−
tan
−
1
(
x
˜
i
(
ϕ
)
x
˜
r
(
ϕ
)
)
for
{
X
˜
−
π
−
tan
−
1
(
x
˜
i
(
ϕ
)
x
˜
r
(
ϕ
)
)
for
{
X
˜
∠
X
˜
(
−
ϕ
)
=
{
r
(
ϕ
)
>
0
size 12{∠ { tilde {X}} \( - ϕ \) =alignl { stack {
left lbrace - "tan" rSup { size 8{ - 1} } \( { { { tilde {x}} rSub { size 8{i} } \( ϕ \) } over { { tilde {x}} rSub { size 8{r} } \( ϕ \) } } \) " " ital "for"" {" tilde ital {X}} rSub { size 8{r} } \( ϕ \)>0 {} #
right none left lbrace - π - "tan" rSup { size 8{ - 1} } \( { { { tilde {x}} rSub { size 8{i} } \( ϕ \) } over { { tilde {x}} rSub { size 8{r} } \( ϕ \) } } \) " " ital "for"" {" tilde ital {X}} rSub { size 8{r} } \( ϕ \)<0 {} #
right no } } lbrace } {}
Therefore,
X
˜
size 12{ { tilde {X}}} {} (φ) is an odd function of φ.
c/ Time shift
{}
x
[
n
−
n
0
]
⇔
F
X
˜
(
ϕ
)
e
−
j2
πϕ
n
0
size 12{x \[ n - n rSub { size 8{0} } \] { size 24{ dlrarrow } } cSup { size 8{F} } { tilde {X}} \( ϕ \) e rSup { size 8{ - j2 ital "πϕ"n rSub { size 6{0} } } } } {}
The proof follows from either the analysis or the synthesis formula,
x
[
n
]
=
∫
−
1
/
2
1
/
2
X
˜
(
ϕ
)
e
j2
πϕ
n
dϕ
size 12{x \[ n \] = Int cSub { size 8{ - 1/2} } cSup { size 8{1/2} } { { tilde {X}} \( ϕ \) e rSup { size 8{j2 ital "πϕ"n} } } dϕ} {}
x
[
n
−
n
0
]
=
∫
−
1
/
2
1
/
2
X
˜
(
ϕ
)
e
j2
πϕ
(
n
−
n
0
)
dϕ
size 12{x \[ n - n rSub { size 8{0} } \] = Int cSub { - 1/2} cSup {1/2} { { tilde {X}} \( ϕ \) e rSup { size 8{j2 ital "πϕ" \( n - n rSub { size 6{0} } \) } } } size 12{dϕ}} {}
x
[
n
−
n
0
]
=
∫
−
1
/
2
1
/
2
X
˜
(
ϕ
)
e
j2
πϕ
n
0
e
j2
πϕ
n
dϕ
size 12{x \[ n - n rSub { size 8{0} } \] = Int cSub { - 1/2} cSup {1/2} { { tilde {X}} \( ϕ \) e rSup { size 8{j2 ital "πϕ"n rSub { size 6{0} } } } e rSup {j2 ital "πϕ"n} } size 12{dϕ}} {}
d/ Frequency shift
x
[
n
]
e
j2
πϕ
0
⇔
F
X
˜
(
ϕ
−
ϕ
0
)
size 12{x \[ n \] e rSup { size 8{j2 ital "πϕ" rSub { size 6{0} } } } { size 24{ dlrarrow } } cSup {F} { tilde { size 12{X}} \( ϕ - ϕ rSub {0} } size 12{ \) }} {}
The proof follows from either the analysis or the synthesis formula,
X
˜
(
ϕ
)
=
∑
n
=
−
∞
∞
x
[
n
]
e
−
j2
πϕ
n
,
size 12{ { tilde {X}} \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x \[ n \] e rSup { size 8{ - j2 ital "πϕ"n} } } ,} {}
X
˜
(
ϕ
)
=
∑
n
=
−
∞
∞
(
x
e
[
n
]
+
x
0
[
n
]
)
(
cos
[
2
πϕ
n
]
−
j
sin
[
2
πϕ
n
]
)
,
X
˜
(
ϕ
)
=
∑
n
=
−
∞
∞
x
e
[
n
]
(
cos
[
2
πϕ
n
]
−
j
∑
n
=
−
∞
∞
x
0
[
n
]
sin
[
2
πϕ
n
]
,
X
˜
(
ϕ
)
=
X
˜
r
(
ϕ
)
+
j
X
˜
i
(
ϕ
)
alignl { stack {
size 12{ { tilde {X}} \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { \( x rSub { size 8{e} } } \[ n \]+x rSub { size 8{0} } \[ n \] \) \( "cos" \[ 2 ital "πϕ"n \]- j"sin" \[ 2 ital "πϕ"n \] \) ,} {} #{ tilde {X}} \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x rSub { size 8{e} } } \[ n \] \( "cos" \[ 2 ital "πϕ"n \]- j Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x rSub { size 8{0} } } \[ n \] "sin" \[ 2 ital "πϕ"n \], {} #
