linear system with random inputs chapter 8 ... - …
TRANSCRIPT
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 1
Chapter 8
LINEAR SYSTEM WITH RANDOM INPUTSHandout by Joon Ho Cho & Kyeongcheol YangEECE 302, Spring 2021 based on P. Z. Peebles, Probability, Random Variables and Random Signal Principles, 4th ed.
8.0 INTRODUCTION
8.1 LINEAR SYSTEM FUNDAMENTALS
8.2 RANDOM SIGNAL RESPONSE OF LINEAR SYSTEMS
8.3 SYSTEM EVALUATION USING RANDOM NOISE
8.4 SPECTRAL CHARACTERISTICS OF SYSTEM RESPONSE
8.5 NOISE BANDWIDTH
8.6 BANDPASS, BAND-LIMITED, AND NARROWBAND PROCESSES
8.7 SAMPLING OF PROCESSES
8.8 DISCRETE-TIME SYSTEMS
8.9 MODELING OF NOISE SOURCES
8.10 INCREMENTAL MODELING OF NOISY NETWORKS
8.11 MODELING OF PRACTICAL NOISY NETWORKS
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 2
8.0 INTRODUCTION
• deterministic signal vs. random signal
• deterministic system vs. random system
• deterministic signal input to a deterministic system
• stochastic input (random signal input) to a deterministic system
• What is the output? How to describe it?
• linear system only
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 3
8.1 LINEAR SYSTEM FUNDAMENTALS
• review of signal and systems
• single-input single-output linear system
• linear time-invariant (LTI) system
The General Linear System
• a system
– A system is a triplet (X ,M,Y)
– X : the set of admissible inputs
– Y : the set of possible outputs
–M: the mapping rule
• a system with vector spaces X = RN and Y = RM over a field R for scalar
multiplication
– Def. When xn ∈ RN and yn =M[xn] ∈ RM , the system (X ,L,Y) is linear if
y = L[∑n
αnxn]] =∑n
αnL[xn] =∑n
αnyn. (8.1-1)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 4
– Lemma. If a system is linear, then there exists H ∈ RM×N such that
y = Hx.
• a system with X and Y subsets of continuous-time (CT) signals
– a continuous-time system = an operation on x(t) to cause y(t)
– Def. The system (X ,L,Y) is linear if
y(t) = L[∑n
αnxn(t)]] =∑n
αnL[xn(t)] =∑n
αnxn(t). (8.1-2)
∗ Lemma. If a system is linear, then there exists h(t, τ ) such that
y(t) =
∫ ∞−∞
h(t, τ )x(τ )dτ,
which is the I/O relation of a general CT linear system.
∗ Proof. based on the sifting property of the Dirac delta function
x(t) =
∫ ∞−∞
x(ξ)δ(t− ξ)dξ
∗ Def. the impulse response of the linear system h(t, ξ)
· the response of the system at time t to the impulse applied at time ξ
L[δ(t− ξ)] = h(t, ξ)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 5
Figure 8.1-1
Linear Time-Invariant Systems
• Def. A CT linear system is time-invariant
h(t, ξ) = h(t− ξ) (8.1-3)
• the I/O relation of a CT LTI system
y(t) =
∫ ∞−∞
x(ξ)h(t− ξ)dξ (8.1-4)
• the convolution integral of x(t) and h(t)
y(t) = x(t) ∗ h(t) (8.1-5)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 6
y(t) =
∫ ∞−∞
h(ξ)x(t− ξ)dξ (8.1-6)
Time-Invariant System Transfer Function
• In the time domain, an LTI system is completely characterized by its impulse
response.
• In the frequency domain,
Y (f ) =
∫ ∞−∞
y(t)e−j2πftdt =
∫ ∞−∞
[∫ ∞−∞
x(ξ)h(t− ξ)dξ
]e−j2πftdt
=
∫ ∞−∞
x(ξ)
[∫ ∞−∞
h(t− ξ)e−j2πf(t−ξ)dt
]e−j2πfξdξ
=
∫ ∞−∞
x(ξ)H(f )e−j2πfξdξ = X(f )H(f ) (8.1-7)
• H(f ): the transfer function of the system
• Def. an alternative definition of the transfer function
x(t) = ej2πft (8.1-8)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 7
H(f ) =h(t) ∗ ej2πft
ej2πft(8.1-9)
>> Example 8.1-1. >> Example 8.1-2.
