Solution Manual For
Upamanyu Madhow Fundamentals of Digital Communication
Chapter 2-8
Solutions to Chapter 2 Problems
Fundamentals of Digital Communication
Problem 2.1: Rather than doing the details of the convolution, we simply sketch the
shapes of the waveforms. For a signal s = sc + jss and a filter h = hc + jhs, the
convolution
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data
science, and technology. The primary goal of mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for
logical reasoning.3.2. Structure of Mathematics ExamsMathematics exams are typically focused on problem-solving and application of formulas, theorems, and mathematical
concepts. These exams may include:Problem Sets: Students are given a series of problems that test various mathematical skills, from basic calculations to more complex word
problems. The problems may require students to apply formulas, solve equations, or prove mathematical theorems.Theoretical Questions: In higher-level mathematics exams,
students may be asked to demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or the explanation of a mathematical
concept.Multiple Choice Questions (MCQs): While less common, some mathematics exams use MCQs to test students' quick recall of formulas, definitions, or theorems.3.3. Skills
Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approach complex problems systematically,
using appropriate methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
y = s ∗ h = (sc ∗ hc − ss ∗ hs) + j(sc ∗ hs + ss ∗ hc)
For h(t) = smf (t) = s∗( t), rough sketches of Re(y), Im(y) and y are shown in
Figure 1. Clearly, the maximum occurs at t = 0.
Re(s*h)
sc*hc −ss*hs
|s* h|
+ =
sc*hs ss*hc Im(s*h)
+ =
Figure 1: The convolution of a signal with its matched filter yields at peak at the origin.
Problem 2.2:
(a) Multiplication in the time domain corresponds to convolution in the frequency
domain. The two sinc functions correspond to boxcars in the frequency domain,
convolving which gives that S(f ) has a trapezoidal shape, as shown in Figure 2.
(b) We have
ej100πt + e−j100πt S(f − 50) + S(f + 50)
u(t) = s(t) cos(100πt) = s(t) ↔ U (f ) =
2 2
The spectrum U (f ) is plotted in Figure 2.
1 = sinc(t) sinc(2t)
sinc(t)
s
(
−1/2 1/2 f
t u(t) = s(t) cos(100 t)
)
, sinc(2t)
1/2
−1 1 f
CONVOLVE S(f)
1/2
1/2 3/2 f
U(f)
1/4
... f
−51.5 −50 −48.5 48.5 50 51.5
Figure 2: Solution for Problem 2.2.
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data science, and technology. The primary goal of
mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for logical reasoning.3.2. Structure of Mathematics ExamsMathematics exams are typically focu
problem-solving and application of formulas, theorems, and mathematical concepts. These exams may include:Problem Sets: Students are given a series of problems that test various mathematical skills, from basic calculations to
more complex word problems. The problems may require students to apply formulas, solve equations, or prove mathematical theorems.Theoretical Questions: In higher-level mathematics exams, students may be asked to
demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or the explanation of a mathematical concept.Multiple Choice Questions (MCQs): While less common, some mathematics e
use MCQs to test students' quick recall of formulas, definitions, or theorems.3.3. Skills Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approac
complex problems systematically, using appropriate methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
Problem 2.3: The solution is sketched in Figure 3.
(a) We have s(t) = I[−5,5] I[−5,5]. Since I[−5,5](t) 10sinc(10f ), we have S(f ) = 100sinc2(10f ).
(b) We have
ej1000πt e−j100πt S(f − 50) − S(f + 50)
u(t) = s(t) sin(1000πt) = s(t) ↔ U (f ) =
2j 2j
s(t) 100
10
1 1
t = ... ...
−10 10 −5 5 t * −5 5 t −0.2 −0.1 0.1 0.2 f
S(f) = 100 sinc2(10f)
10 sinc(10f)
Im (U(f))
50
Re(U(f) = 0
... ... 500
f
−500
0.1
Figure 3: Solution for Problem 2.3.
Problem 2.4: Part (a) is immediate upon expanding s ar||2.
