Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega) = \int_{-\infty}^{\infty} f(t)exp(-j\omega t)dt$ exists.
Which one of the following statements is always TRUE?
Fourier Transform Magnitude Inequality
The Fourier Transform $F(\omega)$ of a continuous-time signal $f(t)$ is defined as:
$F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-j\omega t}dt$We need to determine which statement about the magnitude, $|F(\omega)|$, is always true.
Deriving the Inequality
Consider the magnitude of the Fourier Transform:
$|F(\omega)| = \left| \int_{-\infty}^{\infty} f(t)e^{-j\omega t}dt \right|$Using the triangle inequality property for integrals, which states that $|\int g(t)dt| \leq \int |g(t)|dt$, we can write:
$|F(\omega)| \leq \int_{-\infty}^{\infty} |f(t)e^{-j\omega t}|dt$Since the magnitude of the complex exponential $e^{-j\omega t}$ is always 1 ($|e^{-j\omega t}| = 1$), the inequality simplifies to:
$|F(\omega)| \leq \int_{-\infty}^{\infty} |f(t)| \cdot |e^{-j\omega t}|dt$ $|F(\omega)| \leq \int_{-\infty}^{\infty} |f(t)| \cdot 1 dt$ $|F(\omega)| \leq \int_{-\infty}^{\infty} |f(t)|dt$This inequality states that the magnitude of the Fourier Transform at any frequency $\omega$ is less than or equal to the integral of the absolute value of the time-domain signal. This integral represents the total energy or the L1 norm of the signal, assuming it exists.
Evaluating Other Options
- Options 3 and 4 involve $\int_{-\infty}^{\infty} f(t)dt$, which is equal to $F(0)$. The relationship between $|F(\omega)|$ and $F(0)$ is not a general inequality that holds true for all $\omega$ and all valid signals $f(t)$. For instance, $F(0)$ might be zero or undefined, while $|F(\omega)|$ could be non-zero.
- Option 2 contradicts the derived inequality.
Therefore, the statement $|F(\omega)| \leq \int_{-\infty}^{\infty} |f(t)|dt$ is always TRUE, provided the Fourier Transform and the integral of $|f(t)|$ exist.
Important Questions from Properties of Fourier Transform
The given mathematical representation belongs to:
y(t) = x(t - T)
Which type of property is shown by the following function.
L{K f(t)} = K F(s)
The energy of the signal \(x(t) = \frac{{{\rm{sin}}\left( {4{\rm{\pi t}}} \right)}}{{4{\rm{\pi t}}}}\) is______
Consider the signal x(t) = e-|t|. Let X(jω) = \(\mathop \smallint \limits_{ - \infty }^\infty x\left( t \right){e^{ - j\omega t}}dt\) be the Fourier transform of x(t). The value of X(j0) is
A real-valued signal 𝑥(𝑡) limited to the frequency band \(\left| f \right| \le \frac{W}{2}\) is passed through a linear time-invariant system whose frequency response is
\(H\left( f \right) = \left\{ {\begin{array}{*{20}{c}} {{e^{ - j4\pi f,\;\;\;\left| f \right| \le \frac{W}{2}}}}\\ {0,\;\;\;\;\left| f \right| > \frac{W}{2}} \end{array}} \right.\)
The output of the system is
