counterparts in the fre$uency domain%


4/18

3) Properties of the Fourier transform.The Fourier transform of f&t) is defined by'

FT(f(t))=F()=

+

f(t)e2j tdt

The in#erse Fourier transform is "i#en by'

FT1(F( ))=f( t)=+

F() e2j tdt

Demonstrate the follo(in" properties of the Fourier Transform'

a) imilarity'

FT( f( t))= 1

F( )

FT(f( t))=

+

f( t)e2j tdt

>0 * FT( f( t))= 1

f( t)e2j t dt=

[u= dt

du= dt]=

1

f(u)e2 j

u du +

+ 1

F( )


5/18

e) 0roof the con#olution theorem%

"i# FT( f")=FT(f) 1 FT(")

FT(f")=

(

f()"(t)d)e2j tdt=

f( )(

"(t)e2j tdt)d=[u=tdu=dt] == f( )("(u)e

2j (u)dt d= f() e2j d 1"(u)e

2j u du=FT(f)1 FT(")QD

"ii# FT(f 1 ")=FT(f)FT(")

FT(f 1 ")= f 1 " 1 e2j tdt= (F( -)e2j - td -)"(t)e2j tdt == F( -)("( t)e2j( -)tdt)d=F( -) 1,(2j( -))d -=F(),()

QD


6/18

4) Fourier transform of boxcar function.

o.car function f&t)3 centered at t+4 is "i#en by'

f(t)=

1 T

2


7/18

axis([0 0.1 !10 10])

titl&('&nt&r& oxcar /')

xlab&l('r&u&nc-')

-lab&l(' ,plitu&')

subplot(234)

plot(t%)

axis([!10 10 0 5])titl&('oxcar function')

xlab&l('+i,&')

-lab&l(',plitu&')

subplot(235)

plot(nugur)

axis([0 0.1 !10 10])

titl&('oxcar /')

xlab&l('r&u&nc-')

-lab&l(',plitu&')

subplot(23#)

plot(nuangl&(gur))

axis([0 0.1 !10 10])

titl&('oxcar /')

xlab&l('r&u&nc-')

-lab&l(' ,plitu&')


8/18


1) Sampling theorem

We study the influence of samplin" rate and si"nal len"th% The time si"nal f&t) is "i#en by'f(t)=cos(21t)+cos(2 2t)

(ith fre$uencies 1=12%567 and 2=7567 %a) Discretise the continous function usin" as time inter#als t=2&s3 '&s3(&s respecti#ely3plot these results%

b) Compute the amplitude spectra and display results%

t1=[0:0.002:1]

t2=[0:0.004:1]

t3=[0:0.00:1]

nu1=12.5

nu2=5

f1=cos(2*pi*nu1*t1)6cos(2*pi*nu2*t1)



subplot(311)

plot(t1f1)

xlab&l('+i,&')

-lab&l(',plitu&')

titl& ('+i,& laps& of 2 ,s')

subplot(312)

plot(t2f2)

xlab&l('+i,&')

-lab&l(',plitu&')


subplot(313)

plot(t3f3)

xlab&l('+i,&')

-lab&l(',plitu&')

titl& ('+i,& laps& of ,s')


9/18

c) 8.plain results and compute aliasin" fre$uency% oo! at the amplitude spectra% What does theory

tell you and (hat happened to the side lobes?

)ccordin* to t+e ,-.uist "or sa&/lin*# t+eore&$ t+e &ini&u& fre.uenc- ale to sa&/le it+out

losin* infor&ation is C=M9/=22=2 1 7567=15067 +ic+ -ields a ti&e la/se of tC=%7 ms %

a&/lin* at t1=2 ms and t2=' ms are still o4$ t+is &eans t+at no infor&ation is lost$unli4e t2=( ms $ +ic+ -ields a fre.uenc- under C $ and it is een isile in t+e a&/litudes/ectru& /lot t+at it *ies a /oor dis/la- of t+e function%

d) 8.tend the si"nal len"th by addin" 7eros at the end% 0lot the resultin" amplitude spectra of the

e.tended si"nal and e.plain%

t1=[0:0.002:1]

t2=[0:0.004:1]

t3=[0:0.00:1]

nu1=12.5

nu2=5




1=[f17&ros(15)]

2=[f27&ros(15)]

3=[f37&ros(15)]

+1=[t17&ros(15)]

+2=[t27&ros(15)]

+3=[t37&ros(15)]

subplot(311)

plot(+11)

xlab&l('+i,&')

-lab&l(',plitu&')


subplot(312)

plot(+22)

xlab&l('+i,&')

-lab&l(',plitu&')titl& ('+i,& laps& of 4 ,s')

subplot(313)

plot(+33)

xlab&l('+i,&')

-lab&l(',plitu&')

title ('Time lapse of 8 ms')


10/18

):n e.ercise &) (e ha#e seen that the spectrum of the finite cosine corresponds to t(o sinc

functions centered at < fre$uency of the cosine% :n order to supress the side lobes of the spectral

(indo( different time (indo(s can be used3 arlett or Trian"ular Windo(3 6annin" Windo( or

,aussian Windo(% Compute the spectra for the same si"nal as in e.ercise &) but use the three

(indo(in" functions mentioned abo#e%

t=[0:0.001:1]

nu1=12.5

nu2=5

8=1001

f=cos(2*pi*nu1*t)6cos(2*pi*nu2*t)

=bartl&tt(8)

-=%ann(8)

,=gaussin(8)

='

-=-'

,=,'

%=.*f

=-.*f

s=,.*f

subplot(311)

plot(t%)

xlab&l('+i,&')

-lab&l(',plitu&')titl& ('artl&tt or +riangular ino')

subplot(312)

plot(t)

xlab&l('+i,&')

-lab&l(',plitu&')

titl& ('


11/18


1) !iscrete con"olution.