(b) The minimizing value of a is easily found to be
⟨s, r⟩
amin =
||r||2
Substituting this value into J(a), we obtain upon simplification that
2
⟨s, r⟩
J(amin ) = ||s|| 2
||r||2
, The condition J(amin) 0 is now seen to be equivalent to the Cauchy-Schwartz inequality.
(c) For nonzero s, r, the minimum error J(amin) in approximating s by a multiple of r vanishes if
and only if s is a multiple of r. This is therefore the condition for equality in the Cauchy-Scwartz
inequality. For s = 0 or r = 0, equality clearly holds. Thus, the condition for equality can be
stated in general as: either s is a scalar multiple of r (this includes s = 0 as a special case), or r
is a scalar multiple of s (this includes r = 0 as a special case).
(d) The unit vector in the direction of r is u = r . The best approximation of s as a multiple
||r||
of r is its projection along u, which is given by
r r
ŝ = ⟨s, u⟩u = ⟨s, ⟩
||r|| ||r||
and the minimum error is J(amin) = ||s − ŝ || 2 .
Problem 2.5: We have
∫
y(t) = (x ∗ h)(t) = As(τ − t0)h(t − τ )dτ = A⟨st0 , ht∗⟩
where st0 (τ ) = s(τ − t0) and h∫t(τ ) = h(t − τ ) are functions of τ . (Recall that the complex inner
product is defined as ⟨u, v⟩ = uv∗).
(a) Using the Cauchy-Schwartz inequality, we have
|y(t)| ≤ |A|||st0 ||||ht∗||
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data science, and technology. The primary goal of
mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for logical reasoning.3.2. Structure of Mathematics ExamsMathematics exams are typically focu
problem-solving and application of formulas, theorems, and mathematical concepts. These exams may include:Problem Sets: Students are given a series of problems that test various mathematical skills, from basic calculations t
more complex word problems. The problems may require students to apply formulas, solve equations, or prove mathematical theorems.Theoretical Questions: In higher-level mathematics exams, students may be asked to
demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or the explanation of a mathematical concept.Multiple Choice Questions (MCQs): While less common, some mathematics e
use MCQs to test students' quick recall of formulas, definitions, or theorems.3.3. Skills Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approac
complex problems systematically, using appropriate methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
It is easy to check (simply change variables in the associated integrals) that ||st0 || = ||s|| and
||ht∗|| = ||h||. Using the normalization ||h|| = ||s||, we obtain the desired result that |y(t)| ≤
A||s||2.
(b) Equality is attained for t = t0 if ht∗ = ast0 for t = t0 for some scalar a. Since ||st0 || = ||ht∗|| =
||s||, we must have |a| = 1. Thus, we have
h∗(t0 − τ ) = as(τ − t0) (|a| = 1)
for all τ . (We would get a = 1 if we insisted that y(t0) = A||s||2 rather than |y(t0)| = |A|||s||2.)
Setting a = 1, therefore, we have h(t) = s∗(−t).
Re(Xp (f))
2
−22 −20 20 22 f
Im(Xp (f))
1/ 2
−23 −21
21 23 f
3
, Figure 4: Passband spectrum for Problem 2.6(a).
Re(X(f)) Im(X(f))
2
1
0 2 f 1 3 f
Figure 5: Complex baseband spectrum for Problem 2.6(b).
basic arithmetic and
Problem 2 .√6 (a) The real and imaginary parts of Xp(f ) are sketched in Figure 4.
