Quiz # 03 EEE223 Signals & Systems Fall 2014 14 – 11 – 14

Time Allowed: 40 mins R = CIIT Registration Number, P = daily pocket money Marks: 50

Problem: (Fourier Series and Filtering) (Solution) ( 20- Points)

A) Consider a continuous-time periodic signal, x(t) = P + 2 cos(π/R t) + 4 cos(3π/R t) + 6 sin(5π/R t).

Determine:

i) The fundamental frequency, Ω0 of x(t)

2 2 3 2 5

0 1 2 3 0

2

, , 2 2 2 1

1 2 3T R T R T R

0

x(t) = P + 2 cos(π/R t) + 4 cos(3π/R t) + 6 sin(5π/R t) Pts

T LCM T T T R Pts Ω T R Pt

ii) The Fourier series coefficients, ak

3

; 0

1 ; 1

2 ;

0 0 0 0 0 0

0 0 0

jΩ t -jΩ t j3Ω t -j3Ω t j5Ω t -j5Ω t

k

x(t) = P + 2 cos(π/R t) + 4 cos(3π/R t) + 6 sin(5π/R t)

x(t) = P + 2 cos(Ω t) + 4 cos(3Ω t) + 6 sin(5Ω t)

x(t) = P + e e + 2 e e + 3 j e e Pts

P for k

for k

a for k

3 2

3 ; 5

0 ;

Pts

j for k

otherwise

iii) Express x(t) in terms of exponential Fourier Series

0

5

5

2jk t

k

k

x t a e Pts

iv) and v) Draw the magnitude and phase spectrum of ak

k -5 -4 -3 -2 -1 0 1 2 3 4 5

kΩo -5Ωo -4Ωo -3Ωo -2Ωo -Ωo 0 Ωo 2Ωo 3Ωo 4Ωo 5Ωo

|ak| 3 0 2 0 1 P 1 0 2 0 3

Arg(ak) +900 0 0 0 0 0 0 0 0 0 -900

(3 + 3, Pts)

Now, from this table, we can easily draw the magnitude and phase spectrum of ak by taking kΩ0

along x-axis and |ak| or arg(ak) along y-axis. From the table and plot, we can easily see that |ak|

is an even function of Ω and arg(ak) is an odd function of Ω. 2 – Pts.

B) Now, consider that we formulate another periodic signal, y(t) by applying composite operations

on x(t) as

y(t) = x(2t – R) + x(R – 2t);

Determine: [3 + 2 + 10, Pts]

i) The fundamental frequency, Ω1 of y(t) and the relation between the fundamental

frequencies of x(t) and y(t)

We already know that time shift, time reversal and linearity operations performed on x(t) will

not affect its T0 2- Pts. However, time scaling operation does affect its period. Thus, T1 = T0 / 2

1- Pt and Ω1 = 2Ω0 and Ω0 has been computed above 2- Pts.

ii) The Fourier series coefficients, bk of y(t)

0

0

2 1

2 2 1

2 1

1 2 1

1

k

jk Rkjk

k k k

k

k

jk Rkjk

k k k

k

k

x t a

Pts x t R a e a e a Time shift towards right

Pts x t R a Time scaling no change

Pts x t R a e a e a Time shift towards left

Pt x t R a Time scaling no change

Pt

2 1 1

2 2 2 1

k k

k k

k

k k k

x t R a a Time Reversal

Pts y t x t R x R t a a b Linearity

C) Now x(t) mentioned in A) is applied to a causal continuous-time system whose input x(t) and

output z(t) are related by the following differential equation [15- Pts]

dz tRz t Px t

dt

Determine the frequency response, H(jΩ) and the output, z(t) generated by the above system.

5 5

0

5 5

3

2

3

3

5

k

j t j t j t

j t j t

k k k k

k kc

dz t dt Rz t Px t Pts

d dt H j e R H j e Pe Pts

j H j RH j e P e Pt

Pts R j H j P H j P R j Frequency Response

Output, y t a H jk x t c x t Pts