In matematica , la trasformata di Fourier a tempo discreto , spesso abbreviata con DTFT (acronimo del termine inglese Discrete-Time Fourier Transform ), è una trasformata che a partire da un segnale discreto ne fornisce una descrizione periodica nel dominio della frequenza , analogamente alla trasformata di Fourier tradizionale (definita per funzioni continue).
Si tratta di un caso particolare della trasformata zeta :
X
(
z
)
=
∑
n
=
−
∞
∞
x
[
n
]
z
−
n
{\displaystyle X(z)=\sum _{n=-\infty }^{\infty }x[n]\,z^{-n}}
che si ottiene ponendo
z
=
e
i
ω
{\displaystyle z=e^{i\omega }\,}
ω
{\displaystyle \omega }
|
e
i
ω
|
=
1
{\displaystyle |e^{i\omega }|=1\,}
piano complesso .
La trasformata di Fourier a tempo discreto ha un ruolo rilevante quando si studiano segnali campionati, ovvero segnali a tempo discreto ottenuti da un segnale a tempo continuo considerandone il valore assunto in precisi istanti di tempo, solitamente separati da un intervallo temporale fisso
T
{\displaystyle T}
campionamento , ed è alla base della conversione analogico-digitale (ADC). Essa trasforma una funzione continua
x
(
t
)
{\displaystyle x(t)}
x
[
n
]
=
def
x
(
n
T
)
∀
n
∈
Z
{\displaystyle x[n]\ {\stackrel {\text{def}}{=}}\ x(nT)\qquad \forall \,n\in \mathbb {Z} }
con
1
/
T
{\displaystyle 1/T}
frequenza di campionamento . Il teorema del campionamento pone un limite alla massima frequenza del segnale continuo , che non può essere superiore ad
1
/
(
2
T
)
{\displaystyle 1/(2T)}
aliasing ). La trasformata a tempo discreto fornisce un'approssimazione della trasformata di Fourier
F
{\displaystyle {\mathcal {F}}}
F
(
x
(
t
)
)
(
f
)
=
X
(
f
)
=
∫
−
∞
∞
x
(
t
)
⋅
e
−
i
2
π
f
t
d
t
{\displaystyle {\mathcal {F}}(x(t))(f)=X(f)=\int _{-\infty }^{\infty }x(t)\cdot e^{-i2\pi ft}dt}
Infatti, considerando la formula di sommazione di Poisson , che mostra come ottenere una sommazione periodica di una funzione
X
(
f
)
{\displaystyle X(f)}
X
1
/
T
(
f
)
=
d
e
f
∑
k
=
−
∞
∞
X
(
f
−
k
/
T
)
≡
∑
n
=
−
∞
∞
T
⋅
x
(
n
T
)
⏟
x
[
n
]
e
−
i
2
π
f
T
n
⏟
F
{
δ
(
t
−
n
T
)
}
=
F
{
∑
n
=
−
∞
∞
x
(
t
)
⋅
δ
(
t
−
n
T
)
}
{\displaystyle X_{1/T}(f)\ {\stackrel {\mathrm {def} }{=}}\sum _{k=-\infty }^{\infty }X\left(f-k/T\right)\equiv \sum _{n=-\infty }^{\infty }\underbrace {T\cdot x(nT)} _{x[n]}\ \underbrace {e^{-i2\pi fTn}} _{{\mathcal {F}}\{\delta (t-nT)\}}\ =\ {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }x(t)\cdot \delta (t-nT)\right\}}
dove
X
1
/
T
(
f
)
{\displaystyle \scriptstyle X_{1/T}(f)\,}
X
(
f
)
{\displaystyle \scriptstyle X(f)\,}
f
s
{\displaystyle f_{s}}
f
s
{\displaystyle f_{s}}
K
=
0