(b) Passing 2xp(t) cos 20πt through a lowpass filter yields xc(t), the I component with respect
to fc = 20. In this case, it is easiest to find the Fourier transform (see Figure 5), and then the
time domain expression, for the complex baseband signal x(t), and then take the real part. We
see that
Re(X(f )) ↔ 4sinc(2t)ej2πt
Im(X(f )) ↔ sinc2(t)ej4πt
so that
X(f ) = Re(X(f )) + jIm(X(f )) ↔ x(t) = 4sinc(2t)ej2πt + jsinc2(t)ej4πt
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data science, and technology
primary goal of mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for logical reasoning.3.2. Structure of Mathematics
ExamsMathematics exams are typically focused on problem-solving and application of formulas, theorems, and mathematical concepts. These exams may include:Problem Sets: Students are given a ser
problems that test various mathematical skills, from basic calculations to more complex word problems. The problems may require students to apply formulas, solve equations, or prove mathematical
theorems.Theoretical Questions: In higher-level mathematics exams, students may be asked to demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or
explanation of a mathematical concept.Multiple Choice Questions (MCQs): While less common, some mathematics exams use MCQs to test students' quick recall of formulas, definitions, or theorems.3.
Skills Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approach complex problems systematically, using appropriate
methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
Figure 7: Computing the complex baseband representation in Problem 2.7(c).
~
V(f)/ 2 U(f)
d/df
f
−j2 t
Figure 8: Computing the complex baseband representation in Problem 2.7(d).
Problem 2.7 (a) The passband spectrum Vp(f ) is sketched in Figure 6, where we use the fact
that V (f ) = V ∗(−f ), since v(t) is real-valued.
(b) We have v(t) = v(−t) if V (f ) = V (−f ), which is clearly true here.
(c) The complex baseband spectrum V (f ), with respect to f0√= 100, is sketched in Figure 7. In
order to compute v(t) with minimal work, we break up V (f )/ 2 into two components, A(f ) and
B(f ). Clearly, a(t) = 2sinc(2t). To find b(t), we minimize work further by differentiating in the
frequency domain to obtain C(f ) = d/df B(f ). We have c(t) = 2 cos 2πt 2sinc(2t) = j2πtb(t),
which implies that
√ cos 2πt − sinc(2t)
v(t)/ 2 = a(t) + b(t) = 2sinc(2t) +
−jπt
3
Upamanyu Madhow Fundamentals of Digital Communication
Chapter 2-8
Solutions to Chapter 2 Problems
Fundamentals of Digital Communication
Problem 2.1: Rather than doing the details of the convolution, we simply sketch the
shapes of the waveforms. For a signal s = sc + jss and a filter h = hc + jhs, the
convolution
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data
science, and technology. The primary goal of mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for
logical reasoning.3.2. Structure of Mathematics ExamsMathematics exams are typically focused on problem-solving and application of formulas, theorems, and mathematical
concepts. These exams may include:Problem Sets: Students are given a series of problems that test various mathematical skills, from basic calculations to more complex word
problems. The problems may require students to apply formulas, solve equations, or prove mathematical theorems.Theoretical Questions: In higher-level mathematics exams,
students may be asked to demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or the explanation of a mathematical
concept.Multiple Choice Questions (MCQs): While less common, some mathematics exams use MCQs to test students' quick recall of formulas, definitions, or theorems.3.3. Skills
Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approach complex problems systematically,
using appropriate methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
y = s ∗ h = (sc ∗ hc − ss ∗ hs) + j(sc ∗ hs + ss ∗ hc)
For h(t) = smf (t) = s∗( t), rough sketches of Re(y), Im(y) and y are shown in
Figure 1. Clearly, the maximum occurs at t = 0.
Re(s*h)
sc*hc −ss*hs
|s* h|
+ =
sc*hs ss*hc Im(s*h)
+ =
Figure 1: The convolution of a signal with its matched filter yields at peak at the origin.
Problem 2.2:
(a) Multiplication in the time domain corresponds to convolution in the frequency
domain. The two sinc functions correspond to boxcars in the frequency domain,
convolving which gives that S(f ) has a trapezoidal shape, as shown in Figure 2.
(b) We have
ej100πt + e−j100πt S(f − 50) + S(f + 50)
u(t) = s(t) cos(100πt) = s(t) ↔ U (f ) =
2 2
The spectrum U (f ) is plotted in Figure 2.
1 = sinc(t) sinc(2t)
sinc(t)
s
(
−1/2 1/2 f
t u(t) = s(t) cos(100 t)
)
, sinc(2t)
1/2
−1 1 f
CONVOLVE S(f)
1/2
1/2 3/2 f
U(f)
1/4
... f
−51.5 −50 −48.5 48.5 50 51.5
Figure 2: Solution for Problem 2.2.
