Chapter 1
Continuous-time
Signals and Systems
§ 1.1 Introduction
Any problems about signal analyses and
processing may be thought of letting signals
trough systems.
h(t)f(t) 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
§ 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 most simplest
signals which will pass though the system and
produce a response,Combine the responses of all
simplest signals,which is the system response of
the original signal.
This is the basic method to study the signal
analyses and processing.
§ 1.2 Continue-time Signal
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 the signal causal.
Typical signals and their representation
Unit Step u(t) (in our textbook?(t))
01 00)( tttu
u(t)
1
0 t
u(t- t0)
1
0 tt0
u(t) is basic causal signal,multiply which with
any non-causal signal to get causal signal.
Typical signals and their representation
Sinusoidal Asin(ωt+υ)
f(t) = Asin(ωt+υ)= Asin(2πft+υ)
A - Amplitude
f - frequency(Hz)
ω= 2πf angular frequency (radians/sec)
υ – start phase(radians)
Typical signals and their representation
sin/cos signals may be represented by
complex exponential
)(
2
)c o s (
)(
2
)s in (
)()(
)()(
tjtj
tjtj
ee
A
tA
ee
j
A
tA
Euler’s relation
)s in ()c o s ()( tjte tj
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.
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
Typical signals and their representation
Exponential f(t) = eαt
α is complex α = σ + jω
f(t) = Ae αt = Ae(σ + jω)t
= Aeσ t cos ωt + j Aeσ t sin ωt
σ = 0,sinusoidal
σ > 0,growing sinusoidal
σ < 0,decaying sinusoidal (damped)
Typical signals and their representation
Gate signal
2
||1
2
||0
)(
t
t
tp
The gate signal can be represented by unit step signals:
Pτ(t) = u(t + τ/2) – u(t –τ/2)
-τ/2 τ/2
1
Typical signals and their representation
Unit Impulse Signal
00)(
1)(
tt
dtt
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.
Regular function has exact value at exact time,
Obviously,δ(t) is not a Regular function.
Unit impulse function δ(t)
With a gate signal pτ(t),short the
duration τ and keep the unit area
When τ?0,the amplitude tends to?,which
means it is impossible to define δ(t) by a
regular function.
τ/2-τ/2
1/τ
-τ/4
1/τ
τ/4
2/τ
-τ/8
4/τ
τ/8
Properties of δ(t)
Sampling Property
)0()()( fdtttf?
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:
)0()()0()()0()()( fdttfdttfdtttf
Properties of δ(t)
δ(t) shift
δ(t)
0 t
δ(t- τ)
0 tτ
δ(t) times a constant A:
Aδ(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
)0(')(')( fdtttf?
Integral of δ(t) is unit step signal:
)()()()(
0
0
tudttdttdtt
Properties of δ(t)
δ(t) is a even function,that is
δ(t) = δ(-t)
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
shifted weight sum of δ(t)
f(t)
kΔτ
Approximate f(t) with a serial of rectangles,For the
rectangle near t=kΔτ,the duration is Δτ,the high is
f(kΔτ ),and the area is f(kΔτ )Δτ,So the rectangle can
be represented by f(kΔτ )Δτ δ(t - kΔτ ).
§ 1.3 signal representation based on δ(t)
f(t) can be approximated as follows:
)()()(
ktkftf
k
The smaller of,the higher of the accuracy,
and whend?,k,the above
expression becomes precision representation:
dtftf )()()(?
§ 1.4 Linear time-invariant system
Signal f(t) pass though system,output y(t):
f(t)? y(t)
y(t)f(t)
If satisfy
af(t)? ay(t) and f1(t) + f2(t)? y1(t)+ y2(t)
That is af1(t) +b f2(t)? ay1(t)+ by2(t)
The system is called linear.
§ 1.4 Linear time-invariant system
If satisfy
f(t – t0)? y(t – t0)
The system is called time-invariant.
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.
§ 1.5 System unit impulse response
The system response for unit impulse δ(t) is
called system unit impulse response,and denoted
as h(t)
h(t)δ(t)
h(t) is important figure of linear time-
invariant system,It can be done to get
response for any input signal based on h(t),
because any signal can be represented as
shifted and weighted sum of δ(t).
§ 1.5 Signal pass through
linear time-invariant system
δ(t)?h(t)
aδ(t – t0)? ah(t – t0)
dtftf )()()(?
dthfty )()()(?
