LING124 Digital signal processing 2

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LING124 Digital signal processing (2)

• September 4, 2008

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Class outline

• (Digital) Filters
• Low-pass, band-pass, high-pass filters
• Simple digital filters
• Finite impulse response filter
• Infinite impulse response filter
• Demonstration
• Window

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Recall ...

• Aliasing
• Unable to distinguish two continuous signals with
  different frequencies based on samples
• Frequencies higher than Nyquist frequency
• Anti-aliasing
• Low-pass filter the frequencies above Nyquist
  frequency

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Filters

• Low-pass filter
• Attenuate the frequencies above the cut-off
  frequency
• High-pass filter
• Attenuate the frequencies below the cut-off
  frequency
• Band-pass filter
• Combination of both low-pass and high-pass
  filters
• Attenuate frequencies beyond the specified range
• Demonstration
• http//www.dsptutor.freeuk.com/filter/FIRDigitalFi
  lterDemo.html

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Simple digital filters

• Simple gain filters
• Change input values by a factor of k
• yn kxn
• Amplifier (kgt1)
• Unit gain filter (k1)
• Attenuator (0ltklt1)
• Inverting amplifier (klt0)

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Simple digital filters (2)

• Pure delay filter
• Delay the signal by an interval
• yn xn-1 , xn 0 if nlt0
• Two-term average filter
• Average of the current and previous input
• yn ½(xnxn-1) , xn 0 if nlt0
• Three-term average filter
• Average of the current and two previous inputs
• yn 1/3(xnxn-1xn-2) , xn 0 if nlt0

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Order and coefficients

• Order of a filter
• Number of previous inputs used to calculate the
  current output
• Two for three-term average filter
• yn 1/3(xnxn-1xn-2) , xn 0 if nlt0
• Filter coefficients
• yn b0xnb1xn-1b2xn-2 , xn 0 if nlt0

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Finite impulse response filter (FIR)

• Assume
• Nth order filter
• yn b0xnb1xn-1b2xn-2...bNxn-N
• Unit impulse
• x0 1, xi 0 (igt0)
• The output in response to the unit impulse
  (impulse response) will become zero after finite
  number of intervals (N1 in the example)

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Infinite impulse response filter (IIR)

• Recursive
• Uses previous outputs
• e.g. yn xnyn-1 , yn 0 if nlt0
• yn b0xnb1xn-1b2xn-2...bPxn-P
• -a1yn-1 - a2yn-2 - ... - aQyn-Q
• The output in response to the unit impulse
  (impulse response) will remain non-zero over an
  infinite length of time

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Demonstration

• http//www.falstad.com/dfilter/

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Window

• A function that applies to a chosen interval
• Multiply the sample amplitudes by the
  corresponding value of the window
• Zero for values outside the chosen interval

w(n)1
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Example windows
Rectangular
Hamming