??430 ??F????S ???????O???S

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??430 ??F????S ???????O???S

• Te???a ?e??µat?????a?
• ?/D ?a? D/A µetat??pe??

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??f?a?? pa??stas? ??a??????? s?µ?t??

• ?a a?a?????? s?µata (p.?. ? f???, t? video )
  e??a? s?µata s??e?? st?? ????? ?a? st?? t?µe?
  (amplitude) t???

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??f?a?? pa??stas? ??a??????? s?µ?t?? (2)

• ?e t?? de??µat?????a ta a?a?????? s?µata
  µetat??p??ta? se s?µata d?a???t?? ??????

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??f?a?? pa??stas? ??a??????? s?µ?t?? (3)

• ?e t?? ?ßa?t?sµ? (Quantization) ta de??µata e???
  s?µat?? ?????ta? d?a???t? ?a? ?? p??? t?? t?µ?
  t???

S????t?ta 1/6 S????t?ta de??µat?????a?
16 ?p?peda ?ßa?t?sµ?? 11
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??µ? e??? ??f?a??? ??µp?? ?p????????a?

• S?µa
  e?s?d??

?e??µat?????a
??µp??
?ßa?t?sµ??
??f?a?? ??aµ??f?s?
??peµp?µe?? s?µa
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??f?a?? ?a??stas? ??a??????? S?µ?t??

• ??? ???e? ?p?? p??pe? (s?µf??a µe t? Te???µa t??
  ?e??µat?????a?), ? de??µat?????a de? e?s??e?
  pa?aµ??f?s? st? s?µa.
• ? ?ßa?t?sµ?? e?s??e? p??t?te ??p??a pa?aµ??f?s?.
• ?p??e? ?a ???e? a?ta??a?? µeta?? t??
  pa?aµ??f?s?? ?a? t?? apa?t?se?? se f?sµa ??a t??
  µet?d?s? t?? ?ßa?t?sµ???? s?µat??
• Ta as???????µe a????? µe t?? de??µat?????a
• ?a? ?at?p?? µe t?? ?ßa?t?sµ?

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? s????t?s? delta d(t) -?a??

• ? s????t?s? dirac delta d(t) o???eta? ?? e???
• -???x(t)d(t)dt x(0)
• H d(t) ???µ??eta? ?a? ????st??? s????t?s?
  (impulse function)
• Id??t?te? t?? s????t?s?? delta
• -???d(t)dt 1
• d(-t) d(t)
• d(ut) d(t)/u
• 
  -???x(t)d(t-t)dtx(t)
• x(t) d(t-t) x(t) d(t-t)

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S??????? µe t?? s????t?s? delta

• A? ?p?????s??µe t?? s??????? t?? x(t) µe t??
  d(t-t)
• y(t) x(t)
  d(t-t)
• ?fa?µ????ta? t?? ???sµ? t?? s???????? ????µe
• y(t) -???x(s)d(t-t-s)ds
• ???s?µ?p????µe t?? s?µµet???? ?d??t?ta t?? d(t)
  ?a? ????µe
• y(t) -???x(s)ds-(t-t)ds
• ?fa?µ????µe t?? ßas??? ?d??t?ta t?? d(t), ?p?te
• y(t) x(t) d(t-t) x(t-t)
• ???ad? ? s??e???? µ?a? s??a?t?s?? µe t?? d(t-t)
  µetat?p??e? ??????a t?? s??a?t?s? ?ata t

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Ida???? ?e??µat?????a (Impulse Sampling)

• ?da???? de??µat?????a e??a? ? d?ad??as?a
  p???ap?as?asµ?? e??? s?µat?? x(t) µe µ?a
  pe???d??? a???????a s??a?t?se?? delta ds(t),
• ?? s?µa xs(t) p?? p????pte? ap? t?? ?da????
  de??µat?????a e??a?

