Sampling, DFT and FFT · Definition

Aliasing

signal processingsampling, dft and fft

When we sample a continuous signal, the spectrum of the discrete signal repeats every \(f_s\). If those copies overlap, the original signal cannot be recovered without error.

Definition

Aliasing is the overlap of spectral replicas that occurs when a signal is sampled at a rate below twice its bandwidth, so that different frequencies become indistinguishable from one another.

\[f_{\text{alias}} = |f - k\,f_s|, \quad k \in \mathbb{Z}\]

The spectrum of a signal sampled with period \(T_s\) is a sum of replicas of the original spectrum shifted by multiples of \(f_s = 1/T_s\). As long as these replicas do not overlap, an ideal low-pass filter can reconstruct the original signal. If the maximum frequency \(f_{\max}\) satisfies \(f_s > 2f_{\max}\), there is no aliasing.

Consider a \(7\,\text{Hz}\) sinusoid sampled at \(f_s = 10\,\text{Hz}\). Since \(7 > 5\), aliasing occurs: the apparent frequency is \(|7 - 10| = 3\,\text{Hz}\). The samples of both sinusoids coincide, so they cannot be distinguished from the data.

A typical mistake is to think that increasing the signal amplitude or applying a moving-average filter after sampling removes aliasing. Aliasing occurs during sampling itself, so it is only avoided with an antialiasing filter before sampling that limits the bandwidth.

Think about it

Why does increasing the signal amplitude not remove aliasing?

Because aliasing depends on the frequency \(f\) and on \(f_s\), not on the amplitude: the apparent frequency \(|f - k\,f_s|\) does not change.

Exercises 10

  1. What condition avoids aliasing when sampling a signal with maximum frequency \(f_{\max}\)?
  2. What is the Nyquist frequency for sampling at \(f_s = 200\,\text{Hz}\)?
  3. A signal has a maximum frequency of \(30\,\text{Hz}\). What is the minimum sampling frequency that avoids aliasing?
  4. What phenomenon does the apparent frequency obtained after sampling below the Nyquist rate describe?
  5. What type of filter is used before sampling to limit the bandwidth and avoid aliasing?
  6. A \(8\,\text{Hz}\) sinusoid is sampled at \(f_s = 10\,\text{Hz}\). What is the apparent frequency after sampling?
  7. A signal with components up to \(40\,\text{Hz}\) is sampled at \(f_s = 60\,\text{Hz}\). Does aliasing occur?
  8. A signal contains \(\cos(2\pi \cdot 3t)\) and is sampled at \(f_s = 4\,\text{Hz}\). What apparent frequency is observed?
  9. Two sinusoids of \(2\,\text{Hz}\) and \(8\,\text{Hz}\) are sampled at \(f_s = 10\,\text{Hz}\). Which statement is correct?
  10. A signal \(x(t)=\cos(2\pi \cdot 9t)\) is sampled at \(f_s = 12\,\text{Hz}\) and then reconstructed with an ideal low-pass filter with cutoff \(6\,\text{Hz}\). What is the frequency of the reconstructed signal?

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.