Code a function (hich performs a discrete con#olution bet(een t(o arbitrary se$uences .43.3%%%.n

and .43.3%%%.m%&also called (a#elets)%

a) 0erform direct con#olution in the time domain%

b) 0erform the operation in the =>domainc) Thin! of the se$uences .3 .2and y+con#&.3.2) as #ectors and try to (rite the con#olution as a

matri. multiplication%

d) Chec! all results (ith the intrinsic function conv%

%MASTER PROGRAM EVERYTH!G

"#inp$t('Ente t&e fist avelet "')

inp$t('Ente t&e secon* avelet &')

m#len+t&("),

n#len+t&(&),

-#."/eos(0n)1,

H#.&/eos(0m)1,

fo i#0n2m30

Y(i)#4

fo 5#0m

if(i352064)

Y(i)#Y(i)2-(5)7H(i3520),

else

en*

en*

en*

Y

#m2n30

$#/eos(0)

fo 5#0nfo n#0m

$(52n30)#$(52n30)2&(5)7"(n),

en*

en*

"m#toeplit/(."(0) /eos(0len+t&(&)30) 1 ." /eos(0len+t&(&)30) 1),

+0#&7"m

s$9plot(::0),

stem(Y)

;la9el('Y.n1'),

"la9el('333336n'),

title('


12/18

For t+e aelets6x=[10!12]

-=[5!5401!1]

+ese are t+e /lots t+at are /roduced6

+e last *ra/+s dis/la-s t+e intrinsic conv o/eration%

) !iscrete correlation.Code a function (hich performs a discrete correlation bet(een t(o

arbirtrary (a#elets% The function needs to return both the #alue and the time la"%a) 0erform direct correlation in the time domain%

b) 9pply a #ariable substitution to (rite a correlation as a con#olution% ou can no( chec! your

routine &a) usin" the Matlab intrinsiccon$%

x=input('>nt&r t%& s&u&nc& 1: ');

%=input('>nt&r t%& s&u&nc& 2: ');

-=xcorr(x%);

figur&;

subplot(311);

st&,(x);

xlab&l('n!?');

-lab&l(',plitu&!?');titl&('@nput s&u&nc& 1');

subplot(312);

st&,(fliplr(-));

st&,(%);

xlab&l('n!?');

-lab&l(',plitu&!?');

titl&('@nput s&u&nc& 2');

subplot(313);

st&,(fliplr(-));

xlab&l('n!?');

-lab&l(',plitu&!?');

titl&('Autput s&u&nc&');isp('+%& r&sultant is');

fliplr(-);


13/18

3) #inear filtering.

The (a#elet b is "i#en by @43>343323A3B334% The delay filter f is "i#en by @43434343%

a) 9pply the delay filter to (a#elet b% 0lot the ori"inal and the delayed (a#elet%

b) Compute and #isualise the cross>correlation bet(een b and bf and e.plain ho( the time

difference bet(een b and bf can be estimated%

c) 0erturb f (ith random noise% Compute and #isualise the cross>correlation bet(een the t(o

(a#eforms and determine the time shift%

9#.43040:?>041

f#.444401

t#.400?1

*isp('@EAYE@ BAVEET')

conv(9f)

*isp('


14/18


1) $otch filter.The discrete si"nal f&t) is "i#en by f(t)=cos(21t) and it is perturbed by noisecharacterised by "(t)=cos (2 2t) (ith fre$uencies 1=12%567 and 2=5067 % The

si"nal is recorded at inter#al =' ms 3 use E+2 samples% The noise hast to be eliminated formthe si"nal by filterin" it (ith a notch filter%

a) ho( that for any T: filter the transfer function is "i#en by 6(7)=?(7)

/(7)=

2(7)9(7)

(here the

coefficients of &7) and 9&7) are the filter coefficients b i and aj% Find the filter coefficients of the

notch filter and compute and #isualise the impulse response for G+4343 G+43 and G+>434%

b) Compute the fre$uency response% What other possibility can you thin! of to "et the fre$uency

response?

c) 9pply the filter to the si"nal and in#esti"ate the influence of different #alues for G%

n$0#0:D

n$:#D4

ta$#4C44?t#.44C40DC0:1

n$s#.44C40DC0:1

epsilon#4C4D

f0#cos(:7pi7n$07t)

f:#cos(:7pi7n$:7t)

f#f02f:

