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 | përpunoni burim ] 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 | përpunoni burim ] 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.
Tabele e transformimeve te Furierit në kohë diskrete [ redakto | përpunoni burim ] 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 (bot. 2nd Edition). 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. f. 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)