{\displaystyle K=0}
[
−
f
s
/
2
,
f
s
/
2
]
{\displaystyle [-f_{s}/2,f_{s}/2]}
F
{
∑
n
=
−
∞
∞
T
⋅
x
(
n
T
)
⋅
δ
(
t
−
n
T
)
}
=
F
{
x
(
t
)
⋅
T
∑
n
=
−
∞
∞
δ
(
t
−
n
T
)
}
=
X
(
f
)
∗
F
{
T
∑
n
=
−
∞
∞
δ
(
t
−
n
T
)
}
⏟
∑
k
=
−
∞
∞
δ
(
f
−
k
/
T
)
=
∑
k
=
−
∞
∞
X
(
f
−
k
T
)
{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }T\cdot x(nT)\cdot \delta (t-nT)\right\}&={\mathcal {F}}\left\{x(t)\cdot T\sum _{n=-\infty }^{\infty }\delta (t-nT)\right\}\\&=X(f)*\underbrace {{\mathcal {F}}\left\{T\sum _{n=-\infty }^{\infty }\delta (t-nT)\right\}} _{\sum _{k=-\infty }^{\infty }\delta (f-k/T)}=\sum _{k=-\infty }^{\infty }X\left(f-{\frac {k}{T}}\right)\end{aligned}}}
Calcolando la trasformata di Fourier inversa di entrambi i membri dell'equazione precedente, inoltre, si ottiene il pettine di Dirac modulato:
∑
n
=
−
∞
∞
x
[
n
]
⋅
δ
(
t
−
n
T
)
=
F
−
1
{
X
1
/
T
(
f
)
}
=
d
e
f
∫
−
∞
∞
X
1
/
T
(
f
)
⋅
e
i
2
π
f
t
d
f
.
{\displaystyle \sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)={\mathcal {F}}^{-1}\left\{X_{1/T}(f)\right\}\ {\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }X_{1/T}(f)\cdot e^{i2\pi ft}df.}
con:
x
[
n
]
=
T
∫
1
T
X
1
/
T
(
f
)
⋅
e
i
2
π
f
n
T
d
f
=
1
2
π
∫
2
π
X
(
ω
)
⋅
e
i
ω
n
d
ω
{\displaystyle x[n]=T\int _{\frac {1}{T}}X_{1/T}(f)\cdot e^{i2\pi fnT}df={\frac {1}{2\pi }}\int _{2\pi }X(\omega )\cdot e^{i\omega n}d\omega }
Se la successione in ingresso
x
[
n
]
{\displaystyle x[n]}
serie di Fourier , ottenendo la trasformata discreta di Fourier (DFT):
F
{
∑
n
=
−
∞
∞
x
[
n
]
⋅
δ
(
t
−
n
T
)
}
=
F
{
∑
k
=
−
∞
∞
X
[
k
]
⋅
e
i
2
π
k
N
T
t
⏟
F
o
u
r
i
e
r
s
e
r
i
e
s
}
=
∑
k
=
−
∞
∞
X
[
k
]
⋅
δ
(
f
−
k
N
T
)
⏟
D
T
F
T
{\displaystyle {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right\}={\mathcal {F}}\left\{\underbrace {\sum _{k=-\infty }^{\infty }X[k]\cdot e^{i2\pi {\frac {k}{NT}}t}} _{Fourier\ series}\right\}=\underbrace {\sum _{k=-\infty }^{\infty }X[k]\ \cdot \ \delta \left(f-{\frac {k}{NT}}\right)} _{DTFT}}
Tale relazione mostra che la periodicità nel tempo rende discontinua la trasformata di Fourier a tempo discreto. Si può tuttavia ridurre la formula integrale in una somma di N termini:
X
[
k
]
=
def
1
N
T
∫
N
T
[
∑
n
=
−
∞
∞
x
[
n
]
⋅
δ
(
t
−
n
T
)
]
e
−
i
2
π
k
N
T
t
d
t
=
1
N
T
∑
n
=
−
∞
∞
x
[
n
]
⋅
∫
N
T
δ
(
t
−
n
T
)
⋅
e
−
i
2
π
k
N
T
t
d
t
{\displaystyle X[k]{\stackrel {\text{def}}{=}}\ {\frac {1}{NT}}\int _{NT}\left[\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right]e^{-i2\pi {\frac {k}{NT}}t}dt={\frac {1}{NT}}\sum _{n=-\infty }^{\infty }x[n]\cdot \int _{NT}\delta (t-nT)\cdot e^{-i2\pi {\frac {k}{NT}}t}dt}
