Në analizën e Furierit , Transformimi i Furierit në kohë diskrete (TFKD) (angl: discrete-time Fourier transform (DTFT) ) është një nga format e veçanta të analizës së Furierit . Si i tillë, ai transformon një funksion në një funksion tjetër, i cili jep paraqitjen e sinjalit (funksionit) në fushën e frekuencave fushën kohore ). Duhet thënë se TFKD kërkon si input (funksion eksitues) një funksion që është diskret
Paraqitja e TFKD-së në fushën e frekuencave është gjithmonë një funksion periodik. Meqenëse një periodë e funksionit përmban të gjithë informacionin unik të funksionit, zakonisht thuhet se TFKD-ja është transformimi tek një fushe frekuencash e cila është e fundme (me gjatësinë e një periode), në krahasim me të gjithë vijën reale.
Po te kemi nje bashkesi diskrete te numrave reale ose komplekese :
x
[
n
]
,
n
∈
Z
{\displaystyle x[n],\;n\in \mathbb {Z} }
numra të plotë ), transformimi i Furierit në kohë diskrete (ose TFKD ) i
x
[
n
]
{\displaystyle x[n]\,}
:
X
(
ω
)
=
∑
n
=
−
∞
∞
x
[
n
]
e
−
i
ω
n
.
{\displaystyle X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}.}
Lidhja me marrjen e kampioneve të funksionit (samplimin) [ redakto | redakto tekstin burimor ] Transformimi i meposhtem jep nje sekeuence diskrete ne kohe:
x
[
n
]
{\displaystyle x[n]\,}
=
1
2
π
∫
−
π
π
X
(
ω
)
⋅
e
i
ω
n
d
ω
{\displaystyle ={\frac {1}{2\pi }}\int _{-\pi }^{\pi }X(\omega )\cdot e^{i\omega n}\,d\omega }
=
T
∫
−
1
2
T
1
2
T
X
T
(
f
)
⋅
e
i
2
π
f
n
T
d
f
.
{\displaystyle =T\int _{-{\frac {1}{2T}}}^{\frac {1}{2T}}X_{T}(f)\cdot e^{i2\pi fnT}\,df.}
Integrali eshte mbi nje periode te plote te TFKD, e cila do te thote se kampionet nga x[n] jane ne te njejten kohe koeficentet e zgjerimit të serisë së Furierit te TFKD. Limitet infinite te integrimit ndryshojne transformimin ne tarnsformimin e Furierit me vazhdimësi kohore , [inversi], i se ciles prodhon nje sekuence impulsesh Diraku. Pra:
∫
−
∞
∞
X
T
(
f
)
⋅
e
i
2
π
f
t
d
f
=
∫
−
∞
∞
(
T
∑
n
=
−
∞
∞
x
(
n
T
)
e
−
i
2
π
f
T
n
)
⋅
e
i
2
π
f
t
d
f
=
∑
n
=
−
∞
∞
T
⋅
x
(
n
T
)
∫
−
∞
∞
e
−
i
2
π
f
T
n
⋅
e
i
2
π
f
t
d
f
=
∑
n
=
−
∞
∞
x
[
n
]
⋅
δ
(
t
−
n
T
)
.
{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }X_{T}(f)\cdot e^{i2\pi ft}\,df&=\int _{-\infty }^{\infty }\left(T\sum _{n=-\infty }^{\infty }x(nT)\ e^{-i2\pi fTn}\right)\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }T\cdot x(nT)\int _{-\infty }^{\infty }e^{-i2\pi fTn}\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT).\end{aligned}}}
Diferenca midis TFKD dhe transformimeve të tjera të Furierit [ redakto | redakto tekstin burimor ] TFKD eshte e kuderta e serive te Furierit , per vete faktin se kjo e fundit ka nje input periodik te vazhdueshem dhe nje spekter diskret. Aplikimet e dy transformimeve jane shume te ndryshme megjithate. TDF dhe TFKD mund te shikohen si rezultatet logjike te aplikimit te transformimit standart te Furierit tek bashkesite me informacione diskrete.