=#e"p(:757pi7n$s7ta$)

=4#e"p(3:757n$:7ta$)

=P#(02epsilon)7e"p(3:757n$:7ta$)

A#(=3=4)

#(=3=P)

0:)

stem(filtee*)


15/18

) %utter&orth filter.9 utter(orth filter is a common form of a lo( pass filter defined by the

follo(in" amplitude spectrum%

F( )=F()F()= 1

1+( C)2n

Where HCis the cut>off fre$uency3 n is the number of poles (hich determines the rate of decay of thefilter and F() is the comple. conju"ate of F&H)%a) Use the bilinear transformation to find an e.pression for F&7) for a second order utter(orth

filter &n+2)% 9n e.pression can be obtained in the form F&7)+9&7)5&7)3 (here 9&7) and &7) are

polynomials in 7%

b) Construct a second order &n+2) recursi#e utter(orth filter and "i#e the impulse response%

c) ,i#e the fre$uency response of a second>order utter(orth filter%

d) 9pply the utter(orth filter to remo#e the noise from the si"nal in e.ercise %

F(s )=

c(ss1)(ss2)=

jc2sin3 /'(ss1) +

jc2sin3/ '(ss2)

+e ilinear transfor&ation6

2j =2171+7

* =1

j171+7

+en6

F(s)=

csin3/'

sin (Csin3/')eCcos3/ '7

1

12cos(Csin 3 /')eCcos3/ '7

1+e2Ccos3/ '72=

= C

2 (1+2z1+72)

'' C

cos 3 /'+c

2+((+2 C

2 )71+('+' C

cos 3/'+C

2 )72


16/18


1)The problem is to find the in#erse (a#elet h of len"th 2 of the (a#elet "+&I33)% First find the

in#erse by series e.pansion of its =>transform%

8n t+e ti&e do&ain e +ad6"+&I33)%8ts :ransfor& is t+en6 ,&=)+IJ=J=2

8f e are loo4in* for an6&=)t+at does t+e tric46 ,&=)16&=)+$ t+en6 6(=)= 1,(=)

$ so6

o 6(=)=1

+

5

12= +ic+ in t+e ti&e do&ain is6 +="1;$5;12#

:n a second approach find the optimum in#erse (a#elet% What is the best decon#olution operator in

this case and (hy?

+e est deconolution o/erator ould e t+e


17/18

)Consider the function .&t)+cos &2N4t) (ith N4+4367% ample .&t) e#ery second% Ta!e si. samples

in total% What happens if you sample e#ery t(o seconds? :n (hat follo(s (e only consider the case

of a samplin" inter#al of second? Calculate the autocorrelation of .&t)% Calculate the prediction

filter of len"th 2 (hich "i#es .I3 the Lthsample of .&t)% Compare the prediction to the true #alue% :n

"eneral3 the predicted #alues of .&t) are "i#en by .&t)f&t)% ,i#e the succesi#e errors for these

predictions &r3r2%%%) and comment%

n$#4CD

t0#.40D1

t:#.4:D1

t>#.44C0I01

"0#cos(:7pi7n$7t0)

":#cos(:7pi7n$7t:)

">#cos(:7pi7n$7t>)

+#"co("0"0)

"co(":":)

#"co(">">)

s$9plot(>00)

plot(t0"0)

title('Samplin+ eve; secon*')

s$9plot(>0:)

plot(t:":)

title('Samplin+ eve; : secon*s')

s$9plot(>0>)

plot(t>">)

title('Samplin+ inteval of 0 secon*')

fi+$e

s$9plot(>00)

stem(+)

title('A$tocoelation of "0')

s$9plot(>0:)

stem(&)

title('A$tocoelation of ":')

s$9plot(>0>)

stem()

title('A$tocoelation of ">')


18/18


si+ma#4C0

G#. 4C>>>8 0CIKJ0 4 :C4>4K 4 4 :C4>4K 4 4,

4 :C4004 4 4 :C4004 4 4C?88> 0CD::J 4,

4 :C440: 4 4 :C440: 4 4 :C440: 4,

4 :C440: 4 4 :C440: 4 4 :C440: 4, 4 :C4004 4 4 :C4004 4 4 0CD::J

4C?88>,

4 0CIKJ0 4C>>>8 4 4 :C4>4K 4 4

:C4>4K,

4 4 4 0CIKJ0 4 4 4C>>>8 :C4>4K

:C4>4K,

4 4 4 :C4004 :C4004 0CD::J 4 4 4C?88>

4 4 4 :C440: :C440: :C440: 4 4 4,

4 4 4 :C440: :C440: :C440: 4 4 4,

4 4 4C?88> :C4004 :C4004 0CD::J 4 4 4,

4C>>>8 :C4>4K :C4>4K 0CIKJ0 4 4 4 4 41,m#.D4 I4 D4 I4 D8 I4 D4 I4 D41,

mt#m'

%SOV!G THE ORBAR@ PROEM

*#G7mt

%SOV!G Y EAST SLFARES

GT#tanspose(G)

mls#(inv(GT7G))7GT7*

3777901 dpit exercises

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