Analisa Frekuensi Sinyal dan Sistem
Transcript of Analisa Frekuensi Sinyal dan Sistem
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Analisa FrekuensiSinyal dan Sistem
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Analisis frekuensi sinyal waktukontinu
Analisis frekuensi sinyal waktudiskrit
Sifat-sifat transformasi Fourier Domain frekuensi sistem LTI Sistem LTI sebagai filter
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Peristiwa Dispersi
Newton (1672)
Fraunhofer (1787)
Kirchoff & Bunsen (1800)
Cahaya tampak
Cahaya bintang dan matahari
Bahan kimia
Analisis Frekuensi
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PrismaCahaya Warna
MatematicalTools
Sinyal Sinyal sinusoidal
Instrument
Software program
Speech
ECG
EEG
Pitch
Denyut jantung
, ,
Transformasi Fourier
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Analisis frekuensi sinyal waktukontinu
Deret Fourier untuk sinyal waktu kontinuperiodik
Power spektral density (psd) sinyal periodik Transformasi Fourier untuk sinyal kontinu
aperiodik Energy spectral density (esd) sinyal
aperiodik
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periodaTT
1Fec)t(x p
po
k
tkF2jk
o
Deret Fourier untuk sinyal periodik
komplekscdte)t(xT
1c k
T
0
tkF2j
pk
p
o
*kk ccnyata)t(x
kk jkk
jkk eccecc
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1k
koko )tkF2cos(c2c)t(x
kokoko sin)tkF2sin(cos)tkF2cos()tkF2cos(
)tkF2sin(b)tkF2cos(aa)t(x ok1k
oko
kkkkkkoo sinc2bcosc2aca
k
tkF2jk
oec)t(x
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Power spectral density (psd) dari sinyal periodik
k
2
k
T
0
2
px cdt)t(x
T
1P
p
Energinya tak terbatas, dayanya terbatas
)ba(2
1ac2cP 2
k1k
2k
2o
1k
2
k2ox
2
kc sebagai fungsi dari frekuensi F
psd
Relasi Parseval
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F
Power spectral density dari sinyal periodik
2
kc
-4Fo -3Fo - 3Fo -Fo 0 Fo 2Fo 3Fo 4Fo
2
1c
2
3c
2
2c2
4c
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Contoh Soal 1
Tentukan deret Fourier dan power spectral density darisinyal pulsa persegi panjang di bawah ini.
)t(x
t0
2
2
pTpT
A
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Jawab :
p
2
2p
2
T
2
Tpo T
AAdt
T
1dt)t(x
T
1c
p
p
2
2
tkF2j
op
2
T
2
T
tkF2j
pk
o
p
p
o ekF2j
1
T
AdtAe
T
1c
o
o
p
kFjkFj
pok kF
)kFsin(
T
A
2j
ee
TkF
Ac
oo
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TP tetap berubah tetap TP berubah
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Power spectral density :
,2,1k,kF
)kFsin(
T
A
0k,T
A
c2
o
o
2
p
2
p2
k
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dFe)F(X)t(x Ft2j
Transformasi Fourier untuk sinyal aperiodik
dte)t(x)F(X Ft2j
Energy spectral density (esd) dari sinyal periodik
Energinya terbatas :
dF)F(Xdt)t(xE22
x
2
xx )F(X)F(S esd
Relasi Parseval
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Contoh Soal 2
Tentukan transformasi Fourier dan energy spectral densitydari sinyal yang didefinisikan sebagai :
)t(x
t0
2
2
A
2t,0
2t,A
)t(x
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Jawab :
F
FsinAdtAe)F(X
2
2
Ft2j
2
2xx F
FsinA)F(S
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X(F)
x(t)
1
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Analisis frekuensi sinyal waktu diskrit
Deret Fourier untuk sinyal waktu diskritperiodik
Power spektral density (psd) sinyal diskritperiodik
Transformasi Fourier untuk sinyal diskritaperiodik
Energy spectral density (esd) sinyal diskritaperiodik
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Deret Fourier untuk sinyal diskrit periodik
2
1f
2
1
N
kf
N
k2es
scec)n(x
kk
kknj
k
1N
0kkk
1N
0k
N/kn2jk
k
dasarperiodaN)n(x)Nn(x
kNk
1N
0n
N/kn2j cce)n(xN
1)k(c
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Contoh Soal 3
Tentukan spektrum dari sinyal-sinyal di bawah ini.
