Fourier Transform: Master Frequency Analysis in Signals and Systems

Question 1

For a signal to be Fourier transformable, which requirement must be met?

Accepted Answer

The signal must be absolutely integrable.

Answer

The signal must be periodic.

Answer

The signal must be bounded.

Answer

The signal must be continuous.

Question 2

In the frequency domain, what operation corresponds to convolution in the time domain?

Accepted Answer

Multiplication

Answer

Correlation

Answer

Addition

Answer

Subtraction

Question 3

What is the Fourier transform of a rectangular pulse?

Accepted Answer

A sinc function

Answer

A Gaussian function

Answer

A triangular function

Answer

A trapezoidal function

Question 4

Which of the following is NOT a property of the Fourier transform?

Accepted Answer

Irreversibility

Answer

Linearity

Answer

Time-invariance

Answer

Energy preservation

Question 5

What is the Fourier transform of a unit impulse function?

Accepted Answer

A Dirac delta function

Answer

A unit step function

Answer

A constant function

Answer

A sinusoidal function

Question 6

Which of the following is NOT an application of the Fourier transform?

Accepted Answer

Quantum mechanics

Answer

Signal analysis

Answer

Image compression

Answer

Speech enhancement

Question 7

What is the inverse Fourier transform of a real and even function?

Accepted Answer

A real and even function

Answer

A real and odd function

Answer

An imaginary and odd function

Answer

An imaginary and even function

Question 8

Which of the following is the definition of the Fourier transform of a function x(t)?

Accepted Answer

X(f) = ∫x(t)e^(-j2πft)dt

Answer

X(f) = ∫x(f)e^(-j2πft)dt

Answer

X(t) = ∫x(f)e^(-j2πft)df

Answer

X(t) = ∫x(f)e^(j2πft)df

Question 9

What is the Fourier transform of a periodic function?

Accepted Answer

A periodic function

Answer

An aperiodic function

Answer

A discrete function

Answer

A continuous function

Question 10

Which of the following is the Fourier transform of a Gaussian function?

Accepted Answer

Another Gaussian function

Answer

An exponential function

Answer

A sinusoidal function

Answer

A Dirac delta function

Question 11

What is the effect of convolution in the time domain on the Fourier transforms of the convolved functions?

Accepted Answer

Multiplication

Answer

Addition

Answer

Subtraction

Answer

Division

Question 12

What is the relationship between the Fourier transform and the Laplace transform?

Accepted Answer

The Laplace transform is an extension of the Fourier transform

Answer

The Fourier transform is an extension of the Laplace transform

Answer

They are unrelated transforms

Answer

They both have different definitions

Question 13

The Fourier transform of the derivative of a signal is:

Accepted Answer

jωX(ω)

Answer

-jωX(ω)

Answer

jX'(ω)

Answer

-jX'(ω)

Question 14

Parseval's theorem states that:

Accepted Answer

The energy of a signal is conserved in both time and frequency domains

Answer

The power of a signal is conserved in both time and frequency domains

Answer

The bandwidth of a signal is the same in both time and frequency domains

Answer

The duration of a signal is the same in both time and frequency domains

Question 15

The Fourier transform of a Gaussian function is:

Accepted Answer

Another Gaussian function

Answer

A rectangular function

Answer

A sinc function

Answer

A delta function

Question 16

The discrete-time Fourier transform (DTFT) is used to analyze:

Accepted Answer

Discrete-time signals

Answer

Analog signals

Answer

Digital signals

Answer

Continuous-time signals

Question 17

How can the Fourier transform be used to analyze the frequency response of a linear time-invariant (LTI) system?

Accepted Answer

Apply the Fourier transform to the system's impulse response.

Answer

Apply the inverse Fourier transform to the system's transfer function.

Answer

Apply the Laplace transform to the system's differential equation.

Answer

Apply the z-transform to the system's difference equation.

Question 18

What is the Fourier transform characteristic of a real, even signal?

Accepted Answer

Real and even

Answer

Real and odd

Answer

Imaginary and even

Answer

Imaginary and odd

Question 19

Which of the following operations in the time domain corresponds to multiplication in the frequency domain?

Accepted Answer

Convolution

Answer

Correlation

Answer

Addition

Answer

Subtraction

Question 20

What is the Fourier transform of a Gaussian function?

