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460 9 Fourier analysis
Let the Fourier transforms of x(t) and F(t)be X(ω) and F(ω) respectively. Relate the two
through a convolution, X(ω) = R(ω)F(ω). At what frequency ω c does R(ω) become very
large, exhibiting resonance? What effect does ζ have on the resonance phenomenon? Hint:
Relate first the Fourier transforms of derivatives of x(t) to that of x(t) itself.
κ
9.B.2. For m = 1, K = 1, and ζ = 10 ,κ = 2, 1, 0, −1, −2, −3, plot the response x(t)
κ
to F(t) = cos(ωt) for ω/ω c = 1 ± 10 ,κ =−1, −2, −3.
9.B.3. Let us say that we wish to measure some signal x(t) by a device whose output d(t)
is related to the signal by the convolution D(ω) = R(ω)F(ω). For a device that is only
sensitive near ω dev ,
2
(|ω| − ω dev )
R(ω) = r 0 exp − (9.130)
2σ 2
For r 0 = 1,ω dev = 5,σ = 1, compute the output signals for input pulsed calibration
signals,
1, 0 ≤ t ≤ t pulse
c(t) = (9.131)
0, otherwise
with t pulse = 0.01, 0.1, 0.5, 1. Then, let us say that we did not know the device response
function but rather only the results of these calibration experiments. For each calibration
experiment, estimate R(ω).
9.C.1. In problem 9.B.3, we estimate R(ω) separately from the results of each calibra-
tion experiment. Propose a method that uses all of the data to generate the best estimate
of R(ω).