信号与系统课件(英文)讲解
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信号与系统第一章课件
2. Discrete-Time Signals
—— The independent variable is discrete
11 xn
10 8
5
4
1
1
01 2 34 5 6 7
n is integer number
n
Continuous-time signals
Discrete-time signals
R
R i(t)
+ v(t) -
① t1 t t2
E t2 pt dt 1 t2 v2 t dt
t1
R t1
n1 n n2
n2
E x2 n
nn1
② t
E
ptdt 1
R
v2 t dt
§ 1.1.2 Signal Energy and Power v( t) —— voltage i( t) —— current
7
Chapter 1
Signals and Systems
1. Instantaneous power
瞬时功率 2. Total energy
pt vtit 1 v2t
Chapter 1
Signals and Systems
Chapter 1 Signals and Systems
• The mathematical description and representations of signals and systems.
• Signals and Systems arise in a broad array of application.
信号与系统(英文)chapter 1-信号与系统
1.5.2 Discrete-Time Complex Exponential and Sinusoidal signals(sequences)（序列）: n
x[n] = Cα
Ⅰ.Real Exponential Signals: if C﹑a are real Ⅱ. General Complex Exponential Signals * both C and a are complex numbers
x[n] = x[n + N] for all values of n. In this case, we say that x(t) (x[n]) is periodic with period T(N).
Example 1.1 Determine the fundamental period of the signal x(t) = 2cos(10πt+1)-sin(4πt-1).
1.5 SEVERAL BASIC SIGNALS
1.5.1 Continuous-time Complex Exponential （复 指数函数） and Sinusoidal Signals（正弦信号） at complex exponential signals:
x(t ) = Ce
Ⅰ.Real Exponential Signals: if C﹑a are real Ⅱ. Periodic Complex Exponential and Sinusoidal Signals : * if C is real and a purely imaginary jω 0 t periodic complex exponential
∫
T
−T
信号与系统 双语 奥本海姆 第二章PPT课件
10
Chapter 2 §2.3 卷积的计算 1. 由定义计算卷积积分
例2.6 xte au tt,a0htut
2. 图解法 例2.7 求下列两信号的卷积
xt 1 , 0tT ht
0 , 其余t 3. 利用卷积积分的运算性质求解
LTI Systems
yt
t , 0t2T 0 , 其余t
11
Chapter 2
in Terms of impulses
Example 2
3 xn
2
1
1 01 2
n
xknk
x n x 1 n 1 x 0 n x 1 n 1
xnxknk k 4
Chapter 2
LTI Systems
§2.1.2 The Discrete-Time Unit Impulse Responses and the
LTI Systems
§2.3 Properties of LTI Systems
xt ht ytxtht
xn hn ynxnhn
LTI系统的特性可由单位冲激响应完全描述
Example 2.9 ① LTI system
h n
1
0
n0,1 otherwise
② Nonlinear System
③ Time-variant System
a y n x n x n 1 2 aytco s3 txt
b y n m x n ,x a n 1 x b ytetxt 12
Chapter 2
LTI Systems
§2.3.1 Properties of Convolution Integral and Convolution Sum 1. The Commutative Property (交换律）
Signals and Systems chap2,信号与系统 第二节
cos(1)
e jω a periodic signals because: e jω0 (t +T ) = cos ω0 (t + T ) + j sin ω0 (t + T ) when T=2π/ω0 A closely related signal is the sinusoidal signal: x(t ) = cos(ω0t + φ ) ω0 = 2πf 0 We can always use: A cos(ω0t + φ ) = Aℜ e j (ω0t +φ )
P∞ = limT →∞ P∞ = lim N →∞ 1 2T
∫
T
−T
x(t ) dt
2
1 N 2 x[n] ∑ 2 N + 1 n=− N
Three important classes of signals