Denote y(t) = f(t) * h(t),which is called
convolution of f(t) and h(t)
Frequency domain analyses
of continue-time
signals and systems
Analyses in time domain and
frequency domain
With the time domain analyses,signals are
decomposed the sum of shifted and weighted
δ(t),and only system response for δ(t) is
attracted attention.
With the frequency domain analyses,signals
are decomposed the sum of fundamental sinωt
and harmonious,and the system response for
sinωtcould be attracted attention only.
§ 1.6 Fourier series of periodic signals
Periodic signals is expanded to Fourier series
tnbtnatf
n
n
n
n 0
1
0
0
s inc o s0(
11 )(10 tTt dttfTa The average in one period
11 0c o s)(2 tTtn t d tntfTa?
11 0s in)(2 tTtn t d tntfTb?
§ 1.6 Fourier series of periodic signals
Fourier series in exponential form
tjn
n
n eFtf
0)(
1
1
0)(
1 tT
t
tjn
n dtetfTF
§ 1.6 Spectrum of periodic signals
Periodic rectangles with period T,
duration τ,and amplitude A:
-T T
A
-τ/2 τ/2
f(t)
t
Periodic signals is expanded to
Fourier series
dtetfTF T
T
tjn
n
2/
2/
0)(1?
dtAeT tjn?
2/
2/
01
Tn
n
A
0
0
2
s in2
x
x
T
A s in
2
0nx?
Sampling function Sa(x) = sinx/x
Fn = (Aτ/T)Sa(nω0 τ/2)
§ 1.7 Fourier analyses of non-periodic
signals—Fourier Transform
When the period of a periodic signal is
expanded to ∞,the signal is becoming non-
periodic:
dtetf
T
F
T
T
tjn
n
2/
2/
0)(
1?ω0= 2π/T
But T,Fn and ω0? 0,the amplitude of
lines and the distance between lines tend to 0,
It is unreasonable to describe the signal with
frequency lines,So the concept of spectrum
density is introduced.
dtetfjF tj?
)()(
dejFtf tjn
)(
2
1
)(
Fourier Transform pair
§ 1.8 Fourier Transform of typical signals
Gate
dtetfjF tj )()(
dtttp 0 c o s)(2
dtt 2/0 c o s2
2sin
2
)2( Sa?
F(jω)
τ
2π/ τ 4π/ τ ω
-τ/2 τ/2
1
§ 1.8 Fourier Transform of typical signals
Exponential f(t) = e-at u(t) a>0
1/a
|F(jω)|
ω
e-at u(t)
t
dteejF tjat
0
)(
ja
1
22
1|)(|
a
jF
atg
1)(
§ 1.8 Fourier Transform of typical signals
Unit impulse δ(t)
δ(t)
0 t t
|F(jω)|1
0 ω
Unit impulse has uniform frequency density in whole
frequency range,that means it has infinite wide band.
1)()]([
dtettF tj
§ 1.9 Fourier Transform of typical signals
Constant 1
1 2πδ(ω)
This result could be got directly based on the
symmetry of Fourier Transform.
Constant 1 represents direct current signal,
and its spectrum is non-zero only at ω = 0,
which is a δ(ω)
§ 1.9 Fourier Transform of typical signals
Sin and cos function
Based on the transform pair 1 2πδ(ω) and
δ(t) 1,we have some important conclusions:
F[ejω0t]
F[cosω0t]= F[(ejω0t + e-jω0t)/2]= π[δ(ω + ω0)+ δ(ω - ω0)]
F[sin ω0t]= F[(ejω0t - e-jω0t)/2j]= jπ[δ(ω+ ω0)- δ(ω - ω0)]
-ω0
ω0
(jπ)
(-jπ)
F[sin ω0t]
-ω0 ω0
(π) (π)
F[cos ω0t]
)(2 00 )(0 00 dtedtee tjtjtj
§ 1.9 Fourier Transform of typical signals
Unit impulse sequence
T 2T-T-2T
δT(t)
t0 ω0 2 ω0-ω0-2 ω0
ω0δω0(ω)
0 ω
ω0 = 2π/T
n
T nTtt )()(
n
T ntF )()]([ 00
)(00
§ 1.10 Properties of Fourier Transform
Linear
Fourier Transform is an integral,and it is a
linear operation:
If f1(t)F1(jω) f2(t)F2(jω)
Then af1(t) + bf2(t)aF1(jω) + bF2(jω)
§ 1.10 Properties of Fourier Transform
Time shift
Signal’s shift in time domain equals phase shift
in frequency domain
f (t - t0)F(jω)e-jωt0
Based on the definition of Fourier Transform,the
above result is easy to be shown.
