Fourier Transform

Question 1

What is the main idea behind the Fourier transform?

Answer 1

It expresses a signal as a combination of sinusoidal functions (sines and cosines) with different frequencies.

Question 2

What is the difference between the Fourier series and the Fourier transform?

Answer 2

The Fourier series applies to periodic functions, while the Fourier transform applies to non-periodic functions.

Question 3

What type of signals does the Fourier series apply to?

Answer 3

Periodic signals.

Question 4

What is the formula for the Fourier series of a periodic function f(x)?

Answer 4

f(x) = a₀/2 + Σ[ aₙ cos(nx) + bₙ sin(nx) ]

Question 5

What do the Fourier coefficients aₙ and bₙ represent?

Answer 5

They measure the contribution of each cosine and sine wave to the overall function.

Question 6

What is the formula for aₙ in the Fourier series?

Answer 6

aₙ = (1/π) ∫ from -π to π of f(x)·cos(nx) dx

Question 7

What is the formula for bₙ in the Fourier series?

Answer 7

bₙ = (1/π) ∫ from -π to π of f(x)·sin(nx) dx

Question 8

What kind of functions can be represented using the Fourier series?

Answer 8

Any well-behaved periodic function.

Question 9

What does a Fourier series approximation of a square wave look like?

Answer 9

A sum of odd harmonics of sine waves with decreasing amplitude.

Question 10

What is the intuition behind using sine and cosine waves in Fourier analysis?

Answer 10

They form a complete orthogonal basis for representing any periodic function.

Question 12

What is a complex number?

Answer 12

A number of the form a + bi, where i is the square root of -1.

Question 13

What is the polar form of a complex number?

Answer 13

r(cosθ + i·sinθ), where r is the magnitude and θ is the angle (argument).

Question 14

What is Euler’s formula?

Answer 14

e^{ix} = cos(x) + i·sin(x)

Question 15

Why is Euler’s formula important in Fourier analysis?

Answer 15

It allows sinusoidal functions to be written as complex exponentials, simplifying calculations.

Question 17

What is the formula for the continuous Fourier transform of f(x)?

Answer 17

F(ω) = (1/√(2π)) ∫ from -∞ to ∞ of f(x)·e^{-iωx} dx

Question 18

What is the formula for the inverse Fourier transform?

Answer 18

f(x) = (1/√(2π)) ∫ from -∞ to ∞ of F(ω)·e^{iωx} dω

Question 19

What does F(ω) represent in the Fourier transform?

Answer 19

The amount of frequency ω present in the function f(x).

Question 20

What domain does the Fourier transform operate in?

Answer 20

It transforms a function from the spatial/time domain to the frequency domain.

Question 21

What is the convolution theorem?

Answer 21

Convolution in the spatial domain corresponds to multiplication in the frequency domain.

Question 22

How does the Fourier transform help with convolution?

Answer 22

It allows convolution to be performed as a simple multiplication in the frequency domain, which is faster.

Question 23

What is the role of frequency analysis in image processing?

Answer 23

It helps detect patterns, compress data, and apply filters based on frequency content.

Question 25

What does a high-frequency component represent in an image?

Answer 25

Rapid changes like edges or noise.

Question 26

What does a low-frequency component represent in an image?

Answer 26

Slow variations like smooth gradients or large shapes.

Question 27

What is one use of the Fourier transform in image compression?

Answer 27

It allows removal of high-frequency components that the human eye doesn’t easily notice, as in JPEG compression.

Question 28

What does structured noise look like in the frequency domain?

Answer 28

It appears as sharp spikes or repetitive patterns.

Question 29

Why is the Fourier transform useful for removing structured noise?

Answer 29

Because noise can be identified and removed as specific frequencies.

Question 31

What is the main advantage of viewing an image in the frequency domain?

Answer 31

It reveals repeating patterns and enables efficient filtering and compression.

Question 32

What does the Fourier transform reveal that pixel-based views do not?

Answer 32

The frequency composition of the image — how rapidly it changes in space.

Question 33

What does the linearity property of the Fourier Transform state?

