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EE513 Audio Signals and Systems

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EE513 Audio Signals and Systems Digital Signal Processing (Synthesis) Kevin D. Donohue Electrical and Computer Engineering University of Kentucky * * * * Filters ... – PowerPoint PPT presentation

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Title: EE513 Audio Signals and Systems


1
EE513Audio Signals and Systems
  • Digital Signal Processing (Synthesis)
  • Kevin D. DonohueElectrical and Computer
    EngineeringUniversity of Kentucky

2
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

3
Filter Specifications
  • ExampleLow-pass filter frequency response

4
Filter Specifications
  • Example Low-pass filter frequency response (in
    dB)

5
Filter Specifications
  • Example Low-pass filter frequency response (in
    dB) with ripple in both bands

6
Filter Specification Functions
  • The transfer function magnitude or magnitude
    response
  • The transfer function phase or phase response
  • The group delay (envelope delay)

7
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?

8
Filter Analysis Example
9
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.

10
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.

11
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 .
12
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.

13
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).

14
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.

15
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?

16
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?

17
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.

18
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.
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