Figure 8.1-2
Idealized Systems
• an ideal lowpass system
• an ideal bandpass system
• the 3-dB bandwidth
• the noise bandwidth
Causal and Stable Systems
• Def. An LTI system is causal (physically realizable) if ....
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 8
Figure 8.1-3
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 9
h(t) = 0 for t < 0 (8.1-10)
• Def. An LTI system is bounded-input bounded-output (BIBO) stable if ...
– Lemma. The system is BIBO stable iff∫ ∞−∞|h(t)|dt <∞ (8.1-11)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 10
8.2 RANDOM SIGNAL RESPONSE OF LINEAR SYSTEMS
• temporal characteristic of the random signal response of linear systems
System Response – Convolution
• deterministic signal input
y(t) =
∫ ∞−∞
h(ξ)x(t− ξ)dξ =
∫ ∞−∞
x(ξ)h(t− ξ)dξ (8.2-1)
• stochastic input
Y (t) =
∫ ∞−∞
h(ξ)X(t− ξ)dξ =
∫ ∞−∞
X(ξ)h(t− ξ)dξ, (8.2-2)
which is, at each t, a weighted integration of uncountably infinite number of jointly
distributed random variables.
– We need the notion of Calculus (dealing with continuity, differentiation,
integration) on a random process
– the mean-square Calculus
∗ Def. Given a random process X(t), it is MS continuous at t0 if
limt→t0
E[|X(t)−X(t0)|2
]= 0,
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 11
and denoted by using the limit in the mean (l.i.m) as
X(t0) = l.i.m.t→t0
X(t).
∗ Def. Given a random process X(t), it is MS continuous if it is MS
continuous for all t.
∗ Def. Given a random process X(t), if there exists a random process X ′(t)
such that
limε→0
E
[∣∣∣∣X(t + ε)−X(t)
ε−X ′(t)
∣∣∣∣2]
= 0,
then we call X ′(t) the MS derivative of X(t) and denote it as
X ′(t) , l.i.m.ε→0
X(t + ε)−X(t)
ε.
∗ Def. Given a random process X(t), if there exists a random variable I such
that ..., then I is called the MS integral of X(t) on the interval [a, b]
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 12
Mean and Mean-Squared Value of System Response
• If X(t) is WSS, the mean value of Y (t) is
E[Y (t)] = E
[∫ ∞−∞
h(ξ)X(t− ξ)dξ
]=
∫ ∞−∞
h(ξ)E[X(t− ξ)]dξ
= X
∫ ∞−∞
h(ξ)dξ = Y constant (8.2-3)
• The mean-squared value of Y (t) is
E[|Y (t)|2] = E
[(∫ ∞−∞
h(ξ1)X(t− ξ1)dξ1)∗ ∫ ∞
−∞h(ξ2)X(t− ξ2)dξ2
]=
∫ ∞−∞
∫ ∞−∞
E[X∗(t− ξ1)X(t− ξ2)]h∗(ξ1)h(ξ2)dξ1dξ2 (8.2-4)
which is in general a function of t.
– If X(t) is WSS, i.e.,
E[X∗(t− ξ1)X(t− ξ2)] = RXX(ξ1 − ξ2) (8.2-5)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 13
then the mean-squared value of Y (t) is
E[|Y (t)|2] =
∫ ∞−∞
∫ ∞−∞
RXX(ξ1 − ξ2)h∗(ξ1)h(ξ2)dξ1dξ2 (8.2-6)
which is not a function of t, but a constant.
>> Example 8.2-1.