§ 1.10 Properties of Fourier Transform
Frequency shift(Modulation theorem)
Modulate carrier sinω0t with base band signal f(t)
Based on definition of Fourier Transform:
f(t)ejω0tF[j(ω –ω0)]
With Euler’s relation,it is easy to show:
f(t) cosω0t = 1/2 f(t) (ejω0t +e-jω0t )
f(t) cosω0t1/2 {F[j(ω + ω0)]+ F[j(ω –ω0)]}
f(t) sinω0tj/2 {F[j(ω + ω0)]- F[j(ω –ω0)]}
Spectrum of amplitude modulation
F(jω)τ
Pτ(t) cosω0t
1/2 {F[j(ω + ω0)]+ F[j(ω –ω0)]}
τ/2
ω0-ω0
-τ/2 τ/2
1
Pτ(t)
§ 1.10 Properties of Fourier Transform
Energy theorem
djFdttfW 22 |)(|
2
1)(
W is energy of signal,|F(jω)|2 named signal energy
spectrum is signal energy in unit frequency band that
has similar shape with |F(jω)|,but no phase
information.
§ 1.10 Properties of Fourier Transform
Convolution theorem
convolution theorem in time domain
f1(t)*f2(t)F1(jω) F2(jω)
convolution theorem in frequency domain
f1(t) f2(t)1/(2π)F1(jω) *F2(jω)
Transfer the convolution operation in one domain to the algebra
operation in another domain,Almost all of the properties discussed
above could be shown based on the convolution theorem:
Time shift,f (t - t0) = f (t )*δ (t - t0)F(jω)e-jωt0
Frequency shift:
f(t)ejω0t1/(2π)[F[j(ω )* 2πδ(ω-ω0)]=F[j(ω –ω0)]
§ 1.11 Fourier analyses of linear system
H(jω) is called system function,or transfer function
h(t)
H(jω)
f(t) y(t)
F(jω) Y(jω)
y(t) = f(t) * h(t)
Y(jω) = F(jω) H(jω)
Spectrum analyses is an active research area today,If
system is too complicated to be represented by analytical
expression,it could be done that input sin signals with
different frequencies and measure the system output,
then give the system transfer function,
Continuous-time
Signals and Systems
§ 1.1 Introduction
Any problems about signal analyses and
processing may be thought of letting signals
trough systems.
h(t)f(t) 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
§ 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 most simplest
signals which will pass though the system and
produce a response,Combine the responses of all
simplest signals,which is the system response of
the original signal.
This is the basic method to study the signal
analyses and processing.
§ 1.2 Continue-time Signal
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 the signal causal.
Typical signals and their representation
Unit Step u(t) (in our textbook?(t))
01 00)( tttu
u(t)
1
0 t
u(t- t0)
1
0 tt0
u(t) is basic causal signal,multiply which with
any non-causal signal to get causal signal.
Typical signals and their representation
Sinusoidal Asin(ωt+υ)
f(t) = Asin(ωt+υ)= Asin(2πft+υ)
A - Amplitude
f - frequency(Hz)
ω= 2πf angular frequency (radians/sec)
υ – start phase(radians)
Typical signals and their representation
sin/cos signals may be represented by
complex exponential
)(
2
)c o s (
)(
2
)s in (
)()(
)()(
tjtj
tjtj
ee
A
tA
ee
j
A
tA
Euler’s relation
)s in ()c o s ()( tjte tj
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.
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
Typical signals and their representation
Exponential f(t) = eαt
α is complex α = σ + jω
f(t) = Ae αt = Ae(σ + jω)t
= Aeσ t cos ωt + j Aeσ t sin ωt
σ = 0,sinusoidal
σ > 0,growing sinusoidal
σ < 0,decaying sinusoidal (damped)
Typical signals and their representation
Gate signal
2
||1
2
||0
)(
t
t
tp
The gate signal can be represented by unit step signals:
Pτ(t) = u(t + τ/2) – u(t –τ/2)
-τ/2 τ/2
1
Typical signals and their representation
Unit Impulse Signal
00)(
1)(
tt
dtt
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.
Regular function has exact value at exact time,
Obviously,δ(t) is not a Regular function.
Unit impulse function δ(t)
With a gate signal pτ(t),short the
duration τ and keep the unit area
When τ?0,the amplitude tends to?,which
means it is impossible to define δ(t) by a
regular function.