=
1
N
T
∑
N
x
[
n
]
⋅
e
−
i
2
π
k
N
n
⏟
D
F
T
=
1
N
∑
N
x
(
n
T
)
⋅
e
−
i
2
π
k
N
n
⏟
D
F
T
{\displaystyle ={\frac {1}{NT}}\underbrace {\sum _{N}x[n]\cdot e^{-i2\pi {\frac {k}{N}}n}} _{DFT}={\frac {1}{N}}\underbrace {\sum _{N}x(nT)\cdot e^{-i2\pi {\frac {k}{N}}n}} _{DFT}}
che è periodica in
k
{\displaystyle k}
Se la trasformata di Fourier a tempo discreto è una funzione continua , si usa spesso considerare un numero arbitrario di campioni di un ciclo della funzione periodica
X
1
/
T
{\displaystyle X_{1/T}}
X
1
/
T
(
k
N
T
)
⏟
X
k
=
∑
n
=
−
∞
∞
x
[
n
]
⋅
e
−
i
2
π
k
n
N
=
∑
N
x
N
[
n
]
⋅
e
−
i
2
π
k
n
N
⏟
D
F
T
k
=
0
,
…
,
N
−
1
{\displaystyle \underbrace {X_{1/T}\left({\frac {k}{NT}}\right)} _{X_{k}}=\sum _{n=-\infty }^{\infty }x[n]\cdot e^{-i2\pi {\frac {kn}{N}}}=\underbrace {\sum _{N}x_{N}[n]\cdot e^{-i2\pi {\frac {kn}{N}}}} _{DFT}\qquad \quad k=0,\dots ,N-1}
dove
X
N
{\displaystyle X_{N}}
x
N
[
n
]
=
def
∑
m
=
−
∞
∞
x
[
n
−
m
N
]
{\displaystyle x_{N}[n]\ {\stackrel {\text{def}}{=}}\ \sum _{m=-\infty }^{\infty }x[n-mN]}
La successione
x
N
{\displaystyle x_{N}}
trasformata discreta di Fourier . In questo modo il campionamento così effettuato comporta che la trasformata inversa sia periodica.
Per valutare numericamente un ciclo di
x
N
{\displaystyle x_{N}}
x
[
n
]
{\displaystyle x[n]}
funzione finestra di lunghezza opportuna.[1] [2]
Siano
n
{\displaystyle n}
ω
{\displaystyle \omega }
numero reale in
(
−
π
,
π
)
{\displaystyle (-\pi ,\ \pi )}
u
[
n
]
{\displaystyle u[n]}
gradino di Heaviside tempo-discreto,
sinc
(
t
)
{\displaystyle \operatorname {sinc} (t)}
sinc normalizzata,
δ
(
ω
)
{\displaystyle \delta (\omega )}
delta di Dirac ,
δ
[
n
]
{\displaystyle \delta [n]}
delta di Kronecker ,
rect
(
t
)
{\displaystyle \operatorname {rect} (t)}
funzione rettangolo :
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}}}
e
tri
(
t
)
{\displaystyle \operatorname {tri} (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}}}
Dominio temporale
x
[
n
]
{\displaystyle x[n]\,}
Dominio della frequenza
X
(
ω
)
{\displaystyle X(\omega )\,}
Remarks
δ
[
n
]
{\displaystyle \delta [n]\!}
1
{\displaystyle 1\!}
δ
[
n
−
M
]
{\displaystyle \delta [n-M]\!}
e
−
i
ω
M
{\displaystyle e^{-i\omega M}\!}
M
{\displaystyle 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)\,}
M
{\displaystyle M}
u
[
n
]
{\displaystyle u[n]\!}
1
1
−
e
−
i
ω
+
∑
k
=
−
∞
∞
π
δ
(
ω
−
2
π
k
)
{\displaystyle {\frac {1}{1-e^{-i\omega }}}+\sum \limits _{k=-\infty }^{\infty }{\pi \delta (\omega -2\pi k)}\!}
Il termine
1
/
(
1
−
e
−
i
ω
)
{\displaystyle 1/(1-e^{-i\omega })}
distribuzione .
