Fourier Transform

Question 1

What is the Fourier Series?

Answer 1

The fourier series represents continuous time periodic signals with harmonically related complex exponentials.

Question 2

What is the summation for the fourier series? Name the variables.

Answer 2

Question 3

What is Euler’s formula for representing a complex exponential?

Answer 3

Question 4

What is the formula for the Fourier Transform?

Answer 4

Question 5

How is linearity expressed in the frequency domain?

Answer 5

a₁X₁(ω) + a₂X₂(ω)

Question 6

How is a shift in the time domain represented in the frequency domain?

Answer 6

x(t - t₀) = e^-jωt₀X(ω)

• A multiplication by t₀ for the exponential

Question 7

How is frequency shift in the frequency domain expressed in the time domain?

Answer 7

e^jω₀tx(t) –> X(ω - ω₀)

Question 8

Describe how a signal shift in the time domain affects the frequency domain

Answer 8

Shifting a signal in the time domain results in a linear phase shift in the frequency domain.

Question 9

What is an ideal low pass filter?

Answer 9

Eliminates all frequency components with a frequency whos magnitude is greater than some cutoff frequency, while leaving the remaining frequency components unaffected.

The filter allows signals in the passband and eliminates signals either side of the cutoff frequency (stopband)