Three important (sub)classes of signals:
1. Finite total energy (and therefore zero average power) 2. Finite average power (and therefore infinite total energy) 3. Neither total energy nor average power are finite
So far, considered the real and periodic complex exponential Now consider when C can be complex. Let us express C is polar form and a in rectangular form:
英文版《信号与系统》第3章讲义
Fourier Series
xt ake
k
jk 0t
h t
jk t 0 y t a H jk e k 0 k
j t j H h t e dt
Filter
Frequency-Shaping Filter 频率成形滤波器 Frequency-Selective Filter 频率选择性滤波器
a k real even a k Purely imaginary odd
10
Chapter 3 §3.5.7 Parseval’s Relation（帕兹瓦尔关系式）
1 2 2 x t dt a k T T k
Fourier Series
Average Power of x t
Fourier Series
-4 -2 0
1
FS x t c k
2
4
t
T 4 0
2
-4 -2
1 2 1 ① c tdt 0 x T T 4 2
jk 2
1
1
0
t d2x FS d k 2 dt
2 4 t
11 2 1 k 1 ② d e 1 1 k 2 44 4 1 ck 2 2 d jk c 2 jk / 2 k 0 k 0
1 jk t 0 a x t e dt k T 0 T 0
1 T0 Specially a0 xtdt ——Average value 0 T 0 Example 1 1 a1 j a-1 j 1j 1 t j t 0 0 2 2 1 sin t e e 0 2 j 2 j ak 0 k 1
课件信号与系统奥本海姆.ppt
2. System a process of signals, in which input signals are transformed into output signals
4
Ch1. Signals and Systems
Signal：the carrier of information 信号：信息的载体
1
SIGNALS AND SYSTEMS
• 信号与系统
8
Main content ： Ch1. Signals and Systems
• Continuous-Time and Discrete-Time Signals 〔连续时间与离散时间信号〕
• Transformations of the Independent Variable〔自变量的变换〕
信号是信息的具体物理表现形式，包含了信息的 具体内容。总是1个或多个独立变量的函数。
同一信息可以有不同的物理表现形式，因此对应 有不同的信号，但这些不同的信号都包含同一个信息。 这些不同的信号之间可以相互转换。
例如语音信息用声压表示，可用电压或电流信号 作为载体；也可以用一组数据(01)信号作载体。对应 模拟信号和数字信号，可以AD转换。
2
Ch1. Signals and Systems
控制论创始人维纳认为: 信息是人或物体与外部世界交换内容的名称。内 容是事物的原形，交换是信息载体[信号]将事物原形 [内容]映射到人或物体的感觉器官，人们把这种映射 的结果认为获得了信息。通俗地说，信息指人们得到 的消息。
信息多种多样、丰富多彩，具体的物理形态也千 差万别。
• Basic System Properties (根本系统性质) 9
Ch1. Signals and Systems
4
Ch1. Signals and Systems
Signal：the carrier of information 信号：信息的载体
1
SIGNALS AND SYSTEMS
• 信号与系统
8
Main content ： Ch1. Signals and Systems
• Continuous-Time and Discrete-Time Signals 〔连续时间与离散时间信号〕
• Transformations of the Independent Variable〔自变量的变换〕
信号是信息的具体物理表现形式，包含了信息的 具体内容。总是1个或多个独立变量的函数。
同一信息可以有不同的物理表现形式，因此对应 有不同的信号，但这些不同的信号都包含同一个信息。 这些不同的信号之间可以相互转换。
例如语音信息用声压表示，可用电压或电流信号 作为载体；也可以用一组数据(01)信号作载体。对应 模拟信号和数字信号，可以AD转换。
2
Ch1. Signals and Systems
控制论创始人维纳认为: 信息是人或物体与外部世界交换内容的名称。内 容是事物的原形，交换是信息载体[信号]将事物原形 [内容]映射到人或物体的感觉器官，人们把这种映射 的结果认为获得了信息。通俗地说，信息指人们得到 的消息。
信息多种多样、丰富多彩，具体的物理形态也千 差万别。
• Basic System Properties (根本系统性质) 9
Ch1. Signals and Systems
《信号与系统》第十章课件(英文版)
9The z-transform reduces to the Fourier transform for values of z Unit circle on the unit circle.
