Signals and Systems - A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach - page 6

r f (τ )

( 0 , T )

τ

− T T

FIGURE 1.3. Autocorrelation function r f (τ ) of the rectangular pulse shown in Figure 1.1.

is a real function because r f (τ ) = The energy density spectrum E f (ω) of f ( t )

r f ( − f ) :

∞ ∞

e − j ωτ d

E f (ω) = r f (τ ) τ = r f (τ ) cos (ωτ ) d τ. (1.10)

−∞ −∞

The energy density spectrum of the rectangular pulse f ( t ) is

T

T 2 sin ω 2
ω

2

T 2 sinc 2 ω

T
2 .

E f (ω) =

T (

T − τ) cos (ωτ ) d τ = = (1.11)

T
2

−

The energy density spectrum is depicted in Figure 1.4.

In discrete time the rectangular pulse takes on the form

E f (ω)

1

, 0 ≤ n ≤ N − 1,

( ) =

f n (1.12)

0 , otherwise,

where N is an integer. The Fourier transform of f ( n ) is

N − 1

sin (π fN )

e − j 2 π fn e − j π f ( N − 1 ) . F ( f ) = =

(π )

sin f

n = 0

1.1.5 Random Signals
A random process can be viewed as a mapping of the outcomes of a random
experiment to a set of functions of time—in this context a signal X

( t ) . Such a sig-

nal is stationary if the density functions p describing it are invariant under

translation of time t . A random stationary process is an infinite energy signal, and
therefore its Fourier transform does not exist. The spectral characteristics of a ran-
dom process is obtained according to the Wiener-Khinchine theorem [see, e.g.,
Shanmugan & Breipohl (1988)] by computing the Fourier transform of the ACF.
That is, the distribution of signal power as a function of frequency is given by

( X ( t ))

∞

e − j ωτ d S X (ω) = r X (τ ) τ. (1.13)

−∞