Understanding the Signal
The signal x(t) = e^(-at) A sin(t) combines an exponential decay with a sinusoidal function. Here, 'A' is the amplitude, and 'a' is a positive constant representing the decay rate.
Fourier Transform Definition
The Fourier transform of a continuous-time signal x(t) is defined as:
- X(f) = ∫ x(t) e^(-j2πft) dt
This transforms the signal from the time domain to the frequency domain, allowing analysis of its frequency components.
Applying the Fourier Transform
To find the Fourier transform of x(t), we can use the properties of linearity and the modulation property. The signal can be expressed as:
- x(t) = A e^(-at) sin(t)
Using the modulation property, we can represent the sine function in terms of exponentials:
- sin(t) = (e^(jt) - e^(-jt)) / (2j)
This leads to:
- x(t) = (A/2j) [e^(-at + jt) - e^(-at - jt)]
Fourier Transform Calculation
Now, calculate the Fourier transform for each term:
1. For the term e^(-at + jt):
- X_1(f) = ∫ e^(-at + jt) e^(-j2πft) dt
2. For the term e^(-at - jt):
- X_2(f) = ∫ e^(-at - jt) e^(-j2πft) dt
Both integrals yield results involving the exponential decay term and Dirac delta functions.
Final Result
Combining both transforms, we obtain:
- X(f) = A / 2j * [(1 / (a - j(1 + 2πf))) - (1 / (a + j(1 - 2πf)))]
This expression represents the frequency domain characteristics of the original signal x(t).
In conclusion, the Fourier transform provides a powerful tool for analyzing signals, revealing their frequency components effectively.