EE513 Audio Signals and Systems

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EE513Audio Signals and Systems

• Digital Signal Processing (Synthesis)
• Kevin D. DonohueElectrical and Computer
  EngineeringUniversity of Kentucky

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Filters

• Filter are designed based on specifications
  given by
• spectral magnitude emphasis
• delay and phase properties through the group
  delay and phase spectrum
• implementation and computational structures
• Matlab functions for filter design
• (IIR) besself, butter, cheby1, cheby2, ellip,
  prony, stmcb
• (FIR) fir1, fir2, kaiserord, firls, firpm,
  firpmord, fircls, fircls1, cremez
• (Implementation) filter, filtfilt, dfilt
• (Analysis) freqz, FDAtool, SPtool

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Filter Specifications

• ExampleLow-pass filter frequency response

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Filter Specifications

• Example Low-pass filter frequency response (in
  dB)

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Filter Specifications

• Example Low-pass filter frequency response (in
  dB) with ripple in both bands

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Filter Specification Functions

• The transfer function magnitude or magnitude
  response
• The transfer function phase or phase response
• The group delay (envelope delay)

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Filter Analysis Example

• Consider a filter with transfer function
• Compute and plot the magnitude response, phase
  response, and group delay. Note pole-zero
  diagram. What would be expected for the
  magnitude response?

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Filter Analysis Example
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Basic Filter Design

• The following commands generate filter
  coefficients for basic low-pass, high-pass,
  band-pass, band-stop filters
• For linear phase FIR filters fir1
• For non-linear phase IIR filter besself, butter,
  cheby1, cheby2, ellip
• Example With function fir1, design an FIR
  high-pass filter for signal sampled at 8 kHz with
  cutoff at 500 Hz. Use order 10 and order 50 and
  compare phase and magnitude spectra with freqz
  command. Use grpdelay to examine delay
  properties of the filter. Also use command filter
  to filter a frequency swept signal from 20 to
  2000 Hz over 4 seconds with unit amplitude.

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Basic Filter Design

• Example With function cheby2 design an IIR
  Chebyshev Type II high-pass filter for signal
  sampled at 8 kHz with cutoff at 500 Hz, and
  stopband ripple of 30 dB down. Use order 5 and
  order 10 for comparing phase and magnitude
  spectra with freqz command. Use grpdelay to
  examine delay characteristics. Also use command
  filter to filter a frequency swept signal from 20
  to 2000 Hz over 2 seconds with unit amplitude.
• Example With function butter, design an IIR
  Butterworth band-pass filter for signal sampled
  at 20 MHz with with a sharp passband from 3.5 MHz
  to 9MHz. Use order 5 and verify design of phase
  and magnitude spectra with freqz command. Also
  use command filter and filtfilt to filter an
  ultrasonic signal received from a medical imaging
  probe.

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filtfilt gt Zero Phase Filtering
Consider and N point signal x(n) and filter
impulse response h(n)
Convolve sequences to obtain w(n) and reverse the
order of w(n)
Convolve w(N-n) with h(n) and reverse results to
obtain y(n)
Note the effective filter of x(n) is
, which has zero phase and the magnitude
of .
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Filter Design with Spectral Magnitude Criteria

• The following commands generate filter
  coefficients for an arbitrary filter shape or
  impulse response.
• For linear phase FIR filters fir2, firls
• The frequency points are specified with a vector
  containing the normalized frequency axis and a
  corresponding vector indicating the amplitude at
  each of those points.
• For non-linear phase IIR filter prony, stmcb
• The filter is specified in terms an impulse
  response and the IIR filter model (numerator and
  denominator order specified) is fitted to the
  response to minimize mean square error of the
  impulse response.

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Basic Filter Design

• Example Record a voice repeating random speech
  for about 20 seconds at fs 22050 Hz, and
  compute its average spectrum. From the picture
  of the average spectrum magnitude, determine a
  spectral shape vector for use with function fir2
  to create a filter that approximates the voice
  spectrum. Add white noise to another voice
  signal from the same person to achieve a 6dB SNR
  and use filter to see how well the noise can be
  reduced with the filter you designed.
• Example Generate an approximate impulse
  response of a room and record it with fs11025
  Hz. Use the prony function to attempt to model
  the room distortion as an IIR filter (you have to
  guess some the orders and test to see if it
  works).

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Homework(2)

• Record room noise (or another interesting noise
  process ) for about 10 seconds at fs 8000 Hz,
  and compute its average spectrum (use the pwelch
  function). From the picture of the average
  spectrum magnitude, determine a spectral shape
  vector for use with function fir2 to create a
  filter that matches the noise spectrum. Filter a
  white noise sequence with the FIR filter you
  designed and compare the sound to white noise
  before and after filtering with the room noise
  filter. Briefly describe your observations,
  comment on differences and similarities.

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Useful Filter Functions

• The sinc function and the rectangular pulse form
  a Fourier transform pair.
• Sketch these functions and label the null points
  of the sinc function.
• What would a shift of the rect function in
  frequency do to the sinc function in the time
  domain?
• What would a shift of the sinc function in time
  do to the rect function in the frequency domain?

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Useful Filter Functions

• The sinc function and the rectangular pulse form
  a Fourier transform pair.
• Sketch these function and label the null points
  of the sinc function. What would a shift of the
  rect function in time do to the sinc function in
  the frequency domain?

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Ideal Low-Pass Filter

• The rect function in the frequency domain
  represents the ideal low-pass filter.
• If the ideal low-pass filter were implemented in
  the time domain, what would the convolution
  kernel look like? Comment on the causality of
  this filter.
• This is the filter that is required to perfectly
  reconstruct a bandlimited signal (sampled above
  its Nyquist rate) from its samples.

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Ideal Interpolation Function

• If the rect function in the frequency domain has
  B/2 equal to the Nyquist frequency. Then the
  time domain sinc nulls fall on sampling
  increments and the following convolution becomes
  the reconstruction or interpolation filter to
  restore a sampled band-limited signal from its
  samples
• where T 1/B.