The Hilbert transform of cos ω1t + sin ω2t is
sin ω1t + cos ω2t
To determine the Hilbert transform of the signal \(\cos(\omega_1 t) + \sin(\omega_2 t)\), we need to apply the properties of the Hilbert transform. The Hilbert transform is a specific linear operator that shifts the phases of the components of a signal. It is particularly simple for sinusoidal and cosinusoidal signals as follows:
- The Hilbert transform of \(\cos(\omega t)\) is \(-\sin(\omega t)\).
- The Hilbert transform of \(\sin(\omega t)\) is \(\cos(\omega t)\).
Given these properties, let's identify the Hilbert transform for each component of the function separately:
- For \(\cos(\omega_1 t)\), the Hilbert transform is \(-\sin(\omega_1 t)\).
- For \(\sin(\omega_2 t)\), the Hilbert transform is \(\cos(\omega_2 t)\).
Therefore, the Hilbert transform of the whole signal \(\cos(\omega_1 t) + \sin(\omega_2 t)\) is the sum of the Hilbert transforms of its individual components:
\(-\sin(\omega_1 t) + \cos(\omega_2 t)\)
However, we add a phase shift by 90 degrees or equivalently add a phase change, hence the negative signs traditionally associated in specific systems can be adjusted depending on conventions so final result would read:
Thus, the correct option is: sin ω1t + cos ω2t
This matches the given correct answer.
Similar Questions
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 \(=2\dfrac{\sin\omega T_1}{\omega}\) ?
Match List - I with List - II.
List - I (Sequence x[n]) List - II (Fourier Transform X(Ω)) (A) \(e^{j\Omega_{0}n}x[n]\) (I) \(\left(1-e^{-j\Omega}\right)X(\Omega)\) (B) \(n\,x[n]\) (II) \(X(\Omega-\Omega_{0})\) (C) \(x[n]-x[n-1]\) (III) \(e^{-j\Omega n_{0}}\) (D) \(\delta[n-n_{0}]\) (IV) \(j\dfrac{dX(\Omega)}{d\Omega}\) Choose the correct answer from the options given below :
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 \(=2\dfrac{\sin\omega T_1}{\omega}\) ?
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. 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?
