Exercises

INTEGRAL TRANSFORMS
13.4 Exercises
13.1
Find the Fourier transform of the function f(t) = exp(−|t|).
(a) By applying Fourier’s inversion theorem prove that
∞
π
cos ωt
dω.
exp(−|t|) =
2
1 + ω2
0
(b) By making the substitution ω = tan θ, demonstrate the validity of Parseval’s
theorem for this function.
13.2
Use the general definition and properties of Fourier transforms to show the
following.
(a) If f(x) is periodic with period a then f̃(k) = 0, unless ka = 2πn for integer n.
(b) The Fourier transform of tf(t) is idf̃(ω)/dω.
(c) The Fourier transform of f(mt + c) is
eiωc/m ω .
f̃
m
m
13.3
13.4
Find the Fourier transform of H(x − a)e−bx , where H(x) is the Heaviside function.
Prove that the Fourier transform of the function f(t) defined in the tf-plane by
straight-line segments joining (−T , 0) to (0, 1) to (T , 0), with f(t) = 0 outside
|t| < T , is
T
ωT
,
f̃(ω) = √ sinc2
2
2π
where sinc x is defined as (sin x)/x.
Use the general properties of Fourier transforms to determine the transforms
of the following functions, graphically defined by straight-line segments and equal
to zero outside the ranges specified:
(a) (0, 0) to (0.5, 1) to (1, 0) to (2, 2) to (3, 0) to (4.5, 3) to (6, 0);
(b) (−2, 0) to (−1, 2) to (1, 2) to (2, 0);
(c) (0, 0) to (0, 1) to (1, 2) to (1, 0) to (2, −1) to (2, 0).
13.5
By taking the Fourier transform of the equation
d2 φ
− K 2 φ = f(x),
dx2
show that its solution, φ(x), can be written as
∞ ikx 3
−1
e f(k)
φ(x) = √
dk,
2π −∞ k 2 + K 2
13.6
13.7
where 3
f(k) is the Fourier transform of f(x).
By differentiating the definition of the Fourier sine transform f̃s (ω) of the function
f(t) = t−1/2 with respect to ω, and then integrating the resulting expression by
parts, find an elementary differential equation satisfied by f̃s (ω). Hence show that
this function is its own Fourier sine transform, i.e. f̃s (ω) = Af(ω), where A is a
constant. Show that it is also its own Fourier cosine transform. Assume that the
limit as x → ∞ of x1/2 sin αx can be taken as zero.
Find the Fourier transform of the unit rectangular distribution
1 |t| < 1,
f(t) =
0 otherwise.
460
13.4 EXERCISES
Determine the convolution of f with itself and, without further integration,
deduce its transform. Deduce that
∞
sin2 ω
dω = π,
ω2
−∞
∞
2π
sin4 ω
dω =
.
ω4
3
−∞
13.8
Calculate the Fraunhofer spectrum produced by a diffraction grating, uniformly
illuminated by light of wavelength 2π/k, as follows. Consider a grating with 4N
equal strips each of width a and alternately opaque and transparent. The aperture
function is then
#
A for (2n + 1)a ≤ y ≤ (2n + 2)a,
−N ≤ n < N,
f(y) =
0 otherwise.
(a) Show, for diffraction at angle θ to the normal to the grating, that the required
Fourier transform can be written
2a
N−1
3
exp(−2iarq)
A exp(−iqu) du,
f(q) = (2π)−1/2
r=−N
a
where q = k sin θ.
(b) Evaluate the integral and sum to show that
A sin(2qaN)
3
f(q) = (2π)−1/2 exp(−iqa/2)
,
q cos(qa/2)
and hence that the intensity distribution I(θ) in the spectrum is proportional
to
sin2 (2qaN)
.
q 2 cos2 (qa/2)
(c) For large values of N, the numerator in the above expression has very closely
spaced maxima and minima as a function of θ and effectively takes its mean
value, 1/2, giving a low-intensity background. Much more significant peaks
in I(θ) occur when θ = 0 or the cosine term in the denominator vanishes.
Show that the corresponding values of |3
f(q)| are
2aNA
(2π)1/2
and
4aNA
,
(2π)1/2 (2m + 1)π
with m integral.
Note that the constructive interference makes the maxima in I(θ) ∝ N 2 , not
N. Of course, observable maxima only occur for 0 ≤ θ ≤ π/2.
13.9
By finding the complex Fourier series for its LHS show that either side of the
equation
∞
∞
1 −2πnit/T
δ(t + nT ) =
e
T n=−∞
n=−∞
can represent a periodic train of impulses. By expressing the function f(t + nX),
in which X is a constant, in terms of the Fourier transform f̃(ω) of f(t), show
that
√
∞
∞
2π 2nπ
e2πnit/X .
f(t + nX) =
f̃
X n=−∞
X
n=−∞
This result is known as the Poisson summation formula.
