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Anti-Aliasing Filters

Module by: Justin Romberg

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

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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, ft 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=44.1KHz CD 44.1 KHz .

Many musical instruments (e.g. highhat) contain frequencies above 22KHz 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 |>2π22kHz ω c 2 22 kHz
Figure 2 (anti3.png)

Now the signal is ready to be sampled!

Example 2: Another Example

Speech bandwidth is >±20kHz ± 20 kHz , but it is perfectly intelligible when lowpass filtered to a ±4kHz ± 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 |>2π4kHz ω c 2 4 kHz . The signal we receive from this filter only contains values where |ω|>8πk ω 8 k .

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

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