τ/2-τ/2
1/τ
-τ/4
1/τ
τ/4
2/τ
-τ/8
4/τ
τ/8
Properties of δ(t)
Sampling Property
)0()()( fdtttf?
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:
)0()()0()()0()()( fdttfdttfdtttf
Properties of δ(t)
δ(t) shift
δ(t)
0 t
δ(t- τ)
0 tτ
δ(t) times a constant A:
Aδ(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
)0(')(')( fdtttf?
Integral of δ(t) is unit step signal:
)()()()(
0
0
tudttdttdtt
Properties of δ(t)
δ(t) is a even function,that is
δ(t) = δ(-t)
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
shifted weight sum of δ(t)
f(t)
kΔτ
Approximate f(t) with a serial of rectangles,For the
rectangle near t=kΔτ,the duration is Δτ,the high is
f(kΔτ ),and the area is f(kΔτ )Δτ,So the rectangle can
be represented by f(kΔτ )Δτ δ(t - kΔτ ).
§ 1.3 signal representation based on δ(t)
f(t) can be approximated as follows:
)()()(
ktkftf
k
The smaller of,the higher of the accuracy,
and whend?,k,the above
expression becomes precision representation:
dtftf )()()(?
§ 1.4 Linear time-invariant system
Signal f(t) pass though system,output y(t):
f(t)? y(t)
y(t)f(t)
If satisfy
af(t)? ay(t) and f1(t) + f2(t)? y1(t)+ y2(t)
That is af1(t) +b f2(t)? ay1(t)+ by2(t)
The system is called linear.
§ 1.4 Linear time-invariant system
If satisfy
f(t – t0)? y(t – t0)
The system is called time-invariant.
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.
§ 1.5 System unit impulse response
The system response for unit impulse δ(t) is
called system unit impulse response,and denoted
as h(t)
h(t)δ(t)
h(t) is important figure of linear time-
invariant system,It can be done to get
response for any input signal based on h(t),
because any signal can be represented as
shifted and weighted sum of δ(t).
§ 1.5 Signal pass through
linear time-invariant system
δ(t)?h(t)
aδ(t – t0)? ah(t – t0)
dtftf )()()(?
dthfty )()()(?
Denote y(t) = f(t) * h(t),which is called
convolution of f(t) and h(t)
Frequency domain analyses
of continue-time
signals and systems
Analyses in time domain and
frequency domain
With the time domain analyses,signals are
decomposed the sum of shifted and weighted
δ(t),and only system response for δ(t) is
attracted attention.
With the frequency domain analyses,signals
are decomposed the sum of fundamental sinωt
and harmonious,and the system response for
sinωtcould be attracted attention only.
§ 1.6 Fourier series of periodic signals
Periodic signals is expanded to Fourier series
tnbtnatf
n
n
n
n 0
1
0
0
s inc o s0(
11 )(10 tTt dttfTa The average in one period
11 0c o s)(2 tTtn t d tntfTa?
11 0s in)(2 tTtn t d tntfTb?
§ 1.6 Fourier series of periodic signals
Fourier series in exponential form
tjn
n
n eFtf
0)(
1
1
0)(
1 tT
t
tjn
n dtetfTF
§ 1.6 Spectrum of periodic signals
Periodic rectangles with period T,
duration τ,and amplitude A:
-T T
A
-τ/2 τ/2
f(t)
t
Periodic signals is expanded to
Fourier series
dtetfTF T
T
tjn
n
2/
2/
0)(1?
dtAeT tjn?
2/
2/
01
Tn
n
A
0
0
2
s in2
x
x
T
A s in
2
0nx?
Sampling function Sa(x) = sinx/x
Fn = (Aτ/T)Sa(nω0 τ/2)
§ 1.7 Fourier analyses of non-periodic
signals—Fourier Transform
When the period of a periodic signal is
expanded to ∞,the signal is becoming non-
periodic:
dtetf
T
F
T
T
tjn
n
2/
2/
0)(
1?ω0= 2π/T
But T,Fn and ω0? 0,the amplitude of
lines and the distance between lines tend to 0,
It is unreasonable to describe the signal with
frequency lines,So the concept of spectrum
density is introduced.
dtetfjF tj?
)()(
dejFtf tjn
)(
2
1
)(
Fourier Transform pair
§ 1.8 Fourier Transform of typical signals
Gate
dtetfjF tj )()(
dtttp 0 c o s)(2
dtt 2/0 c o s2
2sin
2
)2( Sa?
