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Digital Filters. Part A characterization and analysis.

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Digital filters

Digital Filters

Part A characterization and analysis


  • “As soon as we started programming, we found out to our surprise that it wasn't as easy to get programs right as we had thought. Debugging had to be discovered. I can remember the exact instant when I realized that a large part of my life from then on was going to be spent in finding mistakes in my own programs.”

    • Maurice Wilkes (a pioneering British computer scientist, winner of the 1967 Turing Award, developed the first stored-program computer in 1949, invented the concept of microprogramming in 1951, credited with originating the fundamental software concepts of symbolic labels, macros, and subroutine libraries; from wikipedia)


What have we seen so far
What have we seen so far? surprise that it wasn't as easy to get programs right as we had thought. Debugging had to be discovered. I can remember the exact instant when I realized that a large part of my life from then on was going to be spent in finding mistakes in my own programs.”

  • So far we have seen…

    • Box filter

      • Moving average filter

      • Example of a lowpass

        • passes low frequencies

          • small, gradual changes in the signal are passed

          • higher frequencies are attenuated (reduced/removed/suppressed)


Applied by cross correlation sum of products of image f and mask h
Applied by cross correlation (sum of products) of image f and mask h

If mask is centered about origin, (x,y) in image:

If origin, (x,y), in image is aligned with (0,0) in mask:




Our moving average box filter
Our moving average (box) filter audio)

Example of a lowpass filter (passes low frequencies, attenuates high frequencies)

y[n] = 1/3 x[n-1] + 1/3 x[n]

+ 1/3 x[n+1]

More generally,

y[n] = h[-1] x[n-1] + h[0] x[n]

+ h[1] x[n+1]






Lowpass filters

Lowpass filters audio)

X

Input: (before)

x(t) = 0.5*sin(t) + sin(3*t+pi/3) + sin(5*t+pi/8)

Output: (after)

y(t) = 0.5*sin(t) + sin(3*t+pi/3)


So how can we determine how our moving average filter behaves
So how can we determine how our moving average filter behaves?

  • 11 point and 51 point moving average filters (on the previous slide) obviously produce different outputs even when given the same input!

  • Answer: By determining how a particular filter responds to an impulse (their impulse response function).


So given h 1 h 0 h 1 how can we plot the impulse response
So given …,h[-1],h[0],h[1],… how can we plot the impulse response?

  • Perform the z-transform (the discrete version of the Laplace transform) of h resulting H.

  • Plot H on the unit circle. The magnitude of H (abs(H) or |H|) is amplitude and the angle of H (arg(H)) is the phase.

  • Say we have a 3 point box filter:

    h[-1] = h[0] = h[1] = 1/3.


So given h 1 h 0 h 1 how can we plot the impulse response1
So given …,h[-1],h[0],h[1],… how can we plot the impulse response?

  • Say we have an 11 point box filter:

    h[-5]=h[-4]=h[-3] =h[-3] =h[-2] =h[-1] =h[0] =h[1] =h[2] =h[3] =h[4] =h[5]=1/11.




Db from http www animations physics unsw edu au jw db htm
dB response?(from http://www.animations.physics.unsw.edu.au/jw/dB.htm)

“The decibel (dB) is used to measure sound level, but it is also widely used in electronics, signals and communication. The dB is a logarithmic way of describing a ratio. The ratio may be power, sound pressure, voltage or intensity or several other things.”

“One decibel is close to the Just Noticeable Difference (JND) for sound level.

Experimentally it was found that a 10 dB increase in sound level corresponds approximately to a perceived doubling of loudness.”



3 point box filter blue vs 3 point gaussian green normal left db below
3 point box filter (blue) vs. 3 point filters.gaussian (green)(normal – left, dB – below)



Other filters other than lowpass
Other filters filters.(other than lowpass)



Spectral inversion how to make a highpass filter the easy way
Spectral inversion: filters.How to make a highpass filter the easy way.

  • Change the sign of each sample in the filter kernel.

  • Add 1 to the sample at the center of symmetry.

    • highpass lowpass

    • lowpass  highpass

    • bandpass bandreject (stopband)

    • bandreject bandpass


Spectral inversion how to make a highpass filter the easy way1
Spectral inversion: filters.How to make a highpass filter the easy way.

  • Change the sign of each sample in the filter kernel.

  • Add 1 to the sample at the center of symmetry.

    Ex. lowpass box  highpass



Lowpass followed by highpass bandpass
Lowpass followed by highpass = bandpass filters.

(more efficient implementation)


Bandreject stopband lowpass highpass lowpass or highpass
Bandreject (stopband) = lowpass + highpass (lowpass or highpass)

(more efficient implementation)


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