x(t)
xs(t)
?
ds(t)
0 t
0 2Ts t
0 ?s 2Ts t
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?da???? ?e??µat?????a (2)

• Te????µe t?? ?da???? de??µat?????a e??? s?µat??
• ? a???????a t?? s??a?t?se?? delta ep????e? t??
  t?µ?? t?? x(t) se ta?t? d?ast?µata p?? ap????? ?s
  seconds.
• H pe???d?? de??µat?????a? e??a? Ts ?a?
• ? s????t?ta de??µat?????a? fs 1/Ts.
• H a???????a s??a?t?se?? delta a?apt?sseta? se
  se??a Fourier
• Ets?

Poisson Sumation formula
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?da???? ?e??µat?????a (3)

• ?a?????µe t?? µ/? Fourier
• ?p?te

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?da???? ?e??µat?????a (4)ß t??p??

• ?? s?µa de??µat?????a? e??a?
• ?a?????ta? t?? µ/? Fourier ?a? µe t?? ???s? t??
  ?e????? µ/?
• e???µe

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? µ/? Fourier t?? s?µat?? e? de??µat?????a?

• ?? ? µ/? Fourier t?? x(t) e?e? t?? µ??f? t??
  s??µat??
• ??te ? µ/? Fourier t?? xs(t) ?a e?e? t?? p?? ?at?
  µ??f?
• ?d? ?p??esaµe ?t? fs ? 2B

X(f)
A
-? 0 ? t
Xs(f)
Xs(ffs)
Xs(f-fs)
Xs(f-2fs)
Xs(f2fs)
A/Ts
-fs -B 0 B fs-B fs
fsB 2fs f
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??a?t?s? t?? a?????? s?µat?? ap? ta de??µata t??

• ?? s?µa ap? t?? de??µat?????a e?e? µ/? Fourier
• ?? pe??aµe µesa ap? e?a ?at?d?aßat? (Low-pass)
  f??t?? t? ?p??? ep?t?epe? t?? d?e?e?s? µ??? t??
  fasµat?? ???? ap? t? f0.

y(t)x(t) X(f)
xs(t) Xs(f)
?at?d?aßat? f??t?? µe e???? fasµat?? W ?p?? ? ? W
? fs-B
?? fs ? 2B t? x(t) a?a?tata? a???ß?? ap? ta
de??µata t??
-W B B W fs-B
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?p?de??µat?????a ?a? aliasimg

• ?? t? s?µa ?p?ste? de??µat?????a µe s????t?ta fs
  ? 2? t?te ?a e???µe ?pe??a???? t?? pe???d??a
  epa?a?aµßa??µe??? fasµat?? X(f-nfs) st? fasµa
  t?? Xs(f), ?p?? fa??eta? st? s??µa
• To s?µa st?? e??d? t?? ?at?d?aßat?? f??t??? ?a
  d?afe?e? ap? t? a????? s?µa
• St?? pe??pt?s? a?t? e???µe t?? eµfa??s? t??
  fa???µe??? p?? ???µa???µe aliasing. ?? sfa?µa
  ???µa?eta? sfa?µa aliasing

?s(f)
-W W fs 2fs
Xa(f)
-W W
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?? Te???µa t?? ?e??µat?????a?

• ?st? s?µa x(t) µe pe?????sµe?? e???? fasµat?? ?a?
  µ/? Fourier
• X(f) 0, ??a f
  gt B
• To x(t) µp??e? ?a a?a?t??e? p????? ap? ta
  de??µata t?? p?? pa?????ta? µe s????t?ta
  de??µat?????a? fs , ea? fs gt 2B.
• ? s????t?ta 2? ???µa?eta? s????t?ta Nyquist.
• ?? fs lt 2B eµfa???eta? t? fa???µe?? t??
  aliasing.
• ?a? t? s?µa de? e??a? a?st??a pe?????sµe???
  fasµat??, t?te p?epe? ?a pe?ase? µesa ap? e?a
  ?at?d?aßat? f??t?? p??? t?? de??µat?????a

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?e??µat?????a s.d. ßas???? ?????

