Correlation and Convolution: Master Signal Analysis Techniques

Question 1

Which of the following is a graphical representation of autocorrelation?

Accepted Answer

Autocorrelation function

Answer

Power spectral density

Answer

Bode plot

Answer

Nyquist plot

Question 2

The cross-correlation of two signals measures their:

Accepted Answer

Similarity over time

Answer

Overlap in frequency

Answer

Energy difference

Answer

Phase shift

Question 3

What is the relationship between the convolution and cross-correlation of two signals?

Accepted Answer

Cross-correlation is the convolution of one signal with the time-reversed version of the other signal.

Answer

Convolution is the cross-correlation of one signal with the time-reversed version of the other signal.

Answer

Cross-correlation is the convolution of one signal with the complex conjugate of the other signal.

Answer

Convolution is the cross-correlation of one signal with the complex conjugate of the other signal.

Question 4

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

Accepted Answer

The product of the Fourier transforms of the two signals

Answer

The sum of the Fourier transforms of the two signals

Answer

The difference of the Fourier transforms of the two signals

Answer

The reciprocal of the product of the Fourier transforms of the two signals

Question 5

What is the convolution of a signal with a unit impulse?

Accepted Answer

The signal itself

Answer

The time-reversed version of the signal

Answer

The integral of the signal

Answer

The derivative of the signal

Question 6

What is the inverse operation of convolution?

Accepted Answer

Deconvolution

Answer

Cross-correlation

Answer

Differentiation

Answer

Integration

Question 7

What is the frequency-domain representation of a convolution between two signals?

Accepted Answer

A product of the Fourier transforms of the two signals

Answer

A sum of the Fourier transforms of the two signals

Answer

A difference of the Fourier transforms of the two signals

Answer

A quotient of the Fourier transforms of the two signals

Question 8

What is the result of correlating a signal with itself?

Accepted Answer

An autocorrelation function that measures the similarity of different parts of the signal.

Answer

A scaled version of the original signal.

Answer

A cross-correlation function that measures the similarity of the signal to another signal.

Answer

A convolution function that combines the signal with itself.

Question 9

What is the main purpose of using cross-correlation in image processing?

Accepted Answer

To detect and identify specific objects or patterns within an image.

Answer

To reduce noise and improve the overall image quality.

Answer

To enhance the edges of objects in an image.

Answer

To compress an image for efficient storage.

Question 10

What is the term for the time-reversed version of a signal?

Accepted Answer

Inverse

Answer

Conjugate

Answer

Reciprocal

Answer

Transpose

Question 11

Which of the following accurately defines the correlation between two signals x(t) and y(t)?

Accepted Answer

The average product of x(t) and y(t) shifted by τ units in time

Answer

The area under the curve formed by the product of x(t) and y(t)

Answer

The difference between the average of x(t) and the average of y(t)

Answer

The sum of the squares of x(t) and y(t)

Question 12

In the frequency domain, what is the representation of the convolution of two signals x(t) and h(t)?

Accepted Answer

X(ω)H(ω)

Answer

X(ω) * H(ω)

Answer

x(t)h(t)

Answer

X(t) * H(t)

Question 13

Which cross-correlation property is valuable for detecting periodic signals?

Accepted Answer

It generates a peak at the signal's period

Answer

It exhibits shift-invariance

Answer

It is commutative

Answer

It is associative

Question 14

How are the autocorrelation of a signal and its power spectral density related?

Accepted Answer

They form a Fourier transform pair

Answer

They form an inverse Fourier transform pair

Answer

They are equal

Answer

They are orthogonal

Question 15

In the context of linear time-invariant (LTI) systems, what is the relationship between correlation and convolution?

Accepted Answer

The convolution theorem states that a convolution in the time domain corresponds to a multiplication in the frequency domain.

Answer

Cross-correlation is the convolution of two signals in the frequency domain.

Answer

Convolution is the cross-correlation of a signal with itself.