Autocorrelation Function of Response
• The autocorrelation function of Y (t) is
RY Y (t, t + τ ) = E[Y ∗(t)Y (t + τ )]
= E
[(∫ ∞−∞
h(ξ1)X(t− ξ1)dξ1)∗ ∫ ∞
−∞h(ξ2)X(t + τ − ξ2)dξ2
]=
∫ ∞−∞
∫ ∞−∞
E[X∗(t− ξ1)X(t + τ − ξ2)]h∗(ξ1)h(ξ2)dξ1dξ2 (8.2-7)
– If X(t) is WSS, then
RY Y (τ ) =
∫ ∞−∞
∫ ∞−∞
RXX(τ + ξ1 − ξ2)h∗(ξ1)h(ξ2)dξ1dξ2 (8.2-8)
i.e.,
RY Y (τ ) = h∗(−τ ) ∗RXX(τ ) ∗ h(τ ) (8.2-9)
Hence, Y (t) is WSS.
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 14
Cross-Correlation Functions of Input and Output
• The cross-correlation function of X(t) and Y (t) is
RXY (t, t + τ ) = E[X∗(t)Y (t + τ )] = E
[X∗(t)
∫ ∞−∞
h(ξ)X(t + τ − ξ)dξ
]=
∫ ∞−∞
E[X∗(t)X(t + τ − ξ)h(ξ)dξ (8.2-10)
– If X(t) is WSS, then
RXY (τ ) =
∫ ∞−∞
RXX(τ − ξ)h(ξ)dξ (8.2-11)
i.e.,
RXY (τ ) = RXX(τ ) ∗ h(τ ). (8.2-12)
Similarly,
RY X(τ ) =
∫ ∞−∞
RXX(τ − ξ)h∗(−ξ)dξ (8.2-13)
i.e.,
RY X(τ ) = h∗(−τ ) ∗RXX(τ ) (8.2-14)
Hence,
– X(t) and Y (t) are jointly WSS.
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 15
– the relation between autocorrelation function and cross-correlation function
RY Y (τ ) =
∫ ∞−∞
h∗(ξ1)RXY (τ + ξ1)dξ1 (8.2-15)
i.e.,
RY Y (τ ) = h∗(−τ ) ∗RXY (τ ) (8.2-16)
Similarly,
RY Y (τ ) =
∫ ∞−∞
RY X(τ − ξ2)h(ξ2)dξ2 (8.2-17)
i.e.,
RY Y (τ ) = RY X(τ ) ∗ h(τ ) (8.2-18)
>> Example 8.2-2.
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 16
8.3 SYSTEM EVALUATION USING RANDOM NOISE
• Suppose we desire to find the impulse response of some LTI system.
– system identification problem
– System Identification is an important topic in Control.
– Among various ways,
∗ What if we have available a broadband (relative to the system) noise source
having a flat power spectrum,
∗ and a cross-correlation measurement device?
Figure 8.3-1
RXX(τ ) ≈(N0
2
)δ(τ ) (8.3-1)
RXY (τ ) ≈∫ ∞−∞
(N0
2
)δ(τ − ξ)h(ξ)dξ =
(N0
2
)h(τ ) (8.3-2)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 17
h(τ ) ≈(
2
N0
)RXY (τ ) (8.3-3)
h(τ ) =
(2
N0
)RXY (τ ) ≈ h(τ ) (8.3-4)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 18
8.4 SPECTRAL CHARACTERISTICS OF SYSTEMRESPONSE
• To find the PSD of the output process
– First, find the autocorrelation function of the output. Then, apply Fourier
transform.
– An alternative approach?
• The input process X(t) is assumed to be WSS.
Power Density Spectrum of Response
• the power density spectrum SY Y (f ) of the response of an LTI system having a
transfer function H(f )
SY Y (f ) = SXX(f )|H(f )|2 (8.4-1)
• the power transfer function of the system |H(f )|2
• proof:
SY Y (f ) =
∫ ∞−∞
RY Y (τ )e−j2πfτdτ (8.4-2)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 19
SY Y (f ) =
∫ ∞−∞
h∗(ξ1)
∫ ∞−∞
h(ξ2)
∫ ∞−∞
RXX(τ + ξ1 − ξ2)e−j2πfτdτdξ2dξ1 (8.4-3)
SY Y (f ) =
∫ ∞−∞
h∗(ξ1)ej2πfξ1dξ1
∫ ∞−∞
h(ξ2)e−j2πfξ2dξ2
∫ ∞−∞
RXX(ξ)e−j2πfξdξ
(8.4-4)
SY Y (f ) = H∗(f )H(f )SXX(f ) = SXX(f )|H(f )|2 (8.4-5)
• the average power in the system’s response
PY Y =
∫ ∞−∞
SXX(f )|H(f )|2df (8.4-6)
>> Example 8.4-1.
