Power Spectral Density Function

Random signals may not always have Fourier transforms. Therefore, it is difficult to study the corre-sponding frequency spectra directly from the integral transformation. In this case, the Fourier trans-forms of their correlation functions are used, which exist for most engineering signals. Therefore, the frequency components of random signals can be studied.

2.3.3.2.1 Definition of Power Spectral Density Function

The Fourier transform of the autocorrelation function defines the auto-power spectral density func-tion, which can be written as

S11

( )

ω ⁼ R11

( )

τ e d^−jωτ τ

−∞

∫

∞ ^(2.237)

Thus,

R11

( )

τ ^↔S11

( )

ω (2.238)

Furthermore, the Fourier transform of the cross-correlation function defines the cross-power spectral density function. This can be written as

S12

( )

ω ⁼ R12

( )

τ e d^−jωτ τ

−∞

∫

∞ ^(2.239)

Thus,

R12

( )

τ ^↔S12

( )

ω (2.240)

Equations 2.238 and 2.240 are referred to as Wiener–Khintchine equations (Silva 2007).

2.3.3.2.2 Properties of Auto-Power Spectral Function

It can be proven that if the signal x(t) is the response obtained through the convolution of h(t)∗f(t), then

Sxx

( )

ω ⁼ H

( )

ω ²Sff

( )

ω (2.241)

Here, Sxx is the autopower spectral density function of the response x(t) and Sff is the autopower spectral density function of the forcing function f(t). From Equations 2.240 and 2.241,

Rxx τ H S d E x X

π ω ω ω

( )

τ₌ ⁼

( ) ( )

_{= ⎡⎣ ⎤⎦ =}

−∞

∫

∞ 0

2 2 2

21 ^ff (2.242)

In addition to Equations 2.241 and 2.242, other important properties of the Fourier pair Rxx(τ) and Sxx(ω) can exist, which are further explored in a later subsection. Here, the proof of Equation 2.241 is obtained, which is very important in the analysis of random signals.

To prove Equation 2.241,

Sxx ω Rxx τ e dj^ωτ τ x σ x σ τ σd e dj^ωτ σ

( )

⁼

( )

^{= ⎡}

( ) (

⁺

)

⎣

⎢⎢

⎤

⎦

⎥⎥

−

−∞

∞

−∞

∞ −

∫

−∞τ

∫

_{( )} ^ττ

∞

∫

( ) ^(2.243)

Here, in order to clearly denote the integrals, the integration symbols are followed by subscripts (τ) and (σ), which stand for the integrations for variables τ and σ, respectively. In the following dis-cussion, similar notations are used to indicate the specific integrations.

Note that x(σ) is the response of time variable σ, which can be expressed as the convolution of the forcing function f(σ) and the unit impulse response function h(σ), that is,

x

( )

σ ⁼ f

(

σ μ⁻

) ( )

h μ μd

−∞

∫

∞ ^(2.244)

Similarly,

x

(

σ τ+

)

⁼ f

(

σ μ τ^{− +}

) ( )

h μ μd

−∞

∫

∞ ^(2.245)

Substituting Equations 2.244 and 2.245 into Equation 2.243 yields

In order to denote clearly the two convolutions described in Equations 2.244 and 2.245, differ-ent variables are used. Namely, for the first convolution integral, variable μ is used, whereas in the second convolution integral, variable ν replaces μ. Such a replacement is only for, convenience of notation; in fact,

Because μ = ν from Equation 2.247, the terms within the braces in Equation 2.248 are rewritten as

Substituting Equation 2.249 into Equation 2.248 yields

Inserting 1 = e^–jωμe^jωμ between h(μ) and h(ν), Equation 2.250 can be written as

and Equation 2.241 is established. Thus,

Sxx

( )

ω ⁼ H

( )

ω ²Sff

( )

ω (2.257)

Next, since the autocorrelation function is an even function of τ (see Equation 2.234), the autopower spectral density function must be real valued, although in general the Fourier transform of a signal x(t) is complex valued.