461
INTEGRAL TRANSFORMS
13.10
In many applications in which the frequency spectrum of an analogue signal is
required, the best that can be done is to sample the signal f(t) a finite number of
times at fixed intervals, and then use a discrete Fourier transform Fk to estimate
discrete points on the (true) frequency spectrum f̃(ω).
(a) By an argument that is essentially the converse of that given in section 13.1,
show that, if N samples fn , beginning at t = 0 and spaced τ apart, are taken,
then f̃(2πk/(Nτ)) ≈ Fk τ where
N−1
1 fn e−2πnki/N .
Fk = √
2π n=0
(b) For the function f(t) defined by
#
f(t) =
1 for 0 ≤ t < 1,
0 otherwise,
from which eight samples are drawn at intervals of τ = 0.25, find a formula
for |Fk | and evaluate it for k = 0, 1, . . . , 7.
(c) Find the exact frequency
spectrum of f(t) and compare the actual and
√
estimated values of 2π|f̃(ω)| at ω = kπ for k = 0, 1, . . . , 7. Note the
relatively good agreement for k < 4 and the lack of agreement for larger
values of k.
13.11
For a function f(t) that is non-zero only in the range |t| < T /2, the full frequency
spectrum f̃(ω) can be constructed, in principle exactly, from values at discrete
sample points ω = n(2π/T ). Prove this as follows.
(a) Show that the coefficients of a complex Fourier series representation of f(t)
with period T can be written as
√
2π
2πn
cn =
.
f̃
T
T
(b) Use this result to represent f(t) as an infinite sum in the defining integral for
f̃(ω), and hence show that
∞
ωT
2πn
sinc nπ −
,
f̃
f̃(ω) =
T
2
n=−∞
where sinc x is defined as (sin x)/x.
13.12
A signal obtained by sampling a function x(t) at regular intervals T is passed
through an electronic filter, whose response g(t) to a unit δ-function input is
represented in a tg-plot by straight lines joining (0, 0) to (T , 1/T ) to (2T , 0) and
is zero
for all other values of t. The output of the filter is the convolution of the
input, ∞
−∞ x(t)δ(t − nT ), with g(t).
Using the convolution theorem, and the result given in exercise 13.4, show that
the output of the filter can be written
∞
∞
1 ωT
y(t) =
e−iω[(n+1)T −t] dω.
x(nT )
sinc2
2π n=−∞
2
−∞
13.13
Find the Fourier transform specified in part (a) and then use it to answer part
(b).
462
13.4 EXERCISES
(a) Find the Fourier transform of
#
f(γ, p, t) =
e−γt sin pt t > 0,
0
t < 0,
where γ (> 0) and p are constant parameters.
(b) The current I(t) flowing through a certain system is related to the applied
voltage V (t) by the equation
∞
K(t − u)V (u) du,
I(t) =
−∞
where
K(τ) = a1 f(γ1 , p1 , τ) + a2 f(γ2 , p2 , τ).
The function f(γ, p, t) is as given in (a) and all the ai , γi (> 0) and pi are fixed
parameters. By considering the Fourier transform of I(t), find the relationship
that must hold between a1 and a2 if the total net charge Q passed through
the system (over a very long time) is to be zero for an arbitrary applied
voltage.
13.14
Prove the equality
∞
e−2at sin2 at dt =
0
13.15
13.16
1
π
∞
0
a2
dω.
4a4 + ω 4
A linear amplifier produces an output that is the convolution of its input and its
response function. The Fourier transform of the response function for a particular
amplifier is
iω
K̃(ω) = √
.
2π(α + iω)2
Determine the time variation of its output g(t) when its input is the Heaviside
step function. (Consider the Fourier transform of a decaying exponential function
and the result of exercise 13.2(b).)
In quantum mechanics, two equal-mass particles having momenta pj = kj and
energies Ej = ωj and represented by plane wavefunctions φj = exp[i(kj ·rj −ωj t)],
j = 1, 2, interact through a potential V = V (|r1 − r2 |). In first-order perturbation
theory the probability of scattering to a state with momenta and energies pj , Ej
is determined by the modulus squared of the quantity
M=
ψf∗ V ψi dr1 dr2 dt.
The initial state, ψi , is φ1 φ2 and the final state, ψf , is φ1 φ2 .
(a) By writing r1 + r2 = 2R and r1 − r2 = r and assuming that dr1 dr2 = dR dr,
show that M can be written as the product of three one-dimensional integrals.
(b) From two of the integrals deduce energy and momentum conservation in the
form of δ-functions.