F(jω)
τ
2π/ τ 4π/ τ ω
-τ/2 τ/2
1
§ 1.8 Fourier Transform of typical signals
Exponential f(t) = e-at u(t) a>0
1/a
|F(jω)|
ω
e-at u(t)
t
dteejF tjat
0
)(
ja
1
22
1|)(|
a
jF
atg
1)(
§ 1.8 Fourier Transform of typical signals
Unit impulse δ(t)
δ(t)
0 t t
|F(jω)|1
0 ω
Unit impulse has uniform frequency density in whole
frequency range,that means it has infinite wide band.
1)()]([
dtettF tj
§ 1.9 Fourier Transform of typical signals
Constant 1
1 2πδ(ω)
This result could be got directly based on the
symmetry of Fourier Transform.
Constant 1 represents direct current signal,
and its spectrum is non-zero only at ω = 0,
which is a δ(ω)
§ 1.9 Fourier Transform of typical signals
Sin and cos function
Based on the transform pair 1 2πδ(ω) and
δ(t) 1,we have some important conclusions:
F[ejω0t]
F[cosω0t]= F[(ejω0t + e-jω0t)/2]= π[δ(ω + ω0)+ δ(ω - ω0)]
F[sin ω0t]= F[(ejω0t - e-jω0t)/2j]= jπ[δ(ω+ ω0)- δ(ω - ω0)]
-ω0
ω0
(jπ)
(-jπ)
F[sin ω0t]
-ω0 ω0
(π) (π)
F[cos ω0t]
)(2 00 )(0 00 dtedtee tjtjtj
§ 1.9 Fourier Transform of typical signals
Unit impulse sequence
T 2T-T-2T
δT(t)
t0 ω0 2 ω0-ω0-2 ω0
ω0δω0(ω)
0 ω
ω0 = 2π/T
n
T nTtt )()(
n
T ntF )()]([ 00
)(00
§ 1.10 Properties of Fourier Transform
Linear
Fourier Transform is an integral,and it is a
linear operation:
If f1(t)F1(jω) f2(t)F2(jω)
Then af1(t) + bf2(t)aF1(jω) + bF2(jω)
§ 1.10 Properties of Fourier Transform
Time shift
Signal’s shift in time domain equals phase shift
in frequency domain
f (t - t0)F(jω)e-jωt0
Based on the definition of Fourier Transform,the
above result is easy to be shown.
§ 1.10 Properties of Fourier Transform
Frequency shift(Modulation theorem)
Modulate carrier sinω0t with base band signal f(t)
Based on definition of Fourier Transform:
f(t)ejω0tF[j(ω –ω0)]
With Euler’s relation,it is easy to show:
f(t) cosω0t = 1/2 f(t) (ejω0t +e-jω0t )
f(t) cosω0t1/2 {F[j(ω + ω0)]+ F[j(ω –ω0)]}
f(t) sinω0tj/2 {F[j(ω + ω0)]- F[j(ω –ω0)]}
Spectrum of amplitude modulation
F(jω)τ
Pτ(t) cosω0t
1/2 {F[j(ω + ω0)]+ F[j(ω –ω0)]}
τ/2
ω0-ω0
-τ/2 τ/2
1
Pτ(t)
§ 1.10 Properties of Fourier Transform
Energy theorem
djFdttfW 22 |)(|
2
1)(
W is energy of signal,|F(jω)|2 named signal energy
spectrum is signal energy in unit frequency band that
has similar shape with |F(jω)|,but no phase
information.
§ 1.10 Properties of Fourier Transform
Convolution theorem
convolution theorem in time domain
f1(t)*f2(t)F1(jω) F2(jω)
convolution theorem in frequency domain
f1(t) f2(t)1/(2π)F1(jω) *F2(jω)
Transfer the convolution operation in one domain to the algebra
operation in another domain,Almost all of the properties discussed
above could be shown based on the convolution theorem:
Time shift,f (t - t0) = f (t )*δ (t - t0)F(jω)e-jωt0
Frequency shift:
f(t)ejω0t1/(2π)[F[j(ω )* 2πδ(ω-ω0)]=F[j(ω –ω0)]
§ 1.11 Fourier analyses of linear system
H(jω) is called system function,or transfer function
h(t)
H(jω)
f(t) y(t)
F(jω) Y(jω)
y(t) = f(t) * h(t)
Y(jω) = F(jω) H(jω)
Spectrum analyses is an active research area today,If
system is too complicated to be represented by analytical
expression,it could be done that input sin signals with
different frequencies and measure the system output,
then give the system transfer function,
