EE 624 Advanced DSP

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EE 624 Advanced DSP

• Lecture Slides 4

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Lets look at an example of a 2-dimensional
function that is separable.
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Now apply the 2D DTFT to this set of nine points
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Alternatively, we could look at desired frequency
responses in the frequency domain and work
backward to find the corresponding impulse
response h(n1, n2) in the time domain. For
instance, we can look at creating 2-dimensional
filters using this approach which would be
similar to how we created 1-dimensional FIR
filters with the windowing technique. The two
common types of 2-D filters are 2-D Separable
Filter 2-D Circularly Symmetric Filter Consider
a typical 2-D separable filter
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The impulse response is obtained from the 2D DTFT
synthesis equation
Since X(?1, ?2) is separable the double integral
becomes
This will simply integrate into two sinc
functions, one for each axis
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Now consider the 2-D circularly symmetric
low-pass filter
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• Some theorems about the 2D DTFT
• The 2-D DTFT of a sequence x(n1,n2) exists if the
  sequence is absolutely summableLet h(n1,n2)
  represent the impulse response of a 2-D LTI
  system. If the input is a 2-D complex
  exponential , then its output is the same
  complex exponential with magnitude and phase
  given by and ,respectively, at (?1, ?2)
  (?x, ?y).
• The 2-D DTFT of any discrete-time sequence is
  periodic with a period of (2?,2?).
• The 2-D DTFT of a separable sequence is the
  product of the 1-D DTFT of the individual
  sequences.