Furthermore, from Equation 2.234, it is proven that the autopower spectral density function must be an even function of ω. This is because

Sxx Rxx e dj R e d

Finally, from the above derivations, it is also realized that the autopower spectral density func-tion must always be greater than zero. That is,

Sxx

( )

ω ^≥0 (2.259)

2.3.3.2.3 Physical Meaning of Autopower Spectral Density Function

In the above, the power spectral function was defined as the Fourier transform of the correlation function. Then, several important properties of the function were examined to emphasize the rela-tionship between the correlation functions and the power spectral density functions. In the following discussion, the physical meaning of the power spectral density function is considered in more detail.

Recall Section 2.2, where it was noted that the existence of the Fourier transform of signal f(t) needs certain conditions. Recall the absolutely integrable condition,

f t dt

( )

^{< ∞}

−∞

∫

∞ ^(2.260)

Then, the time domain signal f(t) can have its Fourier transform pair F(ω).

Now, from the theorem of multiplication, in Equation 2.146, let f₂= f1= f. Then,

f t dt² 1 d ²d

Equation 2.261 is referred to as the Parseval equation. The left side of the equation stands for the total energy within the range of (– ∞, + ∞). On the right side of the equation, the term |F(ω)|² is called the energy spectrum. In this sense, the Parseval theorem can be seen as the energy equation of signal f(t).

It is understandable that many engineering signals within the range of (– ∞, + ∞) have an infinite amount of energy; in addition, the condition in Equation 2.260 may often be violated. The sine function is a simple example of such signals. Random signals also often fall into such a category. To study the frequency components, an alternative approach must be used. For example, instead of the energy, the average power can be written as

_Tlim

T T

T f t dt

→∞2¹

∫

− ²

( )

(2.262)

Now, assume that the signal f(t)|−T ≤ t ≤ T satisfies the Dirichlet conditions, so that it has the Fourier transform F(ω,T) written as

The corresponding Fourier pair satisfies the Parseval energy equation, that is,

f t _{T t T}dt f t _{T t T}dt F T d

Dividing by 2T on both sides of Equation 2.264, and letting T → ∞, results in

It is seen that Equation 2.265 represents the energy density. In fact, the term inside the integration sign on the right side is the average power density function of the transient signal f(t), denoted by

Sff

( )

ω ⁼_Tlim_→∞ 1T F

(

ω,T

)

2

2 (2.266)

In Equation 2.266, the same symbol S( )( )⋅ ⋅ is used and the identical name of “power density func-tion” is used as in Equation 2.237, where the concept of the power density function was introduced by using the Fourier transform of autocorrelation functions. To see the logic in this approach, the essence of Equation 2.266 is examined by further considering random signals, instead of transient signals. From now on, assume that the average power always exists, and for convenience, use xR(t) instead of f(t) in the remaining analysis, where xR(t) represents a stationary random process. As mentioned before, xR(t) is a collection of random signals xi(t). Thus, Equations 2.263 and 2.265 are written as follows:

where X(ω, T) is the Fourier transform of the random process xR(t)|−T ≤ t ≤ T.

Note that the integrations in Equations 2.267 and 2.268 are random. The limit of the mathemati-cal expectation of the left term in Equation 2.267 is considered, which is the average power of the stationary process xR (t), written as

_Tlim R lim Equation 2.212. It is seen that in Equation 2.212, x(t) is a deterministic signal, whereas in Equation 2.269, xR(t) is a random process. Therefore, to study the power of xR(t), the statistical average is needed and the mathematical operation of expectation E[( )]⋅ is used. In addition, the integration range (0, T) in Equation 2.212 and (– T, T) in Equation 2.268 do not actually make an essential dif-ference, especially for engineering signals. Therefore, Equation 2.269 implies that the average power of a stationary process equals the mean square value of that process. Additionally, the follow-ing notation is possible:

Furthermore, exchange the order of operations for the term on the right in Equation 2.268 and consider Equations 2.269 and 2.270. Then,

X²= _→∞ ^⎡_⎣⎢

( )

^⎤_⎦⎥

Examining the integral function in Equation 2.271, and comparing it with Equation 2.266,

S

TE T

xx

( )

ω ⁼T^lim_→∞2^{1 ⎡}_⎣⎢^X

(

ω^,

)

^2⎤_⎦⎥ ^(2.272)

where Sxx(ω) is the autopower spectral density function. Apparently, from Equation 2.272, Sxx(ω) is a real-valued, nonnegative function for the term |X(ω, T)|², similar to the integrand in the integration operation of E[( )]⋅ that is real valued and nonnegative. However, up to now, the term defined in Equation 2.272 as well as in Equation 2.266 has been the power spectral density, which has been used to name the Fourier transform of the autocorrelation function (see the Wiener–Khintchine Equation 2.237).

