matched filter phase

[rutiger]

Has it just been too long since I have looked closely at an actual
phase equation or should there be a quadratic phase on the output of an
LFM matched filter? I know the problem is often worked out in the
frequency domain, but it seems straight forward enough to just carry
out the integration by hand. If alpha is the frequency slope, t is
time, t_tgt round trip delay, and t_0 some reference time, the output
of the mathced filter of n LFM signal is (leaving off the limits for
clarity):

s(t) = integral { exp( j*pi*alpha * ( x - t_tgt )^2 ) * exp(
-j*pi*alpha * ( x - t_0 - t )^2 ) dx }

If I just do the algebra, the phase of this signal is

phase = pi*aplha * ( x^2 + t_tgt^2 - 2*x*t_tgt - x^2 - t_0^2 - t^2 +
2*x*t_0 + 2*x*t - 2*t_0*t_tgt )

The two x^2 terms cancel out, the t_0^2 and t_tgt^2 terms are the RVP,
and the linear terms that stay inside the integral (those that are a
function of x) define the sinc() function that one expects coming out
of the matched filter. My confusion is the t^2 term. He is not a
function of x, so he comes out of the integral and there is nobody to
cancel him out. But that means that the output is a sinc() function
with quadratic phase and I always remembered that the output is a sinc
function with linear phase.

Can anybody help with where I am going wrong?

[rutiger]

Post by rutiger
Has it just been too long since I have looked closely at an actual
phase equation or should there be a quadratic phase on the output of an
LFM matched filter? I know the problem is often worked out in the
frequency domain, but it seems straight forward enough to just carry
out the integration by hand. If alpha is the frequency slope, t is
time, t_tgt round trip delay, and t_0 some reference time, the output
of the mathced filter of n LFM signal is (leaving off the limits for
s(t) = integral { exp( j*pi*alpha * ( x - t_tgt )^2 ) * exp(
-j*pi*alpha * ( x - t_0 - t )^2 ) dx }
If I just do the algebra, the phase of this signal is
phase = pi*aplha * ( x^2 + t_tgt^2 - 2*x*t_tgt - x^2 - t_0^2 - t^2 +
2*x*t_0 + 2*x*t - 2*t_0*t_tgt )
The two x^2 terms cancel out, the t_0^2 and t_tgt^2 terms are the RVP,
and the linear terms that stay inside the integral (those that are a
function of x) define the sinc() function that one expects coming out
of the matched filter. My confusion is the t^2 term. He is not a
function of x, so he comes out of the integral and there is nobody to
cancel him out. But that means that the output is a sinc() function
with quadratic phase and I always remembered that the output is a sinc
function with linear phase.
Can anybody help with where I am going wrong?

A co-worker of mine gave me an article from the old Bell System
Technical Journal that deals with this topic explicity, so I figured
I'd post the answer to my above question since it might be useful for
someone else in the future.

What I have above is correct, that is, there is a residual quadratic
phase on the output of the matched filter. Exactly how it manifests
itself depends on how the chirp and match filter are generated. For
dispersion factors (i.e. time-bandwidth products) of even a small
magnitude, the effect of this residual LFM is mitigated by the envelope
of the sinc function such that it is nearly undetectable; thus, for
systems with decent dispersion factors (like modern radar systems),
this term is ignored.

For details see,

J. R. Klauder, et al, "The Theory and Design of Chirp Radars," The Bell
System Technical Journal, July 1960, pp. 745-808.