Disa çifte transformimesh te zakonshme jepen me poshte. Notacioni i meposhtem aplikohet :
n
{\displaystyle n\!}
ω
{\displaystyle \omega \!}
(
−
π
,
π
)
{\displaystyle (-\pi ,\ \pi )}
Pjesa e transformimit qe ngelet
(
|
ω
|
>
π
)
{\displaystyle (|\omega |>\pi \,)}
:
X
(
ω
+
2
π
k
)
=
X
(
ω
)
{\displaystyle X(\omega +2\pi k)=X(\omega )\,}
u
[
n
]
{\displaystyle u[n]\!}
Funksioni shkallë njësi
sinc
(
t
)
{\displaystyle \operatorname {sinc} (t)\!}
Funksioni sink i normalizuar
δ
(
ω
)
{\displaystyle \delta (\omega )\!}
Funksioni delta i Dirakut
δ
[
n
]
{\displaystyle \delta [n]\!}
Delta e Kronëkerit
δ
n
,
0
{\displaystyle \delta _{n,0}\!}
rect
(
t
)
{\displaystyle \operatorname {rect} (t)}
Funksioni drejtkëndësh për një vlerë reale arbitrare t :
r
e
c
t
(
t
)
=
⊓
(
t
)
=
{
0
if
|
t
|
>
1
2
1
2
if
|
t
|
=
1
2
1
if
|
t
|
<
1
2
{\displaystyle \mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}}
tri
(
t
)
{\displaystyle \operatorname {tri} (t)}
funksioni trekëndësh per një vlere reale arbitrare t :
tri
(
t
)
=
∧
(
t
)
=
{
1
+
t
;
−
1
≤
t
≤
0
1
−
t
;
0
<
t
≤
1
0
otherwise
{\displaystyle \operatorname {tri} (t)=\land (t)={\begin{cases}1+t;&-1\leq t\leq 0\\1-t;&0<t\leq 1\\0&{\mbox{otherwise}}\end{cases}}}
Fusha kohore
x
[
n
]
{\displaystyle x[n]\,}
Fusha e frekuencave
X
(
ω
)
{\displaystyle X(\omega )\,}
Shënime
δ
[
n
]
{\displaystyle \delta [n]\!}
1
{\displaystyle 1\!}
δ
[
n
−
M
]
{\displaystyle \delta [n-M]\!}
e
−
i
ω
M
{\displaystyle e^{-i\omega M}\!}
numer i plote M
∑
m
=
−
∞
∞
δ
[
n
−
M
m
]
{\displaystyle \sum _{m=-\infty }^{\infty }\delta [n-Mm]\,}
∑
m
=
−
∞
∞
e
−
i
ω
M
m
=
1
M
∑
k
=
−
∞
∞
δ
(
ω
2
π
−
k
M
)
{\displaystyle \sum _{m=-\infty }^{\infty }e^{-i\omega Mm}={\frac {1}{M}}\sum _{k=-\infty }^{\infty }\delta \left({\frac {\omega }{2\pi }}-{\frac {k}{M}}\right)\,}
numer i plote M
u
[
n
]
{\displaystyle u[n]\!}
1
1
−
e
−
i
ω
{\displaystyle {\frac {1}{1-e^{-i\omega }}}\!}
e
−
i
a
n
{\displaystyle e^{-ian}\!}
2
π
δ
(
ω
+
a
)
{\displaystyle 2\pi \delta (\omega +a)\,}
numer real a
cos
(
a
n
)
{\displaystyle \cos(an)\!}
π
[
δ
(
ω
−
a
)
+
δ
(
ω
+
a
)
]
{\displaystyle \pi \left[\delta (\omega -a)+\delta (\omega +a)\right]}
numer real a
sin
(
a
n
)
{\displaystyle \sin(an)\!}
π
i
[
δ
(
ω
−
a
)
−
δ
(
ω
+
a
)
]
{\displaystyle {\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]}
numer real a
r
e
c
t
[
(
n
−
M
/
2
)
M
]
{\displaystyle \mathrm {rect} \left[{(n-M/2) \over M}\right]}