To see the term defined in Equation 2.272 as the Fourier transform of the autocorrelation func-tion, substitute Equation 2.267 into Equation 2.272. That is,

Substituting Equation 2.274 into Equation 2.273 yields

Sxx T T R t t exx dt dt

Comparing Equation 2.276 with Equation 2.237, it is realized that the newly defined power tral function is indeed the autopower spectral function. In addition, the reason that the power spec-tral density actually describes the statistics of the distribution of the frequency components of the stationary process x_R(t) can be understood.

2.3.3.2.4 Cross-Power Spectral Density Function

Similar to the definition of the autopower spectral density function, the cross-power density spectral function had been defined in Equation 2.239. Particularly, R12(t) can be seen as the cross-correlation

function of the response x(t) and the forcing function f(t), both of which are random signals.

Additionally, similar to the explanation of the physical meaning of the autopower spectral density function,

Sxf

( )

ω ⁼Tlim_→∞ 1TE_⎡⎣

(

⁻ω,T

) (

ω,T

)

_{⎤⎦ =}Tlim_→∞ TE ^*

(

ω,T

) (

ω,T

)

2 1

X F 2 ⎡⎡⎣X F ⎤⎦ (2.277)

Now, some important properties of the cross-power spectral density function are noted as follows:

i. Sxf

( )

ω ⁼ X F^{* =}Xfx^*

( )

ω (2.278)

ii. Re_⎡⎣Sxf

( )

ω _{⎤⎦ =}Re_⎡⎣Sxf

( )

⁻ω _⎤⎦ ^(2.279)

and

Im_⎡⎣Sxf

( )

ω _{⎤⎦ = −}Im_⎡⎣Sxf

( )

⁻ω _⎤⎦ ^(2.280)

iii. Sxf

( )

ω ^≤Sxx

( ) ( )

ω Sff ω (2.281)

In addition, the autopower and cross-power spectral functions can be used to extract the transfer function from a random excitation F(t) or random response x_R(t) process. Using Equation 2.278 and replacing F by X, results in S_xx(ω) = X^*X and it can be proven that

Suppose that the random forcing functions and responses are measured. Denote Fi(ω) and Xi(ω) as the Fourier transforms of the i^th random forcing function fi(t) and response function xi(t), respectively. (Note that Fi(ω) and Xi(ω) always exist, because once fi(t) and xi(t) are mea-sured, these become deterministic and have finite time duration.) Then, the calculated transfer function is

By multiplying X^*i(ω) in both the numerator and the denominator on the right side of Equation 2.283,

H S mea-sured response and forcing functions can approximate the power density function Sxx and Sxf. That is,

and

S

n S

xf x f

i n

≈ i i

∑

=

1

1

(2.286)

The result is the second part of Equation 2.282. The first part of Equation 2.283 can be proven via a similar methodology.

Example 2.8

Suppose there is significant noise mixed with the measurement of the forcing function, whereas when the signal of the response is picked up, the noise can be ignored. This case can be shown graphically, as in Figure 2.16a.

When the autopower spectral density function of the measured force is used, it is realized that

_⎡⎣^F

( )

^{ω +N}

( )

^ω _⎤⎦⎡⎣^F

( )

^{ω +N}

( )

^ω _{⎤⎦ =}^{* F}

( ) ( )

^{ω F* ω +N}

( ) ( )

^{ω F* ω +F}

( )

^{ω +N*}

( )

^{ω ++}

( ) ( )

=

( )

⁺

( )

⁺

( )

⁺

( )

N N

ff nf fn nn

ω ω

ω ω ω ω

*

S S S S

Here, S_ff(ω) and Snn(ω) are, respectively, the autocorrelation functions of the force and the noise, while Snf(ω) and Sfn(ω) are, respectively, the cross-correlation functions of the force and the noise.

