Fourier series and transform · Definition

Fourier transform properties

signal processingfourier series and transform

Before you start you need to know

Fourier properties let you compute transforms of complex signals from known transforms. This avoids direct integration in many cases.

Definition

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.

\[x(t-t_0) \leftrightarrow e^{-j\omega t_0}X(\omega)\]

Linearity states that the transform of a linear combination is the same linear combination of transforms. A time shift multiplies the transform by \( e^{-j\omega t_0} \), which changes only the phase and leaves the magnitude \( |X(\omega)| \) unchanged. Time differentiation corresponds to multiplying by \( j\omega \) in the frequency domain.

Consider \( x(t) = e^{-2t}u(t) \), with transform \( X(\omega) = \frac{1}{2+j\omega} \). For \( x(t-1) \), the shift property gives \( e^{-j\omega}\frac{1}{2+j\omega} \) without repeating the integral.

A common mistake is forgetting the sign in the exponent when shifting, writing \( e^{j\omega t_0} \) instead of \( e^{-j\omega t_0} \). This reverses the phase and corresponds to shifting in the wrong direction.

Think about it

Why do a signal and its time-shifted copy have the same spectral magnitude even though their time waveforms differ?

Because \( e^{-j\omega t_0} \) has modulus 1 for every real \( \omega \): the shift only adds phase, so each frequency keeps its amplitude.

Exercises 10

  1. By the linearity property, what is the transform of \( a\,x_1(t)+b\,x_2(t) \)?
  2. What is the effect of a time shift \( x(t-t_0) \) on the magnitude \( |X(\omega)| \)?
  3. How is the derivative \( \frac{dx(t)}{dt} \) represented in the frequency domain?
  4. Which property relates \( x(at) \) to \( X(\omega/a) \)?
  5. Which time-domain operation corresponds to multiplying \( X(\omega)Y(\omega) \)?
  6. Let \( x(t)=e^{-3t}u(t) \) with \( X(\omega)=\frac{1}{3+j\omega} \). What is the transform of \( x(t-2) \)?
  7. If \( x(t) \) has transform \( X(\omega) \), what is the transform of \( \frac{dx(t)}{dt} \) when \( x(t)=e^{-t}u(t) \)?
  8. Let \( x(t)=e^{-2t}u(t) \). What is the transform of \( x(3t) \)?
  9. Let \( x(t)=e^{-t}u(t) \) with \( X(\omega)=\frac{1}{1+j\omega} \). Compute the transform of \( y(t)=t\,x(t-1) \) using the shift and frequency-differentiation properties.
  10. Let \( x(t)=\frac{\sin(2t)}{\pi t} \) with transform \( X(\omega)=\mathrm{rect}(\omega/4) \). What is the transform of \( z(t)=x(t)\cos(3t) \)? Check whether the magnitude at \( \omega=0 \) matches that of \( X \).

References

  • Alan V. Oppenheim, Alan S. Willsky, Signals and Systems, 2nd edition, Pearson, 1996.
  • Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing, 3rd edition, Pearson, 2010.