3 (k)
(c) Show that M is proportional to the Fourier transform of V , i.e. to V
where 2k = (p2 − p1 ) − (p2 − p1 ) or, alternatively, k = p1 − p1 .
13.17
For some ion–atom scattering processes, the potential V of the previous exercise
may be approximated by V = |r1 − r2 |−1 exp(−µ|r1 − r2 |). Show, using the result
of the worked example in subsection 13.1.10, that the probability that the ion
will scatter from, say, p1 to p1 is proportional to (µ2 + k 2 )−2 , where k = |k| and k
is as given in part (c) of that exercise.
463
INTEGRAL TRANSFORMS
13.18
The equivalent duration and bandwidth, Te and Be , of a signal x(t) are defined
in terms of the latter and its Fourier transform x̃(ω) by
∞
1
x(t) dt,
Te =
x(0) −∞
∞
1
Be =
x̃(ω) dω,
x̃(0) −∞
where neither x(0) nor x̃(0) is zero. Show that the product Te Be = 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 to x of
(a2 + x2 )−2 , which is
x
x
1
+
tan−1 .
2a2 (a2 + x2 ) 2a3
a
13.19
Calculate directly the auto-correlation function a(z) for the product f(t) of the
exponential decay distribution and the Heaviside step function,
1 −λt
e H(t).
λ
Use the Fourier transform and energy spectrum of f(t) to deduce that
∞
eiωz
π
dω = e−λ|z| .
2
2
λ
−∞ λ + ω
f(t) =
13.20
Prove that the cross-correlation C(z) of the Gaussian and Lorentzian distributions
a
1
t2
1
g(t) =
,
f(t) = √ exp − 2 ,
2τ
π t2 + a2
τ 2π
has as its Fourier transform the function
2 2
τω
1
√ exp −
exp(−a|ω|).
2
2π
Hence show that
2
az 1
a − z2
C(z) = √ exp
cos 2 .
2τ2
τ
τ 2π
13.21
Prove the expressions given in table 13.1 for the Laplace transforms of t−1/2 and
t1/2 , by setting x2 = ts in the result
∞
√
exp(−x2 ) dx = 12 π.
13.22
Find the functions y(t) whose Laplace transforms are the following:
0
(a) 1/(s2 − s − 2);
(b) 2s/[(s + 1)(s2 + 4)];
(c) e−(γ+s)t0 /[(s + γ)2 + b2 ].
13.23
Use the properties of Laplace transforms to prove the following without evaluating any Laplace integrals explicitly:
√ −7/2
(a) L t5/2 = 15
πs
;
8
(b) L (sinh at)/t = 12 ln (s + a)/(s − a) ,
s > |a|;
464
13.4 EXERCISES
(c) L [sinh at cos bt] = a(s2 − a2 + b2 )[(s − a)2 + b2 ]−1 [(s + a)2 + b2 ]−1 .
13.24
Find the solution (the so-called impulse response or Green’s function) of the
equation
dx
+ x = δ(t)
T
dt
by proceeding as follows.
(a) Show by substitution that
x(t) = A(1 − e−t/T )H(t)
is a solution, for which x(0) = 0, of
T
dx
+ x = AH(t),
dt
(∗)
where H(t) is the Heaviside step function.
(b) Construct the solution when the RHS of (∗) is replaced by AH(t − τ), with
dx/dt = x = 0 for t < τ, and hence find the solution when the RHS is a
rectangular pulse of duration τ.
(c) By setting A = 1/τ and taking the limit as τ → 0, show that the impulse
response is x(t) = T −1 e−t/T .
(d) Obtain the same result much more directly by taking the Laplace transform
of each term in the original equation, solving the resulting algebraic equation
and then using the entries in table 13.1.
13.25
This exercise is concerned with the limiting behaviour of Laplace transforms.
(a) If f(t) = A + g(t), where A is a constant and the indefinite integral of g(t) is
bounded as its upper limit tends to ∞, show that
lim sf̄(s) = A.
s→0
(b) For t > 0, the function y(t) obeys the differential equation
d2 y
dy
+a
+ by = c cos2 ωt,
dt2
dt
where a, b and c are positive constants. Find ȳ(s) and show that sȳ(s) → c/2b
as s → 0. Interpret the result in the t-domain.
13.26
By writing f(x) as an integral involving the δ-function δ(ξ − x) and taking the
Laplace transforms of both sides, show that the transform of the solution of the
equation
d4 y
− y = f(x)
dx4
for which y and its first three derivatives vanish at x = 0 can be written as
∞
e−sξ
f(ξ) 4
ȳ(s) =
dξ.
s
−1
0
Use the properties of Laplace transforms and the entries in table 13.1 to show
that
1 x
f(ξ) [sinh(x − ξ) − sin(x − ξ)] dξ.
y(x) =
2 0
465