The general form a sinusoid with a time varying angle can be expressed as x(t) =
ID: 1814295 • Letter: T
Question
The general form a sinusoid with a time varying angle can be expressed as x(t) = Acos(Psi(t)) where Psi(t) represents the angle function versus time. The instantaneous frequency is defined as: fi(t) = (1/2pi)*(d/dt)*Psi(t) (Hz).
Write an equation for a signal x(t) in the form given above such that the instantaneous frequency (in HZ) is as specified below.
(a) A linear sweep:from 250Hz at time 0 seconds to 650Hz at time 8 seconds.
(b) A sinusoid centered on 850Hz:ft(t) = 850 + 300 cos(8*pi*t) for all time t.
(c) An exponential decay: fi(t) = 450e^(-3t) for time 0<t<1
PLEASE SHOW ALL WORK
Explanation / Answer
(a) A linear sweep:from 250Hz at time 0 seconds to 650Hz at time 8 seconds.
f(t) = 250 + 50*t
psi(t) = 2*pi*integration [ f(t) *dt]
= 2*pi*[ 250 t + 25t^2 +C]
X(t) =A*cos[2*pi*(250 t + 25t^2 +C)]
b) A sinusoid centered on 850Hz:ft(t) = 850 + 300 cos(8*pi*t) for all time t.
f(t) = 850 + 300 cos(8*pi*t)
psi(t) = 2*pi*integration [ f(t) *dt]
= 2*pi*( 850t +(37.5/pi) sin(8*pi*t) +c)
X(t) =A*cos[2*pi*( 850t +(37.5/pi) sin(8*pi*t) +c)]
c)
(c) fi(t) = 450e^(-3t) for time 0<t<1
psit) = (-150e^(-3t) +C)
X(t) = A*cos[ (-150e^(-3t) +C)]
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