"""
ECE 09351 - Digital Signal Processing
Lab 1 Question 6

Translated from MATLAB by Aidan Sharpe
Last Modified: January 18, 2025
"""

from sympy import *
from sympy.abc import a, b, t, w


# Please note that the answers are given as piecewise functions.
# The conditions are unnecessary, so take the first part of the piecewise result.
def main():
    # Assume that a and b are positive real numbers
    with assuming(Q.positive(a), Q.positive(b)):
        x_1 = exp(-a*abs(t))                                # x1(t) = e^(-a|t|)
        X_1 = integrate(x_1*exp(-1j*w*t), (t, -oo, oo))     # Fourier transform of x1(t)
        print(simplify(X_1))

        x_2 = x_1*cos(b*t)                                  # x2(t) = x1(t)*cos(bt)
        X_2 = integrate(x_2*exp(-1j*w*t), (t, -oo, oo))     # Fourier transform of x2(t)
        print(simplify(X_2))
    


if __name__ == "__main__":
    main()