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tgoossens
tgoossens
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Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} g(t-u) x(u)$$Y(t) = \int_{-\infty}^{\infty} h(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?

Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} g(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?

Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} h(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?

deleted 2 characters in body; edited title
Source Link
tgoossens
tgoossens
  • 589
  • 1
  • 4
  • 8

Autocovariance Autocorrelation of convolution integral

Autocovariance Autocorrelation of convolution integral

Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} g(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?

Autocovariance of convolution integral

Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} g(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?

Autocorrelation of convolution integral

Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} g(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?

Source Link
tgoossens
tgoossens
  • 589
  • 1
  • 4
  • 8

Autocovariance of convolution integral

Work out the autocorrelation

$r_Y(\tau) = E[Y(t)Y(t+\tau)]$

with $Y(t) = \int_{-\infty}^{\infty} g(t-u) x(u)$ and $X$ a WSS, ergodic process

I always get: $h(t)* h(t+\tau) * r_X(\tau)$ (with $*$ convolution)

My approach

(after multiplication)

$\int h(t-u) ( \int h(t+\tau -u')r_X(u'-u) \text{d}u' ) \text{d}u$

The inner integral is the definition of a convolution hence $= \int h(t-u) ( h(t+\tau)*r_x(t+\tau-u) ) \text{d}u$

This gives again a convolution integral with time variable t. so,

$= h(t)* h(t+\tau) * r_X(\tau) $

Correct solution:

$ r_X(\tau) * h(\tau) * h(-\tau) $

Question

What am I doing wrong?