Frequency Response of First-Order Lag

A first-order lag relation is often used to represent the dynamic
response characteristics of simple systems.  For any input signal
x(t) the output signal y(t) satisfies the ordinary differential 
equation
                     dy
                   T --  +  y  =  x                    (1)
                     dt

where T is the "time constant" of the response.  If we set the input
signal x(t) to a simple sine wave, the steady-state output signal y(t)
will also be a sine wave, although the amplitude and phase of the
signals will generally be different.  One way of describing a first-
order lag is in terms of the amplitude and phase differences it 
produces for sinusoidal input signals of various frequencies.

Let us consider the respone to an input signal given by

                     x(t) = A sin(wt)

where A is the constant amplitude and w is the constant frequency
in radians per second.  (The is equivalent to the frequency f = w/2pi
expressed in cycles per second, also known as Hertz.)  We surmise a
steady-state solution of the form

                    y(t) = kA sin(wt - q)

where k is the amplitude gain factor, and q is the phase shift between
the input and output signals.  The derivative of y(t) is

                dy/dt  =  kAw cos(wt - q)

Substituting into the equation (1) gives

         TkAw cos(wt-q)  +  kA sin(wt-q)  =  A sin(wt)

The constant A can be canceled out, and we can use the trigonometric
identities

              sin(x-y) = sin(x)cos(y) - cos(x)sin(y)

              cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

to expand the above equation into

       Tkw [cos(wt)cos(q) + sin(wt)sin(q)]

           +  k [sin(wt)cos(q) - cos(wt)sin(q)]  =  sin(wt)

Equating the total coefficients of cos(wt) and sin(wt) on both sides
of this equation and dividing through by k gives the two conditions

              cos(q)  +  sin(q) wT  =  1/k

             -sin(q)  +  cos(q) wT  =  0

This is just a plane rotation through the angle q, and can be written
in matrix form as
              _                  _   _    _       _     _
             |   cos(q)  sin(q)   | |  1   |     |  1/k  |
             |                    | |      |  =  |       |
             |_ -sin(q)  cos(q)  _| |_ wT _|     |_  0  _|

Since a rotation doesn't change the length of the vectors, we must
have
                  2          2            2
                 1    +  (wT)    =   (1/k)

and so the amplitude factor k is given by

                               1
                 k  =  ----------------
                       sqrt[1 + (wT)^2]

We can easily solve for the angle  q  by dividing the equation
-sin(q)+cos(q)wT = 0  through by cos(q) to give tan(q) = wT, from
which we get
                     q  =  atan(wT)

The expression for k shows that the amplitude of the output signal
is the square root of a "Cauchy distribution" of the scaled input
frequency wT.  Typically we disregard negative frequencies, and we
say this is a "low pass filter", because for small values of wT the
amplitude factor k is nearly 1 (i.e., very little attenuation of the
input signal), whereas for larger values of wT the amplitude factor
drops.

Likewise the phase lag is small when the frequency wT is low, but it
increases as the frequency increases.  Of course, the inverse tangent
is a multi-valued function, and we typically just take the value from
the principle branch in the range from 0 to pi.

It's interesting that if we set the time constant  T  equal to  i
(the square root of -1) and if we identify the frequency w with the
"velocity"  v  of a moving object, we get

                  1
         k = -----------           q = atan(iv) = i atanh(v)
             sqrt[1-v^2]

Hence the amplitude factor of this transfer function equals the
"gamma" factor of time dilation and length contraction in special 
relativity, and the phase angle is proportional to the additive 
"rapidity", corresponding to the velocity v.  The differential
equation (1) in this case may be written as

                       dy
                     i --   =  x - y
                       dt

It's interesting to compare this with the time-dependent Schrodinger
equation
                        d phi
                     ih -----   =  H phi
                         dt

where h is the (reduced) Planck constant, H is the Hamiltonian operator
for the system, and phi is the quantum wave function of the system.
If we identify the output signal y(t) of our filter with the wave
function phi(t) of the system, and if we identify (x-y)/h   with the
scaled Hamiltonian operator of the system, then the correspondence
between these equations is exact.  Does this suggest an approach to a
quantum mechanical description of spacetime?