Im z=ejω
z-plane
ω
1
Re
5
Sichuan University <Signals and Systems> Ch 10 The z-Transform
Example 10.3 Consider a signal that is the sum of two real
exponentials: x[n] = 7⎜⎛ 1 ⎟⎞ n u[n] − 6⎜⎛ 1 ⎟⎞ n u[n].
⎝3⎠
⎝2⎠
The z-transform is then
∑ ∑ X ( z) = 7 +∞ ⎜⎛ 1 ⎟⎞n u[n]z −n − 6 +∞ ⎜⎛ 1 ⎟⎞ n u[n]z −n
z z-transform expand the application in which Fourier analysis can be used.
2
Sichuan University <Signals and Systems> Ch 10 The z-Transform
10.1 The z-Transform
Transform 2. The Inverse z-Transform 3. Geometric Evaluation of the Fourier Transform from the Pole-
Zero Plot 4. Properties of the z-Transform and some Common z-Transform
英文版《信号与系统》第9章讲义
• Apply Signals and Systems in Engineering Applications: By the end of this chapter, students will have a good understanding of how signals and systems are applied in various engineering fields such as audio processing, image processing, and communication systems.
03
Fourier transform decomposition: Express a nonperiodic signal as an integral of sine and cosine waves of all frequencies.
05
Reconstruct a periodic signal from its Fourier series coefficients.
03
Chapter 9 Key Points and Difficulties
Decomposition and synthesis of signals
01
Decomposition of signals
04
Synthesis of signals
02
Fourier series decomposition: Express a periodic signal as an infinite sum of sine and cosine waves of different frequencies.
02
Nyquist stability criterion: Analyze the stability based on the Nyquist plot of the system.
03
Fourier transform decomposition: Express a nonperiodic signal as an integral of sine and cosine waves of all frequencies.
05
Reconstruct a periodic signal from its Fourier series coefficients.
03
Chapter 9 Key Points and Difficulties
Decomposition and synthesis of signals
01
Decomposition of signals
04
Synthesis of signals
02
Fourier series decomposition: Express a periodic signal as an infinite sum of sine and cosine waves of different frequencies.
02
Nyquist stability criterion: Analyze the stability based on the Nyquist plot of the system.
信号与系统双语课件chapter2.2
solution:
h( ) d
0
1 a e d e a
a
0
当 a<0 时，
1 h( ) d a
stable
当 a0 时，
h( ) d
x(t ) x(t ) h(t ) h1 (t )
We know that
x(t ) x(t ) (t )
So the unit impulse of its inverse system should satisfy
h(t ) h1 (t ) (t )
Similarly ,in discrete time, the impulse response h1[n] of the inverse system for an LTI system with impulse response h[n] must satisfy:
y[n]
k
x[k ]h[n k ] x[n] h[n]
That require the impulse response of a causal discrete-time LTI system satisfy the condition:
h[n] 0 for n 0
y1 (t ) x(t ) h1 (t ) y2 (t ) x(t ) h2 (t ) y(t ) y1 (t ) y2 (t ) x(t ) h1 (t ) x(t ) h2 (t )
The output is :
y(t ) x(t ) [h1 (t ) h2 (t )]
h( ) d
0
1 a e d e a
a
0
当 a<0 时，
1 h( ) d a
stable
当 a0 时，
h( ) d
x(t ) x(t ) h(t ) h1 (t )
We know that
x(t ) x(t ) (t )
So the unit impulse of its inverse system should satisfy
h(t ) h1 (t ) (t )
Similarly ,in discrete time, the impulse response h1[n] of the inverse system for an LTI system with impulse response h[n] must satisfy:
y[n]
k
x[k ]h[n k ] x[n] h[n]
That require the impulse response of a causal discrete-time LTI system satisfy the condition:
h[n] 0 for n 0
y1 (t ) x(t ) h1 (t ) y2 (t ) x(t ) h2 (t ) y(t ) y1 (t ) y2 (t ) x(t ) h1 (t ) x(t ) h2 (t )
The output is :
y(t ) x(t ) [h1 (t ) h2 (t )]
信号与系统-课件-郑君里
School of Computer Science and Information
1.1 Signals
Signals are functions of independent variables that carry information. The independent variables can be continuous or discrete. The independent variables can be 1-D, 2-D, ••• , n-D. For this course: Focus on a single (1-D) independent variable which we call “time”. Continuous-Time signals: x(t), t-continuous values. Discrete-Time signals: x(n), n-integer values only.