sin
[
ω
(
M
+
1
)
/
2
]
sin
(
ω
/
2
)
e
−
i
ω
M
/
2
{\displaystyle {\sin[\omega (M+1)/2] \over \sin(\omega /2)}\,e^{-i\omega M/2}}
numer i plote M
sinc
[
(
a
+
n
)
]
{\displaystyle \operatorname {sinc} [(a+n)]}
e
i
a
ω
{\displaystyle e^{ia\omega }\!}
numer real a
W
⋅
sinc
2
(
W
n
)
{\displaystyle W\cdot \operatorname {sinc} ^{2}(Wn)\,}
tri
(
ω
2
π
W
)
{\displaystyle \operatorname {tri} \left({\omega \over 2\pi W}\right)}
numer real W
0
<
W
≤
0.5
{\displaystyle 0<W\leq 0.5}
W
⋅
sinc
[
W
(
n
+
a
)
]
{\displaystyle W\cdot \operatorname {sinc} [W(n+a)]}
rect
(
ω
2
π
W
)
⋅
e
j
a
ω
{\displaystyle \operatorname {rect} \left({\omega \over 2\pi W}\right)\cdot e^{ja\omega }}
numer real W , a
0
<
W
≤
1
{\displaystyle 0<W\leq 1}
{
0
n
=
0
(
−
1
)
n
n
elsewhere
{\displaystyle {\begin{cases}0&n=0\\{\frac {(-1)^{n}}{n}}&{\mbox{elsewhere}}\end{cases}}}
j
ω
{\displaystyle j\omega }
punon si nje filtër diferencues
W
(
n
+
a
)
{
cos
[
π
W
(
n
+
a
)
]
−
sinc
[
W
(
n
+
a
)
]
}
{\displaystyle {\frac {W}{(n+a)}}\left\{\cos[\pi W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}}
j
ω
⋅
rect
(
ω
π
W
)
e
j
a
ω
{\displaystyle j\omega \cdot \operatorname {rect} \left({\omega \over \pi W}\right)e^{ja\omega }}
numra reale W , a
0
<
W
≤
1
{\displaystyle 0<W\leq 1}
1
π
n
2
[
(
−
1
)
n
−
1
]
{\displaystyle {\frac {1}{\pi n^{2}}}[(-1)^{n}-1]}
|
ω
|
{\displaystyle |\omega |\!}
{
0
;
n
odd
2
π
n
;
n
even
{\displaystyle {\begin{cases}0;&n{\mbox{ odd}}\\{\frac {2}{\pi n}};&n{\mbox{ even}}\end{cases}}}
{
j
ω
<
0
0
ω
=
0
−
j
ω
>
0
{\displaystyle {\begin{cases}j&\omega <0\\0&\omega =0\\-j&\omega >0\end{cases}}}
Transformimi i Hilbertit
C
(
A
+
B
)
2
π
⋅
sinc
[
A
−
B
2
π
n
]
⋅
sinc
[
A
+
B
2
π
n
]
{\displaystyle {\frac {C(A+B)}{2\pi }}\cdot \operatorname {sinc} \left[{\frac {A-B}{2\pi }}n\right]\cdot \operatorname {sinc} \left[{\frac {A+B}{2\pi }}n\right]}
Numra realë A , B C
Kjo tabelë tregon marrëdheniet midis transformimeve të përgjithshme të Furierit në kohë diskrete. Në këtë artikull po përdorim notacionin e mëposhtëm:
Vetitë
Fusha kohore
x
[
n
]
{\displaystyle x[n]\!}
Fusha e frekuencave
X
(
ω
)
{\displaystyle X(\omega )\!}
Shënime
Lineariteti
a
x
[
n
]
+
b
y
[
n
]
{\displaystyle ax[n]+by[n]\!}
a
X
(
e
i
ω
)
+
b
Y
(
e
i
ω
)
{\displaystyle aX(e^{i\omega })+bY(e^{i\omega })\!}
Zhvendosja në kohë
x
[
n
−
k
]
{\displaystyle x[n-k]\!}
X
(
e
i
ω
)
e
−
i
ω
k
{\displaystyle X(e^{i\omega })e^{-i\omega k}\!}
