Fourier series and transform · Definition

Continuous Fourier transform

signal processingfourier series and transform

The Fourier transform extends the Fourier series to non-periodic signals, moving from a discrete spectrum to a continuous one. It allows analyzing finite-energy signals in the frequency domain.

Definition

The continuous Fourier transform \(X(j\omega)\) is a frequency-domain representation of an aperiodic continuous-time signal \(x(t)\), assigning to each frequency \(\omega\) its spectral content.

\[X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\]

If \(x(t)\) is absolutely integrable, its transform exists and the inverse \(x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(j\omega)e^{j\omega t}\,d\omega\) recovers the signal. The transform is linear, and a time shift \(x(t-t_0)\) produces a phase factor \(e^{-j\omega t_0}\) in \(X(j\omega)\) without changing its magnitude. Moreover, time-domain convolution corresponds to frequency-domain multiplication.

Consider the rectangular pulse \(x(t)=1\) for \(|t|<T_1\) and \(0\) otherwise. Direct integration gives \(X(j\omega)=\frac{2\sin(\omega T_1)}{\omega}\), whose value at \(\omega=0\) is \(2T_1\), equal to the area of the pulse. This shows how a short-duration signal has a wide spectrum.

A typical error is forgetting the negative sign in the kernel exponential, using \(e^{j\omega t}\) in the forward transform. This flips the spectrum in frequency and gives wrong results for non-symmetric signals. Always check the sign convention before integrating.

Think about it

What happens to \(X(j\omega)\) when the signal is stretched in time, \(x(at)\) with \(0<a<1\)?

The spectrum is compressed in frequency by the factor \(a\) and its amplitude is multiplied by \(1/a\), since the signal is stretched in time.

Exercises 10

  1. By definition, what is the kernel of the forward continuous Fourier transform?
  2. What factor multiplies the integral in the inverse Fourier transform?
  3. What is the Fourier transform of \(\delta(t)\)?
  4. If \(x(t)\) is real, what symmetry does \(|X(j\omega)|\) satisfy?
  5. What is the value of \(X(j0)\) for integrable \(x(t)\)?
  6. Compute the transform of \(x(t)=e^{-at}u(t)\) with \(a>0\).
  7. If \(X(j\omega)\) is the transform of \(x(t)\), what is the transform of \(x(t-2)\)?
  8. Compute the transform of the convolution \(y(t)=x_1(t)*x_2(t)\) if \(X_1(j\omega)=\frac{1}{1+j\omega}\) and \(X_2(j\omega)=\frac{2}{2+j\omega}\).
  9. What is the transform of \(x(t)=\frac{d}{dt}e^{-t}u(t)\)?
  10. Let \(x(t)=e^{-|t|}\). Compute \(X(j\omega)\) and determine the value of \(X(j1)\) by comparing with the options.

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.