School of Computer Science and Information
Examples
Electrical signals — voltages and currents in a circuit. Acoustic signals — audio or speech signals. Video signals — intensity variations in an image. Biological signals — sequence of bases in a gene.
School of Computer Science and Information
1.3 Types of Signals
1. Certain Signal and Random Signal
1.1 Signals
Signals are functions of independent variables that carry information. The independent variables can be continuous or discrete. The independent variables can be 1-D, 2-D, ••• , n-D. For this course: Focus on a single (1-D) independent variable which we call “time”. Continuous-Time signals: x(t), t-continuous values. Discrete-Time signals: x(n), n-integer values only.
School of Computer Science and Information
Examples
Electrical signals — voltages and currents in a circuit. Acoustic signals — audio or speech signals. Video signals — intensity variations in an image. Biological signals — sequence of bases in a gene.
School of Computer Science and Information
1.3 Types of Signals
1. Certain Signal and Random Signal
1信号与系统英语课件
｜ 信
号 与
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电 子
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罗
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What is a System
｜ 信
A System is formally defined as an entity that manipulates 号 one or more signals to accomplish a function, thereby 与 yielding new signals.
号
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EX3:
x(t )
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t
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｜ 信 1.continuous-time complex exponential and sinusoidal signals
t2
｜ 信
号 与
t1
| x(t ) |2 dt
n2
Over the time interval [n1,n2], the total energy of x[n] is:
E=
n n1 2 | x [ n ] |
系
2
Definitions 能量信号
E lim x(t ) dt E lim n
电子科大信号系统英文课件Chapter1 SignalandSystem-B
1 Signal and System
1. Signals and Systems 1.1 Continuous-time and discrete-time signals 1.1.1 Examples and Mathematical Representation
A. Examples (1) A simple RC circuit
(3) A Speech Signal
1 Signal and System
(4) A Picture
1 Signal and System
(5) Vertical Wind Profile
1 Signal and System
B. Types of Signals (1) Continuous-time Signal
1 Signal and System
Energy over t1 t t2:
t2p(t)dt t2v2 (t)dt t2x2 (t)dt
t1
t1
t1
Total Energy:
lim E T
t2x2 (t)dt
t1
lim Average Power:
P
T
1 2T
T x2 (t)dt
Right shift : x(t-t0) x[n-n0]
Left shift : x(t+t0) x[n+n0]
(Delay) (Advance)
1 Signal and System Examples
1 Signal and System
B. Time Reversal x(-t) or x[-n] : Reflection of x(t) or x[n]
1. Signals and Systems 1.1 Continuous-time and discrete-time signals 1.1.1 Examples and Mathematical Representation
A. Examples (1) A simple RC circuit
(3) A Speech Signal
1 Signal and System
(4) A Picture
1 Signal and System
(5) Vertical Wind Profile
1 Signal and System
B. Types of Signals (1) Continuous-time Signal
1 Signal and System
Energy over t1 t t2:
t2p(t)dt t2v2 (t)dt t2x2 (t)dt
t1
t1
t1
Total Energy:
lim E T
t2x2 (t)dt
t1
lim Average Power:
P
T
1 2T
T x2 (t)dt
Right shift : x(t-t0) x[n-n0]
Left shift : x(t+t0) x[n+n0]
(Delay) (Advance)
1 Signal and System Examples
1 Signal and System
B. Time Reversal x(-t) or x[-n] : Reflection of x(t) or x[n]
信号与系统课件(英文)
or
{ak }
x(t ) ak
are called Fourier Series coefficients or spectral coefficients of .