4N0,0,1,1).b3
ncos)n(x).a
Jawab :
6N6
1f
n6
12cos
3
ncos)n(x).a
o
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5
0n
6/kn2j1N
0n
N/kn2j e)n(xe)n(x)k(c
6/n2j6/n2j e2
1e
2
1n
6
12cos)n(x
1N
0k
6/kn2jk
1N
0k
N/kn2jk ecec)n(x
2
1ccc
0cccc2
1c
2
1c
1615
432o11
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2
1ccc
0cccc2
1c
2
1c
1615
432o11
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2/kj3
0n
4/kn2j e14
1e)n(x
4
1)k(c
4N0,0,1,1).b
1N
0n
N/kn2je)n(xN
1)k(c
)j1(4
1c0c)j1(
4
1c
2
1c 321o
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)j1(4
1c0c)j1(
4
1c
2
1c 321o
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Contoh Soal 4
Tentukan spektrum dari sinyal di bawah ini.
n5
2sinn
3
2cos)n(x
Jawab :
n15
32sinn
15
52cosn
5
2sinn
3
2cos)n(x
j2
ee
2
ee)n(x
n)15/3(2jn)15/3(2jn)15/5(2jn)15/5(2j
n)15/5(2jn)15/5(2jn)15/3(2jn)15/3(2j e2
1e
2
1e
2
je
2
j)n(x
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n)15/5(2jn)15/5(2jn)15/3(2jn)15/3(2j e2
1e
2
1e
2
je
2
j)n(x
14
0k
15/kn2jk
1N
0k
N/kn2jk ecec)n(x
2
1c
2
jc
2
jc
2
1c 5335
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1/2
kc
90o
kc
- 90o
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Power Spectral Density (psd) sinyal diskrit periodik
1N
0k
2
k
21N
0kx c)n(x
N
1P Relasi Parseval
psd
Energi satu perioda
Bila x(n) nyata :
1N
0k
2
k
1N
0k
2
N cN)n(xE
k*k cc kkkk cccc
kNkkNk
kNkNkk
cccc
cccc
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0cccc N0N0
1N11N1 cccc
0ccc 2/N2/N2/N
2/)1N(2/)1N(2/)1N(2/)1N( cccc
Bila N genap
Bila N ganjil
kNkkNk
kNkNkk
cccc
cccc
2/)1N(,2,1,0k,cganjilN
2/N,2,1,0k,cgenapN
k
k
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Contoh Soal 5
Tentukan koefisien deret Fourier dan power spectraldensity dari sinyal diskrit periodik di bawah ini.
Jawab :
1L
0n
N/kn2j1N
0n
N/kn2jk Ae
N
1e)n(x
N
1c
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N/kn2j
N/kL2j
1L
0n
N/kn2jk
e1
e1
N
A
N
AL
eN
Ac
)N/ksin(
)N/kLsin(e
ee
ee
e
e
e1
e1
N/)1L(kj
N/kjN/kj
N/kLjN/kLj
N/kj
N/kLj
N/kn2j
N/kL2j
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lainnyak,
)N/ksin(
)N/kLsin(e
N
A
,N2,N,0k,N
AL
eN
Ac
N/)1L(kj
1L
0n
N/kn2jk
lainnyak,)N/ksin(
)N/kLsin(
N
A
,N2,N,0k,N
AL
cpsd22
2
2
k
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Transformasi Fourier dari sinyal diskrit aperiodik
n
nje)n(x)(X
de)(X2
1)n(x nj
n
nj
n
kn2jnj
n
n)k2(j
)(Xe)n(xee)n(x
e)n(x)k2(X
Bentuk Deret Fourier
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Contoh Soal 6
Tentukan sinyal diskrit yang transformasi Fouriernyaadalah :
Jawab :
c
c
,0
,1)(X
de)(X2
1)n(x nj
cc
c
d2
1)0(x0n
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n
nsin
n
nsin
j2
ee
n
1)n(x
ejn
1
2
1de
2
1)n(x0n
c
cccnjnj
njnj
cc
c
c
c
c
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n
nje)n(x)(X
N
Nn
njcN e
n
nsin)(X
