Question

Which of the following operations performed in the time-domain with any two causal seismic signals result(s) in the \textbf{subtraction of their corresponding phase spectra in the frequency domain? }

• A. Convolution
• B. Crosscorrelation
• C. Deconvolution
• D. Subtraction

Hint:

Phase subtraction occurs in the frequency domain when signals are related via crosscorrelation or deconvolution. Convolution adds phases, and direct subtraction of signals does not correspond to phase subtraction.

Answer 1

The Correct Option is B, C

Step 1: Recall Fourier domain properties.
- Convolution in time domain $\Rightarrow$ Multiplication in frequency domain.
- Crosscorrelation in time domain $\Rightarrow$ Multiplication with the conjugate spectrum in frequency domain.
- Deconvolution in time domain $\Rightarrow$ Division in frequency domain.
- Subtraction in time domain $\Rightarrow$ Subtraction in frequency domain (both magnitude and phase, not phase difference). Step 2: Phase effect of convolution.
If $x(t) * y(t) \leftrightarrow X(\omega) Y(\omega)$, then the phase spectrum is: \[ \angle[X(\omega) Y(\omega)] = \angle X(\omega) + \angle Y(\omega) \] So convolution results in addition of phases, not subtraction. Thus, Option (A) is incorrect. Step 3: Phase effect of crosscorrelation.
If $x(t) \star y(t) \leftrightarrow X(\omega) Y^*(\omega)$, then: \[ \angle[X(\omega) Y^*(\omega)] = \angle X(\omega) - \angle Y(\omega) \] This results in subtraction of phases. Hence, Option (B) is correct. Step 4: Phase effect of deconvolution.
Deconvolution corresponds to division in frequency domain: \[ \frac{X(\omega)}{Y(\omega)} \quad \Rightarrow \quad \angle\left(\frac{X(\omega)}{Y(\omega)}\right) = \angle X(\omega) - \angle Y(\omega) \] This also gives subtraction of phases. Hence, Option (C) is correct. Step 5: Subtraction in time domain.
If $z(t) = x(t) - y(t)$, then in frequency domain: \[ Z(\omega) = X(\omega) - Y(\omega) \] This is not equivalent to phase subtraction, but a direct spectral subtraction. Hence, Option (D) is incorrect. Final Answer: Correct options are (B) and (C). \[ \boxed{\text{Correct Answer: (B), (C)}} \]