信号与系统教程第三章

x(t ) = ∑ a k e
k sk t
x[n] = ∑ a k z k
k
n
then from the eigenfunction property and the superposition property, the output will be:
y (t ) = ∑ a k H ( s k )e sk t
Example 3.1
Consider a real periodic signal x(t), with fundamental frequency 2π, that is expressed in the complex exponential Fourier series as
+3
x(t ) =
CHAPTER 3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS
3.0 INTRODUCTION
• Representation of continuous-time and discretetime periodic signals — Fourier series（傅立叶 级数）. • Use Fourier methods to analyze and understand signals and LTI systems.
k =1
+∞
The relationships between ak and Bk, Ck, Ak, θk are:
a k = Bk − jC k
a k = Ak e jθ k
Construction of the signal x(t) as a linear combination of the harmonically related sinusoidal signals
ak
1
5 2
1/2
argak
arctan(1/2)
-2 1 -1 0 2 3
π/4
-3 -2 -1 0 1 2 3
k
-3
k
Plots of the magnitude spectrum and phase spectrum of the signal x(t)
Example 3.3
The periodic square wave is defined over one period as:
spectrum of periodic square wave
3.3 CONVERGENCE OF THE FOURIER SERIES 傅立叶级数的存在性） （傅立叶级数的存在性） x (t ) = ∑ a e ω An approximation of x(t) is
+N jk
0t
N
k =− N
the Fourier series coefficients are determined by equation:
synthesis equation: analysis equation:
+∞ k=−∞
1 ak = T
+∞
∫
T
x(t )e − jkω0t dt
x(t) = ∑ake jkω0 t = ∑ake jk(2π /T)t ,
3.1 THE RESPONSE OF LTI SYSTEMS TO COMPLEX EXPONENTIALS
Important concept — signal decomposition
basic signals: possess two properties 1. The set of basic signals can be used to construct a broad and useful class of signals. 2. It should be convenient for us to represent the response of an LTI system to any signal constructed as a linear combination of the basic signals. complex exponential signals in continuous time: in discrete time:
k
y[ n] = ∑ a k H ( z k ) z k
k
n
3.2 FOURIER SERIES REPRESENTATION OF CONTINUOUS-TIME PERIODIC SIGNALS
3.2.1 Complex Exponential Fourier Series（指数型傅立叶级数） （指数型傅立叶级数） Given periodic x(t) with fundamental period T , its complex exponential Fourier series is : +∞
1 , x (t ) = 0, t < T1 T1 < t < T 2
…
-T -T/2 -T1 T1 T/2 T
x(t) … t
T1 −T1
Represent it in Fourier series.
2T1 1 a0 = ∫ dt = T −T1 T
T1
1 ak = T
∫
T1
−T1
e
− jkω 0t
dt = −
1 jkω 0T
e − jkω 0t
2 sin (kω0T1 ) sin (kω0T1 ) ak = = , kω0T kπ
T1 sin 2kπ T k ≠0 = kπ
For T = 4T1, the coefficients are:
1 a0 = , 2 ⋮ a 1 = a −1 = 1
k=−∞
1 1 − jkω0t ak = ∫ x(t)e dt = ∫ x(t)e− jk(2π /T )t dt T T T T
Fourier series coefficients or spectrum of x(t): （傅立叶系数） （频谱）
{a k }
arg ak ak magnitude spectrum: phase spectrum: （幅度频谱） （相位频谱） 1 a0 = x ( t )dt constant component or dc of x(t): T T （直流分量）
1 1 1 1 x(t ) = 1 + 1 + e jω0t + 1 − e − jω0t + e j (π 4) e j 2ω0t + e − j (π 4 ) e − j 2ω0t 2j 2j 2 2
x(t ) = 1 + sin ω 0 t + 2 cos ω 0 t + cos( 2ω 0 t +
k = −∞
∑a e
k
jkω0 t
ω0 = 2π / T
{
}
一次谐波分量） （基波分量） 基波分量） （一次谐波分量） 次谐波分量） （ 次谐波分量 a ± N e ± jN ω 0 t : the Nth harmonic components（ N次谐波分量）
a±1e ± jω 0 t: fundamental components or the first harmonic components :
k
There exist error between the original signal x(t) and the approximation xN(t) and with the N increases, the error decreases. The Dirichlet conditions（狄里赫利条件） are as follows: Condition 1: Over any period, x(t) must be absolutely integrable
(
)
(
) (
)
x(t ) = 1 +
1 1 1 ⋅ 2 cos 2πt + ⋅ 2 cos 4πt + ⋅ 2 cos 6πt 4 2 3 1 2 = 1 + cos 2πt + cos 4πt + cos 6πt 2 3
Example 3.2
Consider the signal
4 Plot the magnitude spectrum and phase spectrum of x(t).
e → H ( s )e
st
st
Biblioteka Baidu
z n → H ( z) z n
For a specific value of sk or zk ,
H ( sk )
or H ( z k ) : eigenvalue . （特征值）
If the input to a continuous (discrete) time LTI system is represented as a linear combination of complex exponentials:
x(t ) =
where the coefficients ak is generally a complex function of kω0 . The signals in the set e jkω0 t , k = 0, ±1, ± 2,⋯ are harmonically related complex exponentials.
e st
zn
Eigenfunction（特征函数） （特征函数）
Defining two quantities: H(s) and H(z)
H ( s ) = ∫ h(τ )e
−∞ +∞ − sτ
dτ
H ( z) =
k = −∞
h[k ]z − k ∑
+∞
H(s) or H(z) is in general a function of the complex variable s or z . Eigenfunction( of the system): an input signal for which the system output is a constant times the input.
-8 -2 0 2
k
-4
0
4
k
0
8
k
Plots of the Fourier Series coefficients ak for the periodic square wave with T1 fixed and for several values of T: (a) T=4 T1; (b) T=8 T1; (c) T=16 T1.
π
)
Thus, the Fourier series coefficients for this example are :
a 0 = 1,
1 j (π 4 ) 2 a2 = e = (1 + j ), 2 4 1 1 5 −arctan(1 2) , a1 = 1 + = 1 − j = e 2j 1 − j (π 4 ) 2 2 2 a−2 = e = (1 − j ), 2 4 1 1 5 arctan(1 2) a −1 = 1 − = 1 + j = e , a = 0, k > 2. 2j k 2 2
where
k = −3
a k e jk 2πt ∑
a0 = 1, a1 = a −1 = 1 4 , a 2 = a − 2 = 1 2 , a3 = a −3 = 1 3
Use the trigonometric form to express the signal x(t).
x(t ) = 1 + 1 j 2πt 1 1 e + e − j 2πt + e j 4πt + e − j 4πt + e j 6πt + e − j 6πt 4 2 3
∫
3.2.2 Trigonometric Fourier Series 三角型傅立叶级数） （三角型傅立叶级数）
x(t ) = a 0 + 2∑ [Bk cos kω 0 t + C k sin kω 0 t ]
or
k =1 +∞
x (t ) = a 0 + 2∑ Ak cos(kω 0 t + θ k )
π
,
a3 = a −3
1 =− , 3π
a5 = a −5
1 = , 5π
For T = 8T1, the coefficients are:
1 2 1 , a 2 = a −2 = a0 = , a1 = a −1 = , 4 2π 2π 2 2 a 3 = a −3 = , a 4 = a − 4 = 0, a5 = a −5 = − , 6π 10π ⋮