??431 ??G?S????? ??F???O? ???????O??O?

1
??431 ??G?S????? ??F???O? ???????O??O?

• S?µata
• ?etas??µat?sµ?? Fourier
• ?e??µat?????a

2
??e??e?a ?a? ?s??? s?µat??

• ?????e?a s?µat?? w(t)
• ??a s?µa w(t) e??a? s?µa e?e??e?a? a? 0 lt ? lt ?
• ?s??? s?µat??
• G?a pe???d??a s?µata, µe pe???d? ?, ? ?s???
  µp??e? ?a ?p?????s?e? ???????????ta? µesa se µ?a
  pe???d?.
• ??a s?µa w(t) e??a? s?µa ?s???? a? 0 lt ? lt ? ?a
  s?µata ?s???? de? ?pa????? st?? f?s?.

3
DECIBELS

• ???ada s?????s?? (?????? e?e??e??? ?a? ?s????)
• ???s?µ? ?ta?
• ? e?e??e?a ?a? ? ?s??? µetaßa????ta? ?ata a??ete?
  ta?e?? µe?e????
• ?a? e?d?afe?e? ?????? ? s?es? (? ????? ) d??
  µe?e???
• G?a s?????se?? e?e??e??? ? ?s???? db 10
  log10(P1/P2)
• Me???e? f??e? e??a? ???s?µ? ? s?????s? t?? ?s????
  e??? s?µat?? µe µ?a ?s?? a?af??a? 1Watt (1W) ? 1
  miliWatt (1mW).
• dbW 10 log10(P1/ 1W)
• dbm 10 log10(P1/ 1mW)
• ?a?ade??µata ?11mW 0dbm, ?110mW 10 dbm,
• ?1100mW 20dbm,
  ?11000mW 30dbm,
• ?12mW 3dbm, ?14mW 6dbm, ?18mW 9dbm, ?15mW
  7dbm,
• ?12.5mW 4dbm, ?11.25mW 1dbm

4
O ?etas??µat?sµ?? Fourier

• ? µetas??µat?sµ?? (µ/?) Fourier t?? s?µat?? w(t)
  e??a? ?
• ? a?t?st??f?? µ/? Fourier d?deta? ap? t?? s?es?
• S?µß??????µe e?a ?e???? µ/? Fourier ?? e???
  w(t) ? W(f)
• O µ/? Fourier ?pa??e? ea? t? w(t) e??a? s?µa
  e?e??e?a?

5
???e? µ??fe? t?? µ/? Fourier

• Se µe???a ß?ß??a ß?ep??µe ?a? t?? a???????
  s?µß???sµ?
• ?d? ? ??????? s????t?ta ? 2 p f µet??eta? se
  radians/sec ?a? d?2 p df.
• ?? d?? µ??fe? e??a? e?te??? ?s?d??aµe?

6
F?s??? s?µas?a t?? µ/? Fourier

• O µ/? Fourier µp??e? ?a ?e????e? sa? e?a?
  e??a?e?? µe t? ?p??? ß?ep??µe e?a s?µa ap? µ?a
  a??? ?pt??? ????a
• ???ta?te p?s? d?af??et??? µp??e? ?a fa?e? µ?a
  ?a?e??a ?ta t?? ß?ep??µe ap? d?af??e? ????e?
• ? s????t?ta µet?a t?? ???µ? t?? ????????
  µetaß???? e??? s?µat??
• ? ????? s????t?ta a?t?st???e? st?? ??????e?
  µetaß??e? s??a?t?se? t?? ??????
• ? ?aµ??? s????t?ta a?t?st???e? st?? a??e?
  µetaß??e?

7
?a?ade??µa ?p?????sµ?? Fourier?et?a??????? ?a?µ??

• ?et?a??????? pa?µ??
• ????
• ?p?te

?(t/T)
1
t
-?/2
?/2
8
?/? Fourier ?et?a??????? ?a?µ?? (2)

• E?daµe ?t? ?(t/T) ? ? sinc(pfT)
• ?a?at???se??
• ? d?a??e?a t?? pa?µ?? e??a? a?t?st??f?? a?a????
  t?? e????? fasµat??
• ? as??e?e?a st? ped?? t?? ?????? ?d??e? se
  ape?????st? fasµa

ped?? s????t?t?? ? sinc(fT)
?ed?? ?????? p(t/T)
9
?a?t?t?te? EULER
10
?e???a s?????sµe?a s?µata
1

• ?et?a??????? ?a?µ??
• ?????????? ?a?µ??
• ? s??a?t?s? de??µat?????a?
• ???ad?a?a ß?µat??? s??a?t?s?
• ? µ??ad?a?a ????st??? s??a?t?s?
• ?as??? ?d??t?ta t?? d(t)

-T/2 T/2 t
1
-T/2 T/2 t
1
11
?a?ade??µa µ/? Fourier 2? e??et??? s??a?t?s?

