Signal Processing - MATLAB Simulations for Radar Systems Design - page 628

Completing the integration yields

1
25π

---------

E y = [ atanh 2() – atanh () ] = 0.01799 Joules

Note that an infinite bandwidth would give E y = 0.02 , only 11% larger.

The total power associated with a power signal g () is

T ⁄

∫

2

1
---

g () 2 P = lim d (13.50)

T → ∞

– T ⁄ 2

Define the Power Spectrum Density (PSD) function for the signal g () as

S g () , where

T ⁄

∫

2 ∞

∫

1
---

1
2π

g () 2

------

P = lim d = S g () d

(13.51)

T → ∞

– T ⁄ 2 – ∞

It can be shown that (see Problem 1.13)

G () 2
T

------------------

S g () = lim (13.52)

T → ∞

Let the signals and be two periodic signals with period . The
complex exponential Fourier series expansions for those signals are, respec-
tively, given by

x () g () T

∞
∑

j 2π nt
T

--------------

x () = X n e (13.53)

n = – ∞

∞
∑

j 2π mt
T

---------------

g () = G m e

(13.54)

m = – ∞

The power cross-correlation function R gx () was given in Eq. (13.43), and is

repeated here as Eq. (13.55),

T ⁄

∫

2

1
---

g ∗ () xt τ R gx () = ( + ) d

(13.55)

– T ⁄ 2

Note that because both signals are periodic the limit is no longer necessary.
Substituting Eqs. (13.53) and (13.54) into Eq. (13.55), collecting terms, and
using the definition of orthogonality, we get