MODULE 3

Question 1

CONTINUOUS-TIME FOURIER SERIES

Answer 1

The continuous time Fourier Series (CTFS) refers to a signal processing operation that transforms continuous time signal into discrete frequency signal.
The CTFS is used to represent a periodic non-sinusoidal signal into sum of harmonically related sinusoids.

Question 1

A. TRIGONOMETRIC FOURIER SERIES

Answer 1

x(t)= a_0+∑(n=1)^∞▒〖a_n cos(nω_0 t)+∑(n=1)^∞▒〖b_n sin(nω_0 t)〗〗

Question 2

a_0

Answer 2

1/t_0 ∫_T▒x(t)dt

Question 3

a_n

Answer 3

2/t_0 ∫_T▒x(t)cos(nω_0 t)dt

Question 4

b_n

Answer 4

2/t_0 ∫_T▒x(t)sin(nω_0 t)dt

Question 5

ODD SYMMETRY

Answer 5

{█(a_0=0@a_n=0@b_n≠0)┤

Question 6

EVEN SMMETRY

Answer 6

{█(a_0=0@a_n≠0@b_n=0)┤

Question 7

HALF- WAVE SYMMETRY

Answer 7

{█(a_0=0@a_n≠b_n≠0 ∀ odd n @〖a_n=b〗_n=0∀ even n)┤

Question 8

B. POLAR FOURIER SERIES

Answer 8

x(t)= d_0+∑_(n=1)^∞▒〖d_n cos(nω_0 t+∅_n ) 〗

Question 9

d_0

Answer 9

〖=a〗_0=1/t_0 ∫_T▒x(t)dt

Question 10

d_n

Answer 10

√(〖a_n〗^2+〖b_n〗^2 )

Question 11

∅_n

Answer 11

〖tan〗^(-1) (b_n/a_n )

Question 12

C. EXPONENTIAL FOURIER SERIES

Answer 12

x(t)= c_0+∑_(n=1)^∞▒c_n e^(j〖nω〗_0 t)

Question 13

c_0

Answer 13

〖=a〗_0=1/t_0 ∫_T▒x(t)dt

Question 14

c_n

Answer 14

1/t_0 ∫_T▒〖x(t) e^(-jnω_0 t) dt〗

Question 15

a_n RELATIONSHIP WITH TRIGONOMETRIC FOURIER SERIES COEFFICIENTS

Answer 15

c_n+ c_(-n)

Question 16

b_n RELATIONSHIP WITH TRIGONOMETRIC FOURIER SERIES COEFFICIENTS

Answer 16

j(c_n- c_(-n) )

Question 17

c_n RELATIONSHIP WITH TRIGONOMETRIC FOURIER SERIES COEFFICIENTS

Answer 17

(a_n-jb_n)/2