12
?a?ade??µa µ/? Fourier 3?µ?t???e?d?? S??a?t?s?

• ?e???e? f??e? e??a? e?????te?? ?a ß???µe e?a
  ?e???? µ/? ?p????????ta? t?? a?t?st??f? µ/?. ?ts?
  a??????ta? ap? t?? s?es?
• 
  
  ß??s???µe
• 
  
• d??ad?
• cos(2pfc t) ? (1/2)d(f-fc)
  (1/2) d(ffc)

13
?e??d?? e??es?? ?e???? µ/? Fourier

• ?fa?µ??? t?? ???sµ??
• ??µ??????a p??a?a µe ?e??? µ/? Fourier
• ??µ??????a p??a?a ap??? ?a????? ??a t?? µetat??p?
  t?? ?e???? t?? µ/?

14
S?µa?t??e? ?d??t?te? t?? µ/?

• G?aµµ???t?ta a1x1(t)a2x2(t)
  ?a1X1(f)a2X2(t)
• pa?ade??µa 5?(t/10)3?(t/20) ? 50
  sinc(10f) 30sinc(20f)
• ??????? ?a??ste??s? x(t-Td) ? X(f)exp(-j2pfTd)
• pa?ade??µa sin(2pfct) cos(2pfct p/2)
  cos2pfc(t 1/4fc) ?
• 
  (1/2)d(f-fc)exp(-jpf/2fc) (1/2)
  d(ffc)exp(-jpf/2fc)
• 
  (1/2)d(f-fc)jsin(-p/2) (1/2)d(ffc)jsin(p/2)
• 
  (-j/2)d(f-fc) (j/2)d(ffc)
• A??a?? ???µa?a? x(at) ? (1/a)X(f/a)
• ?e?a?? a gt ste??te?? ??????a s?µa gt
  fa?d?te?? fasµa
• M???? a gt fa?d?te?? ??????a s?µa gt
  ste??te?? fasµa

15
S?µa?t??e? ?d??t?te? t?? µ/? (2)

• ??aµ??f?s? x(t)cos(2pfct) ? (1/2)X(ffc)
  (1/2)X(f-fc)
• pa?ade??µa
• Te???µa t?? Parseval E ?x(t)2dt ?X(f)2df
• H e?e??e?a µp??e? ?a ?p?????s?e? e?te st? ped??
  t?? ?????? e?te st? ped?? s????t?t??
• ?a?ade??µa ?? x(t)sinc(2t) t?te
• E ?sinc2(2t)dt
  ?(1/2)?(f/2)2df 1/2

16
S?µa?t??e? ?d??t?te? t?? µ/? (3)

• S??e???? ???sµ?? x1(t)x2(t) -??? x1(t)
  x2(t-t) dt
• ?d??t?ta x1(t)x2(t) ? X1(f) X2(f)
• ??a p???p???? p?a?? st? ped?? t?? ??????
  a?t?st???e? se ap?? p???ap?as?asµ? st? ped??
  s????t?t??.
• ?ts? ap??p?????ta? t?s? a?a??t???? ?s? ?a?
  a???µ?t???? ?p?????sµ??
• ?fa?µ??? sta ??aµµ??a s?st?µata
• y(t) x(t)h(t) ? Y(f) X(f) H(f)
• h(t) ? H(f) s??a?t?s?
  µetaf??a?
• ? a?t?s?s?et?s? st? ped?? t?? ?????? t?? x(t)
  ????eta? sa?
• Rx(t) -??? x(t) x(t-t) dt x(t)x(-t) ?
  ?(f) X(f) X(f)2

????st??? ap????s? h(t)
y(t) -??? x(t) h(t-t) dt
x(t)
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18
S?µa?t??e? ?d??t?te? t?? µ/? (4)

• ????????s? a? x(t) ? X(f) ?a? y(t) -??
  tx(t)dt t?te
• Y(f) X(f)/(2jpf)
  (1/2) X(0)d(f)
• ?a?ade??µa ?? y(t) -?? t?(t)dt t?te
• Y(f) sinc(f)/(2jpf) (1/2)sinc(0)d(f)
  sinc(f)/(2jpf) (1/2)d(f)
• ?a?a????s? dx(t)/dt ? j2pfX(f) ?a?
• t x(t) ? (j/2p)
  dX(f)/df
• ???sµ?? (duality) ?? x(t) ? X(f) t?te
• ?(-t) ?
  x(f) ?a?
• X(t) ?
  x(-f)
• ?a?ade??µa sinc(t)sin(pt)/(pt) ? ?(-f) ?(f)