Accepted Answer

Another Gaussian function

Answer

Exponential function

Answer

Cosine function

Answer

Sine function

Question 21

Find the Fourier transform of a Gaussian function.

Accepted Answer

Another Gaussian function

Answer

A rectangular function

Answer

A sinc function

Answer

An exponential function

Question 22

Describe the relationship between the bandwidth of a signal and its Fourier transform.

Accepted Answer

The relationship between the bandwidths depends on the specific signal.

Answer

The Fourier transform bandwidth is always equal to the signal bandwidth.

Answer

The Fourier transform bandwidth is always wider than the signal bandwidth.

Answer

The Fourier transform bandwidth is always narrower than the signal bandwidth.

Question 23

State the convolution theorem for the Fourier transform.

Accepted Answer

The Fourier transform of a convolution is equal to the product of the individual Fourier transforms.

Answer

The convolution of two functions in the time domain is equal to the product of their Fourier transforms.

Answer

The convolution of two functions in the frequency domain is equal to the product of their Fourier transforms.

Answer

The convolution theorem is only applicable to functions that are periodic.

Question 24

The Fast Fourier Transform (FFT) offers a computational advantage over the Discrete Fourier Transform (DFT). What is the computational complexity of each algorithm?

Accepted Answer

FFT: O(N log N), DFT: O(N^2)

Answer

FFT: O(N^3), DFT: O(N^2)

Answer

FFT: O(N^2), DFT: O(N)

Answer

FFT: O(N), DFT: O(N^3)

Question 25

Which property of the Fourier transform states that the Fourier transform of a time-reversed signal x(-t) is the complex conjugate of the original Fourier transform X(f)?

Accepted Answer

Time-reversal property

Answer

Conjugation property

Answer

Scaling property

Answer

Convolution property

Question 26

Which of the following is NOT an application of the Fourier transform in signal processing?

Accepted Answer

Causal analysis

Answer

Frequency analysis

Answer

Image processing

Answer

Speech recognition

Question 27

What is the relationship between the bandwidth of a signal and its duration in time?

Accepted Answer

They are inversely proportional.

Answer

They are directly proportional.

Answer

There is no relationship between them.

Question 28

Which of the following is the Fourier transform of a Gaussian function?

Accepted Answer

Another Gaussian function

Answer

A sinusoid

Answer

A Dirac delta function

Answer

A step function

Question 29

Which of the following is the Fourier transform of a periodic signal?

Accepted Answer

A discrete set of impulses

Answer

A continuous function

Answer

A step function

Answer

A Dirac delta function

Question 30

What is the Nyquist frequency?

Accepted Answer

The highest frequency that can be represented without aliasing.

Answer

The lowest frequency that can be represented without aliasing.

Answer

The bandwidth of a signal.

Answer

The duration of a signal.

Question 31

What is Parseval's Theorem for the Fourier transform?

Accepted Answer

The energy of a signal is equal to the integral of the square of its Fourier transform.

Answer

The power of a signal is equal to the integral of the square of its Fourier transform.

Answer

The bandwidth of a signal is equal to the integral of the square of its Fourier transform.

Answer

The duration of a signal is equal to the integral of the square of its Fourier transform.

Question 32

Which of the following is a property of the Fourier transform?

Accepted Answer

Linearity

Answer

Periodicity

Answer

Nonlinearity

Answer

Non-invertibility

Question 33

What is the Fourier transform of a periodic function with period T?

Accepted Answer

A discrete spectrum with frequency spacing of 1/T

Answer

A continuous spectrum

Answer

An impulse train

Question 34

Which of the following applications utilizes the Fourier transform?

Accepted Answer

Speech processing

Answer

Database management

Answer

Word processing

Answer

Email

Question 35

What is the Parseval's theorem for the Fourier transform?

Accepted Answer

The energy of a signal is preserved in the frequency domain.

Answer

The bandwidth of a signal is inversely proportional to its duration.

Answer

The Fourier transform of a real function is real and even.

Question 36

What is the significance of the Nyquist frequency in digital signal processing?

Accepted Answer

It represents the maximum frequency that can be represented without aliasing.

Answer

It is the frequency at which the signal is sampled.

Answer

It is the frequency at which the signal is quantized.

Question 37

Which of the following is NOT a limitation of the Fourier transform?

Accepted Answer

It can analyze non-periodic signals.

Answer

It is computationally intensive for large signals.

