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8/13/2019 3777901 Dpit Exercises
1/18
DATA PROCESSING AND
INVERSE THEORY:
PRACTICAL EXERCISES REPORT
Rafael Fernando Daz Gaztelu
3777901
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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])
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2) Time and frequency domain.in! the ten "i#en functions in the time domain to their
counterparts in the fre$uency domain%
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3) Properties of the Fourier transform.The Fourier transform of f&t) is defined by'
FT(f(t))=F()=
+
f(t)e2j tdt
The in#erse Fourier transform is "i#en by'
FT1(F( ))=f( t)=+
F() e2j tdt
Demonstrate the follo(in" properties of the Fourier Transform'
a) imilarity'
FT( f( t))= 1
F( )
FT(f( t))=
+
f( t)e2j tdt
>0 * FT( f( t))= 1
f( t)e2j t dt=
[u= dt
du= dt]=
1
f(u)e2 j
u du +
+ 1
F( )
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e) 0roof the con#olution theorem%
"i# FT( f")=FT(f) 1 FT(")
FT(f")=
(
f()"(t)d)e2j tdt=
f( )(
"(t)e2j tdt)d=[u=tdu=dt] == f( )("(u)e
2j (u)dt d= f() e2j d 1"(u)e
2j u du=FT(f)1 FT(")QD
"ii# FT(f 1 ")=FT(f)FT(")
FT(f 1 ")= f 1 " 1 e2j tdt= (F( -)e2j - td -)"(t)e2j tdt == F( -)("( t)e2j( -)tdt)d=F( -) 1,(2j( -))d -=F(),()
QD
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4) Fourier transform of boxcar function.
o.car function f&t)3 centered at t+4 is "i#en by'
f(t)=
1 T
2
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axis([0 0.1 !10 10])
titl&('&nt&r& oxcar /')
xlab&l('r&u&nc-')
-lab&l(' ,plitu&')
subplot(234)
plot(t%)
axis([!10 10 0 5])titl&('oxcar function')
xlab&l('+i,&')
-lab&l(',plitu&')
subplot(235)
plot(nugur)
axis([0 0.1 !10 10])
titl&('oxcar /')
xlab&l('r&u&nc-')
-lab&l(',plitu&')
subplot(23#)
plot(nuangl&(gur))
axis([0 0.1 !10 10])
titl&('oxcar /')
xlab&l('r&u&nc-')
-lab&l(' ,plitu&')
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Geophysical Data Processing: Exercise 2
1) Sampling theorem
We study the influence of samplin" rate and si"nal len"th% The time si"nal f&t) is "i#en by'f(t)=cos(21t)+cos(2 2t)
(ith fre$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)
f2=cos(2*pi*nu1*t2)6cos(2*pi*nu2*t2)
f3=cos(2*pi*nu1*t3)6cos(2*pi*nu2*t3)
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&')
titl& ('+i,& laps& of 4 ,s')
subplot(313)
plot(t3f3)
xlab&l('+i,&')
-lab&l(',plitu&')
titl& ('+i,& laps& of ,s')
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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
f1=cos(2*pi*nu1*t1)6cos(2*pi*nu2*t1)
f2=cos(2*pi*nu1*t2)6cos(2*pi*nu2*t2)
f3=cos(2*pi*nu1*t3)6cos(2*pi*nu2*t3)
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&')
titl& ('+i,& laps& of 2 ,s')
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')
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):n e.ercise &) (e ha#e seen that the spectrum of the finite cosine corresponds to t(o sinc
functions centered at < fre$uency of the cosine% :n order to supress the side lobes of the spectral
(indo( different time (indo(s can be used3 arlett or Trian"ular Windo(3 6annin" Windo( or
,aussian Windo(% Compute the spectra for the same si"nal as in e.ercise &) but use the three
(indo(in" functions mentioned abo#e%
t=[0:0.001:1]
nu1=12.5
nu2=5
8=1001
f=cos(2*pi*nu1*t)6cos(2*pi*nu2*t)
=bartl&tt(8)
-=%ann(8)
,=gaussin(8)
='
-=-'
,=,'
%=.*f
=-.*f
s=,.*f
subplot(311)
plot(t%)
xlab&l('+i,&')
-lab&l(',plitu&')titl& ('artl&tt or +riangular ino')
subplot(312)
plot(t)
xlab&l('+i,&')
-lab&l(',plitu&')
titl& ('
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Geophysical Data Processing: Exercise 3
1) !iscrete con"olution.
Code a function (hich performs a discrete con#olution bet(een t(o arbitrary se$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$9plot(::0),
stem(Y)
;la9el('Y.n1'),
"la9el('333336n'),
title('
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For t+e aelets6x=[10!12]
-=[5!5401!1]
+ese are t+e /lots t+at are /roduced6
+e last *ra/+s dis/la-s t+e intrinsic conv o/eration%
) !iscrete correlation.Code a function (hich performs a discrete correlation bet(een t(o
arbirtrary (a#elets% The function needs to return both the #alue and the time la"%a) 0erform direct correlation in the time domain%
b) 9pply a #ariable substitution to (rite a correlation as a con#olution% ou can no( chec! your
routine &a) usin" the Matlab intrinsiccon$%
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(-);
8/13/2019 3777901 Dpit Exercises
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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('
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Geophysical Data Processing: Exercise 4
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*)
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) %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
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Geophysical Data Processing: Exercise 5
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;$5;12#
:n a second approach find the optimum in#erse (a#elet% What is the best decon#olution operator in
this case and (hy?
+e est deconolution o/erator ould e t+e
8/13/2019 3777901 Dpit Exercises
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)Consider the function .&t)+cos &2N4t) (ith N4+4367% ample .&t) e#ery second% Ta!e si. samples
in total% What happens if you sample e#ery t(o seconds? :n (hat follo(s (e only consider the case
of a samplin" inter#al of second? Calculate the autocorrelation of .&t)% Calculate the prediction
filter of len"th 2 (hich "i#es .I3 the Lthsample of .&t)% Compare the prediction to the true #alue% :n
"eneral3 the predicted #alues of .&t) are "i#en by .&t)f&t)% ,i#e the succesi#e errors for these
predictions &r3r2%%%) and comment%
n$#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 ">')
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Geophysical Data Processing: Exercise 6
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*