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data science, and technology. The primary goal of
mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for logical reasoning.3.2. Structure of Mathematics ExamsMathematics exams are typically focu
problem-solving and application of formulas, theorems, and mathematical concepts. These exams may include:Problem Sets: Students are given a series of problems that test various mathematical skills, from basic calculations to
more complex word problems. The problems may require students to apply formulas, solve equations, or prove mathematical theorems.Theoretical Questions: In higher-level mathematics exams, students may be asked to
demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or the explanation of a mathematical concept.Multiple Choice Questions (MCQs): While less common, some mathematics e
use MCQs to test students' quick recall of formulas, definitions, or theorems.3.3. Skills Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approac
complex problems systematically, using appropriate methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
Problem 2.3: The solution is sketched in Figure 3.
(a) We have s(t) = I[−5,5] I[−5,5]. Since I[−5,5](t) 10sinc(10f ), we have S(f ) = 100sinc2(10f ).
(b) We have
ej1000πt e−j100πt S(f − 50) − S(f + 50)
u(t) = s(t) sin(1000πt) = s(t) ↔ U (f ) =
2j 2j
s(t) 100
10
1 1
t = ... ...
−10 10 −5 5 t * −5 5 t −0.2 −0.1 0.1 0.2 f
S(f) = 100 sinc2(10f)
10 sinc(10f)
Im (U(f))
50
Re(U(f) = 0
... ... 500
f
−500
0.1
Figure 3: Solution for Problem 2.3.
Problem 2.4: Part (a) is immediate upon expanding s ar||2.
(b) The minimizing value of a is easily found to be
⟨s, r⟩
amin =
||r||2
Substituting this value into J(a), we obtain upon simplification that
2
⟨s, r⟩
J(amin ) = ||s|| 2
||r||2
, The condition J(amin) 0 is now seen to be equivalent to the Cauchy-Schwartz inequality.
(c) For nonzero s, r, the minimum error J(amin) in approximating s by a multiple of r vanishes if
and only if s is a multiple of r. This is therefore the condition for equality in the Cauchy-Scwartz
inequality. For s = 0 or r = 0, equality clearly holds. Thus, the condition for equality can be
stated in general as: either s is a scalar multiple of r (this includes s = 0 as a special case), or r
is a scalar multiple of s (this includes r = 0 as a special case).
(d) The unit vector in the direction of r is u = r . The best approximation of s as a multiple
||r||
of r is its projection along u, which is given by
r r
ŝ = ⟨s, u⟩u = ⟨s, ⟩
||r|| ||r||
and the minimum error is J(amin) = ||s − ŝ || 2 .
Problem 2.5: We have
∫
y(t) = (x ∗ h)(t) = As(τ − t0)h(t − τ )dτ = A⟨st0 , ht∗⟩
where st0 (τ ) = s(τ − t0) and h∫t(τ ) = h(t − τ ) are functions of τ . (Recall that the complex inner
product is defined as ⟨u, v⟩ = uv∗).
(a) Using the Cauchy-Schwartz inequality, we have
|y(t)| ≤ |A|||st0 ||||ht∗||
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data science, and technology. The primary goal of
mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for logical reasoning.3.2. Structure of Mathematics ExamsMathematics exams are typically focu
problem-solving and application of formulas, theorems, and mathematical concepts. These exams may include:Problem Sets: Students are given a series of problems that test various mathematical skills, from basic calculations t
more complex word problems. The problems may require students to apply formulas, solve equations, or prove mathematical theorems.Theoretical Questions: In higher-level mathematics exams, students may be asked to
demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or the explanation of a mathematical concept.Multiple Choice Questions (MCQs): While less common, some mathematics e
use MCQs to test students' quick recall of formulas, definitions, or theorems.3.3. Skills Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approac
complex problems systematically, using appropriate methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
It is easy to check (simply change variables in the associated integrals) that ||st0 || = ||s|| and
||ht∗|| = ||h||. Using the normalization ||h|| = ||s||, we obtain the desired result that |y(t)| ≤
A||s||2.
