Exercise · ZW-FOURIER-TRANSFORM-PROPERTIES-001 · ★☆☆ Basic
By the linearity property, what is the transform of \( a\,x_1(t)+b\,x_2(t) \)?
Choose one option.
Show answer
- C. \( a X_1(\omega)+b X_2(\omega) \)
Explanation
The transform is linear: the transforms are combined with the same coefficients. Multiplying the transforms corresponds to convolution in time, not to addition.
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
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- 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-005 Which time-domain operation corresponds to multiplying X(ω)Y(ω) ?
- 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…