In article <76aoqh$l4n$1@nnrp1.dejanews.com>, ceballos_c@tsm.es wrote:
> Does anyone know how to simulate an LFM chirp signal on matlab?
> I am very interested in finding out information about radar signal treatment
> using LFM chirp, adapted filter, spectral estimation...and so on > using
> Matlab.
> Could some one give me some guidelines
> Thanks in advance!!!!
> Gracias
> Cristina
>
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Hi Cristina,
I noticed that several people provided you Matlab code for generating LFM
signals in Matlab. I thought it might be helpful for you to see the basic
mathematics involved and how the equations
relate to the real world. I hope you find this helpful.
Cheers,
Jim Luby
Let x(t) be the signal that you want to generate. Define it as follows:
x(t) = w(t) * Real_part( exp( j*phi(t) ) where
w(t) = real valued window function (set it to 1.0 if you don't want any
windowing)
j = square_root(1)
phi(t) = timevarying phase of the complex exponential
Now let phi(t) = a0 + a1*t + 1/2*a2*t*t (i.e. a quadratic in t defined
by the three constants a0, a1, and a2).
The important thing to understand is that the instantaneous frequency is
the time derivative of the phase (and hence phase is the integral of the
instantaneous frequency with respect to time).
Letting f(t) be the timevarying instantaneous frequency we get
f(t) = d/dt(phi(t) = a1 + a2*t
To relate this to the real world let a1 = 2*pi*f0 = center frequency of
LFM sweep. Now note that if a2 = 0 we get f(t) = 2*pi*f0 = constant
frequency (i.e. this is the simple case of a timeinvariant frequency
sinusoid). In other words
x(t) = w(t) * Real_part( exp(j*2*pi*f0*t) ) = w(t) * cos(2*pi*f0*t)
Now consider what happens if a2 is nonzero. In this case we get
phi(t) = 2*pi*f0*t + 1/2*a2*t*t
and
f(t) = 2*pi*f0 + a2*t
Note that for T/2 < t < T/2 (where T is the duration of the waveform)
the instantaneous frequency sweeps from
f(T/2) = 2*pi*f0  a2*T/2
to
f(T/2) = 2*pi*f0 + a2*T/2
In other words, the frequency sweeps linearly over a range of (a2*T/2 ,
+a2*T/2) and hence this defines the LFM bandwidth. Also, note that at
time t = 0, the frequency sweeps through f0.
Usually one knows the desired waveform duration T, the center frequency f0
and the sweep range (a2*T/2 , +a2*T/2). Once you choose these parameters
it is a simple matter to define phi(t) and hence to use the 1st equation I
presented to generate the desired sweep.
The same ideas hold true for nonlinear frequency sweeps except that in
such cases the instantaneous frequency is not linear (and hence the phase
is not quadratic).
I hope you find this helpful.
Good luck and happy holidays,
Jim