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Energy spectral density (esd) sinyal diskrit aperiodik
Relasi Parseval
d)(X2
1)n(xE
2
n
2
x
2
xx )(X)(S
Spektrummagnituda
)(X)()(Xe)(X)(X )(j
Spektrum fasa
x(n) nyata )(X)(X*
)(X)(X)(X)(X
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Contoh Soal 7
Tentukan energy spectral density dari sinyal diskrit :
Jawab :
1a1)n(ua)n(x n
0n
nj
0n
njn
n
nj )ae(eae)n(x)(X
)(X)(X)(x)(Sae1
1)(X *2
xxj
2jjxx acosa21
1
ae1
1
ae1
1)(S
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Contoh Soal 8
Tentukan transformasi Fourier dari sinyal diskrit :
lainnyan,0
1Ln0,A)n(x
Jawab :
)2/sin(
)2/Lsin(Ae
e1
e1AAe)(X
)1L)(2/(j
j
Lj1L
0n
nj
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)(j)1L)(2/(j e)(X)2/sin(
)2/Lsin(Ae)(X
lainnya,
)2/sin(
)2/Lsin(A
0,AL
)(X
)2/sin(
)2/Lsin()1L(
2A)(X)(
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Spektrum fasa
Spektrummagnituda
A = 1
L = 5
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Hubungan transformasi Z dengan transformasi Fourier
n
njn
n
nj
n
z e]r)n(x[)re)(n(xe)n(x)z(X
Transformasi Fourier :
n
nj )(Xe)n(x)z(X1r1z
Transformasi Z
zzrrez j
Transformasi Fourier pada lingkaran satu =
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Contoh Soal 9
Tentukan transformasi Fourier dari : )n(u)1()n(x
Jawab :
1z
z
z1
1)z(X
1
)2/1k(2)2/cos(2
e
)ee)(e(
)e)(e(
1re
re
1z
z
z1
1)(X
2/j
2/j2/j2/j
2/j2/j
j
j
1
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Klasifikasi sinyal dalam domain frekuensi
Sinyal frekuensi rendah :
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Sinyal frekuensi tinggi :
Sinyal frekuensi menengah (bandpass signal) :
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Daerah frekuensi pada beberapa sinyal asliSinyal-sinyal biologi :
Tipe sinyal Daerah frekuensi (Hz)Electroretinogram 0 - 20Electronystagmogram 0 - 20Pneumogram 0 - 40Electrocardiogram (ECG) 0 - 100Electroencephalogram (EEG) 0 - 100Electromyogram 10 - 200Aphygmomanogram 0 - 200Wicara 100 - 4000
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Sinyal-sinyal seismik :
Tipe sinyal Daerah frekuensi (Hz)Wind noise 100 - 1000Seismic exploration signals 10 - 100Earthquake and nuclearexplosion signals
0.01 - 10
Seismic noise 0,1 - 1
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Sinyal-sinyal elektromagnetik :
Tipe sinyal Daerah frekuensi (Hz)Radio broadcast 3x104 – 3x106
Shortwave radio signals 3x106 – 3x1010
Radar, sattellite comunications 3x108 – 3x1010
Infrared 3x1011 – 3x1014
Visible light 3,7x1014 – 7,7x1014
Ultraviolet 3x1015 – 3x1016
Gamma rays and x-rays 3x1017 – 3x1018
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Sifat-sifat transformasi Fourier
Sifat-sifat simetri dari transformasiFourier
Linieritas Pergeseran waktu Pembalikan waktu Teorema konvolusi Pergeseran frekuensi Diferensiasi frekuensi
