数字信号处理第1章
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0 t
Integral of δ(t) is unit step signal:
( )d ( )d ( )d u (t )
0
t
Properties of δ(t)
δ(t) is a even function, that is δ(t) = δ(-t)
f(t)
kΔτ Approximate f(t) with a serial of rectangles. For the rectangle near t = kΔ, the duration is Δ, the height is f(kΔ), and the area is f(kΔ) Δ . So the rectangle can be represented by f(kΔ)Δδ(t - kΔ).
f (t ) (t )dt f (0) (t )dt f (0) (t )dt f (0)
Properties of δ(t)
δ(t) shift
δ(t) δ(t -τ)
0
t
0
τ
t
δ(t) times a constant A:
Aδ(t)
§1.2 Continue-timeBaidu NhomakorabeaSignal
All signals are thought of as a pattern of variations in time and represented as a time function f(t). In the real-world, any signal has a start. Let the start as t = 0 that means f(t) = 0 t<0 Call such signals causal.
Typical signals and their representation sin/cos signals may be represented by complex exponential A j (t ) j (t ) A sin(t ) (e e ) 2j A j (t ) j (t ) A cos(t ) (e e ) 2 Euler’s relation
• Regular function has exact value at exact time. Obviously, δ(t) is not a Regular function, but Singularity function (or Generalized function).
Unit impulse function δ(t)
Typical signals and their representation
Sinusoidal signals: Asin(ωt +φ)
f(t) = Asin(ωt +φ)= Asin(2πft +φ) A - Amplitude
f - frequency(Hz)
ω = 2πf - angular frequency (radians/sec) φ - start phase(radians)
e j (t ) cos(t ) j sin(t )
Typical signals and their representation Sinusoidal is basic periodic signal which is important both in theory and engineering. Sinusoidal is non-causal signal. All of periodic signals are non-causal because they have no start and no end.
With a gate signal pτ(t), short the duration τ and keep the unit area
4/τ 2/τ 1/τ -τ/4 τ/4 -τ/8 τ/8
1/τ -τ/2 τ/2
When τ0, the amplitude tends to , which means it is impossible to define δ(t) by a regular function.
§1.3 Signal representation based on δ(t) f(t) can be approximated as follows:
f (t )
k
f (k ) (t k )
The smaller of , the higher of the accuracy, and when d, k , the above expression becomes precision representation:
y(t)
From f(t) and h(t), find y(t): Signal processing From f(t) and y(t), find h(t): System design From y(t) and h(t), find f(t): Signal reconstruction
f (t )
f ( ) (t )d
§1.4 Linear time-invariant system
Input signal f(t) into system, output y(t):
f(t) y(t) f ( t) If satisfy y(t)
af(t) ay(t) and f1(t) + f2(t) y1(t) + y2(t)
σ> 0, growing sinusoidal σ< 0, decaying sinusoidal (damped)
Typical signals and their representation
Gate signal
1 p (t ) 0 t t
2
1 -τ/2 τ/2
We got δ(t) from a gate signal, and gate signal is an even function. It is also easy to give the math show of the even property.
§1.3 Signal representation based on δ(t) Any signal can be represented by a sum of shifted and weighted δ(t)
§1.1 Introduction
There are so many different signals and systems that it is impossible to describe them one by one.
The best approach is to represent the signal as a combination of some kind of the simplest signals which will pass through the system and produce a response. Combine the responses of all simplest signals, which is the system response for the original signal. This is the basic method to study the signal analyses and processing.
Typical signals and their representation
Exponential f(t) = eαt
•α is complex α = σ + jω
f(t) = Aeαt = Ae(σ + jω)t
= Aeσtcosωt + jAeσtsinωt σ= 0, sinusoidal
2
The gate signal can be represented by unit step signals:
Pτ(t) = u(t +τ/2) – u(t –τ/2)
Typical signals and their representation
Unit Impulse Signal
Review of Continuous-time Signals and Systems
§1.1 Introduction
Any problems about signal analyses and processing may be thought of letting signals trough systems. f ( t) h ( t)
A is called impulse intension which is the area of the integral.
Properties of δ(t)
Differential of δ(t) is also an impulse δ’(t) = dδ(t)/dt and
f (t ) ' (t )dt f ' (0)
Scaling The system is called linear. additivity
That is af1(t) + bf2(t) ay1(t) + by2(t)
§1.4 Linear time-invariant system
If satisfy
f(t – t0) y(t – t0) The system is called time-invariant.
(t )dt 1
t0
(t ) 0
• δ(t) is non-zero only at t = 0,otherwise is 0; • δ(t) could not be represented by a constant even at t = 0, but by an integral;
Typical signals and their representation
Unit Step u(t) [in our textbook (t)]
1 u(t ) 0
u(t)
1 0 t 1 0 t0 t
t 0 t 0
u(t- t0)
u(t) is basic causal signal, multiply which with any non-causal signal to get causal signal.
The reason to study linear time-invariant system is that based on δ(t) shifted and weighted sum representation of f(t), we can only discuss the system response for unit impulse, then make the sum of responses for all shifted and weighted impulses to get the whole response. It can be done only with linear time-invariant systems.
