8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

1/14

Presented by:

K.N.R.A.K.Madhushani

S9235

Computational physics

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

2/14

Signals in digital oscilloscopes

The clock rate of the input signal is well within

Nyquists criteria, the signals edges contain

significant frequency components well beyond

the Nyquist frequency.

Visual distortion

Notice that near the top of the image,

where the checkerboard is very distant,

the image is difficult to recognize and is

not aesthetically appealing.

agon wheel effect

As a wagon accelerates, the wheel picks

up speed as expected, and then the wheel

seems to slow, then stop.

As the wagon further accelerates, the

wheel appears to turn backwards!

In reality, we know the wheel hasn't

reversed like this way.

What causes this phenomenon?The

answer is that the frame rate is not high

enough to accurately capture the spinningof the wheel.

Aliasing

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

3/14

For a limited bandwidth signal with a maximum frequency fMAX,

the equally spaced sampling frequency fSmust be greater than

twice ofthe maximum frequency fMAX, in order to have the signal

be uniquely reconstructed without aliasing.

fS > 2*fMAX

2*fMAX - Nyquist Sampling Rate

fMAX-Nyquist frequency

Dr. Harry Nyquist, 1889-1976Articulated his sampling theorem in 1928

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

4/14

Sampling is the reduction of a continuous signal to decrease signal.

Continuous signal

Discrete signal

Let x(t) be a continuous signal which is to be sampled,

and that sampling is performed by measuring the value of

the continuous signal every Tseconds, which is called the

sampling interval.

Thus, the sampled signal x[n] given by:

x[n] = x(nT), with n = 0, 1, 2, 3.

The sampling frequency or sampling rate fs

is defined as

the number of samples obtained in one second, orfs

=

1/T.

The sampling rate is measured in hertz or in samples persecond.

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

5/14

Nyquist criteria demonstration using matlab

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

6/14

Aliasing

hen the input signal frequency is faster than the sampling frequency,the sampled result will appear to be a low-frequency wave.

In the frequency domain, aliasing is expressed as high-frequency components beingpresent in the low-frequency range.

In the time domain, aliasing is the loss of detail in the signal, and the false perception ofreading a low frequency signal.

Error results are appeared.

Original signal Aliased signal

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

7/14

Best solution sampling at f s > 2.5*f bw

If our sampling hardware is not fast enough or we

dont know the band width of the signal, we shoulduse

An anti-aliasing filter

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

8/14

y increase the resolution

y prefilteringy supersampling or postfiltering

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

9/14

Increase the resolution

A diagonal line on a set of pixels

Aliased pixels

Aliasing at higher resolution

From the above figure it is clear

that the image has jagged effect

and does not represent the

diagonal line clearly.

Since by going for higher number of

samples we have increased the

sampling rate and thus reduced

aliasing effect.

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

10/14

Antialiased edge

This increases the cost of image production.

AA edge higher

resolution

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

11/14

Prefiltering

y Apply a low-pass filter

y Blurs the imagey But ensures that no high frequencies

Prefiltering methods treat a pixel as an area,

compute pixel color based on the overlap of the scene's objects with a pixel's area.

These techniques compute the shades of gray based on how much of a pixel's area is coveredby a object.

We can use any symmetric filters that we like, such as box, Gaussian, or cubic filters

McNamara et al. (2000) developed an efficient prefiltering method

most rendering algorithms generate sampled function directlyy e.g., Z-buffer, ray tracing

For shapes other than polygons, this can be very computationally intensive

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

12/14

Postfiltering

yCreate virtual image at higher resolution

y Apply a low-pass filter

y Resample filtered image

Typical supersampling algorithm:y Compute multiple samples per pixely Combine sample values for pixels value using simple average

There are several types of supersampling algorithms Grid algorithm

Random algorithm Poisson Disc algorithm

Jitter algorithm

Rotated Grid algorithm

y The simplest way to reduce aliasing artifacts is postfiltering.

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

13/14

Supersampling cons

y Doesnt eliminate aliasing, just shifts the Nyquist limithigher

Cant fix some scenes (e.g., checkerboard)

y Badly inflates storage requirements

Supersampling prosy Relatively easy

y Often works all right in practice

8/8/2019 Nyquist Criteria & Anti-Aliasing Filtering_new2

14/14