19
?e?????? µ/? Fourier

• ?a??µat??? p?a?? µe a?st??a ?????µe???? ?a???e?
• F?s??? s?µas?a
• ?e???e? t? fasµat??? pe??e??µe??
  e??? s?µat??
• ??e??????e? t?? ?p?????sµ? p????? p?s?t?t?? ?a?
  pa?aµet???
• Te???µa Parseval (?p?????sµ?? e?e??e?a?)
• Te???µa s??e????? (d?e?e?s? µesa ap? ??aµµ??a
  s?st?µata)

20
S?et??e? e????e??????t?ta Fasµat???? ?s???? ?a?
??e??e?a?

• ?????t?ta fasµat???? ??e??e?a? E(f) X(f)2
• ?????t?ta fasµat???? ?s???? ?st? ?t? t? x(t)
  e??a? s?µa ?s????.
• ??????µe t? truncated s?µa
  xT(t) x(t) ?(t/T)
• O µ/? Fourier t?? xT(t) ?pa??e?
  XT(f) ? xT(t)
• H p????t?ta fasµat???? ?s???? SX(f)
  ????eta? ?? e???
• ?d? ???saµe t?? p????t?te? fasµat???? e?e??e?a?
  ?a? ?s???? ??a µ?-t??a?a s?µata. ?? ?d??? ???sµ??
  ?s????? ?a? ??a ta t??a?a s?µata.

21
S?et??e? ?????e? (2)Se??e? Fourier

• O µ/? Fourier e??a? ???s?µ?? ??a ??a ta f?s??a
  ???p???s?µa s?µata (pepe?asµe??? e?e??e?a?)
• (?a? ep?p?e?? ??a µe???a µ? ???p???s?µa s?µata
  ?p?? t? ?µ?t???e?de? s?µa ?a? ? s??a?t?s? de?ta).
• d(t) ? 1, 1? d(f), cos(2pf0t) ? (1/2)d(f-f0)
  (1/2)d(ff0)
• ?? se??e? Fourier e??a? e?a pa??µ??? e??a?e??
  µetas??µat?sµ?? p?? e??a? ???s?µ? ??a pe???d??a
  s?µata.
• ?a pe???d??a s?µata e??a? s?µata ?s???? ?a? e?
  ???sµ?? µ? ???p???s?µa
• ?a pe???d??a s?µata µp????µe ?a ta pa?ast?s??µe
  µe µ/? Fourier a? de????µe t?? ???s?µ?p???s?
  s??a?t?se?? de?ta.

22
S?et??e? ?????e? (3)Se??e? Fourier - ???sµ??

• ?? t? x(t) e??a? pe???d??? s?µa µe pe???d? ?0 ,
• a? ? a???µ?? t?? µe??st?? ?a? e?a??st?? ?a??? ?a?
  t?? s?µe??? as??e?e?a? se ?a?e pe???d? e??a?
  pepe?asµe??? ?a?
• a? e??a? ????????s?µ? ?at ap???t? t?µ? st? 0,
  ?0, t?te µp??e? ?a ??afe? sa? a?apt??µa se se??a
  Fourier
• ?p??
• ?? s??te?este? xn ???µa???ta? s??te?este? Fourier
  ?a? ? f0 (1/T0) e??a? ? ?eµe?e??d?? s????t?ta
  t?? x(t).
• ?a?ade??µa ?st? x(t) µ?a pe???d??? a???????a
  pa?µ?? ?(t/t), d??.
• (?0 pe???d??)
  

23
S?et??e? ?????e? (4)?et/sµ?? Fourier ?e???d????
S?µat??

• ?st? ?t? t? x(t) e??a? pe???d??? s?µa µe pe???d?
  ?0.
• ?? a?apt??µa se se??a Fourier e??a?
• ?a?????ta? t?? µ/? Fourier aµf?te??? t?? µe???
  ß??s???µe
• ?? s??te?este? Fourier xn ß??s???ta? ?? e???
  ?st? xTo(t) x(t) ?(t/T0)
• 
  
  
• ?p?? xTo(t) ? ?To(f)

24
?a?ade??µata fasµat???? ape?????s??pe???d????
s?µat??
25
?a?µ???af?? ??a??t?? fasµat??
Time domain
Frequency domain
26
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27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30

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

31
??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? µ??e???
  (amplitude) t???

32
??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?? ??????

33
??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? ?? p??? t?? t?µ? t???

