Exercise · ZW-FOURIER-TRANSFORM-PROPERTIES-005 · ★★☆ Intermediate
Which time-domain operation corresponds to multiplying \( X(\omega)Y(\omega) \)?
Choose one option.
Show answer
- C. Convolution of \( x(t) \) and \( y(t) \)
Explanation
The convolution property says that convolution in time equals multiplication of transforms. Pointwise product corresponds to multiplication in time, which is the dual property.
Review the concept: Fourier transform properties
Set of rules describing how the Fourier transform \(X(\omega)\) of a signal \(x(t)\) changes under operations such as shifting, scaling, differentiation, or convolution.
\[x(t-t_0) \leftrightarrow e^{-j\omega t_0}X(\omega)\]
More practice problems: fourier transform properties
- ZW-FOURIER-TRANSFORM-PROPERTIES-001 By the linearity property, what is the transform of a x_1(t)+b x_2(t) ?
- ZW-FOURIER-TRANSFORM-PROPERTIES-002 What is the effect of a time shift x(t-t_0) on the magnitude |X(ω)| ?
- ZW-FOURIER-TRANSFORM-PROPERTIES-003 How is the derivative (dx(t))/(dt) represented in the frequency domain?
- ZW-FOURIER-TRANSFORM-PROPERTIES-004 Which property relates x(at) to X(ω/a) ?
- ZW-FOURIER-TRANSFORM-PROPERTIES-006 Let x(t)=e^-3tu(t) with X(ω)=(1)/(3+jω) . What is the transform of x(t-2) ?
- ZW-FOURIER-TRANSFORM-PROPERTIES-007 If x(t) has transform X(ω) , what is the transform of (dx(t))/(dt) when x(t)=e^-tu(t) ?
- ZW-FOURIER-TRANSFORM-PROPERTIES-008 Let x(t)=e^-2tu(t) . What is the transform of x(3t) ?
- ZW-FOURIER-TRANSFORM-PROPERTIES-009 Let x(t)=e^-tu(t) with X(ω)=(1)/(1+jω) . Compute the transform of y(t)=t x(t-1) using the shift and…
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