Fourier Transform II

Question 1

What does the Fourier Transform (FT) permit for aperiodic signals?

Answer 1

Representation with everlasting complex exponentials

The FT allows for the analysis of signals in the frequency domain.

Question 2

What are the conditions for convergence of x(t) in Fourier Transform?

Answer 2

• x(t) is absolutely integrable
• x(t) is squared integrable

These conditions ensure that signals can be transformed into the frequency domain.

Question 3

What type of signals are Fourier transformable?

Answer 3

• All bounded outputs (including impulse responses)
• All energy signals

Signals that are neither power nor energy are not Fourier transformable.

Question 4

True or false: The LTI system applies a time-scaling factor to input signals expressed as Fourier series.

Answer 4

FALSE

The LTI system scales the amplitude and time-shifts the signal, but does not apply a time-scaling factor.

Question 5

What does the magnitude spectrum represent in the Fourier Transform?

Answer 5

Plot of |X(w)| of f(t)

It allows visualization of the frequency content of a signal.

Question 6

What does the phase spectrum represent in the Fourier Transform?

Answer 6

Plot of ∠X(w) of f(t)

It provides information about the phase shifts of the frequency components.

Question 7

What is Parseval’s relation for the Fourier Transform?

Answer 7

Energy in f(t) is equal to the area under |X(w)|^2 normalized by 2

It indicates that energy is conserved in the Fourier domain.

Question 8

What are power signals characterized by?

Answer 8

• Include periodic signals
• Include constants
• Include the step function

Power signals are neither absolutely nor square integrable.

Question 9

What is the significance of impulses in the Fourier Transform of power signals?

Answer 9

They allow aperiodic, power signals to be Fourier transformable

This is important for analyzing signals that are not energy signals.

Question 10

What is the energy spectral density in the context of Parseval’s relation?

Answer 10

Reflects the energy per unit frequency

It is derived from the Fourier Transform and indicates how energy is distributed across frequencies.

Question 11

What is the energy of a signal f(t) given by?

Answer 11

∫ |f(t)|^2 dt

This integral calculates the total energy of the signal over time.

Question 12

What does the FT of power signals not exist in?

Answer 12

The conventional sense

Power signals require special consideration in Fourier analysis.

Question 13

What is the plot of the frequency spectrum used for?

Answer 13

Visualization of the frequency content or ‘spectral’ content of a signal

It helps in understanding how different frequencies contribute to the overall signal.