linear system with random inputs chapter 8 ... - …

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


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


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)


– 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)


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)


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)


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 ....


Figure 8.1-3


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)


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,


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]


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)


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.


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.


– 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.


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)


h(τ ) ≈(

2

N0

)RXY (τ ) (8.3-3)

h(τ ) =

(2

N0

)RXY (τ ) ≈ h(τ ) (8.3-4)


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)


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)


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)


– 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


• When X(t) is a bandpass process,

Figure 8.4-2

– The power meter output is


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.


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)


– 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)

linear system with random inputs chapter 8 ... - …

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