Signals and Systems: Fourier Transform

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Questions and Answers

What does an impulse in the time domain correspond to in the frequency domain?

A constant value (correct)

A variable signal

An exponentially decaying function

A sinusoidal wave

In the Fourier Transform, what does a constant signal in the time domain yield?

A continuous waveform

A constant function

An impulse at zero frequency (correct)

A periodic signal

Why does the integral of a constant function not converge when performing Fourier Transform?

The amplitude is too high

It oscillates indefinitely

It has infinite area under the curve (correct)

It has a variable frequency

What ensures that the inverse operation of a Fourier Transform recovers the original time-domain signal exactly?

The scaling factor of 2π Signup and view all the answers

What can be said about the frequency content of an impulse signal?

It is useful for system analysis Signup and view all the answers

Which of the following best describes a DC signal?

Represents zero-frequency content Signup and view all the answers

What happens to the representation of a constant in the time domain during Fourier Transform?

It results in a flat spike in frequency domain Signup and view all the answers

What characteristic does an impulse function in the time domain have?

It exists as a sharp spike Signup and view all the answers

What is the relationship shown by the duality property in the context of forward and inverse Fourier transforms?

The forward and inverse transforms are similar. Signup and view all the answers

Which Fourier transform pair represents the function rect(t/τ)?

τ sinc(ωτ/2) Signup and view all the answers

For the function y(t) = f(t) cos(ω0t), what is the outcome when applying the modulation property?

The outcome is a convolution of F(ω) and M(ω). Signup and view all the answers

What happens to the function F(t) when applying the duality property to f(t)?

It transforms to 2π f(−ω). Signup and view all the answers

Which property is demonstrated when applying the Fourier transform to an even function like rect(·)?

It maintains the even characteristics. Signup and view all the answers

In the context of Fourier transforms, what does the term 'sinc' represent when derived from the rectangular function?

The Fourier transform of a box-like function. Signup and view all the answers

When rect(ω/τ) is obtained from the time domain function τ sinc(tτ/2), what does this indicate about the nature of the transform?

It suggests linear scaling in frequency. Signup and view all the answers

If a signal f(t) is processed through a Fourier transform yielding F(ω), what does M(ω) represent in the context of modulation property?

It is the Fourier transform of a modulating signal. Signup and view all the answers

What does the Fourier Transform reveal about a real-valued signal?

It has conjugate symmetry. Signup and view all the answers

If a signal $f(t)$ is compressed in the time domain, what happens to its frequency spectrum?

It becomes narrower. Signup and view all the answers

Which of the following statements about the linearity property of Fourier Transform is true?

It asserts that $ax_1(t) + bx_2(t)$ transforms to $aX_1( au) + bX_2( au)$. Signup and view all the answers

What is the effect of a scaling factor $|a| < 1$ on a signal in the time domain?

It compresses the frequency axis and stretches the time axis. Signup and view all the answers

The Fourier Transform of a cosine signal is represented as which of the following?

$rac{1}{2}[ ext{δ}( au - au_0) + ext{δ}( au + au_0)]$ Signup and view all the answers

Which property of the Fourier Transform indicates the time and frequency extent relationship?

Time Scaling Signup and view all the answers

In the context of Fourier Transform, what does a function represented as $A e^{j heta}$ signify?

It combines both amplitude and phase information. Signup and view all the answers

Which of the following results occurs if a signal undergoes a Fourier Transform and is subsequently shifted in time?

The frequency spectrum is modified by a phase shift. Signup and view all the answers

Study Notes

Signals and Systems Course Information

Course: Signals and Systems
Level: Software Engineering & IT, 2nd year
Instructor: Dr. Eng. Iyad M. Abuhadrous
Affiliation: Associate professor in Computer & Control (Robotics), Egyptian Chinese University
Email: [email protected]

The Fourier Transform

Lecture Objective: Understanding the Fourier Transform's role in signal analysis and its engineering applications.
Key Concepts:
- Signal representation in both time and frequency domains.
- Analyzing periodic and non-periodic signals using the Fourier Transform.
- Examples of non-periodic signals, including u(t), exp[-t]u(t), and rect(t/T).
Applications:
- Audio signal processing
- Image compression, such as JPEG format

The Continuous Fourier Transform

Function: The Fourier Transform decomposes a signal into its constituent frequency components (spectral content).
Wide Use: Widely used in signal processing, communications, and image analysis.
Time-to-Frequency Conversion: Used to convert time-domain waveforms into their frequency-domain equivalents.
Mathematical Formula: Provides a mathematical formula for analyzing a signal's frequency content, using the formula: X(ω) = ∫_-∞^∞ x(t)e^-jωt dt
Inverse Fourier Transform: Used to convert back to the time domain. x(t) = (1/2π) ∫_-∞^∞ X(jω)e^jωt dω

Fourier Transform Examples

Rectangular pulse:
- A rectangular pulse in the time domain results in a sinc function in the frequency domain.
- The pulse width is inversely proportional to the spread of its frequency components.
Impulse:
- An impulse function in the time domain corresponds to a constant in the frequency domain.
- A constant signal in the time domain results in an impulse in the frequency domain.
Cosine Wave:
- A cosine wave, mathematically represented and transformed, corresponds to two impulses in the frequency domain.

Fourier Transform Properties

Linearity: Fourier transform of a linear combination of signals is equivalent to a linear combination of the respective transforms.
Symmetry: If the signal is real-valued, its Fourier Transform satisfies the conjugate symmetry property (X(-ω) = X*(ω), where * denotes complex conjugate).
Time Scaling: Scaling in the time domain affects the frequency domain in an inverse proportional manner.
Time Shifting: Shifting a signal in the time domain results in a phase shift in the frequency domain.
Frequency Shifting: Modulation of the signal in the time domain adds a phase shift in the frequency domain.
Convolution: Convolution in time is equivalent to multiplication in frequency, and vice versa.
Differentiation: The Fourier transform of the derivative of a signal in the time domain corresponds to multiplying by jω.
Integration: The Fourier transform of the integral of a signal is given by a function over which F(ω) is considered.
Duality: A duality concept that states there is symmetry between the time-domain and frequency-domain for functions that are transformed.

Fourier Transform Applications

Amplitude Modulation (AM): Used to understand and work with the signal spectrum for AM radio, analog TV, and aircraft communication.
Multiplexing: A technique for efficiently using bandwidth by transmitting multiple signals simultaneously over the same channel, (e.g. in various radio bands)

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Description

This quiz covers the key concepts of the Fourier Transform in signal analysis, focusing on its application in engineering. Students will explore both time and frequency domain representations and how to analyze various signals. Examples include audio processing and image compression techniques like JPEG.