Answer 33

The transform of a linear combination of functions equals the linear combination of their transforms.

Question 34

What is the formula for linearity in the Fourier Transform?

Answer 34

F[af + bg] = aF[f] + bF[g]

Question 36

What happens when a signal is shifted in time?

Answer 36

Its Fourier transform is multiplied by a complex exponential phase shift.

Question 37

What is the time shift property formula?

Answer 37

f(x - x₀) ↔ e^{-iωx₀}F(ω)

Question 38

What is the frequency shift property?

Answer 38

Multiplying a signal by e^{iω₀x} shifts its spectrum: f(x)e^{iω₀x} ↔ F(ω - ω₀)

Question 40

What is the convolution theorem?

Answer 40

Convolution in the time/spatial domain corresponds to multiplication in the frequency domain.

Question 41

What is the formula for the convolution theorem?

Answer 41

f * g ↔ F(ω) · G(ω)

Question 43

What is Parseval’s Theorem?

Answer 43

It states that the total energy of a signal is conserved between the time and frequency domains.

Question 44

What is the formula for Parseval’s Theorem?

Answer 44

∫|f(x)|² dx = ∫|F(ω)|² dω

Question 46

What does the scaling property of the Fourier Transform state?

Answer 46

Compressing a function in time stretches it in frequency, and vice versa.

Question 47

What is the formula for the scaling property?

Answer 47

f(ax) ↔ (1/|a|)F(ω/a)

Question 49

What is the Discrete Fourier Transform (DFT)?

Answer 49

A method to transform a sequence of values into its frequency components.

Question 50

What is the formula for the DFT?

Answer 50

Xₖ = Σₙ xₙ · e^{-i2πkn/N} for k = 0 to N-1

Question 51

What is the inverse DFT formula?

Answer 51

xₙ = (1/N) Σₖ Xₖ · e^{i2πkn/N}

Question 53

What is the time complexity of the naive DFT?

Answer 53

O(N²)

Question 54

What problem does the Fast Fourier Transform (FFT) solve?

Answer 54

It computes the DFT efficiently using a divide-and-conquer algorithm.

Question 55

What is the time complexity of the FFT?

Answer 55

O(N log N)

Question 56

Why is the FFT important in image processing?

Answer 56

It enables real-time or large-scale Fourier-based filtering and analysis.

Question 58

What is frequency-based filtering in images?

Answer 58

The use of Fourier transforms to isolate or suppress specific frequency components.

Question 59

What is a low-pass filter used for?

Answer 59

To remove high-frequency noise and smooth the image.

Question 60

What is a high-pass filter used for?

Answer 60

To remove low-frequency components and enhance edges.

Question 61

What does a band-pass filter do?

Answer 61

It preserves a specific range of frequencies while removing others.

Question 63

What is structured noise in the frequency domain?

Answer 63

Unwanted repetitive patterns that appear as spikes in the frequency spectrum.

Question 64

How can structured noise be removed?

Answer 64

By identifying and suppressing specific frequency components in the Fourier domain.

Question 66

How does JPEG use frequency-based compression?

Answer 66

It applies a Discrete Cosine Transform (DCT), keeping low-frequency components and discarding high frequencies.

Question 67

Why are high-frequency components often removed in compression?

Answer 67

Because they contribute less to perceived image quality and take up more space.

Question 69

What does a repetitive image pattern look like in the frequency domain?

Answer 69

As sharp peaks indicating dominant frequencies.

Question 70

How is Fourier analysis used in texture classification?

Answer 70

By analyzing the frequency content of repeating patterns.

Question 72

What is the main advantage of the convolution theorem in image processing?

Answer 72

It allows faster filtering by multiplying in the frequency domain instead of convolving in the spatial domain.

Question 74

What is Euler’s identity and why is it important in Fourier analysis?

Answer 74

e^{ix} = cos(x) + i sin(x); it simplifies expressions using complex exponentials.

Question 76

What does the DFT do for digital signals and images?

Answer 76

It decomposes them into discrete frequency components for analysis or filtering.