a
n
u
[
n
]
{\displaystyle a^{n}u[n]\!}
1
1
−
a
e
−
i
ω
{\displaystyle {\frac {1}{1-ae^{-i\omega }}}\!}
0
<
|
a
|
<
1
{\displaystyle 0<|a|<1}
e
−
i
a
n
{\displaystyle e^{-ian}\!}
2
π
δ
(
ω
+
a
)
{\displaystyle 2\pi \delta (\omega +a)\,}
a
∈
R
{\displaystyle a\in \mathbb {R} }
cos
(
a
n
)
{\displaystyle \cos(an)\!}
π
[
δ
(
ω
−
a
)
+
δ
(
ω
+
a
)
]
{\displaystyle \pi \left[\delta (\omega -a)+\delta (\omega +a)\right]}
a
∈
R
{\displaystyle a\in \mathbb {R} }
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]}
a
∈
R
{\displaystyle a\in \mathbb {R} }
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}\!}
M
{\displaystyle M}
sinc
[
(
a
+
n
)
]
{\displaystyle \operatorname {sinc} [(a+n)]}
e
i
a
ω
{\displaystyle e^{ia\omega }\!}
a
∈
R
{\displaystyle a\in \mathbb {R} }
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)}
0
<
W
≤
0.5
∈
R
{\displaystyle 0<W\leq 0.5\quad \in \mathbb {R} }
W
⋅
sinc
(
W
n
)
{\displaystyle W\cdot \operatorname {sinc} (Wn)}
rect
(
ω
2
π
W
)
{\displaystyle \operatorname {rect} \left({\omega \over 2\pi W}\right)}
0
<
W
≤
1
∈
R
{\displaystyle 0<W\leq 1\quad \in \mathbb {R} }
{
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 }
Filtro differenziatore
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 }}
0
<
W
≤
1
∈
R
a
∈
R
{\displaystyle 0<W\leq 1\quad \in \mathbb {R} \qquad a\in \mathbb {R} }
1
π
n
2
[
(
−
1
)
n
−
1
]
{\displaystyle {\frac {1}{\pi n^{2}}}[(-1)^{n}-1]\!}
|
ω
|
{\displaystyle |\omega |\!}
{
0
;
n
even
2
π
n
;
n
odd
{\displaystyle {\begin{cases}0;&n{\mbox{ even}}\\{\frac {2}{\pi n}};&n{\mbox{ odd}}\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}}}
Trasformata di Hilbert
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]}
A
,
B
∈
R
B
∈
C
{\displaystyle A,B\in R\quad B\in \mathbb {C} }
^
Charles Constantine Gumas, Window-presum FFT achieves high-dynamic range, resolution , in Personal Engineering & Instrumentation News , luglio 1997, pp. 58–64.
^
Richard G. Lyons, DSP Tricks: Building a practical spectrum analyzer eetimes.com , EE Times, giugno 2008.
Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing ISBN 0-13-754920-2 . William McC. Siebert, Circuits, Signals, and Systems , MIT Electrical Engineering and Computer Science Series. Cambridge, MA, MIT Press, 1986. Boaz Porat, A Course in Digital Signal Processing ISBN 0-471-14961-6 . 
