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Questions and Answers
What does an impulse in the time domain correspond to in the frequency domain?
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?
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?
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?
What ensures that the inverse operation of a Fourier Transform recovers the original time-domain signal exactly?
What can be said about the frequency content of an impulse signal?
What can be said about the frequency content of an impulse signal?
Which of the following best describes a DC signal?
Which of the following best describes a DC signal?
What happens to the representation of a constant in the time domain during Fourier Transform?
What happens to the representation of a constant in the time domain during Fourier Transform?
What characteristic does an impulse function in the time domain have?
What characteristic does an impulse function in the time domain have?
What is the relationship shown by the duality property in the context of forward and inverse Fourier transforms?
What is the relationship shown by the duality property in the context of forward and inverse Fourier transforms?
Which Fourier transform pair represents the function rect(t/Ï„)?
Which Fourier transform pair represents the function rect(t/Ï„)?
For the function y(t) = f(t) cos(ω0t), what is the outcome when applying the modulation property?
For the function y(t) = f(t) cos(ω0t), what is the outcome when applying the modulation property?
What happens to the function F(t) when applying the duality property to f(t)?
What happens to the function F(t) when applying the duality property to f(t)?
Which property is demonstrated when applying the Fourier transform to an even function like rect(·)?
Which property is demonstrated when applying the Fourier transform to an even function like rect(·)?
In the context of Fourier transforms, what does the term 'sinc' represent when derived from the rectangular function?
In the context of Fourier transforms, what does the term 'sinc' represent when derived from the rectangular function?
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?
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?
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?
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?
What does the Fourier Transform reveal about a real-valued signal?
What does the Fourier Transform reveal about a real-valued signal?
If a signal $f(t)$ is compressed in the time domain, what happens to its frequency spectrum?
If a signal $f(t)$ is compressed in the time domain, what happens to its frequency spectrum?
Which of the following statements about the linearity property of Fourier Transform is true?
Which of the following statements about the linearity property of Fourier Transform is true?
What is the effect of a scaling factor $|a| < 1$ on a signal in the time domain?
What is the effect of a scaling factor $|a| < 1$ on a signal in the time domain?
The Fourier Transform of a cosine signal is represented as which of the following?
The Fourier Transform of a cosine signal is represented as which of the following?
Which property of the Fourier Transform indicates the time and frequency extent relationship?
Which property of the Fourier Transform indicates the time and frequency extent relationship?
In the context of Fourier Transform, what does a function represented as $A e^{j heta}$ signify?
In the context of Fourier Transform, what does a function represented as $A e^{j heta}$ signify?
Which of the following results occurs if a signal undergoes a Fourier Transform and is subsequently shifted in time?
Which of the following results occurs if a signal undergoes a Fourier Transform and is subsequently shifted in time?
Flashcards
Duality Property (FT)
Duality Property (FT)
A property of the Fourier Transform that relates the transform of a function to the transform of its time-reversed/inverse time-scaled version.
Forward Fourier Transform
Forward Fourier Transform
Calculates the frequency spectrum (F(ω)) from a time-domain signal (f(t)).
Inverse Fourier Transform
Inverse Fourier Transform
Calculates the time-domain signal (f(t)) from a frequency-domain representation (F(ω)).
Modulation Property (FT)
Modulation Property (FT)
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Convolution
Convolution
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Fourier Transform
Fourier Transform
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Rectangular function
Rectangular function
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Sinc function
Sinc function
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Impulse in Time Domain
Impulse in Time Domain
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DC Signal in Time Domain
DC Signal in Time Domain
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Fourier Transform of Impulse
Fourier Transform of Impulse
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Fourier Transform of DC Signal
Fourier Transform of DC Signal
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Impulse Function Symbol
Impulse Function Symbol
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Frequency Domain Representation
Frequency Domain Representation
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Time Domain Representation
Time Domain Representation
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Frequency Domain for Impulse
Frequency Domain for Impulse
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Linearity Property (FT)
Linearity Property (FT)
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Symmetry Property (FT)
Symmetry Property (FT)
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Time Scaling (FT)
Time Scaling (FT)
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Time Scaling: Compressed Time
Time Scaling: Compressed Time
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Time Scaling: Expanded Time
Time Scaling: Expanded Time
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Time Scaling Effect (FT)
Time Scaling Effect (FT)
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Cosine Wave Spectrum (FT)
Cosine Wave Spectrum (FT)
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What happens to the Fourier Transform of f(t) when it undergoes time scaling?
What happens to the Fourier Transform of f(t) when it undergoes time scaling?
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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)ejωt 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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