In many cases, the noise N(ω) is not correlated with the forcing function and is not correlated with itself. Therefore, roughly,

^Snf

( )

^{ω =Sfn}

( )

^{ω =Snn}

( )

^{ω =}0

In this way, if the first part of Equation 2.282 is used to calculate the transfer function, the noise from the input sides can be eliminated, that is, with

H S

S

1 ω fx ω

( )

⁼

( )

ω

( )

ff (2.287a)

F(ω)

F(ω) F(ω)+N(ω)

F(ω) H(ω)

H(ω)

X(ω)

N(ω) X(ω)

F(ω)+N(ω) N(ω)

(a)

(b) +

+

FIGURE 2.16  Noises and transfer function calculation: (a) input with significant noise and (b) output with significant noise.

If the measured response is contaminated with significant noise but the noise is relatively small from the input side, as in Figure 2.16b, then the second part of Equation 2.282 should be used to calculate the transfer function. Namely,

H S

S

xx

2 ω xf ω

( )

⁼

( )

ω

( )

^(2.287b)

Similar to the first case of very large input noise, it can be proven that the noise from the output side will be greatly reduced by using [X(ω) + N(ω)][X(ω) + N(ω)]^* to obtain S_xx(ω).

The above two equations represent the transfer functions. Subscripts 1 and 2 are used because in the literature, the computation of a transfer function that reduces input noise (see Figure 2.16a) is referred to as an H1 transfer function. On the other hand, the computation of a transfer function that reduces the output noise (see Figure 2.16b) is referred to as an H2 transfer function.

2.3.4  correlation between Forcing Function and imPulSe reSPonSe Function

In this subsection, the relationship between the Laplace transform of a temporal signal, which can be seen as a forcing function, is introduced, and the correlation functions between the forcing function and the unit impulse response function are examined. In this way, the essence of the Laplace variables and Laplace transform can be better understood, and the effect of damping can be further examined.

In Equation 2.154, the variable υ > 0 in many cases, which can be denoted as

υ ξω= n (2.288)

And the variable ω is now denoted as

ω= −ωd (2.289)

In Equations 2.288 and 2.289, ξ, ωn and ωd can be seen as the damping ratio, natural and damped frequencies of a SDOF system. Thus, the Laplace transform of a temporal function that is seen as a general forcing function f(t) can be written as

_⎡⎣f t

( )

_{⎤⎦ =}F s

( )

⁼

∫

0^∞f t e dt

( )

^{−st =}

∫

0^∞f t e

( )

^{(− n+j d)t}dt

ξω ω

(2.290)

Multiplying by (e− τ^s)/(mωd)T on both sides of Equation 2.290 yields

e

m TF s e

m T f t e dt

T f t e m dt

s s

st s t

− −

∞ − ∞ −( )+

( )

⁼

∫ ( )

⁼

∫ ( )

τ τ τ

ωd ωd 0 0 ωd

1 (2.291)

This operation is equivalent to making a time shift τ of the forcing function f(t). In addition, the operation in Equation 2.290 also divides the Laplace transform F(s) by the factor mωd.

Next, taking the complex conjugate of the above equation results in

e

m T⁻ω^s*d^τ F s

( )

^{* =} T

∫

^∞f t e

^{( )}

m^{−s t*}ω^{( )+}d^τ dt 1

0 (2.292)

Dividing both sides of Equations 2.291 and 2.292 by 2j and subtracting the resulting second equation from the first one yields

From the previous section, it is known that the unit impulse response function with time shift τ, h t( )+ τ of a linear SDOF vibration system (m-c-k) system can be written as,

h t e

m ^t t

(

+^τ

)

