Math Problem Statement
Find the Fourier transform of given signals. Also plot magnitude and phase. a) 𝑥(𝑛) = 𝛿(𝑛) b) 𝑥(𝑛) = 𝑎 𝑛 𝑢(𝑛) (L2/CO2) 6 B
Solution
To find the Fourier Transform of the given signals, let's handle each one step by step.
a)
The Fourier Transform of the discrete-time signal , where is the Kronecker delta function, is a well-known result:
This is because is the identity signal in the discrete-time domain, and its Fourier transform is constant (1) across all frequencies.
- Magnitude of : for all frequencies.
- Phase of : for all frequencies.
b)
Here, , where is the unit step function. The Fourier Transform of this signal can be computed using the standard formula for the discrete-time signal , which is:
Where is the Fourier Transform, and the magnitude and phase of this expression can be computed as follows:
- Magnitude of :
- Phase of :
Now, let’s plot the magnitude and phase of these signals.
Would you like me to go ahead and plot the graphs of the magnitude and phase for both of these functions (for a specific value of in case of the second signal), or would you prefer to focus on the theoretical details?
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Math Problem Analysis
Mathematical Concepts
Fourier Transform
Discrete-Time Signals
Kronecker Delta Function
Unit Step Function
Formulas
X(ω) = 1 for x(n) = δ(n)
X(ω) = 1 / (1 - a * e^(-jω)) for x(n) = a^n u(n)
Theorems
Fourier Transform of Kronecker Delta
Fourier Transform of a^n u(n)
Suitable Grade Level
University
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