• ?? ? X(t) e??a? µ?a stat??? s.d. pe?????sµe???
  fasµat??, d??. SX(f) 0 ??a f ? B, t?te µp??e?
  ?a pa?asta?e? ap? ta de??µata t?? p?? pa?????ta?
  µe s????t?ta de??µat?????a? fs , ea? fs gt 2B. H
  pa?astas? ???eta? µe t?? e????a t?? p?? ?at?
  s?es??

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???e? µ??fe? t?? Te???µat?? ?e??µat?????a?

• St? ß?ß??? a?afe???ta? ?a? a???? t??p??
  de??µat?????a?
• ?e de??µata pepe?asµe??? d?a??e?a? (a?t? ??a
  a???????a s??a?t?se?? delta ???s?µ?p?????ta?
  pa?µ?? pepe?asµe??? d?a??e?a?)
• ?e de??µata µe ep?ped? ????f? ?p?? ?a? pa??
  ???s?µ?p?????ta? pa?µ?? pepe?asµe??? d?a??e?a? µe
  ???? ?s? ? t?µ? t?? s?µat?? ?ata t?? a??? t??
  pa?µ?? (sample-and-hold)
• S????????? st?? ?da???? de??µat?????a ?ta? ?
  d?a??e?a µ???a??e?.

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?a?ade??µa efa?µ???? t?? Te???µat?? t??
?e??µat?????a?

• Te????µe t? a??????? s?µa
• x(t) sinc(10000t)2 sin(10000pt)/1000
  0pt2
• ?fa?µ????ta? t?? d???? ?d??t?ta t?? µ/? Fourier
• sinc(10000t) ?(1/10000) ?(f/10000)
• sinc(10000t)2 ? (1/10000) ? (f/10000) X(f)
• ?? e???? fasµat?? a?t?? t?? s?µat?? e??a?
• ? 10000 ?z
• ?p?µe??? ta de??µata p?epe? ?a pa?????ta? µe
  ???µ?
• fs gt 2B 20000
  samples/sec

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?a?ade??µa 2 Raised Cosine Pulse

• ??deta?

?0.5 r 0.35 ? ? 1.35
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To fasµa t?? s?µat?? x(t) µeta ap?
?pe?de??µat?????a

• fs 20 gtgt 2B 2x1.352.70

-20 -15 -10 -5
0 5 10 15
20 25 30 35
40 f
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To f?sµa t?? s?µat?? x(t) µet? ap? de??µat?????a
µe ?aµ???te?? ???µ?

• fs 6 gt 2B 2.70

-6
0
6
12 f
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To f?sµa µet? ap? ?p?de??µat?????a?liasing

• fs 2.5 lt 2B 2.70

-2.5
0
2.5
5 f
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??p??e? s????t?te? de??µat?????a?

• S?µata f????
• ???ef?????? p???t?ta? f??? e?e? e???? fasµat??
  300 Hz e?? 4000 Hz
• ?a pe??ss?te?a s?t?µata ??f?a??? t??ef???a?
  ?a???? de??µat?????a µe 8000 samples/ sec.
• ????st??a s?µata
• ? ?????te?? s????t?ta p?? a?t??aµßa?eta? t?
  a????p??? a?t? e??a? pe??p?? 15 kHz.
• Sta CDs ? s????t?ta de??µat?????a? e??a? 44.000
  samples/sec.
• S?µata Video
• ?? µat? ??e?a?eta? de??µata µe ???µ? t???a??st?
  20 p?a?s?a/sec ??a ?a d?µ???????e? ? e?t?p?s?
  ?µa??? ????s??

