數(shù)字信號處理部分課堂講義第三章

(DiscreteFourier1§3-1axa(t),Xa(a

(j)

xa(t)

jt1xa(t)1

Xa(j

jtd

Xa( 2

X

xa(t)

m1

(m)

jmX(m)T

x x

T mT3

2X(

j )

n

x(n)

x(n)

X

j

4

X(eTnTn)2)2結(jié)論：非周期的離散時間函數(shù)對應(yīng)于周期性連續(xù)頻率函數(shù)變 5~(n)

X2(k2k ~(n)周期 XkX散 6

離

已知x(n0≤n≤N-1有限長,X(e )

Nn

x(n)

j

2~e

j(~(

N

)x(n)e

~7

2NN ~X(k)~

j

2N

~(n)e2Nknn0

DFS顯顯 X

N)

N

~(n)

j2(kN)n

N ~(n)n

jN

(k

j

N

~(

N N~

(x(n

) k

k nNn

N N~(n ejNk

k(rn8N ej

1

ej(rn

sin(rk

1

en)

N

(rn) r

Nk

rX~(X

j2kr

N~() ~()

N1~ 1~ k

(k

j2nk

j記 ~X(k)~

N(n)

(n)WN

N~(n)

X(k)~N~

k

X(k)W

9X~(kX義x(n)N

~

0nN1N X(z)x(n)znn0

~(n)zn0

X(k)X

X

j2k

xn)的一個周期所得序列的ZX(X ~(k)(XjIjIm2NZeRe

(n)3a

(n)1b

2(n

(k)~~

a

(k)X1X

2(kNN

mk

m)

Nn

n

m

令NN1N

N1NNNN上式NN

i

~(i

W

i

~(iW

NN

Ni

~(

N

~(k)

~XkX

l)W

ln~( ~(n)~212

~(n)

X2(k~ X1(k

~(n)

~(n而

x2

m

x1(m)x2 m

m

x2(m) m證：

(n)

N1Nn0m

~(

2(n

m)W1N1m

(m

N22n

(nx1Nx1N

(nm)N~ N1 ~rnm

X1(k)

r

x2(r)W~(x2(r),W~

)

~X1(k)~

Nr

~2(r)W

X1(k)

X2(k12~(n~(n)12

N下1...

01

1

(1)m

~(n)1)21

2

nm

-3-2-10 2 4~(n)~(n)3 1~

1

N1 X3(k)

X1(k)

X2(k)

Nl

X1(l)X2(kl

N1

N1N1

3(n)

k

X3(k

X1 X kX1 X

X1(l)X2(kl)W11Nl

(l)

nlN

k

(kl)

~(n) 2)§ 一、定義設(shè)有限長序列

0nN~ x(n

r

x(nrN~~(n)x((

(n)~(n)

N(n

X((

X(k)R

N(k~(n~NX(k)x(n)WNknn0N x(n)1 X(k

0kN0nN-

列長

X(k)DFTx X2(k) 若x3(nax1(nbx2則X3k)aX1(kbX2x(n)

1N k

X(k)W

x(n)

1NNk

X(

DFTX(k)Nx(n)X(k)則DFTX(n)Nx((kN

(k)Nx(Nk x(n)N

Nk

X(k

NN

NNx((nN

(n)

k

X(k)W

(n)

Nx((k)) R

(k

x(n)X(k

mn

Rx((n))NRN(n)X((k))NRN(k)X(k

k0

Nn

x(n

k0

Nn

x(n)

X(0NN

x(0)

N X k

(k (((N

失，移位時應(yīng)先將x(n周期延拓成~(

N

x(n N~(n N

-3-2-1

~(

m)R

(n

x~(nm)

N

~(X)X知DFTx((nmN

(n)

(kN

X(kxnmNRN(n)x(nm)線性移位IDFTX((kl (k) nl 若X3(k)X1(k)X2Nx3nx1nx2n)x1mx2nmm

RN(nNx2(m)x1((nm))NRN(nm X (k)X

(k)

(kXXXX

N1

x1(m)x2(nm)N

x(n)

x(m)x((n R

m0

x2

11

~

(1(1...... ......

