Digital Signal Processing
Lecture3
Frequency-Domain Reprentation
Tesheng Hsiao,Associate Professor
1Fourier Transform
Definition
The Fourier reprentation for a quence x,or Fourier transform of x is defined as
X(e jω)=F{x[n]}=
∞
∑
n=−∞
x[n]e−jωn,∀ω∈R
The inver Fourier transformation is
x[n]=F−1{X(e jω)}=
1
2π
∫π
−π
X(e jω)e jωn dω,∀n∈Z
The Fourier transform can be decompod into rectangular form
X(e jω)=X R(e jω)+jX I(e jω)
or in polar form
X(e jω)=|X(e jω)|e j∠X(e jω)
The quantities|X(e jω)|and∠X(e jω)are the magnitude(spectrum)and pha(spec-trum),respectively,of the Fourier transform.
Remark1The pha spectrum is not uniquely specified since any integer multiple of 2πmay be added to∠X(e jω)at any value ofωwithout affecting the result of the complex exponentiation.We u the notation ARG[X(e jω)]to denote the pha restricted in the range of−π<ω≤πwhile arg[X(e jω)]is referred to as a continuous pha function. Convergence
Theorem1If x∈l1,then ∑∞
n=−∞
x[n]e−jωn converges uniformly
Proof:Since|x[n]e−jωn|≤|x[n]|,∀ω∈R,∀n∈Z,and ∑∞
n=−∞
|x[n]|<∞.Apply
Weierstrass M-test to both the positive ries ∑∞
n=0
x[n]e−jωn and the negative ries
∑−1
n=−∞
x[n]e−jωn,the uniform convergence of the Fourier transform can be established.
Q.E.D.发芽大蒜能吃吗
Theorem 2If x ∈l 1,then x [n ]=F −1
{F{x [n ]}}
.回族的风俗
Proof:
F −1{F{x [n ]}}=
12π
∫
π
−π
(
∞∑
m =−∞
x [m ]e −jωm )
e jωn dω=∞∑m =−∞
x [m ](1
2π
∫
π−π
e jω(n −m )dω
)
=∞∑m =−∞x [m ]神通广大的意思
(
sin π(n −m )π(n −m )
)洛克的教育思想
=
∞∑m =−∞
x [m ]δ[n −m ]
=x [n ]
Note that the integration term by term of the infinite ries in the 2nd line is implied by the uniform convergence of the infinite ries.
Q.E.D.
Theorem 3If x ∈l 2,then its Fourier transform converges almost everywhere (a.e.)Remark 2Let x ∈l 2and define X M (e jω)=∑M
n =−M x [n ]e −jωn
for every M ∈N .Then it can be shown that
lim M →∞
∫π−π
|X (e jω)−X M (e jω)|2dω=0(1)
We said that X (e jω)converges in the mean-square n .As M approaches infin-ity,the t of ωsuch that Eq.(1)does not hold form a t of ”measure zero”.Roughly
speaking,the ”volumn”of such a t is zero.Hence we also said that X (e jω)converges almost everywhere .
Remark 3If x is a periodic quence,x belongs to neither l 1nor l 2.However Fourier transform can be applied to x provided that the Dirac delta function
is ud.Since x is periodic,it can be expanded as a Fourier ries x [n ]=∑k a k e jωk n ,n ∈Z .Then its Fourier transform is
X (e jω)=
∞∑r =−∞
∑k
2πa k δ(ω−ωk +2πr )
This is becau
12π∫π−πX (e jω)e jωn dω=12π∫π−π∑k
2πa k δ(ω−ωk )e jωn
dω=∑k a k e jωk n =x [n ]
Example 1Let x [n ]=a n u [n ],|a |<1.Then the Fourier transform of x [n ]is
X (e jω
)=F{x [n ]}=
∞∑n =0
a n e
−jωn
=
∞∑n =0
(ae
−jω)n
=
1
1−ae −jω
Example2Consider the following ideal lowpassfilter
H lp(e jω)={
1,|ω|<ωc
0,ωc<|ω|≤π
where0<ωc<πis called the cutofffrequency of H lp(e jω).Then the inver Fourier transform of H lp(e jω)is
h lp=F−1{
H lp(e jω)
}
=
1
2π
∫ω
c
−ωc
e jωn dω=
1
2π
1
jn
[
e jωc n−e−jωc n
]
=
sin(ωc n)
nπ
Properties
Let x[n]F
←→X(e jω)denote the Fourier transform pairs.We then explore the proper-ties of the Fourier transform.
•Periodicity
For any integer r,we have
X(e j(ω+2πr))=
∞
∑
n=−∞
x[n]e−j(ω+2πr)n
=
∞
∑
动力造句n=−∞
《鸿门宴》x[n]e−j(ωn)e−j2πrn
=
∞
∑
n=−∞
x[n]e−j(ωn)
=X(e jω)
Hence X(e jω)is periodic with period2π.