{ tilde {X}} \( ϕ \) = { tilde {X}} rSub { size 8{r} } \( ϕ \) +j { tilde {X}} rSub { size 8{i} } \( ϕ \) {}} } {}
X
˜
(
ϕ
−
ϕ
0
)
=
∑
n
=
−
∞
∞
x
[
n
]
e
−
j2π
(
ϕ
−
ϕ
0
)
n
,
size 12{ { tilde {X}} \( ϕ - ϕ rSub { size 8{0} } \) = Sum cSub {n= - infinity } cSup { infinity } {x \[ n \] e rSup { size 8{ - j2π \( ϕ - ϕ rSub { size 6{0} } \) n} } } size 12{,}} {}
X
˜
(
ϕ
−
ϕ
0
)
=
∑
n
=
−
∞
∞
x
[
n
]
e
j2
πϕ
0
n
e
−
j2
πϕ
n
size 12{ { tilde {X}} \( ϕ - ϕ rSub { size 8{0} } \) = Sum cSub {n= - infinity } cSup { infinity } {x \[ n \] e rSup { size 8{j2 ital "πϕ" rSub { size 6{0} } n} } e rSup { - j2 ital "πϕ"n} } } {}
e/ Multiplication of sequences
What is the discrete time Fourier transform of z[n] = x[n]y[n]?
Z
˜
(
ϕ
)
=
∑
n
x
[
n
]
y
[
n
]
e
−
j2
πϕ
n
,
size 12{ { tilde {Z}} \( ϕ \) = Sum cSub { size 8{n} } {x \[ n \] y \[ n \]e rSup { size 8{ - j2 ital "πϕ"n} } } ,} {}
=
∑
n
x
[
n
]
(
∫
−
1
/
2
1
/
2
Y
˜
(
η
)
e
−
j2
πη
n
dη
)
e
−
j2
πϕ
n
,
size 12{ {}= Sum cSub { size 8{n} } {x \[ n \] \( Int rSub { size 8{ - 1/2} } rSup { size 8{1/2} } { { tilde {Y}} \( η \) } } e rSup { size 8{ - j2 ital "πη"n} } dη \) e rSup { size 8{ - j2 ital "πϕ"n} } ,} {}
=
∫
−
1
/
2
1
/
2
dη
Y
˜
(
η
)
∑
n
x
[
n
]
e
−
j2π
(
ϕ
−
η
)
n
size 12{ {}= Int rSub { size 8{ - 1/2} } rSup { size 8{1/2} } {dη { tilde {Y}} \( η \) } Sum cSub { size 8{n} } {x \[ n \] e rSup { size 8{ - j2π \( ϕ - η \) n} } } } {}
=
∫
−
1
/
2
1
/
2
X
˜
(
ϕ
−
η
)
Y
˜
(
η
)
dη
=
∫
1
>
X
˜
(
ϕ
−
η
)
Y
˜
(
η
)
d
η
size 12{ {}= Int rSub { size 8{ - 1/2} } rSup { size 8{1/2} } { { tilde {X}} \( ϕ - η \) { tilde {Y}} \( η \) } dη= Int rSub { size 8{<1>} } { { tilde {X}} \( ϕ - η \) { tilde {Y}} \( η \) d} η} {}
This convolution over one period is called circular convolution and we use the symbol ⊗ as follows
X
˜
(
ϕ
)
⊗
Y
˜
(
ϕ
)
=
∫
1
>
X
˜
(
ϕ
−
η
)
Y
˜
(
η
)
d
η
=
∫
η
0
η
0
+
1
X
˜
(
ϕ
−
η
)
Y
˜
(
η
)
d
η
size 12{ { tilde {X}} \( ϕ \) ⊗ { tilde {Y}} \( ϕ \) = Int rSub { size 8{<1>} } { { tilde {X}} \( ϕ - η \) { tilde {Y}} \( η \) d} η= Int rSub { size 8{η rSub { size 6{0} } } } rSup {η rSub { size 6{0} } +1} { { tilde {X}} \( ϕ - η \) { tilde {Y}} \( η \) d} size 12{η}} {}
f/ Summary of simple properties
Most proofs of DT Fourier transform properties are simple and similar to those of CT Fourier transform properties. Some important properties are summarized here.
g/ Plotting the DTFT
Because the DTFT of a real DT function
has a period of 1,
has conjugate symmetry (real part and magnitude of the DTFT are even functions of φ, the imaginary part and angle are odd functions of φ),
the DTFT need only be plotted over the frequency range 0 ≤ φ ≤ 0.5. Nevertheless, we shall plot several DTFTs over a larger range of φ to reinforce the periodicity and symmetry properties.
