信号与系统 奥本·海姆 课件

periodic signal, as the period increases the
fundamental frequency decreases and the
harmonically related components become
closer in frequency. As the period becomes
For aperiodic signal x(t) :
T x(t ) lim ~(t ) or ~(t ) x(t ) x x T
When T 0 (
2 T
) 0
ak 0
For periodic signal ~ (t ) , we x have its Fourier series:
~ jk 0 t x (t ) ak e k 1 ~ (t )e jk 0t dt a k T x T
How to represent the aperiodic signal x(t) ?
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CTFT
As T ,
T
Ta k

T /2
T / 2
~ x (t )e
jk 0 t
dt
d , k0 )
lim T a k


x (t )e x (t )e
jk 0 t
dt
( 0
2 T
j t
dt 0
Use X(jω) to denote this integral, then we have:
X ( j )



x (t )e
j t
dt
Fourier transform of x(t)
spectrum of x(t)
ak
1 T
X ( jk 0 )
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周期信号的频谱就是与它相对应的非周期信号频谱的样本
Since
X ( j ) lim T a k
T T , f0 0
(非周期信号的表示—连续时间傅立叶变换)
4.1.1 Development of the Fourier transform Representation of an Aperiodic signal (1) example:
Over one period of the continuous-time periodic square wave:
series should apply to the convergence of
Fourier transform.
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Two sets of conditions：
1. If



x ( t ) dt then X ( j ) exist .
2
This shows that if the energy is finite then the Fourier transform must exist. 2. Dirichlet conditions (1) x(t) is absolutely integrable.
T=8T1
T=16T1
As T increases with T1 fixed, the fundamental frequency decreases, then: A. the fundamental frequency decreases, the Fourier series coefficients multiplied by T becomes more and more closely spaced samples of the envelope, so that the set of Fourier series coefficients approaches the envelope function as T→∞
1

ak
---equally spaced samples of this envelope
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T=4T1
As T increases with T1 fixed, the fundamental frequency decreases. we have two observations: A. The harmonics are packed more and more closer to each other; B. However, the shape (the envelope) of the spectrum remains the same . What do we know from the above graph?
lim
ak f0
X(jω) is in fact spectrum-density function (频谱密度函数:具有频谱随频率分布的物理含义) Note: Although X(jω) is often abbreviated as “spectrum”, it is different from ak, which is the spectrum of periodic signals.
transform (傅立叶级数与傅立叶变换之间的关系);
3. Properties of the continuous-time Fourier
transform (傅立叶变换的性质);
4. Frequency response and frequency analysis (系
统的) dt
(2) x(t) have a finite number of maxima and minima within any finite interval. (3) x(t) have a finite number of discontinuity within any finite interval. Furthermore, each of these discontinuities must be finite.
Example:
periodic square wave:
x(t ) ak
FS
2sin(k0T1 ) k0T
So aperiodic signal x(t):
x(t ) X ( j )
F
2sin(T1 )

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4.1.2 . Convergence of Fourier Transforms The derivation of the Fourier transform suggests that the convergence of Fourier
2
In practice, a rather large class of signals are
aperiodic signals. How the aperiodic signals
should be decomposed, what is the spectrum of
aperiodic signals, how should we get the LTI
第4章 连续时间傅立叶变换
CHAPTER 4 THE CONTINUOUS-TIME FOURIER TRANSFORM
1
Main content ：
1. The continuous-time Fourier transform (连续时间
傅立叶变换);
2. Relationship between Fourier series and Fourier
1 ak X ( j ) | k0 T X ( j ) T ak |k 0
( Periodic signal ) (Aperiodic signal)
or
周期信号的频谱是对应的非周期信号频谱的样本； 而非周期信号的频谱是对应的周期信号频谱的包络。
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6
T 1 不变 T 0
时
2 T1 T0

1 2
2 T1 T0

1 4
2 T1 T0

1 8
7
We can regard Tak as samples of an envelope function, specifically
Tak 2 sin T1

k0
---continuous variable 2 sin T ---the envelope of Tak
0
T : 0 d , k 0 ,

j t
Thus, we obtain
x (t ) 1 2



X ( j )e
d
Inverse Fourier transform
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Fourier transform pair :
X ( j ) x(t )e jt dt 1 jt x(t ) X ( j )e d 2
infinite, the frequency components form a
continuum
and
the
Fourier
series
sum
becomes an integral.
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4.1 Representation of Aperiodic Signals: The Continuous-Time Fourier Transform
system response to an aperiodic signal are the issues to be solved in this chapter .
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4.0 Introduction
In the time domain, when a periodic signal with an infinite period, then the periodic signal