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S?µata ?as???? ????? ?a? ????pe?ataBaseband and
Bandpass Signals

• ??a s?µa x(t) ?as???? ????? µe e???? fasµat?? ?
  e??a? e?a s?µa ??a t? ?p??? ? µ/? Fourier X(f)
  e??a? µ? µ?de????? ??a f ? B, ?a? e??a?
  µ?de????? X(f) 0 ??a f gt B.
• ??a ????pe?at? s?µa x(t) µe e???? fasµat?? ? f2
  f1 e??a? e?a s?µa ??a t? ?p??? ? X(f) e??a? µ?
  µ?de????? ??a 0 ? f1 ? f ? f2 , ?a? e??a?
  µ?de????? a????

X(f)
-? ? f
X(f)
B
-f2 -f1
f1 f2
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?p???t? e???? fasµat??Absolute Bandwidth
?p???t? e???? fasµat?? e??a? ? pe????? t??
fasµat?? µesa st?? ?p??a ? p????t?ta fasµat????
?s???? e??a? ? 0
?a?ade??µa tet?a??????? pa?µ??
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????? fasµat?? 3-dB 3-dB bandwidth
To e???? fasµat?? 3-dB e??a? ? pe????? t??
fasµat?? µesa st?? ?p??a ? p????t?ta fasµat????
?s???? e??a? µe?a??te?? ap? t?? t? µ?s? (1/2)
t?? µe??st?? t?µ?? t??
?a?ade??µatet?a??????? pa?µ?? d?a??e?a? ?s
f1
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Bounded Spectrum Bandwidth
To Bounded Spectrum Bandwidth e??a? ? pe?????
t?? fasµat?? µesa st?? ?p??a ? p????t?ta
fasµat???? ?s???? e??a? a f??e? ? µe??st? t?µ?
t??
?a?ade??µatet?a??????? pa?µ?? d?a??e?a? ?s
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Power Bandwidth
To Power Bandwidth e??a? ? pe????? t??
fasµat?? µesa st?? ?p??a pe??e?eta? t? R t??
s???????? ?s???? t?? s?µat??
To Power Bandwidth e??a? ? pe????? t??
fasµat?? µesa st?? ?p??a pe??e?eta? t? R t??
s???????? ?s???? t?? s?µat??
?
?
?
?
?a?ade??µatet?a??????? pa?µ?? d?a??e?a? ?s
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????? fasµat?? ?s?d??aµ?? ????ß??
To e???? fasµat?? ?s?d??aµ?? ????ß?? e???
s?µat?? s(t) e??a? t? e???? fasµat?? Beq e???
e???????? ?da????? ????d?aßat?? f??t???. ?ta?
? e?s?d?? t?? f??t??? e??a? ?e???? ????ß??, µe
PSD ?p?? st? s??µa,? ?s??? t?? s?µat?? e??d??
e??a? ?s? µe t?? ?s?? t?? s(t)
?
?
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????? fasµat?? ?s?d??aµ?? ????ß???a?ade??µa ??a
tet?a?????? pa?µ? d?a??e?a? ?s
?? e??????? ?da???? BP f??t??. ? ep?fa?e?a ?at?
ap? t?? ?aµp??? ?s??ta? ep?s?? µe t?? s???????
?s?? t?? s?µat?? s(t).
H p????t?ta fasµat???? ?s???? t?? s?µat??
s(t). ? ep?fa?e?a ?at? ap? t?? ?aµp??? e??a? ?
s??????? ?s??? t?? s?µat?? s(t).
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Null to Null Bandwidth
To Null-to-Null e???? fasµat?? e??a? ? pe?????
t?? fasµat?? µeta?? t?? s?µe??? µ?de??sµ?? de??a
?a? a??ste?a t?? ?????? ??ß??
?a?ade??µatet?a??????? pa?µ?? d?a??e?a? ?s
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???sµ?? t?? e????? fasµat?? s?µat?? ßas???? ?????

• ?? ? db e???? fasµat?? ? ??a??p??e? t?? s?es?

S?????? t?µ? ?3 db
? db
-B B

• Y t?? ?s???? e???? fasµat??