numer i plote k
Zhvendosja në frekuencë (modulimi)
x
[
n
]
e
i
a
n
{\displaystyle x[n]e^{ian}\!}
X
(
e
i
(
ω
−
a
)
)
{\displaystyle X(e^{i(\omega -a)})\!}
numer real a
Pasqyrimi kohor
x
[
−
n
]
{\displaystyle x[-n]\!}
X
(
e
−
i
ω
)
{\displaystyle X(e^{-i\omega })\!}
Konjugimi kohor
x
[
n
]
∗
{\displaystyle x[n]^{*}\!}
X
(
e
−
i
ω
)
∗
{\displaystyle X(e^{-i\omega })^{*}\!}
Pasqyrimi kohor & konjugimi
x
[
−
n
]
∗
{\displaystyle x[-n]^{*}\!}
X
(
e
i
ω
)
∗
{\displaystyle X(e^{i\omega })^{*}\!}
Derivati në frekuencë
n
i
x
[
n
]
{\displaystyle {\frac {n}{i}}x[n]\!}
d
X
(
e
i
ω
)
d
ω
{\displaystyle {\frac {dX(e^{i\omega })}{d\omega }}\!}
Integrali në frekuencë
i
n
x
[
n
]
{\displaystyle {\frac {i}{n}}x[n]\!}
∫
−
π
ω
X
(
e
i
ϑ
)
d
ϑ
{\displaystyle \int _{-\pi }^{\omega }X(e^{i\vartheta })d\vartheta \!}
Konvulimi në kohë
x
[
n
]
∗
y
[
n
]
{\displaystyle x[n]*y[n]\!}
X
(
e
i
ω
)
⋅
Y
(
e
i
ω
)
{\displaystyle X(e^{i\omega })\cdot Y(e^{i\omega })\!}
Shumëzimi në kohë
x
[
n
]
⋅
y
[
n
]
{\displaystyle x[n]\cdot y[n]\!}
1
2
π
X
(
e
i
ω
)
∗
Y
(
e
i
ω
)
{\displaystyle {\frac {1}{2\pi }}X(e^{i\omega })*Y(e^{i\omega })\!}
Korrelacioni
ρ
x
y
[
n
]
=
x
[
−
n
]
∗
∗
y
[
n
]
{\displaystyle \rho _{xy}[n]=x[-n]^{*}*y[n]\!}
R
x
y
(
ω
)
=
X
(
e
i
ω
)
∗
⋅
Y
(
e
i
ω
)
{\displaystyle R_{xy}(\omega )=X(e^{i\omega })^{*}\cdot Y(e^{i\omega })\!}
Transformimi i Furierit mund të dekompozohet në pjesën reale dhe imagjinare ose në pjesë çift dhe tek.
X
(
e
i
ω
)
=
X
R
(
e
i
ω
)
+
i
X
I
(
e
i
ω
)
{\displaystyle X(e^{i\omega })=X_{R}(e^{i\omega })+iX_{I}(e^{i\omega })\!}
X
(
e
i
ω
)
=
X
E
(
e
i
ω
)
+
X
O
(
e
i
ω
)
{\displaystyle X(e^{i\omega })=X_{E}(e^{i\omega })+X_{O}(e^{i\omega })\!}
Fusha kohore
x
[
n
]
{\displaystyle x[n]\!}
Fusha e frekuencave
X
(
e
i
ω
)
{\displaystyle X(e^{i\omega })\!}
x
∗
[
n
]
{\displaystyle x^{*}[n]\!}
X
∗
(
e
−
i
ω
)
{\displaystyle X^{*}(e^{-i\omega })\!}
x
∗
[
−
n
]
{\displaystyle x^{*}[-n]\!}
X
∗
(
e
i
ω
)
{\displaystyle X^{*}(e^{i\omega })\!}
Alan V. Oppenheim and Ronald W. Schafer (1999). Discrete-Time Signal Processing (2nd Edition ed.). Prentice Hall Signal Processing Series. ISBN 0-13-754920-2 . William McC. Siebert (1986). Circuits, Signals, and Systems . MIT Electrical Engineering and Computer Science Series. Cambridge, MA: MIT Press. Boaz Porat. A Course in Digital Signal Processing . John Wiley and Sons. pp. pp. 27–29 and 104–105. ISBN 0-471-14961-6 .