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Sifat-sifat simetri dari transformasi Fourier
nj1
n
nj
e)(X2
1)}(X{F)n(x
e)n(x)}n(x{F)(X
)(X)n(xF
nsinjncosesinjncose njnj
![Page 53: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/53.jpg)
]nsin)n(xncos)n(x[)(X
]nsin)n(xncos)n(x[)(X
Rn
II
nIRR
)(jX)(X)(X
)n(jx)n(x)n(x
R
IR
x(n) dan X () kompleks
d]ncos)(Xnsin)(X[2
1)n(x
d]nsin)(Xncos)(X[2
1)n(x
I
2
0 RI
I
2
0 RR
![Page 54: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/54.jpg)
x(n) nyata 0)n(x)n(x)n(x IR
)(X)(Xnsin)n(x)(X
)(X)(Xncos)n(x)(X
IIn
I
Rn
RR
nsin)nsin(ncos)ncos(
)(X)(X)(X)(X IIRR
)(X)(X*
![Page 55: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/55.jpg)
)(X
)(Xtg)(X
)(X)(X)(X
I
I1
2I
2R
)(X)(X
)(X)(X
d]nsin)(Xncos)(X[1
)n(x
ganjilnsindan)(Xgenapncosdan)(X
d]nsin)(Xncos)(X[2
1)n(x
I0 R
IR
I
2
0 R
![Page 56: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/56.jpg)
x(n) nyata dan fungsi genap
dncos)(X1
)n(x
0)(Xncos)n(x2)0(x)(X
)n(x)n(x
0 R
I1n
R
x(n) nyata dan fungsi ganjil
dnsin)(X1
)n(x
0)(Xnsin)n(x2)(X
)n(x)n(x
0 I
R1n
I
![Page 57: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/57.jpg)
x(n) imajiner murni
d]ncos)(Xnsin)(X[1
)n(x
ncos)n(x)(X
nsin)n(x)(X
)n(jx)n(x0)n(x
0 IRI
nII
nIR
IR
![Page 58: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/58.jpg)
x(n) imajiner murni dan genap
dnsin)(X1
)n(x
0)(Xnsin)n(x2)(X
)n(x)n(x
0 RI
I1n
IR
II
x(n) imajiner murni dan ganjil
dncos)(X1
)n(x
0)(Xncos)n(x2)0(X)(X
)n(x)n(x
0 II
R1n
III
II
![Page 59: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/59.jpg)
Contoh Soal 10
Tentukan dan buat sketsa XR(), XI(), X() dan X(dari transformasi Fourier :
Jawab :
1a1ea1
1)(X
j
22jj
j
j
j
j
acosa21
sinjacosa1
a)ee(a1
ea1
ea1
ea1
ea1
1)(X
![Page 60: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/60.jpg)
2R acosa21
cosa1)(X
2I acosa21
sina)(X
2
2
2
2222
2I
2R
a)cos(a21
cosa2a1
a)cos(a21
)(sinacosa2)(cosa1
)(X)(X)(X
cosa1
sinatg)(X 1
![Page 61: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/61.jpg)
Linieritas
)(Xa)(Xa)(X)}n(x{F
)n(xa)n(xa)n(x
)(X)}n(x{F)(X)}n(x{F
2211
2211
2211
Contoh Soal 11
Tentukan transformasi Fourier dari : 1a1a)n(x n
0n,0
0n,a)n(x
0n,0
0n,a)n(x
)n(x)n(x)n(xn
2
n
1
21
Jawab :
![Page 62: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/62.jpg)
j
0n
nj
0n
njn
n
nj11
ae1
1
)ae(eae)n(x)(X
j
j
1k
kj
1
n
nj1
n
njn
n
nj22
ae1
ae)ae(
)ae(eae)n(x)(X
2
2
2jj
2jj
j
j
j21
acosa21
a1
a)aeae(1
aaeae1
ae1
ae
ae1
1)(X)(X)(X
![Page 63: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/63.jpg)
Pergeseran waktu
)(Xe)}n(x{F)kn(x)n(x
)(X)}n(x{F
1kj
1
11
Pembalikan waktu
)(X)}n(x{F)n(x)n(x
)(X)}n(x{F
11
11
![Page 64: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/64.jpg)
Teorema konvolusi
)(X)(X)}n(x{F)n(x*)n(x)n(x
)(X)}n(x{F)(X)}n(x{F
2111
2211
Jawab :
Contoh Soal 12
Tentukan konvolusi antara x1(n) dan x2(n), dengan :
x1(n) = x2(n) ={1, 1, 1}