(b) Equality is attained for t = t0 if ht∗ = ast0 for t = t0 for some scalar a. Since ||st0 || = ||ht∗|| =
||s||, we must have |a| = 1. Thus, we have
h∗(t0 − τ ) = as(τ − t0) (|a| = 1)
for all τ . (We would get a = 1 if we insisted that y(t0) = A||s||2 rather than |y(t0)| = |A|||s||2.)
Setting a = 1, therefore, we have h(t) = s∗(−t).
Re(Xp (f))
2
−22 −20 20 22 f
Im(Xp (f))
1/ 2
−23 −21
21 23 f
3
, Figure 4: Passband spectrum for Problem 2.6(a).
Re(X(f)) Im(X(f))
2
1
0 2 f 1 3 f
Figure 5: Complex baseband spectrum for Problem 2.6(b).
basic arithmetic and
Problem 2 .√6 (a) The real and imaginary parts of Xp(f ) are sketched in Figure 4.
(b) Passing 2xp(t) cos 20πt through a lowpass filter yields xc(t), the I component with respect
to fc = 20. In this case, it is easiest to find the Fourier transform (see Figure 5), and then the
time domain expression, for the complex baseband signal x(t), and then take the real part. We
see that
Re(X(f )) ↔ 4sinc(2t)ej2πt
Im(X(f )) ↔ sinc2(t)ej4πt
so that
X(f ) = Re(X(f )) + jIm(X(f )) ↔ x(t) = 4sinc(2t)ej2πt + jsinc2(t)ej4πt
basic arithmetic and algebra to more advanced areas like calculus, statistics, and abstract algebra. Mathematics is crucial in many industries, including engineering, finance, data science, and technology
primary goal of mathematics exams is to test students' understanding of mathematical concepts, their problem-solving abilities, and their aptitude for logical reasoning.3.2. Structure of Mathematics
ExamsMathematics exams are typically focused on problem-solving and application of formulas, theorems, and mathematical concepts. These exams may include:Problem Sets: Students are given a ser
problems that test various mathematical skills, from basic calculations to more complex word problems. The problems may require students to apply formulas, solve equations, or prove mathematical
theorems.Theoretical Questions: In higher-level mathematics exams, students may be asked to demonstrate their understanding of theoretical concepts, such as the proof of a mathematical theorem or
explanation of a mathematical concept.Multiple Choice Questions (MCQs): While less common, some mathematics exams use MCQs to test students' quick recall of formulas, definitions, or theorems.3.
Skills Tested in Mathematics ExamsProblem-Solving Ability: The core skill tested in mathematics exams is problem-solving. Students must approach complex problems systematically, using appropriate
methods to reach the correct solution.Logical Reasoning: Mathematics is rooted in logical structures, and students are expected to
Figure 7: Computing the complex baseband representation in Problem 2.7(c).
~
V(f)/ 2 U(f)
d/df
f
−j2 t
Figure 8: Computing the complex baseband representation in Problem 2.7(d).
Problem 2.7 (a) The passband spectrum Vp(f ) is sketched in Figure 6, where we use the fact
that V (f ) = V ∗(−f ), since v(t) is real-valued.
(b) We have v(t) = v(−t) if V (f ) = V (−f ), which is clearly true here.
(c) The complex baseband spectrum V (f ), with respect to f0√= 100, is sketched in Figure 7. In
order to compute v(t) with minimal work, we break up V (f )/ 2 into two components, A(f ) and
B(f ). Clearly, a(t) = 2sinc(2t). To find b(t), we minimize work further by differentiating in the
frequency domain to obtain C(f ) = d/df B(f ). We have c(t) = 2 cos 2πt 2sinc(2t) = j2πtb(t),
which implies that
√ cos 2πt − sinc(2t)
v(t)/ 2 = a(t) + b(t) = 2sinc(2t) +
−jπt
3
