Signal Processing: Power Spectral Density

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Questions and Answers

What is the value of the limit of R(τ) as τ approaches infinity?

36 (correct)

What is the value of μx² given RXX(τ) = 18 + (1 + 4cos(12τ))/6 + τ²?

59 (correct)

What is the expression for the autocorrelation function RXX(τ)?

36 + 4/1 + 3τ²

6 + 3τ²

18 + (1 + 4 cos(12τ)) (correct)

40 + 1/1 + 3τ²

What is the condition for RXY(τ) to be a function of τ?

The processes X(t) and Y(t) are jointly wide sense stationary Signup and view all the answers

What is the value of E[X(t)]?

6 Signup and view all the answers

What is the value of E[X²(t)]?

40 Signup and view all the answers

What is the relationship between RXY(-τ) and RYX(τ)?

RXY(-τ) = RYX(τ) Signup and view all the answers

What is the value of μx given RX(0) = 18 + [1 + 4]?

18 Signup and view all the answers

What is the expression for the variance of X(t)?

E[X²(t)] - E[X(t)]² Signup and view all the answers

What is the value of the variance of X(t)?

4 Signup and view all the answers

What is the condition for RXY(τ) to be zero?

The processes X(t) and Y(t) are orthogonal Signup and view all the answers

What is the relationship between RXX(τ) and RXX(0)?

RXX(0) is the limit of RXX(τ) as τ approaches infinity Signup and view all the answers

What is the formula for the cross correlation function RXY(τ)?

E[X(t)Y(t - τ)] Signup and view all the answers

What does RXY(τ) represent?

The cross correlation function of X(t) and Y(t) Signup and view all the answers

What is the meaning of the symbol τ in the given equation?

Delay Signup and view all the answers

What is the property of RXY(τ) when the processes X(t) and Y(t) are orthogonal?

RXY(τ) = 0 Signup and view all the answers

What is the formula for the power spectral density S XX (ω)?

1 / (4 + ω²) Signup and view all the answers

What is the integral of e^(iωτ) from -∞ to ∞?

2π δ(τ) Signup and view all the answers

What is the value of R(τ)?

e^(-2τ) Signup and view all the answers

What is the average power of the process?

1/4 Signup and view all the answers

What is the contour used to evaluate the integral?

A closed contour consisting of the real axis from -R to R and the upper half of the circle z = R Signup and view all the answers

What is the value of the integral of 1 / (4 + z²) from -∞ to ∞?

π Signup and view all the answers

What is the value of the integral of e^(izτ) / (4 + z²) from -∞ to ∞?

π e^(-2τ) Signup and view all the answers

What is the value of R(0)?

1/4 Signup and view all the answers

What is the formula for R(τ)?

e^(-2τ) Signup and view all the answers

What is the value of the integral of e^(izτ) / (z - 2i) from -∞ to ∞?

2πi e^(-2τ) Signup and view all the answers

What is the given autocorrelation function?

RXX(τ) = e^(-λτ) Signup and view all the answers

What is the expression for the power spectral density S(ω)?

S(ω) = ∫RXX(τ)e^(-iωτ)dτ Signup and view all the answers

What is the result of integrating e^(-λτ)cos(ωτ) from -∞ to ∞?

2/(λ^2 + ω^2) Signup and view all the answers

Why is the second integral in the expression for S(ω) equal to zero?

Because the integrand is an odd function Signup and view all the answers

What is the final expression for the power spectral density S(ω)?

S(ω) = 2/(λ^2 + ω^2) Signup and view all the answers

What is the relationship between the autocorrelation function RXX(τ) and the power spectral density S(ω)?

RXX(τ) is the inverse Fourier transform of S(ω) Signup and view all the answers

What is the purpose of the Fourier transform in the expression for S(ω)?

To convert the autocorrelation function to the frequency domain Signup and view all the answers

What is the condition for the second integral in the expression for S(ω) to be zero?

The integrand is an odd function Signup and view all the answers

What is the significance of the parameter λ in the autocorrelation function RXX(τ)?

It represents the decay rate of the signal Signup and view all the answers

What is the relationship between the power spectral density S(ω) and the autocorrelation function RXX(τ)?

S(ω) is the Fourier transform of RXX(τ) Signup and view all the answers

What is the autocorrelation function of Y(t) in terms of RXX(τ)?

2 RXX(τ) - RXX(τ + 2a) - RXX(τ - 2a) Signup and view all the answers

What is the expression for Y(t) in terms of X(t) and a?

X(t + a) - X(t - a) Signup and view all the answers

What is the autocorrelation function of a stationary random process X(t)?

24τ^2 + 36 Signup and view all the answers

What is the mean of X(t) for the given autocorrelation function?

0 Signup and view all the answers

What is the variance of X(t) for the given autocorrelation function?

36 Signup and view all the answers

What is the relationship between RYY(τ) and RXX(τ)?

RYY(τ) = 2 RXX(τ) - RXX(τ + 2a) - RXX(τ - 2a) Signup and view all the answers

What is the expression for RYY(t) in terms of X(t) and a?

E[(X(t + a) - X(t - a))(X(t - τ + a) - X(t - τ - a))] Signup and view all the answers

What is the property of the random process X(t) that is used to derive the expression for RYY(τ)?

Wide-Sense Stationarity Signup and view all the answers

Study Notes

Autocorrelation Functions and Power Spectral Density

Autocorrelation function: RXX(τ) = E[X(t)X(t-τ)]
Power spectral density: SXX(ω) = ∫RXX(τ)e^{-iωτ}dτ

Random Signal with Autocorrelation Function

Given RXX(τ) = e^{-λτ}, find the power spectral density SXX(ω)
Solution: SXX(ω) = 2/(λ^2 + ω^2)

Random Process with Autocorrelation Function

Given RXX(τ) = 36 + 1/(1 + 3τ^2), find the mean and variance of X(t)
Solution: μx = 6, E[X^2(t)] = 40, variance = 4

Cross Correlation Function

Definition: RXY(τ) = E[X(t)Y(t-τ)]
Properties:
- RXY(-τ) = RYX(τ)
- If X(t) and Y(t) are orthogonal, then RXY(τ) = 0

Power Spectral Density and Average Power

Given SXX(ω) = 1/(4 + ω^2), find the average power of the process
Solution: R(τ) = 1/4e^{-2τ}, average power = 1/4

Wide Sense Stationary (WSS) Process

Definition: E[X(t)] and E[X(t)X(t-τ)] are finite and independent of t
If X(t) is a WSS process, then Y(t) = X(t+a) - X(t-a) is also a WSS process
RYY(τ) = 2RXX(τ) - RXX(τ+2a) - RXX(τ-2a)

Stationary Random Process

Given RXX(τ) = 24τ^2 + 36, find the mean and variance of X(t)
Solution: μx = ?, E[X^2(t)] = ?, variance = ? (NOT GIVEN IN THE TEXT)

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Description

This quiz is about finding the power spectral density of a random signal with an exponential autocorrelation function. Calculate the PSD given the autocorrelation function.