信号与系统英文课件chapter 4

∫
T
0
x(t )e
− jkω 0t
dt
Review: CTFS
x(t ) = ∑k = −∞ ak e
∞ jkω 0t
Synthesis equation
The synthesis equation tells us: a periodic signal x(t) can be viewed as a linear combination of harmonically related periodic complex exponentials.
CONTENTS
The Fourier transform for periodic signals Formula and Characteristics Properties of CTFT Examples The convolution property Examples The multiplication property
ω where {ejkω0t} are the harmonically related periodic exponentials, and {ak} describe the complex amplitudes of these components.
4.1 Representation of Aperiodic Signals: CTFT
Let T→∞
Relation of periodic signal and nonperiodic signal
Under this idea, lets verify the Fourier Series coefficient ak: 1 T /2 − jkω0t ak = ∫ x(t )e dt T −T / 2
4.1 Representation of Aperiodic Signals: CTFT
It has a Fourier series representation given by ~ (t ) = ∞ a e jkω 0t x
∑
k = −∞
k
−T
− T1 0 T1
T
2 sin(kω 0T1 ) ak = kω 0T
∞ 1 − jkω0t ak = ∫−∞ x(t )e dt T →∞
Because T→∞, thus ak→0, in this case, the discussion for ak has no meanings.
Relation of periodic signal and nonperiodic signal
Similarly, for a nonperiodic signal x(t), X(jω) called the spectrum of x(t) ω describes the amplitudes and phases relatively. It provides us with the information needed for describing x(t) as a linear combination of sinusoidal signals at different frequencies.
T=2s
The envelope
T=4s
The envelope function is defined by
X ( jω ) =
T /2
T=8s
−T / 2
x(t )e − jωt dt ∫
4.1 Representation of Aperiodic Signals: CTFT
So the Fourier series coefficients of the periodic signal can be expressed as T /2 1 1 − jkω 0t ak = ∫/ 2x(t )e dt = T X ( jω ) ω =kω0 T −T 1 = X ( jkω 0 ) T The above equation describe the relationship between the envelope and the coefficients.
CONTENTS
LTI system description in the frequency-domain Frequency Response from the linear constant coefficient differential equation Summary
Review: CTFS
Review: CTFS
1 T − jkω 0t ak = ∫ x(t )e dt Analysis equation T 0 The Fourier Series coefficient ak shows the complex amplitude of each frequency component in a periodic signal.
We see, when T→∞,
2π ω0 = →0 T
This means kω0 comes to a continuous variable, we denote it as ω. And ak T = ∫ x(t )e
−∞ ∞ − jkω0t
dt may be nonzero.
Goback
4.1 Representation of Aperiodic Signals: CTFT
T=2s
The envelope
T /2
Tak =
T=4s
−T / 2
∫ x(t )e
− jkω 0t
dt
T=8s
2 sin(kω 0T1 ) Tak = kω 0
=
2 sin(ωT1 )
ω
ω = kω 0
4.1 Representation of Aperiodic Signals: CTFT
−T
− T1 0 T1
ω0
T
Goback
Relation of periodic signal and nonperiodic signal
In this and the following chapter, we extend these concepts to apply to signals that are not periodic. A nonperiodic signal x(t) can be viewed as a periodic signal with an infinite period:
ω 0 = 2π / T
ω0
4.1 Representation of Aperiodic Signals: CTFT
Now lets focus on T·ak.
ak T = ∫
T /2 −T / 2
x(t )e
− jkω0 t
2 sin(kω0T1 ) dt = kω0
When the period T is chosen as 2s, 4s and 8s respectively, the corresponding T·aks are depicted in the following figures. ctft_develope.m
Chapter 4
The Continuous-Time Fourier Transform
CONTENTS
Review: CTFS The Continuous-Time Fourier Transform Relation between a periodic signal and a nonperiodic signal Development of CTFT, frequency spectra of continuous-time signals Existence condition of CTFT Examples of CTFT
4.1 Representatioห้องสมุดไป่ตู้ of Aperiodic Signals: CTFT
4.1.1 Development of The Fourier Transform Representation of an Aperiodic Signal Again consider the continuous-time periodic square wave: ~ (t ) = x(t + kT ) x where 1, t < T1 x(t ) = { T 0, T1 < t < −T T − T1 0 T1 2
4.1 Representation of Aperiodic Signals: CTFT
The physical meanings of the Fourier transform. For a periodic signal, its Fourier series representation is ~ (t ) = ∞ a e jkω 0t x ∑k =−∞ k
where
X ( jω ) =
T /2
−T / 2
∫ x(t )e
− jωt
dt
1 ak = X ( jkω0 ) T
4.1 Representation of Aperiodic Signals: CTFT
As indicated before, when T→∞ the →∞, →∞ periodic signal approaches to an aperiodic signal, and the function Tak becomes to the envelope function. That is T /2
X ( jω ) = lim Tak = lim
T →∞ T →∞ −T / 2