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.
8 -1.0
S????t?ta 1/6 S????t?ta de??µat?????a?
20/6 ?p?peda ?ßa?t?sµ?? 11
34
??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
35
??f?a?? epe?e??as?a a?a??????? s?µat??
36
?eta??pe?? A/D ?a? D/A

• ? µetat??pea? ??a???????-se-??f?a?? (Analog to
  Digital - A/D) ??d???p??e? t?? t?µ? t?? de??µat??
  e??? s?µat?? se d?ad??? a???µ? a?a???? t?? t?µ??
  a?t??.
• ? µetat??pea? ??f?a???-se-??a?????? (Digital to
  Analog (D/A) µetat?epe? e?a? d?ad??? a???µ? se
  tas? (? e?tas?) a?a???? t?? t?µ?? t?? a???µ??
  a?t??.

37
A/D
38
??f?a?? ?a??stas? ??a??????? S?µ?t??

• ??? ???e? ?p?? p??pe? ( µe t?? s?st? s????t?ta
  de??µat?????a?), ? de??µat?????a a?t? ?a? ea?t?
  de? e?s??e? pa?aµ??f?s? st? s?µa.
• ? ?ßa?t?sµ?? ?µ?? e?s??e? p??t?te ??p??a
  pa?aµ??f?s?.
• ? pa?aµ??f?s? µe???eta? a? a????e? ? a???µ?? t??
  ep?ped?? ?ßa?t?sµ?? ( ? a???µ?? t?? apa?t??µe???
  bits ??a t?? ??d???p???s?)
• ?p??e? ?a ???e? a?ta??a?? µeta?? t?? pa?aµ??f?s??
  ?a? t?? ???µ?? pa?a????? bits/sec ( 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?

39
? s????t?s? delta d(t) - ?a??

• ? s????t?s? dirac delta d(t) e?e? t?? e???
  ßas??? ?d??t?ta
• -???x(t)d(t)dt x(0)
• H d(t) ???µ??eta? ?a? ????st??? s????t?s?
  (impulse function)
• ???e? ?d??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)

40
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?? ?d??t?ta µetat?p?s?? st? ?????
  t?? d(t), ?p?te
• y(t) x(t) d(t-t) x(t-t)

41
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
  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 ?s 2Ts t
42
?da???? ?e??µat?????a

• Te????µe to s?µa ?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
43
?da???? ?e??µat?????a (2)

• ?a?????µe t?? µ/? Fourier
• ?p?te

44
? 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)
-? 0 ? t
Xs(f)
-fs -B 0 B fs-B fs
fsB 2fs f
45
??a?t?s? t?? a?????? s?µat?? ap? ta de??µata t??

• ?? s?µa e? 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 ? B t? x(t) a?a?tata? a???ß?? ap? ta
de??µata t??
-W B B W fs-B
46
?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
47
?? 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?? ?aµßa???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

48
???e? µ??fe? t?? Te???µat?? ?e??µat?????a?

• St?? p?a?? ? de??µat?????a ???eta?
• ?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??
• S????????? st?? ?da???? de??µat?????a ?ta? ?
  d?a??e?a µ???a??e?.

49
?a?ade??µa Raised Cosine Pulse

• ??deta?

?0.5 r 0.35 ? ? 1.35
50
To fasµa t?? s?µat?? x(t) µeta ap?
?pe?de??µat?????a

• fs 20 gtgt 2B 2x1.352.70

51
To f?sµa t?? s?µat?? x(t) µet? ap? de??µat?????a
µe ?aµ???te?? ???µ?

• fs 6 gt 2B 2.70

52
To f?sµa µet? ap? ?p?de??µat?????a?liasing

• fs 2.5 lt 2B 2.70

53
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54
(No Transcript)
55
Aliasing
56
(No Transcript)
57
(No Transcript)
58
??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??

59
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
60
???sµ?? t?? e????? fasµat?? s?µat?? ßas???? ?????

• ?p???t? e???? fasµat??
• ?(f) 0, ??a f gt B
• To e???? fasµat?? p??t?? µ?de??sµ?? e??a? ?
  e?a??st? s????t?ta ??a t?? ?p??a µ?de???eta? ?
  µ/? Fourier.
• min f gt 0 ? X(f) 0

-B B f
?????? ??ß??
??a???? ??ß??
-B B
61
???sµ?? t?? e????? fasµat?? s?µat?? ßas???? ?????
(2)

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

S?????? t?µ? ?3 db d??ad? µe??s? t??
p????t?ta? fasµat???? ?s???? st? µ?s?,
? db
-B B

• Y t?? ?s???? e???? fasµat??

