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Two Dimensional Digital Signal
Processing
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Many signals are inherently two dimensional
Photographic data, weather photos, X-rays
Generally spatial signals
Basic ideas of 1D signal processing can be
extended to 2D case
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2D Signals
Notation used to represent 2D variables
where are integer variables
is short hand version of 2D signal
Perspective plots to represent 2D signalsgraphically
Third dimension represents magnitude
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2D Signals
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2D Signals
Useful 2D sequences
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2D Signals
2D Digital Impulse Or Unit Sample
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2D Signals
2D Digital Step
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2D Signals
2D Step is related to 2D Impulse by the
following relation
For 1D signal the relation is
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2D Signals
Exponential and Sinusoidal signals
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2D Signals
2D Exponential signal
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2D Systems
Convolution Theorem
For LTI Systems Convolution theorem is valid
Relation Input & Impulse response to output is
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2D Systems
For a system with Impulse response and input
Output is
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2D Systems
Direct evaluation of Convolution is difficult since
cannot be factored into
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2D Systems
Causality and Separability
A 2D filter is causal or realizable if its impulse
response satisfies the property
And separable if its impulse response can be
factored into a product of 1D responses
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2D Systems
Advantage of separable filters - 2D convolution can
be carried out as a sequence of 1D convolutions
[ ] for each value of
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2D Systems
If input sequence is also separable
where
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2D Systems
A 2D filter is stable only if the impulse
response satisfies the constraint
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2D Systems
2D linear Difference Equations
To describe an LTI 2D filter
The constant coefficients
plus a set of initial conditions specify the filter
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2D Systems
Recursion relation for assuming
is
-
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1D Fourier series relations
With sinusoidal input to a 2D system
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Fourier series & frequency domain analysis
Output of the system is
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Frequency response of the system is a Fourier
series representation
with the coefficients given by
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The frequency response is doubly periodic
Find the Fourier series coefficients, of
the filter with frequency response
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Frequency response = 1 in hatched
area
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If the frequency response can be decomposed
to the product of a term inimpulse response will be a product of a term
in
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