^{= −ξω}_ω^{n(+τ) ω}

(

^+τ

)

d sin d

In Equation 2.293, the operation of subtraction causes the left side term in Equation 2.292 to be real valued. Thus, Equation 2.293, can be rewritten as

e

Note that the term on the right side of Equation 2.293 is the cross-correlation function of f(t) and h(t + τ), denoted by Rfh(τ), Thus,

And note that, similar to the theorem of time shift for Fourier transform (see Equation 2.143), multiplying F(s) by e^−sτ gives the Laplace transform of f(t − τ). Namely,

_⎡⎣αf t

(

−τ

)

_{⎤⎦ +⎡⎣}αf t

(

⁻τ

)

_{⎤⎦ =}^* Rfh

( )

τ ⁼Rhf

( )

⁻τ ^(2.297) Furthermore, denote

F s F e( ) = 0 ^jθ (2.298)

Here F₀ and θ are the real-valued amplitude and phase angle of the Laplace transform F(s). Both are functions of s, namely, the function of the eigenvalue λ of the SDOF vibration system described in Equation 1.2, namely,

− = = −s λ ξωn±jωd

and

F F0= 0

( )

^λ

θ θ λ=

( )

With the help of Equation 2.298, the right side of Equation 2.297 becomes

F e

m T e e e e

j F e

m T t

j j j j

0 0

2

−^{ξω τ ω τ θ}− −^{ω τ} −^θ = −^{ξω τ}

(

+

)

ω ^{n d d} ω ⁿ ω θ

d d sin d (2.299)

Therefore,

R F e

m T t

fh τ

ω ω θ

( )

^{= 0 −}ξω τⁿ

(

⁺

)

d sin d (2.300)

Equation 2.300 implies that the cross-correlation function equals the amplitude of the corre-sponding Laplace transform F(s) times a function, which has the identical form of the unit impulse response h(t), except with a phase shift θ. Here, θ is the phase of F(s).

It is also seen that such a correlation function, obtained through the above manipulation of the Laplace transform of the forcing function, is the product of two functions. The function sin(ωdτ + θ) is a sinusoidal function with the damped natural frequency ωd and the phase shift θ. The function e^{−ξω τn} makes a decaying envelope with respect to time τ. It is seen that as the damping ratio ξ increases, the decay occurs more quickly.

It is also known that when the Laplace variable s is chosen, there is always a corresponding SDOF system with the solution λ of its characteristic to be – s = – ξωn± jωd. The impulse response function used above is exactly determined through this SDOF system.

The above discussion implies that the sum of the Laplace transform and its complex conjugate of a forcing function with time delay τ, denoted by f(t – τ) and a proportional factor α equals the cross correlation of an impulse response function h(t) of a SDOF system and the function of f(t − τ). Due to the aforementioned orthogonality of sine and cosine functions, it is seen that the correlation inte-gration will eliminate all uncorrelated frequency components in f(t) and only the frequency equal to the natural frequency ωn of the SDOF system remains and the damping ratio of this system is ξ.

Therefore, the operation of this correlation integral can be seen as a result of filtering by a SDOF system with eigenvalue λ. Thus, Equation 2.297 further implies that the Laplace transform of a signal f(t) can be seen as the filtered result through the corresponding SDOF systems, which act as a series of mechanical filters.

Example 2.9

Let the forcing function be a unit delta function δ(t), then

_⎡⎣δ

( )

t_{⎤⎦ = 1}

Thus,

This equation implies that the cross-correlation function of the unit impulse with a unit impulse response function is proportional to the unit impulse response function itself.

Example 2.10

Let the forcing function be a unit step u(t) and

u t t

The above should be equal to Rfh(τ), which can be proven to have the following form:

2.3.5  baSic aPProach to dealing with random vibrationS

As a brief summary, the basic approach to dealing with random vibrations is averaging with proper methods. For example, both mean value and standard deviation are arithmetic averages in dealing with random variables; the correlation functions are averages of the products of two sets of signals taken from certain time points.

The main effort in averaging is the summation for discrete signals, or integration for continuous signals, which helps distinguish specific frequency components based on the orthogonality of the vibration signals. The summation and/or integration will also help eliminate unwanted noises and uncertainties and indicate the major trends of certain random sets.

In document Structural Damping (Page 142-155)