III. DT FOURIER TRANSFORM PAIRS
1/ Unit samples in time and their relatives
From the definition
X
˜
(
ϕ
)
=
∑
n
=
−
∞
∞
x
[
n
]
e
−
j2
ϕπ
n
size 12{ { tilde {X}} \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x \[ n \] e rSup { size 8{ - j2 ital "ϕπ"n} } } } {}
it is apparent that
δ
[
n
]
⇔
F
1
size 12{δ \[ n \] { size 24{ dlrarrow } } cSup { size 8{F} } 1} {}
Shifting and combining unit samples in time yields
δ
[
n
]
⇔
F
1
size 12{δ \[ n \] { size 24{ dlrarrow } } cSup { size 8{F} } 1} {}
δ
[
n
]
⇔
F
1,
size 12{δ \[ n \] { size 24{ dlrarrow } } cSup { size 8{F} } 1,} {}
δ
[
n
−
n
0
]
⇔
F
e
−
j2
πϕ
n
0
,
δ
[
n
−
n
0
]
⇔
F
e
−
j2
πϕ
n
0
,
size 12{δ \[ n - n rSub { size 8{0} } \] { size 24{ dlrarrow } } cSup { size 8{F} } e rSup { size 8{ - j2 ital "πϕ"n rSub { size 6{0} } } } ,δ \[ n - n rSub {0} size 12{ \]{ size 24{ dlrarrow } } cSup {F} } size 12{e rSup { - j2 ital "πϕ"n rSub { size 6{0} } } } size 12{,}} {}
1
2
(
δ
[
n
+
n
0
]
+
δ
[
n
−
n
0
]
)
⇔
F
cos
(
2
πϕ
n
0
)
,
1
2
(
δ
[
n
+
n
0
]
+
δ
[
n
−
n
0
]
)
⇔
F
cos
(
2
πϕ
n
0
)
,
size 12{ { {1} over {2} } \( δ \[ n+n rSub { size 8{0} } \] +δ \[ n - n rSub { size 8{0} } \]\) { size 24{ dlrarrow } } cSup { size 8{F} } "cos" \( 2 ital "πϕ"n rSub { size 8{0} } \) , { {1} over {2} } \( δ \[ n+n rSub { size 8{0} } \] +δ \[ n - n rSub { size 8{0} } \]\) { size 24{ dlrarrow } } cSup { size 8{F} } "cos" \( 2 ital "πϕ"n rSub { size 8{0} } \) ,} {}
1
2j
(
δ
[
n
+
n
0
]
−
δ
[
n
−
n
0
]
)
⇔
F
sin
(
2
πϕ
n
0
)
,
size 12{ { {1} over {2j} } \( δ \[ n+n rSub { size 8{0} } \] - δ \[ n - n rSub { size 8{0} } \]\) { size 24{ dlrarrow } } cSup { size 8{F} } "sin" \( 2 ital "πϕ"n rSub { size 8{0} } \) ,} {}
1
2
(
δ
[
n
+
n
0
]
−
δ
[
n
−
n
0
]
)
⇔
F
j
sin
(
2
πϕ
n
0
)
,
size 12{ { {1} over {2} } \( δ \[ n+n rSub { size 8{0} } \] - δ \[ n - n rSub { size 8{0} } \]\) { size 24{ dlrarrow } } cSup { size 8{F} } j"sin" \( 2 ital "πϕ"n rSub { size 8{0} } \) ,} {}
Note that all the DTFTs are periodic in φ with period φ = 1. [Note symmetry properties for imaginary time functions.]