Properties of δ(t)
Sampling Property
f (t ) (t )dt f (0)
Briefly understanding:
• When t 0, δ(t) = 0, then f(t)·δ(t) = 0 • When t = 0, f(t) = f(0) is a constant. Based on the definition of δ(t), it is easy to get:
f (t) = f(t + mT)
m = 0, ±1, ±2, · · · , ±
Typical signals and their representation
Exponential f(t) = eαt
•α is real
α < 0 decaying
α = 0 constant
α 0 growing
Integral of δ(t) is unit step signal:
( )d ( )d ( )d u (t )
0
t
Properties of δ(t)
δ(t) is a even function, that is δ(t) = δ(-t)
f(t)
kΔτ Approximate f(t) with a serial of rectangles. For the rectangle near t = kΔ, the duration is Δ, the height is f(kΔ), and the area is f(kΔ) Δ . So the rectangle can be represented by f(kΔ)Δδ(t - kΔ).
f (t ) (t )dt f (0) (t )dt f (0) (t )dt f (0)
Properties of δ(t)
δ(t) shift
δ(t) δ(t -τ)
0
t
0
τ
t
δ(t) times a constant A:
Aδ(t)
§1.2 Continue-timeBaidu NhomakorabeaSignal
All signals are thought of as a pattern of variations in time and represented as a time function f(t). In the real-world, any signal has a start. Let the start as t = 0 that means f(t) = 0 t<0 Call such signals causal.
Typical signals and their representation sin/cos signals may be represented by complex exponential A j (t ) j (t ) A sin(t ) (e e ) 2j A j (t ) j (t ) A cos(t ) (e e ) 2 Euler’s relation
• Regular function has exact value at exact time. Obviously, δ(t) is not a Regular function, but Singularity function (or Generalized function).
Unit impulse function δ(t)
Typical signals and their representation
Sinusoidal signals: Asin(ωt +φ)
f(t) = Asin(ωt +φ)= Asin(2πft +φ) A - Amplitude
f - frequency(Hz)
ω = 2πf - angular frequency (radians/sec) φ - start phase(radians)
e j (t ) cos(t ) j sin(t )
Typical signals and their representation Sinusoidal is basic periodic signal which is important both in theory and engineering. Sinusoidal is non-causal signal. All of periodic signals are non-causal because they have no start and no end.
With a gate signal pτ(t), short the duration τ and keep the unit area
4/τ 2/τ 1/τ -τ/4 τ/4 -τ/8 τ/8
1/τ -τ/2 τ/2
When τ0, the amplitude tends to , which means it is impossible to define δ(t) by a regular function.
§1.3 Signal representation based on δ(t) f(t) can be approximated as follows:
f (t )
k
f (k ) (t k )
The smaller of , the higher of the accuracy, and when d, k , the above expression becomes precision representation:
y(t)
From f(t) and h(t), find y(t): Signal processing From f(t) and y(t), find h(t): System design From y(t) and h(t), find f(t): Signal reconstruction
f (t )
f ( ) (t )d
§1.4 Linear time-invariant system
Input signal f(t) into system, output y(t):
f(t) y(t) f ( t) If satisfy y(t)
af(t) ay(t) and f1(t) + f2(t) y1(t) + y2(t)
σ> 0, growing sinusoidal σ< 0, decaying sinusoidal (damped)
Typical signals and their representation
Gate signal
1 p (t ) 0 t t
2
1 -τ/2 τ/2
We got δ(t) from a gate signal, and gate signal is an even function. It is also easy to give the math show of the even property.
§1.3 Signal representation based on δ(t) Any signal can be represented by a sum of shifted and weighted δ(t)
§1.1 Introduction
There are so many different signals and systems that it is impossible to describe them one by one.
The best approach is to represent the signal as a combination of some kind of the simplest signals which will pass through the system and produce a response. Combine the responses of all simplest signals, which is the system response for the original signal. This is the basic method to study the signal analyses and processing.
Typical signals and their representation
Exponential f(t) = eαt
•α is complex α = σ + jω
f(t) = Aeαt = Ae(σ + jω)t
= Aeσtcosωt + jAeσtsinωt σ= 0, sinusoidal
2
The gate signal can be represented by unit step signals:
Pτ(t) = u(t +τ/2) – u(t –τ/2)
Typical signals and their representation
Unit Impulse Signal
Review of Continuous-time Signals and Systems
§1.1 Introduction
Any problems about signal analyses and processing may be thought of letting signals trough systems. f ( t) h ( t)
A is called impulse intension which is the area of the integral.
Properties of δ(t)
Differential of δ(t) is also an impulse δ’(t) = dδ(t)/dt and
f (t ) ' (t )dt f ' (0)
Scaling The system is called linear. additivity
That is af1(t) + bf2(t) ay1(t) + by2(t)
§1.4 Linear time-invariant system
If satisfy
f(t – t0) y(t – t0) The system is called time-invariant.
(t )dt 1
t0
(t ) 0
• δ(t) is non-zero only at t = 0,otherwise is 0; • δ(t) could not be represented by a constant even at t = 0, but by an integral;
Typical signals and their representation
Unit Step u(t) [in our textbook (t)]
1 u(t ) 0
u(t)
1 0 t 1 0 t0 t
t 0 t 0
u(t- t0)
u(t) is basic causal signal, multiply which with any non-causal signal to get causal signal.
The reason to study linear time-invariant system is that based on δ(t) shifted and weighted sum representation of f(t), we can only discuss the system response for unit impulse, then make the sum of responses for all shifted and weighted impulses to get the whole response. It can be done only with linear time-invariant systems.
Properties of δ(t)
Sampling Property
f (t ) (t )dt f (0)
Briefly understanding:
• When t 0, δ(t) = 0, then f(t)·δ(t) = 0 • When t = 0, f(t) = f(0) is a constant. Based on the definition of δ(t), it is easy to get:
f (t) = f(t + mT)
m = 0, ±1, ±2, · · · , ±
Typical signals and their representation
Exponential f(t) = eαt
•α is real
α < 0 decaying
α = 0 constant
α 0 growing