MODULE 4

Question 1

CONTINUOUS-TIME FOURIER TRANSFORM

Answer 1

The continuous time Fourier transform (CTFT) is a signal processing operation that transforms a continuous-time signal into a continuous frequency signal

Question 2

WELL-DEFINED FOURIER TRANSFORM

Answer 2

The FT does not contain undefined and/or Infinite values.

Question 3

MATHEMATICAL FORMULA (CTFT)

Answer 3

X(ω)=∫_(-∞)^∞▒〖x(t) e^(-jωt) dt〗

Question 4

EXISTENCE OF FOURIER TRANSFORM

Answer 4

A Continuous Time Signal has CTFT if it is either an Energy or a Power Signal.
In example, NENP signal (except for δ(t)) has no CTFT.

Question 5

e^(-at) u(t)

Answer 5

1/(a+jω)

Question 6

e^(-a|t| )

Answer 6

2a/(a^2+ω^2 ),a>0

Question 7

〖t ∙e〗^(-at) u(t)

Answer 7

1/(a+jω)^2

Question 8

Sgn(t)

Answer 8

2/jω

Question 9

Sa(at)

Answer 9

π/a rect (ω/2a)

Question 10

rect(t/T)

Answer 10

T Sa(ωT/a)

Question 11

1. LINEARITY

Answer 11

x_1 (t)↔X_1 (ω)
x_2 (t)↔X_2 (ω)
〖ax〗_1 (t)+bx_2 (t)↔ax_1 (ω)+bx_2 (ω)

Question 12

2. TIME SHIFTING

Answer 12

x(t-t_0 )↔e^(-jωt_0 ) x(ω)
x(t+t_0 )↔e^(jωt_(0 ) ) x(ω)

Question 13

3. TIME SCALING

Answer 13

x(at)↔1/|a| X(ω/a)
NOTE: Divide all ω by a

Question 14

4. TIME FOLDING

Answer 14

x(-t)↔ X(-ω)
NOTE: Put negative sign in all ω

Question 15

5. FREQUENCY SHIFT

Answer 15

e^jωt x(t)↔ X(ω-ω_0 )
e^(-jωt_ ) x(t)↔ X(ω+ω_0 )

Question 16

6. DUALITY

Answer 16

x(t)↔X(ω)
X(t)↔2πx(-ω)

Question 17

7. TIME DIFFERENTATION

Answer 17

d^n/〖dt〗^n [x(t)]=〖(jω)〗^n X(ω)

Question 18

8. FREQUENCY DIFFERENTATION

Answer 18

tx(t)↔j d/dω [X(ω)]

Question 19

9. MULTIPLICATION PROPERTY

Answer 19

〖x_1 (t)x〗_2 (t)↔X_1 (ω)∙X_2 (ω)
〖x_1 (t)∙x〗_2 (t)↔1/2π (X_1 (ω)X(ω))

Question 20

10. AREA PROPERTY

Answer 20

X(ω)=∫(-∞)^∞▒x(t) e^(-jωt) dt
X(0)=∫(-∞)^∞▒x(t) dt=AREA
INVERESE FT
x(t)=1/2π ∫(-∞)^∞▒x(ω) e^jωt dω
x(0)=1/2π ∫(-∞)^∞▒x(ω) dω=AREA

Question 21

PARSEVAL’S THEOREM

Answer 21

∫(-∞)^∞▒|x(t)|^2 dt=1/2π ∫(-∞)^∞▒|X(ω)|^2 dω

Question 22

1

Answer 22

2πδ(ω)

Question 23

cos(ωt)

Answer 23

π[δ(ω-ω)+δ(ω+ω)]

Question 24

sin(ωt)

Answer 24

jπ[δ(ω-ω)-δ(ω+ω)]