Question 16

For any two discrete-time signals x[n] and y[n], their convolution is mathematically defined as:

Accepted Answer

(x * y)[n] = ∑_(k=-∞)^∞ x[k]y[n-k]

Answer

x[n] * y[n]

Answer

x[n] + y[n]

Answer

x[n] - y[n]

Question 17

Which property of convolution states that the order of the convolution operation does not affect the result?

Accepted Answer

Commutative property

Answer

Associative property

Answer

Distributive property

Answer

Shifting property

Question 18

Which technique is widely used to reduce the computational complexity of convolution?

Accepted Answer

Fast Fourier Transform (FFT)

Answer

Direct convolution

Answer

Circular convolution

Answer

Overlap-save method

Question 19

The direct convolution algorithm has a time complexity of:

Accepted Answer

O(N^2)

Answer

O(N)

Answer

O(N log N)

Answer

O(N^3)

Question 20

Clearly define correlation in the context of signals.

Accepted Answer

A measure of correspondence between two signals over time

Answer

A measure of the distinction between two signals over time

Answer

A measure of energy contained in a signal over time

Answer

A measure of frequency distribution in a signal over time

Question 21

What is the outcome of convolving two signals?

Accepted Answer

A new signal representing the overlap and interaction of the original signals

Answer

A new signal representing the summation of the original signals

Answer

A new signal representing the subtraction of the original signals

Answer

A new signal representing the multiplication of the original signals

Question 22

What is the practical significance of the cross-correlation function?

Accepted Answer

Determining the time delay between two signals

Answer

Determining the frequency difference between two signals

Answer

Determining the amplitude difference between two signals

Answer

Determining the phase difference between two signals

Question 23

Which signal type exhibits an autocorrelation function that resembles an impulse?

Accepted Answer

White noise

Answer

Sine wave

Answer

Square wave

Answer

Gaussian noise

Question 24

Describe the time-domain interpretation of convolution.

Accepted Answer

Output signal is the summation of shifted and multiplied inputs

Answer

Output signal is the difference of shifted and multiplied inputs

Answer

Output signal is the product of shifted and added inputs

Answer

Output signal is the quotient of shifted and multiplied inputs

Question 25

Explain the relationship between the Fourier transforms of two signals and their convolution.

Accepted Answer

Multiplication of their Fourier transforms

Answer

Addition of their Fourier transforms

Answer

Subtraction of their Fourier transforms

Answer

Division of their Fourier transforms

Question 26

What is the key benefit of employing the Fast Fourier Transform (FFT) for convolution calculations?

Accepted Answer

Substantial reduction in computational complexity

Answer

Enhanced accuracy of the convolution operation

Answer

Faster convolution execution in real-time applications

Answer

Elimination of circular convolution

Question 27

What is the impulse response of an LTI system?

Accepted Answer

The output when the input is a unit impulse

Answer

The frequency response at zero frequency

Answer

The inverse Fourier transform of the transfer function

Answer

The integral of the output over all time

Question 28

What is the inverse operation of convolution?

Accepted Answer

Deconvolution

Answer

Correlation

Answer

Differentiation

Answer

Integration

Question 29

Describe the relationship between the autocorrelation function and the power spectral density.

Accepted Answer

Fourier transform pairs

Answer

Inversely proportional

Answer

Identical

Answer

Unrelated

Question 30

Explain the primary difference between convolution and cross-correlation.

Accepted Answer

Similarity between two signals vs. similarity between a signal and its shifted version

Answer

Time domain vs. frequency domain operation

Answer

Commutative vs. non-commutative operation

Answer

Linear vs. non-linear operation

Question 31

How does the Fast Fourier Transform (FFT) algorithm enhance the computational efficiency of convolution?

Accepted Answer

Reduces the computational complexity from O(N^2) to O(N log N)

Answer

Increases the computational complexity from O(N) to O(N^2)

Answer

Has no effect on the computational complexity

Question 32

In image processing, which technique utilizes convolution to highlight the edges of an image?

Accepted Answer

Edge Detection

Answer

Smoothing

Answer

Noise Removal

Answer

Image Sharpening

Question 33

How does the Fourier transform relate to convolution operations?