Cross-Power Density Spectrums of Input and Output
SXY (f ) = SXX(f )H(f ) (8.4-7)
SY X(f ) = H∗(−f )SXX(f ) (8.4-8)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 20
Measurement of Power Density Spectrums
• the practical measurement of a power density spectrum
• Spectral Estimation: an important topic in Signal Processing
• Measurement of a power density spectrum of a real-valued lowpass, stationary, and
ergodic process X(t)
– The power meter output is
PY Y (fIF) =
∫ ∞−∞
SXX(f )|H(f )|2df
= 2
∫ ∞0
SXX(f )|H(f )|2df
≈ 2SXX(fIF)
∫ ∞0
|H(f )|2df
= 2SXX(fIF)|H(fIF)|2WN (8.4-9)
where the last form uses a quantity called noise bandwidth defined as
WN ,
∫ ∞0
|H(f )|2df
|H(fIF)|2(8.4-10)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 21
– Hence, we have an approximation to SXX(f ) at f = fIF as
SXX(fIF) ≈ PY Y (fIF)
2WN |H(fIF)|2(8.4-11)
– By varying fIF, the system can measure the power spectrum for various f .
Figure 8.4-1
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 22
• When X(t) is a bandpass process,
Figure 8.4-2
– The power meter output is
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 23
PY Y (fIF) =
∫ ∞−∞
A20
4[SXX(f − f0 − fIF − fc) + SXX(f + f0 + fIF + fc)]|H(f )|2df
=A2
0
2
∫ ∞0
SXX(f − f0 − fIF − fc)|H(f )|2df
≈ A20
2SXX(f0 + fc)
∫ ∞0
|H(f )|2df
=A2
0
2SXX(f0 + fc)WN |H(fIF)|2 (8.4-12)
where WN is the filter’s noise bandwidth. Thus,
SXX(f0 + fc) ≈2PY Y (fIF)
A20WN |H(fIF)|2
(8.4-13)
– For proper performance of the system, WN � W is required so that
SXX(f ) ≈ SXX(fIF) for all frequencies near fIF that are in the passband of the
filter.
– Furthermore fIF must not be chosen too small.
>> Example 8.4-2.
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 24
8.5 NOISE BANDWIDTH
• Consider a lowpass transfer function H(f )
– the total average power
PY Y =
∫ ∞−∞
(N0
2
)|H(f )|2df (8.5-1)
– assuming the system impulse response is real
PY Y = N0
∫ ∞0
|H(f )|2df (8.5-2)
– Consider an idealized system that is equivalent to the actual system
– in the sense that both produce the same output average power
– when they both are excited by the same white noise source,
– and both have the same value of power transfer function at midband; that is
|H(0)|2 is the same in both systems.
– a rectangularly shaped power transfer function
|HI(f )|2 =
{|H(0)|2 |f | < WN
0 |f | > WN
(8.5-3)
LINEAR SYSTEM WITH RANDOM INPUTS 8 – 25
– WN is selected to make output powers equal.
– the output power in the idealized system
∫ ∞−∞
(N0
2
)|HI(f )|2df = N0
∫ WN
0
|H(0)|2df =N0|H(0)|2WN
2π(8.5-4)
– the noise bandwidth of the system
WN =
∫ ∞0
|H(f )|2df
|H(0)|2(8.5-5)
>> Example 8.5-1.
• Consider a bandpass transfer function H(f ) with a centerband frequency fIF
–
WN =
∫ ∞0
|H(f )|2df
|H(fIF)|2(8.5-6)
PY Y = N0|H(fIF)|2WN (8.5-7)