FS
x(t )
3 Fourier Series Representation of Periodic Signals 1 (t ) Note:About orthogonality
e
e
j 0t
,e
j 0t
: Fundamental components
j 20t
,e
j 20t
: Second harmonic components
e
jN0t
,e
jN 0t : Nth harmonic components
So, arbitrary periodic signal can be represented as ( Fourier series )
e
jk 0T1
]
3 Fourier Series Representation of Periodic Signals
sin k0T1 sin k0T1 ak 2 k0T k
(3.44)
0T 2
1 a0 T
2T1 dt T T1
T1
So,Fourier Series Representation of
Discrete time LTI system:
x[n] ak zk
k 1
N
n
y[n] ak H ( zk ) zk
k 1
N
n
H ( z)
3 Fourier Series Representation of Periodic Signals Example 3.1
(完整版)奥本海姆信号与系统第二版英文课件第一章
For Discrete-time period signal: x[n]=x[n+N] for all n N Fundamental Period
1 Signal and System Examples of periodic signal
1 Signal and System
1.2.3 Even and Odd Signals
1.2.1 Examples of Transformations A. Time Shift
Right shift : x(t-t0) x[n-n0]
Left shift : x(t+t0) x[n+n0]
(Delay) (Advance)
1 Signal and System Examples
Even signal: x(-t) = x(t) or x[-n]= x[n] Odd signal : x(-t)= -x(t) or x[-n]= -x[n]
Even-Odd Decomposition:
Ev{x(t )}
xe (t)
1 [x(t) 2
x(t )]
Od {x(t )}
xo (t)
1 Signal and System
B. Time Reversal x(-t) or x[-n] : Reflection of x(t) or x[n]
1 Signal and System
C. Time Scaling
x(at) ( a>0 )
Stretch
if a<0
Compressed if a>0
T x2 (t)dt
T
1 Signal and System
1 Signal and System Examples of periodic signal
1 Signal and System
1.2.3 Even and Odd Signals
1.2.1 Examples of Transformations A. Time Shift
Right shift : x(t-t0) x[n-n0]
Left shift : x(t+t0) x[n+n0]
(Delay) (Advance)
1 Signal and System Examples
Even signal: x(-t) = x(t) or x[-n]= x[n] Odd signal : x(-t)= -x(t) or x[-n]= -x[n]
Even-Odd Decomposition:
Ev{x(t )}
xe (t)
1 [x(t) 2
x(t )]
Od {x(t )}
xo (t)
1 Signal and System
B. Time Reversal x(-t) or x[-n] : Reflection of x(t) or x[n]
1 Signal and System
C. Time Scaling
x(at) ( a>0 )
Stretch
if a<0
Compressed if a>0
T x2 (t)dt
T
1 Signal and System
英文版《信号与系统》第678章讲义
Frequency-Domain:
X j
H j
ht
yt Y j
Y j X j H j §6.1 The Magnitude-Phase Representation （ 幅度-相位） of the Fourier Transform
xt
Sampling
xt
x p t
pt
n
t nT
x0 xT x2T
-3T -2T
-T
0
T
2T
3T
4T
12
t
Chapter 7 Sampling Theorem:
Sampling
Let xt be a band-limited signal with X j 0 , M Then xt is uniquely determined by its samples xnT , n 0,1, 2 if where s s 2 M T
to obtain a signal g t with Fourier transform G j . Determine the maximum value of 0 for which it is guaranteed that
y t cos t / 3 1/ 3 cos t 1 cos
3t 3 y t cos t 1 / 3 cos t 1 cos 3 t 1
4
Chapter 6 2. Nonlinear Phase
§6.2 The Magnitude-Phase Representation of the Frequency Response of LTI Systems
信号与系统英文课件
y(t ) cos[4(t 3)] cos[7(t 3)]
3.2 The Response of LTI Systems to Complex Exponentials
From this example, we see that if an input applied to an LTI system is a linear combination of complex exponentials, then the corresponding output response is also a linear combination of the same complex exponentials. sk t y(t ) ak H (sk )e sk t x(t ) ak e
k k
x[n] ak zk
k
n
y[n] ak H ( zk ) zk
k
n
3.2 The Response of LTI Systems to Complex Exponentials
If the input is a linear combination of sinusoids, then the output response is also a combination of the same sinusoids.