6的乘法口诀教案•Linearity
ax1[n]+bx2[n]F
←→aX1(e jω)+bX2(e jω),∀a,b∈R •Symmetry
Definition1A conjugate-symmetric quence x e[n]is defined as a quence such
that x e[n]=x∗
e [−n].A conjugate-antisymmetric quence x o[n]is defined as a
quence such that x o[n]=−x∗
o
[−n],where∗denotes complex conjugate.
If x is a real-valued quence,then x e is called an even quence while x o is called an odd quence.
Any quence x can be decompod as a sum of a conjugate-symmetric and conjugate-antisymmetric quence.
x[n]=x e[n]+x o[n]
where
x e[n]=1
2
(
x[n]+x∗[−n]
)
and
x o[n]=1
2
(
x[n]−x∗[−n]
)
Similarly,we can decompo Fourier transform X(e jω)into a sum of conjugate-symmetric and conjugate-antisymmetric parts.
X(e jω)=X e(e jω)+X o(e jω)(2)
where
X e(e jω)=1
2
(
X(e jω)+X∗(e−jω)
)
and
X o(e jω)=1
2
(
X(e jω)−X∗(e−jω)
)
Remark4Eq.(2)just says that X(e jω)can be decompod as the sum of the conjugate-symmetric and conjugate-antisymmetric parts.It does not mean that x e[n]F
←→X e(e jω)or x o[n]F←→X o(e jω).
Theorem4
x∗[n]F
←→X∗(e−jω)(3)
x∗[−n]F
←→X∗(e jω)(4)
Re(x[n])F
←→X e(e jω)(5)
jIm(x[n])F
←→X o(e jω)(6)
x e[n]F
←→Re(X(e jω))(7)
x o[n]F
←→jIm(X(e jω))(8) Proof:We show only Eq.(3).
F {
x∗[n]
}
=
∞
∑
n=−∞
x∗[n]e−jωn
=
∞
∑
n=−∞
(
x[n]e−j(−ω)n
)∗
=
(
∞
∑
n=−∞
x[n]e−j(−ω)n
)∗
=X∗(e−jω)
Corollary1If x is a real-valued quence and X(e jω)=F{x[n]}.Then|X(e jω)|= |X(e−jω)|,and∠X(e jω)=−∠X(e−jω),i.e.The magnitude spectrum is an even function,while the pha spectrum is an odd function.
Proof:Since x[n]is real,x[n]=x∗[n].We have X(e jω)=X∗(e−jω).or |X(e jω)|e j∠X(e jω)=|X(e−jω)|e−j∠X(
e−jω).Therefore,|X(e jω)|=|X(e−jω)|,and ∠X(e jω)=−∠X(e−jω).
•Time shifting and frequency shifting
x[n−n d]F
←→e−jωn d X(e jω),∀n d∈Z
e jω0n x[n]F
←→X(e j(ω−ω0)),∀ω0∈R
•Time reversal x [−n ]F
←→X (e −jω)If x is real-valued,then
x [−n ]F
←→X ∗(e jω)
•Differentiation in frequency
nx [n ]F
←→j
dX (e jω)
dω
•Convolution theorem
If x [n ]F ←→X (e jω)and h [n ]F
←→H (e jω),Then
x [n ]∗h [n ]F
←→X (e jω)H (e jω)
•Modulation or windowing theorem
Suppo x,w ∈l 1.Let x [n ]F ←→X (e jω)and w [n ]F
←→W (e jω).If y [n ]=x [n ]·w [n ]
and y [n ]F
←→Y (e jω),then
Y (e jω)=
1
2π∫π−πX (e jθ)W (e j (ω−θ))dθProof:
12π∫π−πX (e jθ)W (e j (ω−θ))dθ=12π∫π−π[∞∑n =−∞
刘德华公司
x [n ]e −jθn
][∞∑m =−∞
w [m ]e −j (ω−θ)m ]dθ
=∞∑n =−∞[x [n ]∞∑m =−∞
(w [m ]e −jωm 1
2π∫π−π
e jθ(m −n )dθ
)]=∞∑n =−∞[x [n ]∞∑m =−∞
(w [m ]e −jωm
sin(π(m −n ))
π(m −n ))]=∞∑n =−∞
[x [n ]∞∑
m =−∞
w [m ]e −jωm δ[m −n ]
]=
∞∑n =−∞
x [n ]w [n ]e −jωn
=F {x [n ]w [n ]}=Y (e jω)
Remark 5Y (e jω)is a periodic convolution of X (e jω)and H (e jω),i.e.a convo-lution of two periodic functions with the limits of integration extending over only one period.
•Parval’s Theorem
If x [n ]F
←→X (e jω),then
∞∑
n =−∞
|x [n ]|2=
1
2π∫
π
−π
|X (e jω)|2dω
