Computer Algebra Recipes

332 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
8.2.1 Frank N. Stein's Heartbeat
The more powerful and original a mind, the more it will incline
towards the religion of solitude.
Aldous Huxley, British writer (1894{1963)
Let's ¯rst illustrate how the power spectrum is calculated for a simple example.
We are given Frank N. Stein's steady heartbeat described by
2X
x = Ai sin(2 ¼fit); i=1
with fundamental frequency f1 = 1 beat per second (60 beats per minute) and
a small second harmonic f2 = 2 beats per second. The suitably normalized
amplitudes are A1 =1andA2 =0:4. Pretending that we do not know the
frequencies, we will extract them from the power spectrum. (We could be
dealing with experimental data, not a known analytic form.)
To calculate the discrete Fourier transform, the DiscreteTransforms package
must ¯rst be loaded. The number of sampling points is taken to be N = 1000.
The numerical factor d = 10 will be used to determine the sampling time.
> restart: with(DiscreteTransforms): N:=1000: d:=10:
The parameter values are entered and the time T2 =1=f2 is calculated.
> A[1]:=1: A[2]:=0.4: f[1]:=1: f[2]:=2.0: T[2]:=1/f[2];
T2 := 0:5000000000
An operator is formed to calculate x at a speci¯ed time t.
> x:=t->add(A[i]*sin(2*Pi*f[i]*t),i=1..2):
Then, x(t) is plotted over the time interval t =0todT2.
> plot(x(t),t=0..d*T[2],labels=["t","x"]);
x
1
0.5
0
–0.5
–1
1 2 3 4 5
t
Figure 8.8: Frank N. Stein's heartbeat.
If one were given only the plot shown in Figure 8.8, it would not be obvious
exactly what frequencies are contained in Frank's heartbeat. The power spec-