Chapter 3-Discrete-Time Signals in the frequency Domain

n

x[n]


• Then for all values of w
X (e ) x[n]e
n j jn
x[n]
n
• Thus, the absolute summability of x[n] is a sufficient condition（充分条件） for the existence of the DTFT X(ej)
• But, x[n] is not absolutely summable
§3.2.4 Convergence Condition
• The DTFT can also be defined for a certain class of sequences which are neither absolutely summable nor square summable • Examples of such sequences are μ[n], cos(0 n+f) and Aan • For this type of sequences, a DTFT representation is possible using the Dirac delta function（狄拉克δ函数） δ(ω)，也 称理想冲激函数
§3.2.1 Discrete-Time Fourier Transform
• A Dirac delta function d() is a function of with infinite height, zero width, and unit area
• It is the limiting form of a unit area pulse function p() as goes to zero satisfying
§3.2.4 Convergence Condition
• Example - The sequence
1 / n, n 1 x[n] 0, n 0 has a finite energy equal to 2 1 2 p Ex 6 n 1 n
§3.2.2 Basic Properties
• The phase function () cannot be uniquely specified for any DTFT • Unless otherwise stated, we shall assume that the phase function () is restricted to the following range of values: -p () p called the principal value（主值区间）
X(ej) = Xre(ej) + j Xim(ej)
§3.2.1 Definition
• Xre(ej) and Xim(ej) are real functions of
• X(ej) can alternately be expressed as X(ej) = | X(ej) |ej() magnitude spectrum where Real phase j) } () = arg{X(e Real
1
§3.2.1 Definition——周期性
• The DTFT X(ej) of a sequence x[n] is a continuous function of • It is also a periodic function of with a period 2p:
Commonly Used DTFT Pairs
Sequence DTFT
d[n] 1
1 e
jo n

k
2pd( 2p k) 2pd( o 2p k)

1 [n] p d( 2p k) j 1 e k 1 n [n], ( 1) 1 e j
Chapter 3 Discrete-Time Signals in the Frequency-Domain
Chapter 3
3.1 The Continuous-Time Fourier Transform 3.2 The Discrete-Time Fourier Transform 3.3 Discrete-Time Fourier Transform Theorems 3.4 Energy Density Spectrum of a Discrete-Time Sequence 3.5 Band-Limited Discrete-Time Signals 3.6 DTFT Computation Using MATLAB 3.7 The Unwrapped Phase Function
X (e
j ( o 2 pk )
) x[n]e
n

j ( o 2 pk ) n
x[n]e
n
jo n j 2 p k n
e
x[n]e jon X (e jo )
n
§3.2.1 Definition ——周期性
1 p j jn x[n] X ( e )e d 2p p
DTFT and IDTFT
x[n] X (e )
X (e )
j n

j
x[n]e
j

jn
1 x[n] 2p

p X (e
p
)e
jn
d
§3.2.2 Basic Properties
DTFT的重要性质
• X(ej) 是关于的连续函数，即序列的
频谱是连续谱。
• X(ej) 是关于 的周期函数，周期为2p
• |X(ej)| 和Xre(ej) 是的偶函数。 • () 和Xim(ej)是的奇函数。 • x[n]绝对可和是DTFT存在的充分条件
§ 3.2.3 Symmetry Relations Symmetry Relations can simplify the computational complexity.
§3.2.1 Definition
• DTFT of a sequence x[n] is given by
X (e
j
)
n
x[n]e
j n
In general, X(ej) is a complex function of the real variable and can be written as
spectrum
§3.2.1 Definition
• Example - The DTFT of the unit sample sequence d[n] is given by
(e ) d[n]e
n
j

jn
d[0] 1
• Example - Consider the causal sequence
• The function

X (e ) 2pd( o 2p k)
k
j
is a periodic function of with a period 2p and is called a periodic impulse train （周期冲激串） or impulse train
For a real sequence x[n] • |X(ej)| and Xre(ej) are even functions of • () and Xim(ej) are odd functions of
DTFT的相位函数不是唯一的。
Note: X(ej) = | X(ej) |ej(+2pk) = | X(ej) |ej() for any integer k
d ( ) lim p ( )
0
1
p ( )
lim
0

p

( )d d ( )d

0 2 2

§3.2.4 Convergence Condition
• Example - Consider the complex exponential sequence
§3.2.4 Convergence Condition
• Thus
1 p jn x[n] 2pd( o 2p k)e d 2p p k d( o )e
p p jn
d e
jo n
where we have used the sampling property of the impulse function d()
x[n] e Its DTFT is given by
j k
jon
X (e ) 2pd( o 2p k)
where d() is an impulse function of and
p o p
§3.2.4 Convergence Condition
• Therefore
j j n
X (e ) x[n]e
n
represents the Fourier series representation of the periodic function As a result, the Fourier coefficients x[n] can be computed from X(ej) using the Fourier integral
x[n] [n],
n
1
§3.2.1 Definition
• Its DTFT is given by