Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

13.18 The equivalent duration and bandwidth,TeandBe, of a signalx(t) are defined
in terms of the latter and its Fourier transform ̃x(ω)by

Te=

1

x(0)

∫∞

−∞

x(t)dt,

Be=

1

̃x(0)

∫∞

−∞

x ̃(ω)dω,

where neitherx(0) nor ̃x(0) is zero. Show that the productTeBe=2π(this is a

form of uncertainty principle), and find the equivalent bandwidth of the signal

x(t)=exp(−|t|/T).

For this signal, determine the fraction of the total energy that lies in the frequency

range|ω|<Be/4. You will need the indefinite integral with respect toxof

(a^2 +x^2 )−^2 ,whichis

x

2 a^2 (a^2 +x^2 )

+

1

2 a^3

tan−^1

x

a

.

13.19 Calculate directly theauto-correlation functiona(z) for the productf(t)ofthe
exponential decay distribution and the Heaviside step function,

f(t)=

1

λ

e−λtH(t).

Use the Fourier transform and energy spectrum off(t) to deduce that

∫∞

−∞

eiωz

λ^2 +ω^2

dω=

π

λ

e−λ|z|.

13.20 Prove that the cross-correlationC(z) of the Gaussian and Lorentzian distributions

f(t)=

1

τ

√

2 π

exp

(

−

t^2

2 τ^2

)

,g(t)=

(a

π

) 1

t^2 +a^2

,

has as its Fourier transform the function

1

√

2 π

exp

(

−

τ^2 ω^2

2

)

exp(−a|ω|).

Hence show that

C(z)=

1

τ

√

2 π

exp

(

a^2 −z^2

2 τ^2

)

cos

(az

τ^2

)

.

13.21 Prove the expressions given in table 13.1 for the Laplace transforms oft−^1 /^2 and
t^1 /^2 , by settingx^2 =tsin the result
∫∞

0

exp(−x^2 )dx=^12

√

π.

13.22 Find the functionsy(t) whose Laplace transforms are the following:

(a) 1/(s^2 −s−2);

(b) 2s/[(s+1)(s^2 + 4)];

(c) e−(γ+s)t^0 /[(s+γ)^2 +b^2 ].

13.23 Use the properties of Laplace transforms to prove the following without evaluat-
ing any Laplace integrals explicitly:
(a) L

[

t^5 /^2

]

=^158

√

πs−^7 /^2 ;

(b)L

[

(sinhat)/t

]

=^12 ln

[

(s+a)/(s−a)

]

,s>|a|;