![{\displaystyle x[n],\;n\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b783e3bb4f41250a3bce26399b1885f0e2c975)
![{\displaystyle x[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b575dce543a2cf10a5a3e108204b928c2c9aaa54)
![{\displaystyle X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c9ad19546738b0bdd7606437609877a442b1ce)
![{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }X_{T}(f)\cdot e^{i2\pi ft}\,df&=\int _{-\infty }^{\infty }\left(T\sum _{n=-\infty }^{\infty }x(nT)\ e^{-i2\pi fTn}\right)\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }T\cdot x(nT)\int _{-\infty }^{\infty }e^{-i2\pi fTn}\cdot e^{i2\pi ft}\,df\\&=\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/decfa7832a2d257a08b027c3605593b3e80ab57f)
![{\displaystyle u[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2911183bef1ad935d3bc8b5c1be97bac439b7f)
![{\displaystyle \delta [n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1553d02b03c2a79a43f7862ebfb8352705b0b87c)
![{\displaystyle \mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3585cde90bc1dfbce7b14531690022ad0a7b3a6)
![{\displaystyle \delta [n-M]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd610c46bdeb2682e719b6a0b445c1cf1893639)
![{\displaystyle \sum _{m=-\infty }^{\infty }\delta [n-Mm]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ce55d69101c8a851f1b8f09c0492852e8e9e49)
![{\displaystyle \pi \left[\delta (\omega -a)+\delta (\omega +a)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b949fed13f25129a9f8634cb4c0aa34354b0320a)
![{\displaystyle {\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b48e8676730fb37bcadaea9b1bb342d107cf2f9)
![{\displaystyle \mathrm {rect} \left[{(n-M/2) \over M}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa40f5391166dc137b30a45584c1de1a33b2f8c)
![{\displaystyle {\sin[\omega (M+1)/2] \over \sin(\omega /2)}\,e^{-i\omega M/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/883ef01f74f9697fd3a68706785bc6e702c8f961)
![{\displaystyle \operatorname {sinc} [(a+n)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3526652071fa8be34f6bccac88dc94260441a47f)
![{\displaystyle W\cdot \operatorname {sinc} [W(n+a)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4185f56193a63b426f87329e9d7c0d1ae50fced)
![{\displaystyle {\frac {W}{(n+a)}}\left\{\cos[\pi W(n+a)]-\operatorname {sinc} [W(n+a)]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d4cd8e41d9e483f5d90da18cff3661d914ce9b)
![{\displaystyle {\frac {1}{\pi n^{2}}}[(-1)^{n}-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4114640d8ab7e22e07fe380574400fb036941530)
![{\displaystyle {\frac {C(A+B)}{2\pi }}\cdot \operatorname {sinc} \left[{\frac {A-B}{2\pi }}n\right]\cdot \operatorname {sinc} \left[{\frac {A+B}{2\pi }}n\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38083aff1d4f22b4848dfafdc988b43142f7c472)
![{\displaystyle x[n]^{*}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49908fe3dfbd435e478d58b4c10c47e3a609c96)
![{\displaystyle \rho _{xy}[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/358049d59c74edde52765a758324ab97be7ee3a5)
![{\displaystyle x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91384637c5188ffed9b7929f145a78fb314c4141)
![{\displaystyle ax[n]+by[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c30c8f75093b9bafb5c5bde1392348701ce3be0)
![{\displaystyle x[n-k]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bfc2b24164de1192ea1e17a90312a3e045911e)
![{\displaystyle x[n]e^{ian}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f540caf14c269c45b6b5d32ceb555ce8841bae)
![{\displaystyle x[-n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61499a6c6544d338cf6ac340b27ad734c4daba05)
![{\displaystyle x[-n]^{*}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb208f439fc83acd99c3db8d63304b65890993c4)
![{\displaystyle {\frac {n}{i}}x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2eb71af662480e119b9bd16ff2e4c49151ab4b9)
![{\displaystyle {\frac {i}{n}}x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/384ec2e6f1fb841111d00c800196000a4d406964)
![{\displaystyle x[n]*y[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3b1fd5ad225be66dfa022f43a3333b08fb3fb7)
![{\displaystyle x[n]\cdot y[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e208ac041ff192194184d3c708d7789f9c9178)
![{\displaystyle \rho _{xy}[n]=x[-n]^{*}*y[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37d82c9624d4b16650c5652c000d0bb459f794c6)
![{\displaystyle x^{*}[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4793a5489c4d67f5ffcc700e9ba06acd87e231c3)
![{\displaystyle x^{*}[-n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34e0bcbaa111fee79bfcf34b83f9f3b6303ddaed)
