Nyquist Criteria & Anti-Aliasing Filtering_new2
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Transcript of Nyquist Criteria & Anti-Aliasing Filtering_new2
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Presented by:
K.N.R.A.K.Madhushani
S9235
Computational physics
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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
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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
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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.
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Nyquist criteria demonstration using matlab
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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
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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
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y increase the resolution
y prefilteringy supersampling or postfiltering
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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.
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Antialiased edge
This increases the cost of image production.
AA edge higher
resolution
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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
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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.
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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
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