2/ Unit impulse trains in frequency and their relatives
From the definition
we can derive the time sequences that correspond to periodic impulse trains in φ. Since (φ) has period φ = 1, the impulse trains have the same period.
x
[
n
]
=
∫
−
1
/
2
1
/
2
X
˜
(
ϕ
)
e
j2
πϕ
n
dϕ
size 12{x \[ n \] = Int rSub { size 8{ - 1/2} } rSup { size 8{1/2} } { { tilde {X}} \( ϕ \) e rSup { size 8{j2 ital "πϕ"n} } } dϕ} {}
Shifting and combining unit impulses in frequency yields
∑
k
=
−
∞
∞
δ
(
ϕ
+
k
)
⇔
F
1
size 12{ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {δ \( ϕ+k \) { size 24{ dlrarrow } } cSup { size 8{F} } } 1} {}
∑
k
=
−
∞
∞
δ
(
ϕ
+
k
)
⇔
F
1
size 12{ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {δ \( ϕ+k \) { size 24{ dlrarrow } } cSup { size 8{F} } } 1} {}
∑
k
=
−
∞
∞
δ
(
ϕ
−
ϕ
0
+
k
)
⇔
F
e
j2
πϕ
0
n
,
size 12{ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {δ \( ϕ - ϕ rSub { size 8{0} } +k \) { size 24{ dlrarrow } } cSup { size 8{F} } } e rSup { size 8{j2 ital "πϕ" rSub { size 6{0} } n} } ,} {}
∑
k
=
−
∞
∞
1
2
(
(
δ
(
ϕ
−
ϕ
0
+
k
)
+
δ
(
ϕ
+
ϕ
0
+
k
)
)
)
⇔
F
cos
(
2
πϕ
0
n
)
,
size 12{ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } { { {1} over {2} } \( \( δ \( ϕ - ϕ rSub { size 8{0} } +k \) +δ \( ϕ+ϕ rSub { size 8{0} } +k \) \) \) { size 24{ dlrarrow } } cSup { size 8{F} } } "cos" \( 2 ital "πϕ" rSub { size 8{0} } n \) ,} {}
∑
k
=
−
∞
∞
1
2j
(
(
δ
(
ϕ
−
ϕ
0
+
k
)
−
δ
(
ϕ
+
ϕ
0
+
k
)
)
)
⇔
F
sin
(
2
πϕ
0
n
)
,
size 12{ Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } { { {1} over {2j} } \( \( δ \( ϕ - ϕ rSub { size 8{0} } +k \) - δ \( ϕ+ϕ rSub { size 8{0} } +k \) \) \) { size 24{ dlrarrow } } cSup { size 8{F} } } "sin" \( 2 ital "πϕ" rSub { size 8{0} } n \) ,} {}
3/ Rectangular pulse in time
x
[
n
]
=
∑
m
=
−
M
M
δ
[
n
−
m
]
size 12{x \[ n \] = Sum cSub { size 8{m= - M} } cSup { size 8{M} } {δ \[ n - m \]} } {}
We can take the DTFT of this equation as follows.
X
˜
(
ϕ
)
=
∑
m
=
−
M
M
e
−
j2
πϕ
n
0
=
1
+
∑
m
=
14
M
2
cos
(
2πmϕ
)
size 12{ { tilde {X}} \( ϕ \) = Sum cSub {m= - M} cSup {M} {e rSup { size 8{ - j2 ital "πϕ"n rSub { size 6{0} } } } } size 12{ {}=1+ Sum cSub {m="14"} cSup {M} {2"cos" \( 2πmϕ \) } }} {}
A closed form solution is obtained by using the formula for the sum of a finite geometric series,
X
˜
(
ϕ
)
=
∑
m
=
−
M
M
(
e
−
j2
πϕ
)
m
=
e
j2
πϕ
M
−
e
−
j2
πϕ
(
M
+
1
)
1
−
e
−
j2
πϕ
size 12{ { tilde {X}} \( ϕ \) = Sum cSub { size 8{m= - M} } cSup { size 8{M} } { \( e rSup { size 8{ - j2 ital "πϕ"} } } \) rSup { size 8{m} } = { {e rSup { size 8{j2 ital "πϕ"M} } - e rSup { size 8{ - j2 ital "πϕ" \( M+1 \) } } } over {1 - e rSup { size 8{ - j2 ital "πϕ"} } } } } {}
A factor of e−jπφ can be factored out of the numerator and the denominator and the resulting expressions reduced to
X
˜
(
ϕ
)
=
(
sin
(
π
(
2M
+
1
)
ϕ
)
sin
(
πϕ
)
)
size 12{ { tilde {X}} \( ϕ \) = \( { {"sin" \( π \( 2M+1 \) ϕ \) } over {"sin" \( ital "πϕ" \) } } \) } {}
A single unit sample has a DTFT that is 1. Addition of a pair of unit samples at ±1 adds a cosine wave of frequency 1 to the DTFT. Addition of a pair of unit samples at ±2 adds a cosine of frequency 2 to the DTFT. As more unit samples are added, x[n] → 1 and the DTFT approaches a periodic impulse train of frequency 1.