cos21ee1
ee)n(x)(X
jj
n
1
1n
njnj11
![Page 65: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/65.jpg)
)ee()ee(23
2cos2cos43
2
2cos14cos41
cos4cos41
)cos21()(X)(X)(X
cos21)(X)(X
2j2jjj
2
221
21
2jjj2j
n
nj ee23e2ee)n(x)(X
}12321{)n(x
![Page 66: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/66.jpg)
Pergeseran frekuensi
)(X)}n(x{F)n(xe)n(x
)(X)}n(x{F
o11nj
11
o
Teorema modulasi
ncos)n(x)n(x)(X)}n(x{F o111
)n(xe2
1)n(xe
2
1)n(x)ee(
2
1)n(x 1
nj1
nj1
njnj oooo
)(X2
1)(X
2
1)(X)}n(x{F o1o1
![Page 67: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/67.jpg)
Diferensiasi frekuensi
)n(nx)n(x)(X)}n(x{F 111
d
)(dXj)}n(x{F 1
)}n(nx{jFe)n(nxj
ed
d)n(xe)n(x
d
d
d
)(dX
e)n(x)(X
1n
nj1
n
nj1
n
nj1
1
n
nj11
![Page 68: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/68.jpg)
Domain frekuensi sistem LTI
Fungsi respon frekuensi Respon steady-state dan respon
transien Respon terhadap sinyal input periodik Respon terhadap sinyal input aperiodik Hubungan antara fungsi sistem dan
fungsi respon frekuensi Komputasi dari fungsi respon
frekuensi
![Page 69: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/69.jpg)
Fungsi respon frekuensi
k
njkj
k
)kn(j
nj
k
e]Ae)k(h[AAe)k(h)n(y
Ae)n(xkompleksInput
)kn(x)k(h)n(y
nj
k
kj e)(AH)n(ye)k(h)(H
Eigen function
Eigen value
![Page 70: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/70.jpg)
Contoh Soal 13
Respon impuls dari suatu sistem LTI adalah :
)n(u2
1)n(h
n
Jawab :
Tentukan outputnya bila mendapat input : 2/njAe)n(x
21
j1
1
e21
1
1)(H
e21
1
1)(H
e2
1e
2
1)(H)n(hF
2/jj
n
nj
n
njn
![Page 71: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/71.jpg)
)6,262/n(2/nj6,26j
nj
6,26j
oo
o
e5
A2ee
5
2A
e)(AH)n(y
e5
2
21
j1
1)(H
Amplituda
Frekuensi
Fasa
3
2
21
1
1
e21
1
1)(HAe)n(x
j
nj
njAe3
2)n(y
![Page 72: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/72.jpg)
)sin)(cos()(
)()()(
kjkkhekh
jHHH
kk
kj
IR
)()(sin)()(
)()(cos)()(
IIk
I
RRk
R
HHkkhH
HHkkhH
)(
)()()(
)()()(
1
22
I
I
IR
H
HtgH
HHH
![Page 73: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/73.jpg)
njj
njjnj
njjnj
eeHA
eeHAnyAenx
eeHAnyAenx
)(
)(22
)(11
)(
)()()(
)()()(
)](cos[)()]()([2
1)(
cos][2
1)]()([
2
1)(
21
21
nHAnynyny
nAAeAenxnxnx njnj
)](sin[)()]()([2
1)(
sin][2
1)]()([
2
1)(
21
21
nHAnynyj
ny
nAAeAej
nxnxj
nx njnj
![Page 74: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/74.jpg)
Contoh Soal 14
Respon impuls dari suatu sistem LTI adalah :
)n(u2
1)n(h
n
Jawab :
Tentukan outputnya bila mendapat input :
nnnx
cos202
sin510)(
jeH
21
1
1)(
![Page 75: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/75.jpg)
3
2)(
5
2)2/(
2
21
1
1)0(
21
1
1)(
6,26
H
eH
H
eH
o
j
nnnx
cos3
40
2sin
5
1020)(
![Page 76: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/76.jpg)
Contoh Soal 15
Suatu sistem LTI dinyatakan dengan persamaan beda :
10)()1()( anbxnayny
)4
cos(202
sin125)(
nnnx
9,01)().