Conclusion: for an LTI system, if the input signal is a complex exponential, then the output response is the same complex exponential modified by H(s).
Compare it with y (t ) e H ( s)
3.2 The Response of LTI Systems to Complex Exponentials
From this example, we see that if an input applied to an LTI system is a linear combination of complex exponentials, then the corresponding output response is also a linear combination of the same complex exponentials. sk t y(t ) ak H (sk )e sk t x(t ) ak e
k k
x[n] ak zk
k
n
y[n] ak H ( zk ) zk
k
n
3.2 The Response of LTI Systems to Complex Exponentials
If the input is a linear combination of sinusoids, then the output response is also a combination of the same sinusoids.
Conclusion: for an LTI system, if the input signal is a complex exponential, then the output response is the same complex exponential modified by H(s).
Compare it with y (t ) e H ( s)
信号与系统的概念ppt课件
完整版ppt课件
29
2. 离散时间信号的展宽和压缩
设离散时间信号 f [ n ] 的波形如图1.3.4(a)所示， 其时间展宽 倍的N 情况可表示为
f1[n]f[Nn], 0，
n为N整倍数 其它
完整版ppt课件
30
1.4 信号的基本运算 1.4.1 两信号相加
两信号相加，是指两信号对应时刻的信号值（函数 值）相加，得到一个新的信号。
例：
25 24
23 22
f[n] 21
123 4
31
n
图1.2.2 某地7月份日平均温度是离散时间信号
完整版ppt课件
15
计算机只能处理离散时间信号，因此，将日常的连续时 间信号(如语音信号等)送给计算机处理之前，应先将其 转换为离散时间信号。简单的方法如图1.2.3所示，以时 间 T 为间隔对连续时间信号 f ( t ) 进行取样，则可得到 一数组{ ,f ( 2 T ) ,f ( T ) ,f ( 0 ) ,f ( T ) ,f ( 2 T )} ，可表为
和 Plim 1
N
f[n]2
N2N1nN
完整版ppt课件
25
能量信号(finite-energy signal)：若信号的能量有界，平
均功率趋于零，E,P0,则称该信号为能量信号。
功率信号(finite-power ,signal)：若信号的平均功率有界 能量趋于无穷大, P , E ，则称该信号为功率信号。
图1.4.1 两信号的相加
完整版ppt课件
31
1.4.2 两信号相乘
两信号相乘，是指两信号对应时刻的信号值相乘， 得到一个新的信号。
f(t)f1(t)f2(t) 或 f [ n ] f 1 [ n ] f 2 [ n ]
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balance --- y[n] net deposit --- x[n] interest --- 1% so y[n]=y[n-1]+1%y[n-1]+x[n] or y[n]-1.01y[n-1]=x[n]
x[n] Balance in bank y[n]
(sytem
x(t)
t1 y(t)
t2
1 Signal and System
1.6.4 Stability
x[n]
Discrete-time y[n]
System
SISO system
MIMO system?
1 Signal and System
1.5.1 Simple Example of systems
Example 1.8:
RC Circuit in Figure 1.1 : Vc(t) Vs(t)
Memoryless system: It’s output is dependent only on the input at the same time. Features: No capacitor, no conductor, no delayer.
Examples of memoryless system: y(t) = C x(t) or y[n] = C x[n]
Representation of System: (1) Relation by the notation
x(t) L y(t)
x[n] L y[n]
1 Signal and System
(2) Pictorial Representation
x(t) Continous-time
System
y(t)
dvc (t) dt
1 RC
vc (t)
1 RC
vs (t)
vs(t)
RC Circuit vc(t)
(system)
KVL rule (physically)
1 Signal and System
Example
dv(t) v(t) f (t)
dt
Newton rule (physically)
(3) Feed-back interconnection
1 Signal and System Example of Feed-back interconnection
1 Signal and System
1.6 Basic System Properties
WHY ?
A. Important practical/physical implications
depends only on values of the input at the present time and in the past.