Accepted Answer

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

Answer

Convolution in the time domain corresponds to addition in the frequency domain.

Answer

Convolution and Fourier transform are unrelated operations.

Answer

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

Question 34

What is the significance of the impulse response in convolution?

Accepted Answer

It serves as the weighting function in the convolution operation, shaping the output signal.

Answer

The impulse response is the system's output to an arbitrary input signal.

Answer

Convolution is the inverse operation of the impulse response.

Question 35

Which operation is employed to determine the output of a linear time-invariant (LTI) system?

Accepted Answer

Convolution

Answer

Cross-correlation

Answer

Differentiation

Answer

Integration

Question 36

Describe the graphical representation of a convolution operation.

Accepted Answer

One of the signals is flipped and shifted along the time axis.

Answer

The two signals are superimposed on top of each other.

Answer

The two signals are multiplied point-by-point.

Question 37

What is a key advantage of using circular convolution?

Accepted Answer

It eliminates edge effects when dealing with periodic signals.

Answer

It allows for real-time signal processing.

Answer

It reduces the computational complexity.

Answer

It produces a higher quality output signal.

Question 38

What is the computational complexity of performing convolution using the direct method?

Accepted Answer

O(N^2)

Answer

O(N log N)

Answer

O(log N)

Answer

O(N)

Question 39

Which of the following statements about the cross-correlation function is **incorrect**?

Accepted Answer

It always has a maximum value of 1.

Answer

It is symmetric about the origin.

Answer

It can be periodic if both signals are periodic.

Answer

It measures the similarity between two signals.

Question 40

What is the result of convolving a signal with a unit impulse?

Accepted Answer

The original signal.

Answer

A scaled version of the original signal.

Answer

A shifted version of the original signal.

Answer

A signal with no relationship to the original signal.

Question 41

Which frequency domain operation corresponds to convolution in the time domain?

Accepted Answer

Multiplication.

Answer

Subtraction.

Answer

Addition.

Answer

Division.

Question 42

What is the effect of convolving a signal with a rectangular window of length N?

Accepted Answer

Smoothing the signal over N samples.

Answer

Inverting the signal over N samples.

Answer

Sharpening the signal over N samples.

Answer

Delaying the signal by N samples.

Question 43

What is the primary application of convolution in signal processing?

Accepted Answer

Filtering and smoothing signals.

Answer

Generating noise.

Answer

Amplifying signals.

Answer

Detecting periodic signals.

Question 44

Which property of convolution demonstrates that the order of the signals does not change the result?

Accepted Answer

Commutative property.

Answer

Distributive property.

Answer

Identity property.

Answer

Associative property.

Question 45

Convolving a signal with itself results in what operation?

Accepted Answer

Autocorrelation.

Answer

Cross-correlation.

Answer

Impulse response.

Answer

Fourier transform.

Question 46

How are the Fourier transforms of two signals related to their convolution?

Accepted Answer

The Fourier transform of their convolution is the product of their individual Fourier transforms.

Answer

The Fourier transform of their convolution is the sum of their individual Fourier transforms.

Answer

The Fourier transform of their convolution is the difference of their individual Fourier transforms.

Question 47

Which of the following signals has an impulse response that is a delta function?

Accepted Answer

Ideal low-pass filter

Answer

Gaussian filter

Answer

Moving average filter

Answer

Ideal high-pass filter

Question 48

What is the effect of convolving a signal with a rectangular window?

Accepted Answer

Smoothing the signal and reducing its bandwidth.

Answer

Sharpening the signal and increasing its bandwidth.

Answer

Introducing noise into the signal.

Answer

Shifting the signal in time.

Question 49

Which of the following is an application of cross-correlation in image processing?

Accepted Answer

Template matching

Answer

Edge detection

Answer

Denoising

Answer

Histogram equalization

Question 50

True or False: Convolution is a linear operation.

Accepted Answer

True

Answer

Not applicable

Answer

False

Answer

Depends on the signals involved