Conference rusure::math

	In which domain? The impulse response?  The frequency response?
	More information is needed in any case, including 3dBpoints or
	resonant frequency.
Tom

    The frequency response.

>>	More information is needed in any case, including 3dBpoints or
>>	resonant frequency.
    Yes, but won't those be part of the equation?

    thanks,
    -dave

    what is its "Q" ? what is the "Q" here?
    

    I don't know the exact definition of Q, but as I understand it, the Q
    (for quality, I think) of a system gives an indication of the shape of
    the response curve at resonance.  For loudspeakers, a high Q system has
    a large peak at the resonance frequency, with the response falling rapidly
    above and below that frequency.  A low Q system has a smaller (or no)
    peak, and a gentler rolloff.

    -dave

    Q is an old term that origionally refered to the "qality factor" of a
    resonant circuit.  It more generally measures the ratio of stored energy
    to dissapated energy in a complex pole pair.

    That is, if you shock excite a simple resonant circuit it will
    give an exponentially decaying ringing transient.  The sinusoidal
    ringing is due to potential and kinetic energy swapping between the
    capacitor and inductor, while the decay is due to power being bled
    off in the resistive losses. (If the q is very low, then you won't
    see any ringing since the energy will be dissapated before any
    ringing has a chance to happen.)

    The base note asks about the low frequency behaviour of direct
    radiator loudspeaker systems, which can be approximately modeled
    by lumped LRC circuits.  For example a sealed box system (with no
    passive radiators or other equalization) acts like a second order
    high pass filter with one complex pole pair, with a response like

                     s^2
	H(s) = ---------------  
		1 + s/q + s^2

    s = normalized frequency = 2*pi*i*f/f0
    f = frequency in Hz
    f0 = cutoff frequency in Hz
    q = quality factor

    The higher the q, the more closely the complex pole pair approaches the
    imaginary axis and the sharper the resonant peak. Write a little program
    and try graphing the magnitude of H(s) to see this.

    Nowadays, direct radiator speaker system design is done in terms of
    a set of normalized electroacoustic parameters that are easy to measure
    physically meaningful.  For the scoop on this, you're best getting ahold
    the papers written by Thiele, Small, and Benson.  The Audio Engineering
    Society's Loudspeaker Anthology Vol I has most of the important papers.
    Benson's papers appeard in the AWA Technical Review and are available from
    the British Lending Library, they give a very leasurely and thorough
    discussion of the theory.

    There are some popular books out (Weems is one author I've heard of)
    but they really gloss over the theory.  You really have to be able to
    understand what's happening from first principles and be able to measure
    things since the drivers you can buy as a hobbyist are rarely within
    spec and you'll need to be able to modify your construction to account
    for it.

    For a general intro to network response, look thru a network
    theory book, such as Zverev & Blinchikoff's _Filtering in the Time and
    Frequency Domain_.

    - Jim

    >                 s^2
    >   H(s) = ---------------  
    >		1 + s/q + s^2

    in time domain, this becomes

       Y(s)         s^2
      ----- = ---------------
       U(s)     1+s/q + s^2

     2                         2
    d y         dy            d u
    ----- + 1/q --  + y(t) =  ---
    dt^2        dt            dt^2

    in discrete time it is

    y[n-2]+1/q y[n-1] + y[n] = u[n-2]

    y[n]= u[n-2]- y[n-2] - 1/q y[n-1]

    which means output at time n depends on input at time n-2 and on output
    at time n-1 and n-2, this means there is some feedback loop, hence we
    can control the output at time n by adjusting the "gain" 1/q.

    we can draw this as

                 u-----------------------
                                         |
                       -1 |\ -1/q |\     V
                     -----| |-----| |--o----> y(t)
                      ^   |/      |/   |
                      |                |
                      -----------------

                         
    where the triangle represent an integrator. u is the input, y is the
    output.
         
    we can also find what q values make this system stable, by finding the
    roots of (1+s/q+s^2) and making sure to adjust q so that the roots lie
    in the left hand side of the complex plane.

    /nasser

    Thanks for the formula and discussion.  Now I just need to find some
    spare time to work with it :-)

    -dave