L.P. Butterworth filter gain at 3wc <= -50 dB

Discussion in 'Homework Help' started by J_Rod, Nov 14, 2015.

  1. J_Rod

    Thread Starter Member

    Nov 4, 2014
    109
    6
    Good morning,
    I have some questions about this problem about a L.P. (low pass) filter:

    "Determine n, the order of a lowpass Butterworth filter, and the corresponding cutoff frequency wc, required to satisfy the following lowpass filter specifications. Find both the values of wc, the one that oversatisfies the passband specifications, and the one that oversatisfies the stopband specifications.

    The gain at 3wc is required to be no greater than -50 dB."

    For the lowpass filter, I think that for frequencies w < wc, the gain is 1 (no amplification or attenuation of the signal), and after that, the gain quickly drops to 0 (unity). I need to identify the minumum passband gain (at frequency wp) and maximum stopband gain (at frequency ws) to use the equations in my text for n and wc, but I'm not certain how those were given in the problem. Specifically, where is 3wc in relation to wp and ws? Intuitively, would 3wc = ws since the gain there is no greater than -50 dB? But then, what simplifying assumption(s) determine(s) where wp is?

    Thank you for the help,
    Sincerely,
    J_Rod
     
  2. crutschow

    Expert

    Mar 14, 2008
    13,033
    3,241
    The filter rolloff rate is determined by the filter order.
    Each step in order increases the rolloff by an additional 6dB per octave or 20dB per decade, i.e. 1st order is -6dB/octave, 2nd order is 12dB/octave, etc.
    This rolloff starts after the corner frequency (-3dB point).
    So from that you should be able to determine the filter order required to get -50dB gain at 3wc.

    Note that 1 is unity gain, not 0.
    And the gain never reaches 0 at the output of a filter, it just gets arbitrarily small.
     
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  3. Papabravo

    Expert

    Feb 24, 2006
    10,152
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    You need to find the transfer function of a Butterworth lowpass filter. From the transfer function you should determine if the solution is unique, or might be satisfied bay a range of solutions. The gain at DC/lowfrequency should be close to 1, or 0 dB. As you approach the corner frequency the gain will drop to -3 dB. From \omega_c to 3 \cdot \omega_c is the transition band and the gain can be in the range [-3 dB to -50 dB.]. For frequencies greater than than 3 \cdot \omega_c the gain must be less than -50 dB.
     
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  4. Papabravo

    Expert

    Feb 24, 2006
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  5. J_Rod

    Thread Starter Member

    Nov 4, 2014
    109
    6
    Would the filter have to be at least 3rd order since 3*-20 dB = -60 dB <= -50 dB? Is it possible for the filter to have a higher order, though, since that still satisfies the inequality?
     
  6. J_Rod

    Thread Starter Member

    Nov 4, 2014
    109
    6
    Is wc the corner (or cutoff) frequency? It is defined as the point where the signal loses 1/2 power. Would this be the same frequency as the maximum passband frequency in this problem?
     
  7. Papabravo

    Expert

    Feb 24, 2006
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    Yes. The reason it is called the corner frequency is because it is the intersection of the passband gain of 0 dB and the asymptote of the rolloff which depends on the order.

    Be careful.
    The expression for voltage gain is

    20 \cdot log(\frac{v_o}{v_i})

    The expression for power gain is

    10 \cdot log(\frac{p_o}{p_i})

    Make sure you know which -3 dB point you are referring to.

    For voltage:

    10^{(-3/20)} \approx \frac{1}{\sqrt{2}}= 0.707

    For power

    10^{(-3/10)} \approx \frac{1}{2}
     
    Last edited: Nov 14, 2015
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  8. crutschow

    Expert

    Mar 14, 2008
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    But the -3dB point is always the half power point.
    It's just that, due to the V^2 relation between power and voltage, the voltage is .707 down at the -3dB point.
     
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  9. Papabravo

    Expert

    Feb 24, 2006
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    Yes and you need to make sure you don't get confused when working with this stuff. It can easily happen.
     
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  10. Veracohr

    Well-Known Member

    Jan 3, 2011
    552
    76
    Given how it's worded, I think yes ws=3wc for this question. As for wp I think it could be any frequency sufficiently far below wc that its amplitude equals the minimum passband gain, unaffected by the filter attenuation. Try wp=0.01wc.
     
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