Answer

It does not provide information about the phase of the signal.

Answer

It assumes the signal is of infinite duration.

Question 38

What is the Fourier transform of a Dirac delta function?

Accepted Answer

1

Answer

cos(t)

Answer

sin(t)

Answer

t

Question 39

The Fourier transform of an aperiodic signal is:

Accepted Answer

A continuous spectrum

Answer

A complex spectrum

Answer

A discrete spectrum

Question 40

In the frequency domain, the operation that corresponds to convolution in the time domain is:

Accepted Answer

Multiplication

Answer

Division

Answer

Addition

Answer

Subtraction

Question 41

What does the Fourier transform of a sinusoidal signal look like?

Accepted Answer

Two impulses located at the positive and negative frequencies corresponding to the sinusoidal signal.

Answer

A single impulse at the corresponding frequency.

Answer

A cosine function at the corresponding frequency.

Answer

A step function at the corresponding frequency.

Question 42

In the context of the Fourier transform, what operation in the frequency domain corresponds to convolution in the time domain?

Accepted Answer

Multiplication.

Answer

Addition.

Answer

Division.

Question 43

If a signal is real-valued in the time domain, what kind of signal will its inverse Fourier transform produce?

Accepted Answer

A real-valued signal.

Answer

An even signal.

Answer

A periodic signal.

Answer

A complex-valued signal.

Question 44

What is the Fourier transform of a Dirac delta function?

Accepted Answer

A constant function with a value of 1.

Answer

0.

Answer

-1.

Answer

Infinity.

Question 45

What operation in the frequency domain corresponds to shifting a signal in the time domain?

Accepted Answer

Multiplication by a complex exponential.

Answer

Addition of a constant.

Answer

Division by a complex exponential.

Question 46

What does the Fourier transform of a rectangular pulse resemble?

Accepted Answer

A sinc function.

Answer

A trapezoidal function.

Answer

A triangular function.

Answer

A Gaussian function.

Question 47

Which type of signal has a Fourier transform that is band-limited (i.e., has a finite bandwidth)?

Accepted Answer

Signal with finite duration.

Answer

Signal with a non-zero mean.

Answer

Periodic signal.

Answer

Signal with infinite duration.

Question 48

What is the Fourier transform of the convolution of two signals?

Accepted Answer

The product of their individual Fourier transforms.

Answer

The quotient of their Fourier transforms.

Answer

The difference of their Fourier transforms.

Answer

The sum of their Fourier transforms.

Question 49

What is the Fourier transform of a rectangular pulse of width T and amplitude A?

Accepted Answer

A sinc function with a width inversely proportional to T.

Answer

A delta function with a strength of AT.

Answer

A constant function with a value of A.

Question 50

A signal has a spectrum with a dominant peak at 1kHz. What does this indicate about the signal in the time domain?

Accepted Answer

The signal has a strong component at a frequency of 1kHz.

Answer

The signal has a period of 1kHz.

Answer

The signal has a bandwidth of 1kHz.

Question 51

What is the primary advantage of using the Fourier transform in signal processing?

Accepted Answer

It allows for analyzing the frequency content of signals.

Answer

It simplifies the process of signal generation.

Answer

It improves the signal-to-noise ratio.

Question 52

Consider a signal x(t) and its Fourier transform X(f). What is the Fourier transform of x(t-T), where T is a constant?

Accepted Answer

X(f) * exp(-j2πfT)

Answer

X(f) * T

Answer

X(f - T)

Answer

X(f)/T

Question 53

A system has a frequency response H(f). What information does this provide about the system?

Accepted Answer

How the system affects different frequency components of the input signal.

Answer

The energy content of the system.

Answer

The time-domain characteristics of the system.

Question 54

What is the purpose of the inverse Fourier transform?

Accepted Answer

Reconstructing a signal from its frequency spectrum.

Answer

Finding the frequency response of a system.

Answer

Analyzing the energy content of a signal.

Question 55

A signal is sampled at a rate of 10kHz. What is the maximum frequency component that can be accurately represented in its Fourier transform?

Accepted Answer

5kHz

Answer

1kHz

Answer

10kHz

Answer

20kHz

Question 56

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

Accepted Answer

Fourier series represents periodic signals, while the Fourier transform represents non-periodic signals.

Answer

Fourier series is used for continuous signals, while the Fourier transform is used for discrete signals.