In this section the DTFT transform of a number of interested signals are given.
(a) Unit sample
δ(n)δ(n) size 12{δ \( n \) } {}
X(ω)=∑n=−∞∞δ(n)e−jωn=e−jω0=1X(ω)=∑n=−∞∞δ(n)e−jωn=e−jω0=1 size 12{X \( ω \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( n \) e rSup { size 8{ - jωn} } } =e rSup { size 8{ - jω0} } =1} {}
(1)
This is very special, the transform has real amplitude of 1 and zero phase at all frequencies. Since the transform is
2π2π size 12{2π} {}-periodic, all frequencies just means a period
[−π,π][−π,π] size 12{ \[ - π,π \] } {}.
(b) Delayed unit sample
δ(n−n0)δ(n−n0) size 12{δ \( n - n rSub { size 8{0} } \) } {}
The transform is
X(ω)=∑n=−∞∞δ(n−n0)e−jωn=e−jωn0X(ω)=∑n=−∞∞δ(n−n0)e−jωn=e−jωn0 size 12{X \( ω \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {δ \( n - n rSub { size 8{0} } \) e rSup { size 8{ - jωn} } } =e rSup { size 8{ - jωn rSub { size 6{0} } } } } {}
(2)
The magnitude and phase spectra are respectively
|X(ω)|=|
e
−jω
n
0
|=1
Φ(ω) ∠
e
−jω
n
0
=−ω
n
0
|X(ω)|=|
e
−jω
n
0
|=1
Φ(ω) ∠
e
−jω
n
0
=−ω
n
0
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaaiiFaiaadIfacaGGOaGaeqyYdCNaaiykaiaacYhacqGH9aqpcaGG8bGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3jaad6gadaWgaaadbaGaaGimaaqabaaaaOGaaiiFaiabg2da9iaaigdaaeaacqqHMoGrcaGGOaGaeqyYdCNaaiykaiaaysW7cqGHGic0caaMe8UaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3jaad6gadaWgaaadbaGaaGimaaqabaaaaOGaeyypa0JaeyOeI0IaeqyYdCNaamOBamaaBaaaleaacaaIWaaabeaaaaaa@5D0E@
The phase is proportional to frequency, but remember that the phase spectrum is understood to stay within the range
[−π,π][−π,π] size 12{ \[ - π,π \] } {}.
(c) Decaying exponential
x(n)=
a
n
u(n), |a| <1
x(n)=
a
n
u(n), |a| <1
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWGHbWaaWbaaSqabeaacaWGUbaaaOGaamyDaiaacIcacaWGUbGaaiykaiaacYcacaaMf8UaaGzbVlaacYhacaWGHbGaaiiFaiaaysW7cqGH8aapcaaIXaaaaa@49A7@
The transform is
X(ω)=
∑
n=−∞
∞
a
n
u(n)
e
−jωn
=
∑
n=0
∞
a
n
e
−jωn
X(ω)=
∑
n=−∞
∞
a
n
u(n)
e
−jωn
=
∑
n=0
∞
a
n
e
−jωn
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaiabg2da9maaqahabaGaamyyamaaCaaaleqabaGaamOBaaaakiaadwhacaGGOaGaamOBaiaacMcacaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqyYdCNaamOBaaaaaeaacaWGUbGaeyypa0JaeyOeI0IaeyOhIukabaGaeyOhIukaniabggHiLdGccqGH9aqpdaaeWbqaaiaadggadaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoakiaadwgadaahaaWcbeqaaiabgkHiTiaadQgacqaHjpWDcaWGUbaaaaaa@5D43@
Resulting in
X(
ω
)=
1
1−a
e
−jω
X(
ω
)=
1
1−a
e
−jω
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfadaqadaqaaiabeM8a3bGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcaWGHbGaamyzamaaCaaaleqabaGaeyOeI0IaamOAaiabeM8a3baaaaaaaa@4334@
(3)
For a = 1 we have the unit step u(n). The unit step does not satisfy the existence condition
Equation hence, in principle, has no DTFT. However there is this function
X(ω)=
1
1−
e
−jω
+ πδ(ω)
X(ω)=
1
1−
e
−jω
+ πδ(ω)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIfacaGGOaGaeqyYdCNaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHsislcaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqyYdChaaaaakiaaysW7caaMe8Uaey4kaSIaaGjbVlabec8aWjabes7aKjaacIcacqaHjpWDcaGGPaaaaa@4E39@
(4)
whose inverse transform is u(n)
(d) Symmetric rectangular pulse
The digital pulse (see
Example) consists of 2N + 1 samples of amplitude 1:
x
(
n
)
=
1
,
−
N
≤
n
≤
N
0
,
otherwise
x
(
n
)
=
1
,
−
N
≤
n
≤
N
0
,
otherwise
alignl { stack {
size 12{x \( n \) =1 matrix {
{} # {}
} , matrix {
{} # {}
} - N <= n <= N} {} #
size 12{ matrix {
{} # {} # {}
} 0 matrix {
{} # {}
} , matrix {
{} # {}
} ital "otherwise"} {}
} } {}
X(ω)=1+2∑n=1NcosnωX(ω)=1+2∑n=1Ncosnω size 12{X \( ω \) =1+2 Sum cSub { size 8{n=1} } cSup { size 8{N} } {"cos"nω} } {}
(5)
With relation (
Equation) this result can be put in another the form of a radio of two simusoidal functions.
The magnitude spectrum ansists of the main lobe having peak value of 2N + 1 and decreasing sidelobes. The origin and the zero-crossing points are separated evenly with distance of
2π/(2N+1)2π/(2N+1) size 12{2π/ \( 2N+1 \) } {} (
Figure).
(e) Complex exponential and sine, cosine
The complex exponential is
x
(
n
)
=
e
jω
0
n
−
∞
<
n
<
∞
x
(
n
)
=
e
jω
0
n
−
∞
<
n
<
∞
size 12{x \( n \) =e rSup { size 8{jω rSub { size 6{0} } n} } matrix {
{} # {} # {}
} - infinity <n< infinity } {}
Its DTFT transform is
X(ω)=2π∑k=−∞∞δω−ω0−2πkX(ω)=2π∑k=−∞∞δω−ω0−2πk size 12{X \( ω \) =2π Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {δ left (ω - ω rSub { size 8{0} } - 2πk right )} } {}
(6)
The cosin and sine have transforms
cos
ω
0
n↔πδ(ω−
ω
0
)+πδ(ω+
ω
0
)
cos
ω
0
n↔πδ(ω−
ω
0
)+πδ(ω+
ω
0
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacogacaGGVbGaai4CaiabeM8a3naaBaaaleaacaaIWaaabeaakiaad6gacqGHugYQcqaHapaCcqaH0oazcaGGOaGaeqyYdCNaeyOeI0IaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaaiykaiabgUcaRiabec8aWjabes7aKjaacIcacqaHjpWDcqGHRaWkcqaHjpWDdaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@538F@
(7)
sin
ω
0
n↔−jπδ(ω−
ω
0
)+jπδ(ω+
ω
0
)
sin
ω
0
n↔−jπδ(ω−
ω
0
)+jπδ(ω+
ω
0
)
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaaIWaaabeaakiaad6gacqGHugYQcqGHsislcaWGQbGaeqiWdaNaeqiTdqMaaiikaiabeM8a3jabgkHiTiabeM8a3naaBaaaleaacaaIWaaabeaakiaacMcacqGHRaWkcaWGQbGaeqiWdaNaeqiTdqMaaiikaiabeM8a3jabgUcaRiabeM8a3naaBaaaleaacaaIWaaabeaakiaacMcaaaa@565F@
(8)