62
?a?astas? s?µat?? st? MATLAB

• H s????t?ta de??µat?????a? p?epe? ?a e??a? a??eta
  µe?a?? ?ste ?a ap?fe??eta? t? fa???µe?? aliasing
• ?a?ade??µa
• Te???t??a e?a? ?????????? pa?µ?? e?e? ape???
  e???? fasµat??
• ?a? ?a???µe a??eta p???? de??µat?????a t?
  aliasing ?a e??a? as?µa?t?
• MATLAB script ??a t?? s?ed?as? e??? tet?a???????
  pa?µ??
• gtgt x ones(1,10) zeros(1,10)
• gtgt plot(00.11.9, x, b)
• gtgt axis(0 2 0 2)
• gtgt ylabel(x(t))xlabel(t(usec))

63

• QUANTIZATION

64
??ad??as?a µetat??p?? a?a??????? se ??f?a?? (ADC)

??a?????? s?µa
?eta t?? de??µat?????a
?ßa?t?sµe?? s?µa
65
Kßa?t?sµ?? - Quantization

• ?a s?µata s??e???? ?????? ?f?sta?ta?
  de??µat?????a ?ata ta?ta ??????a d?ast?µata
• ? de??µat?????a de? e?sa?e? ?p??ad?p?te
  pa?aµ??f?s? a? ???e? µe ???µ? µe?a??te?? ap? t??
  ???µ? Nyquist.
• ?a a?a?????a de??µata e???? t?µe? se e?a s??e?e?
  d?ast?µa t?µ?? ?a? ??e?a?eta? ape???? a???µ??
  bits ??a t?? pa?astas? t??? µe te?e?a a???ße?a.
• ?ßa?t?sµ?? (Quantization) e??a? ? d?ad??as?a t??
  p??se???s?? e??? a?a??????? (s??e????) de??µat??
  µe e?a pepe?asµe?? a???µ? bits.
• O ?ßa?t?sµ?? e?sa?e? pa?t?te ?ap??a pa?aµ??f?s?

66
?ßa?t?st?? 3 bit , 8 ep?ped??
101 100 011 010 001 000
67

?ßa?t?st?? 3 bit , 8 ep?ped??

x8
Sfa?µa

x1
S?µa ??s?d?? x
68
???a?a? ?a?a?t???st???? pa?aµet??? ?ßa?t?st?

69
G?af??? pa?astas? t?? ?ßa?t?sµ??

M8
x8
3.5

3
-3
x5
x1
x8
x7
x0

x1
-3.5
70
?ßa?t?st?? ?26 64 ep?ped??
71
S?µß???sµ?? s?et???? µe t?? ?ßa?t?sµ?

• ?st? ? µ?a t??a?a µetaß??t? p?? pa??sta?e? t?
  de??µa e??? a?a??????? s?µat?? t? ?p??? ?f?stata?
  de??µat?????a.
• ??te e??a? ? ?ßa?t?sµe?? t?µ?
  t?? ?.
• ??a? ?ßa?t?st?? e?e? ? ep?peda ?ßa?t?sµ??
• ?a ???a t?? ????? ?ßa?t?sµ?? ??????ta? ap? t?? ?
  1 t?µe?
• x0, x1, , xM, ?p?? x0 -?, xM ?
• ?ts? ??a

72
???p???s? t?? ?ßa?t?st?

• ??a? µetat??pea? a?a???????-se-??f?a?? (ADC
  Analog to Digital Converter) p??se????e? t? s?µa
  (de??µa) st?? e?s?d? t?? µe e?a ap? ta ? ep?peda
  ?ßa?t?sµ??.
• ??a ?????µa se???a??? e??d?? (SIO Serial Input
  Output) µetat?epe? t? ep?ped? ?ßa?t?sµ?? se
  a???????a ap? nlog2M bits.

n bits
x
ADC
SIO
Quantizer
73
??a?t???? t??p?? ?atas?e??? ADCs

• Counting or Ramp ADC
• ??a tas? a?af??a? a??a?e? ?ata sta?e?a ß?µata
  µe??? ?a ???e? µe?a??te?? ap? t? de??µa
• Serial or Successive Approximation ADC (Se???a???
  ADC ? d?ad?????? p??se???se?? ADC)
• ???s?µ?p??e? d?ad??? e?e??a (µe??? ?a f?ase? st??
  ep???µ?t? a???ße?a ) ??a ?a p??sd????se? t?? ????
  ?ßa?t?sµ?? t?? de??µat?? e?s?d??.
• Parallel or Flash ADC (?a?a?????? ?
  ast?ap?a???)
• ?? de??µa e?s?d?? s??????eta? ta?t?????a µe ??a
  ta d??ata ep?peda ?ßa?t?sµ??

?e?a??te?? ta??t?ta
M????te?? p???p????t?ta
74
??a e?a??a?t??? ?e???s? t?? ?ßa?t?sµ??

