Chapter 5: Fourier Transforms
Rules of Thumb for Calculating the Fourier Transform of a Signal x(t):
(page 337)
\cdot If $x(t)$ has a finite time support and in that support $x(t)$ is bounded, its Fourier
transform exists. To find it use the integral definition or the Laplace transform of $x(t)$.
\cdot If $x(t)$ has infinite time support and a Laplace transform $X(s)$ with a region of
convergence including the $j\Omega$-axis, its Fourier transform is $X(s)|_{s=j\Omega}$.
\cdot If $x(t)$ is periodic, its Fourier transform is obtained using the signal Fourier series.
\cdot If $x(t)$ is none of the above, if it has discontinuities (e.g., $x(t) = u(t)$), or it is has disconti-
nuities and it is not finite energy (e.g., $x(t) = cos(\Omega_0 t)u(t)$) or it has possible discontinuities
in the frequency domain even though it has finite energy (e.g., $x(t) = sinc(t)$) use properties
of the Fourier transform.
Q1. Compute $X(\Omega)$ using the appropriate method as explained in above rules of thumb
and sketch the Magnitude line spectrum for the following signals:
(a) Real exponential signal $x_1(t) = e^{-4t} u(t)$.
(b) Rectangular pulse $x_2(t) = u(t)-u(t-2)$
(c) Signal contraction $x_3(t) = x_2(2t)$
(d) Periodic signal $x_4(t) = cos(2t)$, $-\infty < t < \infty$