![{\displaystyle X(z)=\sum _{n=-\infty }^{\infty }x[n]\,z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a599aa526823646d32469385b73490b60e28ea7)
![{\displaystyle x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d)
![{\displaystyle X(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/553acaf0fb714decb09013e12275ca65bed638d7)
![{\displaystyle x[n]={\frac {1}{2\pi }}\int _{2\pi }X(\omega )\cdot e^{i\omega n}d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdec9f866497246173ad9c633f12839a4a019ca7)
![{\displaystyle x[n]\ {\stackrel {\text{def}}{=}}\ x(nT)\qquad \forall \,n\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/565f01b488cc520b3aecdf0a1d58370501a21cbd)
![{\displaystyle X_{1/T}(f)\ {\stackrel {\mathrm {def} }{=}}\sum _{k=-\infty }^{\infty }X\left(f-k/T\right)\equiv \sum _{n=-\infty }^{\infty }\underbrace {T\cdot x(nT)} _{x[n]}\ \underbrace {e^{-i2\pi fTn}} _{{\mathcal {F}}\{\delta (t-nT)\}}\ =\ {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }x(t)\cdot \delta (t-nT)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/361a8cddde0bb7b75c7134e2376ba760b2dadcdb)
![{\displaystyle [-f_{s}/2,f_{s}/2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b709f5f5354894fddd10bd415574d7f9483d742)
![{\displaystyle \sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)={\mathcal {F}}^{-1}\left\{X_{1/T}(f)\right\}\ {\stackrel {\mathrm {def} }{=}}\int _{-\infty }^{\infty }X_{1/T}(f)\cdot e^{i2\pi ft}df.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67126209df0c938694ada453169eb06f1cfd05a2)
![{\displaystyle x[n]=T\int _{\frac {1}{T}}X_{1/T}(f)\cdot e^{i2\pi fnT}df={\frac {1}{2\pi }}\int _{2\pi }X(\omega )\cdot e^{i\omega n}d\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a31004995be0386670d4d29f9b58f1d3322a79b7)
![{\displaystyle {\mathcal {F}}\left\{\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right\}={\mathcal {F}}\left\{\underbrace {\sum _{k=-\infty }^{\infty }X[k]\cdot e^{i2\pi {\frac {k}{NT}}t}} _{Fourier\ series}\right\}=\underbrace {\sum _{k=-\infty }^{\infty }X[k]\ \cdot \ \delta \left(f-{\frac {k}{NT}}\right)} _{DTFT}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63d78c24d3d7aac90cdb8429638024c24ccba9c3)
![{\displaystyle X[k]{\stackrel {\text{def}}{=}}\ {\frac {1}{NT}}\int _{NT}\left[\sum _{n=-\infty }^{\infty }x[n]\cdot \delta (t-nT)\right]e^{-i2\pi {\frac {k}{NT}}t}dt={\frac {1}{NT}}\sum _{n=-\infty }^{\infty }x[n]\cdot \int _{NT}\delta (t-nT)\cdot e^{-i2\pi {\frac {k}{NT}}t}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f09ef0aa3f461cd85205eaffd860e8eebef4bef)
![{\displaystyle ={\frac {1}{NT}}\underbrace {\sum _{N}x[n]\cdot e^{-i2\pi {\frac {k}{N}}n}} _{DFT}={\frac {1}{N}}\underbrace {\sum _{N}x(nT)\cdot e^{-i2\pi {\frac {k}{N}}n}} _{DFT}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a10ba17e54c790564936b4b64349e619310a0c59)
![{\displaystyle \underbrace {X_{1/T}\left({\frac {k}{NT}}\right)} _{X_{k}}=\sum _{n=-\infty }^{\infty }x[n]\cdot e^{-i2\pi {\frac {kn}{N}}}=\underbrace {\sum _{N}x_{N}[n]\cdot e^{-i2\pi {\frac {kn}{N}}}} _{DFT}\qquad \quad k=0,\dots ,N-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f7f4d0909123bde1a57ef6745469c9cbe7e812)
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![{\displaystyle u[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2911183bef1ad935d3bc8b5c1be97bac439b7f)
![{\displaystyle a^{n}u[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e544c0a61b9fba33dac991628cf707c2d38ad1eb)
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![{\displaystyle {\frac {\pi }{i}}\left[\delta (\omega -a)-\delta (\omega +a)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b48e8676730fb37bcadaea9b1bb342d107cf2f9)
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![{\displaystyle {\frac {1}{\pi n^{2}}}[(-1)^{n}-1]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b97d9986fb5b086ebd0c5302cfbd88c6e4d9e236)
![{\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 x[n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91384637c5188ffed9b7929f145a78fb314c4141)
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![{\displaystyle x[n-k]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bfc2b24164de1192ea1e17a90312a3e045911e)
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![{\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 E=\sum _{n=-\infty }^{\infty }{x[n]\cdot y^{*}[n]}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4441b0f623efac9e5784561b56bb587551ffd43d)
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![{\displaystyle x^{*}[-n]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34e0bcbaa111fee79bfcf34b83f9f3b6303ddaed)
