1. No category

18-290 Signals & Systems
Homework 5
Spring 2015
Total Points: 100
Submit your Solutions in three separate parts.
Make sure to write your Name and Andrew ID on each part.
Part 1 (Problem 1); Part 2 (Problem 2); Part 3 (Problems 3-4)
1. (26 pts) Fourier Series Analysis and Synthesis
∞
X
x(t) =
k=−∞
1
ak =
T
Z
T
∞
X
ak ejkω0 t =
2π
ak ejk( T )t ,
(1)
k=−∞
1
x(t)e−jkω0 t dt =
T
Z
2π
x(t)e−jk( T )t dt,
(2)
T
The above pair of equations defines the Fourier Series representation of a periodic continuous
time signal. Equation (1) is referred to as the synthesis equation and Equation (2) is referred
to as the analysis equation. We refer to the sequence ak as the Fourier Series coefficients of
x(t).
(a) (14 pts) Analysis
Find the Fourier Series coefficients ak for the periodic continuous time signal x(t) given in
each of the problems below. Note that the the first step should be identifying the period
of the continuous time signal.
i. (3 pts) x(t) is shown in figure below.
ii. (3 pts) x(t) =
P∞
m=−∞
δ(t −
m
3)
+ δ(t −
2m
3 )
iii. (5 pts) x(t) = sin(3πt) + cos(4πt)
Though we can always use the Analysis equation of Fourier Series to find the Fourier
Series coefficients, it is sometimes easier to express x(t) in the form of the Synthesis
equation and infer the Fourier Series coefficients by observation.
A. Find the period of x(t)
B. Express sin(3πt) as a sum of complex exponentials
C. Express cos(4πt) as a sum of complex exponentials
D. Using parts (b) and (c), express x(t) as a sum of complex exponentials, and find
the Fourier Series coefficients of x(t)
iv. (3 pts) This is a particularly important Fourier Series pair. x(t) is periodic with
period T, where:

1, |t| < T1
x(t) =
0, T ≤ |t| ≤ T /2
1
(b) (12 pts) Synthesis
Find x(t) given the Fourier Series coefficients in the following problems. Note that in
these problems, you should be given the period of the corresponding continuous time
signal.
i. (4 pts) You can use result of part(d) of above problem for this. You are given that
x(t) has a period of 4. Find x(t) in each of these cases.
A. ak =
B. ak =
sin(kπ/4)
kπ
sin(kπ/8)
2kπ
for k 6= 0; a0 = 0.5
for k 6= 0; a0 =
1
8
ii. (4 pts) Fundamental frequency of x(t) is ω0 = 4π
ak = jδ[k − 1] − jδ[k + 1] + δ[k − 3] + δ[k + 3]
iii. (4 pts) In this problem, you are given the magnitude and phase of ak . You are given
that ω0 = 2π
2
2. (38 pts) Properties of Continuous-Time Fourier Series (CTFS)
F ,T
F ,T
In all of the following questions, assume that x ←→ {ak } and y ←→ {bk }, unless mentioned
otherwise.
(a) (6 pts) Inner product of complex exponentials and consistency:
i. (3 pts) Show that
1
T
where
∗
Z
ejkω0 t (ejlω0 t )∗ dt = δk,l
T
refers to complex conjugation, ω0 =
2π
T ,
and δk,l is the Kronecker delta,
defined by:
δk,l

0,
=
1,
k 6= l
k=l
ii. (3 pts) Hence show that the synthesis and analysis equations of CTFS are consistent.
This can be done in one of two ways: by plugging in the formula for x(t) from
the synthesis equation into the analysis equation, and showing that the expression
simplifies to ak , or by plugging in the formula for ak from the analysis equation into
the synthesis equation and showing that the expression simplifies to x(t).
Hint: For the former method, use a different dummy index for the summation, i.e. if you have ak on
the LHS of your equation, use a new index, l, inside your summation. Exchange the summation and
integration. Then use part the result from part (i) above, to introdue the Kronecker delta function.
Finally, use the sifting property of the Kronecker delta (for discrete l) to arrive at the result.
If you choose the latter method, use a different dummy index for the integration, t0 . Exchange
integration and summation. Then use the result derived in class and in the recitation session, for the
Fourier series coefficients of an impuse train, to introduce the Dirac delta function. Finally, use the
sifting property of the Dirac delta function (for continuous t0 ) to arrive at the result.
F ,T 0
(b) (4 pts) Changing the CTFS period: Show that x ←→ {ck }, where T 0 = nT (n ∈ Z) is
also a period of x, and