33~3301

~33

x1(n)x2線卷列長為5（2N-1或N+M-

1 1

(1

1 1 3n x3nx1(nx2

n x1nx2n)x1nx2n),若x(n)很長，要先將x(n)分段，再進行DFT。設(shè) x(n)分段，每段長為N點，則第K段xk(n (n)

x(n

n(k

1)

而xn)

k

xk(ny(n)x(n)h(n)xk(n)k由于xk(n與h(n)N+M-1即先要將xk(nh(n)均補零到N+M-1,因此每一段卷積的結(jié)果都多出M-1 hh(m) 卷積的后M-1點，即前后有M-1。匕匕

若X(k)

(k)

(k

N 則x(n)IDFTX(k)

*(l)x((l

證明：x(nIDFTX(k

1X(k)X

k

N1N

N N

x

(m)W

Wk01N1N

l

m

(mnl)k

(l

x

W

x1(l)x 1Nk0l1

N

N

l0

N k N1N

N

當m當m 時當m 時即mlnx3(n)

Nl

(l)x((l2

N

NNx3(n) x1(l)x2((ln))NRN(而：x3n)

m

x1mx2nm) 2

1N

X(k

x(n)

NkNl

x(l)x(l)N

Nk

X(k)X(k NNN x2(n)n

1NNk

X(k) x(n)xe(n)xox2 (n)1x(n)x2 x(n) (n)1x(n) 2x(n)xep(n)xop(n)

~(n)1n)~( 2

(n)

1n)~(2 (n)22x((n))

x

(n)1x(n)x(N (n)

xR

2 2

N x(n)x(n)

e(n)R

(n)

(n)

xep(n)和xop(n分別稱為列長為N的序列x(n)<2>對稱性質(zhì)

X(k)

則：(a)x(n)X((k ) (kmRex(n) m(d

op(k(e)

(n)RX(f

(n)

m X(m)

N

x

nk

NN

X((k

(k

Nn

x

RN(n)W

Nn

x((n))

N

nk

X(kNN

nk

1

kn1

X(k

X

RN

n0

W(d

N1m

N1 2 2

R

N

W

11

N2N

1

NNNN

n0

x

N(n)

X(k22

X(k

ReX ((

nk

()x

N

X(k)X

jIm(f

op

xn0

NReX(k)ReX((k N

N(kmm X(k) mm

R(kNNRNNX(k)X((k))NRN(k) R(k x(n)

x(n)xX(k)X((k ) X(z)在單位圓上進行等間隔取樣。設(shè)MX(k)X

j

n式中

e2x(n)

MX(k

MM

M

x(m

mk

Mk

k0m m

x(m)

MWk

(mn)k1M

mnrM(mnrM Mk

x(n

r

x(nrM

即x’(n)是原序列x(n)以M為周期的周期延拓序列對列長為N的有限長序列,頻率取樣不失真的條件：M二、X(z)X(k)--->X(z),即N個X(k如何表達整個X(z)x(n)

1NNk

X(k)W

X(Z)

N

x(n)z

N1

N

X(k

NN

z

n0

k 1N

N1

1

1N

1

N

z k

X(k)W n

k

X(k

1

zNz 1z

N

X(k:W

ejN

X(z

k

kzNN

X()

x(t)e將(,)分為無限個小段，

長T。在T即將dtTtnT

X()x()

n

x(nT)e

j X()

n

x(

)ej據(jù)時域取樣，X()是一個以s2 (即s2TT0

TNTN

1 1

F0

T0 0T20

T0F00表示非周期信號的頻譜經(jīng)離散化后樣點之間的頻率間隔離散化后k

k

N

X(k

0)n

x(n 或X(kFT

k0,1,2·Nx(n)而x(n)

1 X( 0

x(tetu(t)

X(