Power Spectral Density - Telecommunications - Lecture Slides, Slides of Telecommunication electronics

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Power Spectral Density

Summary of Random Variables

• Random variables can be used to form models of a communication system
• Discrete random variables can be described using probability mass functions
• Gaussian random variables play an important role in communications - Distribution of Gaussian random variables is well tabulated using the Q-function - Central limit theorem implies that many types of noise can be modeled as Gaussian

Terminology

• A stationary random process has statistical properties which do not change at all with time.
• A wide sense stationary (WSS) process has a mean and autocorrelation function which do not change with time.
• Unless specified, we will assume that all random processes are WSS and ergodic.

Spectral Density

Although Fourier transforms do not exist for random processes (infinite energy), but does exist for the autocorrelation and cross correlation functions which are non-periodic energy signals. The Fourier transforms of the correlation is called power spectrum or spectral density function (SDF).

∫

∞

−∞

ℑ{ x ( t )}= X (ω )= x ( t ) e − j^ ω tdt

The Fourier transform of a non-periodic energy signal x(t) is

The original signal can be recovered by taking the inverse Fourier transform

∫

∞

−∞

ℑ−^1 { X (ω)}= x ( t ) = X (ω ) ej ω^ td ω

x ( t ) ↔ X ( ω )

Remarks and Properties

X (ω ) = X (ω) ej φ^ ω

The Fourier transform is a complex function in ω having amplitude and phase, i.e.

Autocorrelation

• Autocorrelation measures how a random process changes with time.
• Intuitively, X(1) and X(1.1) will be more strongly related than X(1) and X(100000).
• Definition (for WSS random processes):
• Note that Power = RX(0)

R (^) X (τ ) = E [ X ( t ) X ( t − τ)]

Power Spectral Density

• P( ω ) tells us how much power is at each

frequency

• Wiener-Klinchine Theorem:
  • Power spectral density and autocorrelation are a Fourier Transform pair!

P (ω ) = ℑ { R ( τ )}

Gaussian Random Processes

• Gaussian Random Processes have several special properties: - If a Gaussian random process is wide-sense stationary, then it is also stationary. - Any sample point from a Gaussian random process is a Gaussian random variable - If the input to a linear system is a Gaussian random process, then the output is also a Gaussian process

Linear System

• Input: x(t)
• Impulse Response: h(t)
• Output: y(t)

x(t) h(t) y(t)

Power Spectrum or Spectral Density Function (PSD)

• For deterministic signals, there are two ways to

calculate power spectrum.

• Find the Fourier Transform of the signal, find magnitude squared and this gives the power spectrum, or
• Find the autocorrelation and take its Fourier transform
• The results should be the same.
• For random signals, however, the first approach can not be used.

Let X(t) be a random with an autocorrelation of R (^) xx ( τ ) (stationary), then

and

S XX ω RXX τ e^ − j ωτ^ d τ

∞

−∞

( ) = (^) ∫ ( )

R XX τ SXX e^ − j ωτ^ d

∞

−∞

= (^2) ∫ ( )

Special Case

For white noise,

Thus,

R XX (τ ) =σ X^2 δ ( τ )

( )^2 ( )^2 X

j t

SXX ω σ X δ τ e dt σ

= − ω^ =

∞

−∞

∫

τ

R (^) XX( τ )

σ X^2 δ(τ)

SXX( ω ) σ X^2

ω

←→

Example 1

Random process X(t) is wide sense stationary and has a autocorrelation function given by:

Find S (^) XX.

τ τ σ

−

R XX ( ) = X^2 e