a ,
k/n
ck =
0,
k/n ∈ Z
otherwise
Hint: Remember that the ω0 in the synthesis and analysis equations contains a T .
F ,T
(c) (4 pts) Linearity: Show that ux + vy ←→ {ck }, where ck = uak + vbk , and u and v are
complex constants.
F ,T
(d) (4 pts) Time-shift: Show that δt0 ∗ x ←→ {e−jkω0 t0 ak }, where δt0 (t) = δ(t − t0 ).
F ,T
(e) (4 pts) Time-reversal: Let y(t) = x(−t). Show that y ←→ {bk }, where bk = a−k , ∀k ∈
Z.
F ,T /u
(f) (4 pts) Time-scaling: Let y(t) = x(ut), u ∈ (0, ∞). Show that y ←→ {ak }.
3
F ,T
(g) (4 pts) Multiplication in time-domain: Show that x · y ←→ (a ∗ b)k where (x · y)(t) =
P
x(t)y(t) and (a ∗ b)k = ∞
l=−∞ al bk−l .
Hint: Substitute for x(t) and y(t) using the synthesis equation. Remember to use different dummy indices,
l and m say, for each synthesis equation. Switch summation and integration, and use the result from
question 2(a) part (i) to introduce the Kronecker delta again. Finally, use the sifting property to arrive
at the result.
(h) (4 pts) Real and even: Show that if x is real and even, then the CTFS coefficients ak
are also real and even.
(i) (4 pts) Parseval’s relation: Prove that
1
T
Z
|x(t)|2 dt =
T
∞
X
|ak |2
k=−∞
Hint: Replace |x(t)|2 with the product of x(t) and its conjugate. Then expand each using the synthesis
equation. Remember to use different dummy indices, k and l say, for each. Interchange integration and
summation, and use the result from question 2(a) part (i) to introduce the Kronecker delta. Finally, use
the sifting property to arrive at the result.
4
3. (12 pts) Application of CTFS properties
(a) (6 pts)
i. (3 pts) Use the period-change, linearity and time-shift properties to compute the
CTFS coefficients of x(t) = 2 sin(4πt) + cos(6πt + π/4).
ii. (3 pts) Plot the real and imaginary parts of the coefficients {ak } you computed, as
functions of k.
(b) (3 pts) Use the multiplication property to compute the CTFS coefficients of x(t) =
cos(2πt) cos(2πt). What is a0 ? What does this tell you about the signal?
(c) (3 pts) Use Parseval’s relation to compute the total energy of a discrete sinc waveform,
defined by
ak =

 sin(πk) ,
k 6= 0
1,
k=0
πk
5
4. (24 pts) LTI Systems and the Continuous Time Fourier Series
A. (6 pts) System Input CTFS
Compute the CTFS coefficients (Xi )k =
1
T0
R T0
0
xi (t)e−jkω0 t dt for each of the following
signals xi where T0 is the fundamental period of xi and ω0 =
2π
T0 .
Use these coefficients
to write xi (t) in CTFS form.
i. (3 pts) Sinusoid: x1 (t) = 4 sin( 2π
t + π3 ) + 1
(3
A 0 ≤ t + 4n < 2 for some n ∈ Z
ii. (3 pts) Square Wave: x2 (t) =
−A 2 ≤ t + 4n < 4 for some n ∈ Z
B. (6 pts) System Frequency Response
Compute the frequency response Hi (jω) =
R∞
−jωt dt
−∞ hi (t)e
for each of the following LTI
systems.
1
i. (3 pts) System H1 with h1 (t) = e− 5 t u(t)
ii. (3 pts) System H2 with h2 (t) = u(t + 1) − u(t − 1)
C. (6 pts) System Output CTFS Coefficients
Compute the output CTFS coefficients (Yi )k where yi (t) = Hi {xi }(t) using the CTFS
coefficients of xi (t) and system frequency response Hi (jω) computed in previous parts.
(Recall that H{gω }(t) = H(jω)gω (t) for gω (t) = ejωt and LTI system H.) Do not worry
about simplifying your answer.
i. (3 pts) Compute (Y1 )k
ii. (3 pts) Compute (Y2 )k
D. (6 pts) System Output via CTFS Coefficents
Using the CTFS coefficients (Yi )k from the previous part, write the series form for yi (t).
Do not worry about simplifying your answer. Note that the coefficients for part i are
those of a sinusoid with a dc component. Also, note that the coefficients for part ii are
those of a triangle waveform. (You do not need to write these waveforms explicitly.)
i. (3 pts) Find y1 (t) = H1 {x1 }(t) from (Y1 )k
ii. (3 pts) Find y2 (t) = H2 {x2 }(t) from (Y2 )k
6