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Questions and Answers
What does the equation $Y(ω) = H(ω0) imes 2 ext{π} imes ext{δ}(ω - ω0)$ represent?
What does performing an inverse FT on $Y(ω)$ yield?
When a complex exponential is input into a system, what is the primary effect on its output?
What does the frequency response of a system demonstrate?
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What is implied by shifting a delta function in frequency analysis?
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What does the input signal x(t) represent in the context of frequency response?
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How is the Fourier Transform X(ω) of the input signal x(t) characterized?
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What is the role of H(ω0) when the system responds to a shifted delta function?
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What is the nature of the output Y(ω) when the input X(ω) is a shifted delta function?
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Which of the following statements correctly describes the action of the system on a complex sinusoid?
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Study Notes
Frequency Response: System's Action on Input
- Input: A complex exponential signal with a fixed frequency ω0.
- Frequency Domain Representation: The Fourier Transform (FT) of the input signal x(t) is a shifted delta function at ω0.
- System Response: The system's frequency response H(ω) represents how the system affects different frequency components.
- Output: The FT of the output signal Y(ω) is the input frequency response multiplied by the system's frequency response at ω0.
- Time Domain Output: The inverse Fourier Transform (IFT) of Y(ω) gives the time domain output y(t), which is the same complex exponential as the input, scaled by H(ω0).
Real LTI System
- Input: A real cosine signal.
- Output: The same cosine signal, scaled by the magnitude of the frequency response at the input frequency and phase-shifted by the angle of the frequency response at that frequency.
Introduction to Filters
- Ideal Filter: A filter that eliminates all frequencies outside a specific band.
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Practical Filter: Real-world filters have limitations:
- Causality: They cannot be implemented as non-causal filters.
- Realizability: The impulse response of a practical filter cannot be infinitely long or have oscillations.
One-Sided Exponential Impulse Response
- Frequency Response: This type of impulse response exhibits a lowpass filter characteristic.
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Parameter 'a': Controls the width of the filter's frequency response.
- Larger 'a' results in a broader frequency response.
- Smaller 'a' results in a narrower frequency response.
Computing Outputs Using Frequency Response
- General Setup: The frequency response provides a way to determine the output of an LTI system for any input signal.
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Steps:
- Take the Fourier Transform of the input signal.
- Multiply the input frequency response by the system's frequency response.
- Take the inverse Fourier Transform of the product to obtain the output signal in the time domain.
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Description
This quiz explores the principles of frequency response in linear time-invariant (LTI) systems. It covers the effects of input signals on system outputs and the significance of the Fourier Transform in analyzing these interactions. Understand how different frequency components are processed and represented in both the frequency and time domains.
