Natural Science Forum / Physics / Acoustics / March 2004

, I have implemented inverse filters on the

You have a source signal that is applied to a filter that has a propagation
delay.  You measure the filter output and apply it to a causal inverse
filter.  At the output of the inverse filter you should see the source
signal delayed.  That is exactly what you say you see.    

For example the filter

That too is correct, but the filter (1+s)/(1+2s) does not have a
propagation delay and does not model the situation that you previously
described.  

The proper model would be H(s)=[(1+s)/(1+2s)]exp(-sT), which would
represent a filter with a propagation delay.

Now, if you cascaded an inverse filter (1+2s)/(1+s), you would still see
the delay at the output.  In order to back out the delay, the inverse
filter would have to contain the multiplicative term exp(sT), which
corresponds to a negative delay and which is non-causal.

I used pspice to create a 10-section LCLCLCL line.  I calculated the
component vales using the equations Zo=sqrt(L/C) and v=1/sqrt(LC).  For Zo,
I used 92 cgs ohms (which is the characteristic impedance of the human ear
canal) and for v I used 34000 cm/sec, which is the approximate speed of
sound in air.  With the above values for Zo and v, the calculated value for
L was 2.71m, and the calculated value for C was 319n.  At the front and
rear of the line I used L/2.  

When the line was terminated with a resistance equal to Zo (92 ohms), the
output response magnitude was virtually flat up to the cutoff frequency
(approx 10KHz) and the output response phase was linear up to about 5KHz.  
The linear phase lag corresponded to a constant time delay of 0.297msec.  
Using this delay and the velocity of sound, I calculated that the effective
length of the 10-section line is 10.1cm.  Using this length, I then
calculated quarter wave and three quarter wave resonancet frequencies of
842Hz and 2526Hz.  

I then mismatched the line by increasing the terminating resistance from 92
ohms to 200 ohms.  This resulted in resonances in the output, the first two
of which occurred at frequencies of 845 and 2520, in excellent agreement
with the calculated quarter wave and three quarter wave resonant
frequencies. Additionally, the phase response of the mismatched line showed
the same average delay as the Zo terminated line, but contained ripples
associated with the resonances and antiresonances in the magnitude
response.  

In conclusion, the 10-section LCLCLCL line showed BOTH the expected delay
(linear phase) and the expected resonances that are associated with a
continuous transmission line.   This, for me, conclusively rules out
candidate 1.