Ch3.5: FIR??????

1
Ch3.5 FIR??????
2
FIR????????

• H(z)?z-1?N-1????,
• ?z???(N-1)???,
• z0?(N-1)?(?)??, ?????????

3
FIR??????

• FIR?????
• FIR filter?h(n)???,????,???,??????
• FIR filter????????,???????????
• ????????????
• ??????????,
• ?????????,
• FIR?????DFT???

4
????

• ???? ??????, ??????????,
• ?? ??????????LTI????????????????,
• ?????
• ?? ?????????????????,
• ?? ?????????,
• ??????,

5
?????

• ???????????????,
• ??????
• ?????,????????hd(n)???????h(n)??????hd(n)
• ?????
• ?????,????Hd(j?)???H(k), ????????H(z)
• ?????(????????)
• ???,???????,?????

6
??????

• ??????(????)
• ?????x(n)????????????????
• ????(????)
• ????x(n)??????????????????
• ???
• ??????????, ???????????????,
• ???????????, ??
• ????????,
• ????, ???????????,

7
Linear phase

• ???????,??????????????????y(n)Ax(n-k)
• ??????????
• ????????,
• ??????,why?
• Suppose argH(?)-k?, ??????????

8
An example

• A signal having two sinusoids is filtered with
  phase ß1 and ß2.
• What is the condition when all sinusoids have the
  same time delay L?

9
FIR??????????

• Question condition for undistorted transfer?
• ??FIR????h(n)???, ??????????, ?????????????,
• ??????,h(n)h(N-1-n), f(?)-(N-1) ? /2
• ??????,h(n)-h(N-1-n), f(?)-(N-1) ? /2 pi/2
• ?????n(N-1)/2?,

10
??????FIR???
???
???
h(n)
h(n)
N11
N11
0
n
0
n
N10
h(n)
N10
h(n)
0
n
0
n
11
???????
?(?)
?(?)
?/2
?
2?
0
?
2?
0
?
?
-(N-1)?
-(N-3/2)?
h(n)??? (Both Ns)
h(n)??? (Both Ns)
12
h(n)????H(?)
13
h(n)????H(?)
14
Four different filters

• h(n), evenly symmetric, odd N,
• h(n), evenly symmetric, even N,
• h(n), oddly symmetric, odd N,
• h(n), oddly symmetric, even N,
• The above four filters have different frequency
  response and are thus good for different
  purposes.

15
????

• ??????????,
• h(n)????????????

16
??????(??????)

• ???? ???Hd(ej?)--gt hd(n), FIR?h(n)??hd(n),
• ?????
• ?????????Hd(ej?), hd(n)?????, ?????,
• ???????????w(n)???hd(n), h(n)w(n)hd(n),
  H(ej?)Hd(ej?)W(ej?)

17
????????

• ??????????, ????????N????w(n),
• w(n)??????????????,

18
???
Hd(ej?)
H(ej?)
????
?
?c
-?c
0
W(ej?)
?
??
-?c
?c
??????
?
0
???? 4pi/N
19
??????

• ????????N, ????????, ???1/N???, 2pi/N,
• ?????????????,
• ???H(ej?)??????, ???????????, ????, ?????,
• ??????, ?????????????, ?????????????,

20
???????

• ????
• ????,?????????,
• ??????????????,????????,????????,
• ????????,
• ?????
• ???????????????,??????????????????,

21
????????

• ????????13dB, ????????,
• ?????????hd(n)?????, ??Gibbs??,
• ??N, ?????????, ?????, ??????????,

22
??????

• ???(Boxcar)
• ????(Bartlett)
• ???(Hanning)
• ???(Hamming)
• ????(Blackman)
• ???(Kaiser)

23
??????
Name ????? ????? ??????
Boxcar 4?/N 1.8?/N 13dB
Bartlett 8?/N 6.1?/N 27dB
Hanning 8?/N 6.2?/N 32dB
Hamming 8?/N 6.6?/N 43dB
Blackman 12?/N 11?/N 58dB
24
??????
25
??

• ????N13???FIR?????,?c0.2pi, ??????????????,

26
??(Boxcar Hanning)
27
?????????(LP)

• ????????????, ?s, ?p, ?????h(n),
• ???1, ?2??????????w(n)?????h(n)???N,
• h(n)w(n), ??????????????,
• h(n) ? H(z)

28
Matlab???

• fir1(N, Wn) FIR LP filter design using the
  window method, Wn cut-off frequency.
• fir2(N, F,M) FIR filter design using the window
  method - arbitrary filter shape (F,M).
• firls(N,F,A) Linear-phase FIR filter design
  using least-squares error minimization.
• freqz() frequency response

29
???????

• ????, ?Hd(ej?)???????, ?0-2???N?????????,
  Hd(ej?)--gt H(k)
• H(k)--gt h(n) (?DFT)
• h(n)--gt H(z)?H(ej?)

30
???????(2)

• ????Hd(ej?), ?????????N,
• ?Hd(ej?)?0-2???N?????????
• H(k)--gt h(n) (?DFT)
• h(n)--gt H(z)?H(ej?)

31
H(k)?????

• ???H(k)1, ???H(k)0, ????????????,
• ???H(k)??????h(n)???,
• ?h(n)???H(ej?)???????,
• ???????h(n)?????H(k)?

32
H(k)??
33
????

• ?????Hd(j?), ?????????M,
• ?Hd(j?),?0-2pi??M??????H(k), ???H(k)0,
• ???H(k) ???,??H(z),

34
??

• ????????????FIR DF, ???????????1/10, ?N20,

35
Features of the filter
36
MatLab

• FIR design using frequency sapling method?

37
?????

• ??????,???????????????????????,??????????????
• ????????(?????????????), ??Parks-McClellan??,
  remez() in MATLAB

38
FIR??????

• ??FIR????
• ??????????,
• ????????, ????,
• ???, ???DFT,
• ?????FIR????
• ???, ???????, ?????
• ???????, ?????????,
• ???M????, ?????M/2???,

39
IIR?FIR????????

• IIR???????FIR????, ??, ????-20dB, FIR?33?,
  IIR??4-5??Butterworth, FIR????5-10?,
• FIR???????????, IIR???????,
• FIR??????, ????,
• FIR???FFT, IIR???????,
• IIR??AF?????, ????, FIR???????????,
• FIR????,????filter????

40
??????

• Filter, h(n) or H(z),h(n)?H(w) freqz
• x(n) ? y(n)?
• conv() in matlab
• filter() in matlab, Y FILTER(B,A,X)
• X(O) ? Y(O)
• Y(O)X(O)H(O)