• ? ?ßa?t?sµ?? p??s?ete? e?a t??a?? ????ß? nQ x
  - x st?? p?a?µat??? t?µ? x t?? de??µat??.
• ?ts? ? pa?aµ??f?s? D ? t? MSE EnQ2 µp??e? ?a
  ?e????e? sa? ?s??? ????ß??
• ?p????µe ?a ???s??µe µ?a s?µat?????ß??? s?es?
  (SNR Signal to Noise Ratio) ??a ?a
  ?a?a?t???s??µe t?? s?µpe??f??a t?? ?ßa?t?st?.

?ßa?t?st??
x
x fQ(x) x nQ


nQ x - x
75
SNR ??a ?µ???µ??f? ?ßa?t?st? (s?????)

• Ge???? ap?te?esµa (µp??e? ?a ap?de???e? e????a)
• ????p??ese??
• ?µ???µ??f?? ?ßa?t?st?? ? ep?ped??
• ?a de??µata e?s?d?? e???? ?µ???µ??f? ?ata??µ?, µe
  pe????? t?µ?? ?s? µe t?? pe????? t?µ?? p??
  ?a??pte? ? ?ßa?t?st??.
• ?µpe?????? ?a???a?
• ?a?e p??s?et? bit ( d?p?as?asµ?? ep?ped??)
  
• a??a?e? t? SNR ?ata 6 db (tet?ap?as?asµ??)

76
?a?µ???d??? ??aµ??f?s? (PCM)

• ? ?a?µ???d??? ??aµ??f?s? pa?a?e? e?a s?µa
  ?as???? ????? t? ?p??? pa???e? ?at e??e?a? ap?
  t?? se???a?? e??d? e??? ?ßa?t?st?.
• ?e???e? f??e? ? ???? PCM ???s?µ?p??e?ta?
  e?a??a?t??a ??a t?? ??? ?ßa?t?sµ??.

77
????? ????? s?µat?? PCM

• ?µ???µ??f?? ?ßa?t?st?? n bits ?a? 2n M
  ep?ped??
• ? ???µ?? de??µat?????a? e??a? fs
  de??µata/sec.
• ? ???µ?? pa?a????? d?ad???? s?µß???? st? e??d?
  t?? ?ßa?t?st? e??a? fs log2M fs n bits/sec.
• To e?a??st? apa?t??µe?? e???? fasµat?? (µe
  ße?t?st? µ??f? pa?µ??) e??a? fs n/2 Hz.
• To e???? fasµat?? p??t?? µ?de??sµ?? (µe
  ?????????? µ??f? pa?µ??) e??a? fs n Hz.
• Ta a?afe????µe se ep?µe?a µa??µata st?? ep?d?as?
  t?? µ??f?? t?? pa?µ?? ep? t?? e????? fasµat?? t??
  s?µat?? PCM

78
?a?ade??µa ?p?????sµ?? ??a s?µa PCM

• ???ß??µa ??a a?a?????? µ??s??? s?µa e?e? e????
  fasµat?? 15 kHz ?a? ta de??µata t?? µp????? ?a
  ?e??????? ?t? e???? ?µ???µ??f? pdf.
• ?a ß?e?e? t? e?a??st? e???? fasµat?? ??a t??
  µetad?s? t?? s?µat?? µe d?aµ??f?s? PCM ?a? µe
  µes? pa?aµ??f?s? ?a???te?? ap? 58db.
• ??s?
• S????t?ta de??µat?????a? fs ? 2B 30.000
  de??µata/sec.
• 10log10M2 ? 58db ? M ? 102.9?103 ?210 ? n ? 10
  bits/sample
• ??a??st?? ???µ?? pa?a????? ded?µe??? R fs n ?
  300.000 b/s
• ??a??st? e???? fasµat?? BW R/2 ? 150 kHz.
  (?p?t??eta? ?t? ???s?µ?p????µe pa?µ? ße?t?st??
  µ??f??)

79
S?es? ???µ?? R ?a? pa?aµ??f?s?? D

• S?ed?ast???? st???? ?ßa?t?st??
• G?a ded?µe?? ???µ? ??d???p???s?? R, ?a
  e?a??st?p????e? ? pa?aµ??f?s? D, G?a ded?µe??
  pa?aµ??f?s? D, ?a e?a??st?p????e? ? ???µ??
  ??d???p???s?? R
• ?pe???µ?s? R fs n bits/sec.
• Teµe???de? ???? ? ?e???a p????f????? ap?de????e?
  ?t? ??a e?a ded?µe?? µet?? pa?aµ??f?s?? D,
  ?pa??e? µ?a s??a?t?s? ???µ??-pa?aµ??f?s?? R(D)
  ? ?p??a d??e? e?a p??se???s?µ? ?at?te?? ???? st??
  µ????te?? ???µ? R.
• ?ts? a?
• t?te

