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Anti-Aliasing Filters

Module by: Justin Romberg. E-mail the author

Summary: This module discusses Anti-Aliasing and provides examples of filters that can be used to avoid aliasing.

Note: You are viewing an old version of this document. The latest version is available here.

Introduction

The idea of aliasing has been described as the problem that occurs if a signal is not sampled at a high enough rate (for example, below the Nyquist Frequency). But exactly what kind of distortion does aliasing produce?

Figure 1
(a)
Figure 1(a) (anti1.png)
(b)
Figure 1(b) (anti2.png)

High frequencies in the original signal "fold back" into lower frequencies.

High frequencies masquerading as lower frequencies produces highly undesirable artifacts in the reconstructed signal.

Warning:

We must avoid aliasing anyway we can.

Avoiding Aliasing

What if it is impractical/impossible to sample at Ω s >2⁢ Ω B Ω s 2 Ω B ?

Filter out the frequencies above Ω s 2 Ω s 2 before you sample. The best way to visualize doing this is to imagine the following simple steps:

  1. Take the CTFT of the signal, f⁢t f t .
  2. Send this signal through a lowpass filter with the following specification, ω c = Ω s 2 ω c Ω s 2 .
  3. We now have a graph of our signal in the frequency domain with all values of |ω|> Ω s 2 ω Ω s 2 equal to zero. Now, we take the inverse CTFT to get back our continuous time signal, f a ⁢t f a t .
  4. And finally we are ready to sample our signal!

Example 1

Sample rate for CD=44.1⁢KHz CD 44.1 KHz .

Many musical instruments (e.g. highhat) contain frequencies above 22⁢KHz 22 KHz (even though we cannot hear them).

Because of this, we can filter the output signal from the instrument before we sample it using the following filter:

Figure 2: This filter will cutoff the higher, unnecessary frequencies, where | ω c |>2⁢π⁢22⁢kHz ω c 2 22 kHz
Figure 2 (anti3.png)

Now the signal is ready to be sampled!

Example 2: Another Example

Speech bandwidth is ±⁢20⁢kHz ± 20 kHz , but it is perfectly intelligible when lowpass filtered to a ±⁢4⁢kHz ± 4 kHz range. Because of this, we can take a normal speech signal and pass it through a filter like the one shown in Figure 2, where we now set | ω c |>2⁢π⁢4⁢kHz ω c 2 4 kHz . The signal we receive from this filter only contains values where |ω|>8⁢π⁢k ω 8 k .

Now we can sample at 16⁢π⁢k=8⁢kHz 16 k 8 kHz -- standard telephony rate.

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Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

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Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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