)().
adanHUntukb
HTentukana
maks
Tentukan y(n) bila inputnya :
![Page 77: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/77.jpg)
Jawab :)()()()1()( nubanhnbxnayny n
jn
nj
ae
benhH
1
)()(
cos1
sin1
cos211
sin)cos1(1
1
2
a
atgae
aaae
jaaae
j
j
j
cos1
sin)(
cos21)(
1
2
a
atgb
aa
bH
![Page 78: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/78.jpg)
aba
bHH
maks
11
1)0()(
cos1
sin)(
cos21
1)( 1
2 a
atg
aa
aH
0)(1)0( H
otgH 429,0)(074,09,01
1,0)2/( 1
2
0)(053,09,1
1,0
1
1)(
a
aH
![Page 79: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/79.jpg)
)4
cos(202
sin125)(
nnnx
)](4
cos[)(20
)]2/(2
sin[)2/(12)0(5)(
nH
nHHny
]4
cos[06,1]422
sin[888,05)(
nnny o
![Page 80: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/80.jpg)
Respon steady-state dan respon transien
)n(x)1n(ay)n(y
n
0k
k1n )kn(xa)1(ya)n(y)n(x
n
0k
)kn(jk1n
nj
eaA)1(ya)n(y
0nAe)n(x
njkn
0n
j1n
nj
e)ae(A)1(ya)n(y
0nAe)n(x
![Page 81: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/81.jpg)
njkn
0n
j1n
nj
e)ae(A)1(ya)n(y
0nAe)n(x
njj
)1n(j1n1n
nj
eae1
ea1A)1(ya)n(y
0nAe)n(x
njj
njj
)1n(j1n1n e
ae1
Ae
ae1
eaA)1(ya)n(y
![Page 82: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/82.jpg)
Respon transien
Respon steady state
njj
njj
)1n(j1n1n e
ae1
Ae
ae1
eaA)1(ya)n(y
1aStabil
njnjj
nss e)(AHe
ae1
A)n(y)n(y lim
njj
)1n(j1n1n
tr eae1
eaA)1(ya)n(y
![Page 83: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/83.jpg)
Respon steady state terhadap sinyal input periodik
1N
0k
N/kn2jkec)n(xFourierDeret
N/kn2jkk
N/kn2jkk e
N
k2Hc)n(yecx
N
k2)(HN
k2H
1N
0k
N/kn2jk
1N
0kk e
N
k2Hc)n(y)n(y
N
k2Hcded)n(y kk
1N
0k
N/kn2jk
![Page 84: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/84.jpg)
Respon steady state terhadap sinyal input aperiodik
)(XH)(YkonvolusiTeori
)(X)(H)(Y)(XH)(Y
)(SH)(S)(XH)(Y xx
2
yy
222
d)(SH
2
1E:Energi xx
2
y
![Page 85: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/85.jpg)
Contoh Soal 16
Suatu sistem LTI mempunyai respon impuls :
)n(u2
1)n(h
n
Tentukan spektrum dan esd-nya bila mendapat input :
)n(u4
1)n(x
n
Jawab :
j0n
njn
e21
1
1e
2
1)(H
je41
1
1)(X
![Page 86: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/86.jpg)
jj e
41
1
1
e21
1
1)(XH)(Y
)e161
e41
1(
1
)e41
e1(
1)(S
j2jj2jy
22
yy )(XH)(S
cos21
1617
1
cos45
1)(Sy
![Page 87: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/87.jpg)
Hubungan antara fungsi sistem dan fungsi responfrekuensi
n
nj
ez
j e)n(hzH)(Hez j
)(H)(H)(H)(HH *2
jez
12)z(H)z(HH
![Page 88: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/88.jpg)
Contoh Soal 17
Suatu sistem LTI dinyatakan dengan :
)1n(x)n(x)2n(y2,0)1n(y1,0)n(y
Tentukan2
)(H
Jawab :
21
1
z2,0z5,01
z1)z(H
221
11
z2,0z5,01
z1
z2,0z5,01
z1)z(H)z(H
![Page 89: Analisa Frekuensi Sinyal dan Sistem](https://reader036.fdocument.pub/reader036/viewer/2022082213/5876dc3e1a28ab206f8b9d95/html5/thumbnails/89.jpg)
221
11
z2,0z5,01
z1
z2,0z5,01
z1)z(H)z(H
)zz(2,0)zz(08,005.1
zz2)z(H)z(H
221
11
)ee(2,0)ee(08,005.1
ee2)(Hez
2j2jjj
jj2j
2cos4,0cos16,005.1
cos22)(H
2
2
22
cos8,0cos16,045.1
)cos1(2)(H1cos22cos