2
R(D)
1
D
80
?e????e? ??d???p???s?? ??a f???

• ??e? ?? µe??d?? ??d???p???s?? f???? ???s?µ?p?????
  ?ßa?t?sµ?.
• ????e? ?a???? ep?p?e?? ???s? ?d?a?te???
  ?a?a?t???st???? t?? f????
• ??d???p???s? ???????? ??µat?µ??f?? p??spa?e?a
  pa?astas?? t?? de??µat?? t?? ??µat?µ??f?? t??
  f???? st? ped?? t?? ??????.
• Fasµat??? ?e??d?? ??d???p???s?? p??spa?e?a
  pa?astas?? t?? fasµat???? ?a?a?t???st???? t??
  ??µat?µ??f?? t?? f????.
• ??d???p???s? ßas?sµe?? st? µ??te?? t?? f????.
  p??spa?e?a a?apa?a????? t?? µ??te??? t??
  d?ad??as?a? pa?a????? f????

81
??d???p???s? t?? ??µat?µ??f?? t?? f??????af?????
PCM

• ?? s?µa t?? ?µ???a? e?e? µe?a?? s?s?et?s? ??a
  ?e?t????e? ??????e? st??µe?.
• ? d?af?????? pa?µ???d???p???t?? (DPCM-
  Differential PCM) ?ßa?t??e? t?? d?af??a µeta??
  e??? de??µat?? ?a? t?? p??ß?ep?µe??? t?µ?? t?? ?
  ?p??a p??ß?epeta? µe ??aµµ??? s??d?asµ?
  p??????µe??? de??µat??.
• S?????? ? d?af??a a?t? e??a? µ???? ?a? µe
  ????te?a bits ep?t???a?eta? µ???? pa?aµ??f?s?
• ? a?t???pt? p???t?ta f???? e?a?tata? ap? t??
  s?et??? t?µ? t?? sfa?µat?? ?ßa?t?sµ?? ?a? t??
  e?tase?? (?s????) t?? f????.
• ?? p??sa?µ???µe??? DPCM (ADPCM Adaptive DPCM)
  µeta-ßa????? t? ß?µa ?ßa?t?sµ?? a?a???a µe t??
  e?tas? t?? f????.
• ?? ADPCM µe 32kbits/sec e??a? p??? ????? ?a?
  e????? st?? ???p???s?, ???s?µ?p??e?ta? de e??tata
  (p.?. st? DECT)

82
??µ? t?? DPCM

xn
en
rn


en
Quantizer
S
S

-
Delay T1
Delay T1
rn-1
xn-1
S?ss??e?s? t?? sfa?µat?? ?ßa?t?sµ??

yn
yn
xn


yn
Quantizer

S
xn
-




yn

yn-1
Delay T1
Delay T1
S
S



xn-1
??-s?ss??e?s? t?? sfa?µat?? ?ßa?t?sµ??
83
Predictive DPCM
O predictor ???s?µ?p??e? ta p p??????µe?a
?ßa?t?sµe?a de??µata ??a ?a p??ß?e?e? t? pa???
de??µa. ?ßa?t??eta? ? d?af??a t???. ?a ai
ep??e???ta? ets? ?ste ?a e?a??st?p??e?ta? t? µes?
tet?a?????? sfa?µa p??ß?e??? Tet??µe

• O predictor ???s?µ?p??e? ta p p??????µe?a
  ?ßa?t?sµe?a de??µata ??a ?a p??ß?e?e? t? pa???
  de??µa. ?ßa?t??eta? ? d?af??a t???.
• ?a ai ep??e???ta? ets? ?ste ?a e?a??st?p??e?ta?
  t? µes? tet?a?????? sfa?µa p??ß?e???
• Tet??µe

84
??aµ??f?s? DELTA (DM)

yn
yn1
xn


yn
1 bit Quantizer

• ???a?a pe??pt?s? ??d???p???t? PCM st?? ?p??? t?
  s?µa ?f?stata? ?pe?-de??µat?????a ?a?
  ??d???p??e?ta? µe 1 bit/sample.
• ? ??aµ??f?t?? Delta µp??e? ?a ???p????e? µe e?a?
  e?a??et??a ap?? ?ßa?t?st? 1 ep?ped??.
• H Adaptive DM µe 16 kbits/sec µp??e? ?a pa?a?e?
  ??? ?a??? p???t?ta?