For causal system, if x(t)=0 for t<t0, there must be y(t)=0 for t<t0. ( nonanticipative )
1.5.2 Interconnections of System
(1) Series(cascade) interconnection
1 Signal and System
(2) Parallel interconnection
Series-Parallel interconnection
1 Signal and System
when
1 1
RC
(number)
Observation: Very different physical systems may be modeled mathematically in very similar ways.
1 Signal and System
Example 1.10:
Balance in a bank account from month to month:
system,yields an output equal to the input to the original system.
1 Signal and System
1 Signal and System
1.6.3 Causality
Definition: A system is causal If the output at any time
B. They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply.
1.6.1 Systems with and without Memory
1 Signal and System
1.5 Continuous-time and Discrete-time System Definition: (1) Interconnection of Component,device, subsystem…. (Broadest sense) (2) A process in which signals can be transformed. (Narrow sense)
For example, y(t)= x(t+1) and y(t)= x(-t) are both noncausal . y(t)= x(t)cos(t+1) is a causal system. Note:Memoryless systems are causal.
1 Signal and System
Examples of memory system:
dy(t) 2y(t) x(t) dt
or y[n]-0.5y[n-1]=2x[n]
1 Signal and System
1.6.1 Invertibility and Inverse Systems
Definition: (1) If a system is invertible,then an inverse system exists. (2) An inverse system cascaded with the original
x[n] Balance in bank y[n]
(sytem
x(t)
t1 y(t)
t2
1 Signal and System
1.6.4 Stability
x[n]
Discrete-time y[n]
System
SISO system
MIMO system?
1 Signal and System
1.5.1 Simple Example of systems
Example 1.8:
RC Circuit in Figure 1.1 : Vc(t) Vs(t)
Memoryless system: It’s output is dependent only on the input at the same time. Features: No capacitor, no conductor, no delayer.
Examples of memoryless system: y(t) = C x(t) or y[n] = C x[n]
Representation of System: (1) Relation by the notation
x(t) L y(t)
x[n] L y[n]
1 Signal and System
(2) Pictorial Representation
x(t) Continous-time
System
y(t)
dvc (t) dt
1 RC
vc (t)
1 RC
vs (t)
vs(t)
RC Circuit vc(t)
(system)
KVL rule (physically)
1 Signal and System
Example
dv(t) v(t) f (t)
dt
Newton rule (physically)
(3) Feed-back interconnection
1 Signal and System Example of Feed-back interconnection
1 Signal and System
1.6 Basic System Properties
WHY ?
A. Important practical/physical implications
depends only on values of the input at the present time and in the past.
For causal system, if x(t)=0 for t<t0, there must be y(t)=0 for t<t0. ( nonanticipative )
1.5.2 Interconnections of System
(1) Series(cascade) interconnection
1 Signal and System
(2) Parallel interconnection
Series-Parallel interconnection
1 Signal and System
when
1 1
RC
(number)
Observation: Very different physical systems may be modeled mathematically in very similar ways.
1 Signal and System
Example 1.10:
Balance in a bank account from month to month:
system,yields an output equal to the input to the original system.
1 Signal and System
1 Signal and System
1.6.3 Causality
Definition: A system is causal If the output at any time
B. They provide us with insight and structure that we can exploit both to analyze and understand systems more deeply.
1.6.1 Systems with and without Memory
1 Signal and System
1.5 Continuous-time and Discrete-time System Definition: (1) Interconnection of Component,device, subsystem…. (Broadest sense) (2) A process in which signals can be transformed. (Narrow sense)
For example, y(t)= x(t+1) and y(t)= x(-t) are both noncausal . y(t)= x(t)cos(t+1) is a causal system. Note:Memoryless systems are causal.
1 Signal and System
Examples of memory system:
dy(t) 2y(t) x(t) dt
or y[n]-0.5y[n-1]=2x[n]
1 Signal and System
1.6.1 Invertibility and Inverse Systems
Definition: (1) If a system is invertible,then an inverse system exists. (2) An inverse system cascaded with the original