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
Which one of the following is the Fourier transform of the signal given in Fig. II, if the Fourier transform of the signal in Fig. I is \(=2\dfrac{\sin\omega T_1}{\omega}\) ?
The Hilbert transform of cos ω1t + sin ω2t is
Consider two continuous time signals $x(t)$ and $y(t)$ as shown below
If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is _________Let $X(\omega)$ be the Fourier transform of the signal
$x(t) = e^{-t^4} \cos t$, $-\infty < t < \infty$.
The value of the derivative of $X(\omega)$ at $\omega=0$ is ________ (rounded off to 1 decimal place).- Let $f(t)$ and $g(t)$ represent continuous-time real-valued signals. If $h(t)$ denotes cross-correlation between $f(t)$ and $g(-t)$, its continuous-time Fourier transform $H(j\omega)$ equals
Note: $F(j\omega)$ and $G(j\omega)$ denote the continuous-time Fourier transforms of $f(t)$ and $g(t)$, respectively.