S
xn
-
?
X
X


yn


yn-1

Delay T1
Delay T1
S
S


85
Fasµat??e? µe??d?? ??d???p???s??Subband Coding

• H a?t????? t?? a???st???? p???t?ta? e?a?tata? ap?
  t?? ???? s????t?t??. St?? p??? ?aµ??e? ? p???
  ????e? s????t?te? µp????? ?a ?????? a?e?ta
  µe?a??te?a MSE.
• ?? Subband Coders d?a???????? t? s?µa se p???e?
  ???e? µe ?ata????a f??t?a.
• ?a s?µata se ?a?e ???? µetat?ep??ta? se ??f?a?a
  µe t? standard PCM a??a µe d?af??et??? ???µ? R se
  ?a?e ????.

86
Fasµat??e? µe??d?? ??d???p???s??Adaptive
Transform Coding

• ?a de??µata t?? s?µat?? ?µad?p?????ta? se
  p?a?s?a.
• ?a s?s?et?sµe?a st?? ????? de??µata µetat?ep??ta?
  se (e?p????µe...) as?s?et?sta fasµat??a de??µata
  ?a???ta? FFT ? DCT (Discrete Cosine Transform).
• ?a fasµat??a de??µata pa??sta???ta? s?µf??a µe
  t?? ßa??? t??? st?? a?t????? t?? f????
• ? ße?t?st?? ??d???p???t?? f???? half-rate (4
  kbits/sec) ???s?µ?p??e? Adaptive Transform Coding
  ( µa?? µe d?a??sµat??? ?ßa?t?sµ? ?a? G?aµµ???
  p??ß?e??)

87
?e????e? ??d???p???s?? ßas???µe?e? se µ??te?a

• ??d???p???te? f???? G?aµµ???? p??ß?e??? Linear
  Predictive Speech Coders (LPC)
• ?? s?µa t?? f???? d?a??e?ta? se p?a?s?a d?a??e?a?
  20 ms.
• ? ?µ???a µ??te??p??e?ta? sa? ????ß?? (ae?a? ap?
  t??? p?e?µ??e?) p?? d?e?e??e? e?a ??aµµ??? f??t??
  (?a????a?, f???t??e? ???de?, ?a? st?µa ??????
  ?????t?ta).
• ? LPC ?ßa?t??e? t?? a???????a d?e?e?s??, t??
  pa?aµet???? t?? f??t???, ?a? t? ?e?d?? t??
  f??t???, ?a? µetad?de? ta d?ad??a ded?µe?a st??
  de?t?. S???a ???s?µ?p??e?ta? d?a??sµat????
  ?ßa?t?st??

????????a ??e?e?s??
???d?? F????
F??t?? ???ß?e???
?
?e?d?? F??t???
?a?aµet??? F??t???
88
Vector Sum Excited Linear Prediction (VSELP)

• ???s?µ?p??e?ta? st? ??f?a?? ????t? t??ef???a t??
  ?. ?µe????? (p??t?p? IS-136)
• ??a?s?a t?? 20 ms. ?a?e p?a?s?? pa??sta?eta? µe
  159 bits. ? ???µ?? ded?µe??? e??a? pe??p?? 8
  kbits/sec.
• ??a? d?a??sµat???? ?ßa?t?st?? d?? ßa?µ?d??
  ???s?µ?p??e?ta? ??a t?? ?ßa?t?sµ? t?? a???????a?
  d?e?e?s??.
• ?e???a bits (?p?? t? ?e?d?? t?? f??t???) e??a?
  p?? s?µa?t??a ??a t?? p???t?ta t?? f???? ap? ?t?
  a??a. ??ta p??state???ta? µe ??d???p???s?
  d?????s?? ?a???.

89
S?????s? µe??d?? ??d???p???s?? f????

• ??p?? ???µ?? (kb/s) ????p????t?ta(MIPS)
  ?a??ste??s? (ms) ????t?ta
• PCM 64
  0.01 0
  ?????
• ADPCM 32 0.1
  0
  ?????
• Delta Mod. 16 0.1
  0
  ?????
• Sub-Band 16 1
  25
  ?????
• VSELP 8
  100 35
  ?a??
• Te???a 1
  ?? ??
  ?????

90
??d???p???s? e????a? ?a? Video

• ?????a 1000 x 1000 pixels µe 8 bits ??a ?a?e e?a
  ap? ta t??a ???µata ??e?a?eta? 24 Mbits ??a ?a
  ??d???p????e?.
• ?? video ??e?a?eta? pe??p?? 20 frames/sec.
• ?a standards s?µp?es?? e??a? ßas??a ??a t??
  a?apt??? t?? ??f?a??? video.
• JPEG S?µp?es? e????a? ?ata 20 f??e? t???a??st??
• MPEG S?µp?es? video ?ata 100 f??e? ? pe??ss?te??