Consider the frequency demodulation scheme shown in Fig. below in whic...
Frequency Demodulation Scheme
The frequency demodulation scheme shown in the given figure is used to demodulate FM waves.
Delay Line
The incoming FM signal is passed through a delay line that produces a delay of T such that 2π fcT= π/2.
Subtraction
The delay-line output is subtracted from the incoming FM signal.
Envelope Detection
The resulting output is envelope detected.
Calculation of a(t)
Given, s(t) = Ac cos(2π fct βsin(2π fmt))
Let x(t) be the output of the delay line.
x(t) = s(t - T) = Ac cos[2π fc(t-T) + βsin(2π fm(t-T))]
On simplification, we get
x(t) = Ac cos[2π fcT + 2π fc(t/T - 1) + βsin(2π fm(t-T))]
x(t) = Ac cos[2π fcT + 2π f(t/T - 1) + βsin(2π fm(t-T))]
On subtracting x(t) from s(t), we get
y(t) = s(t) - x(t)
y(t) = Ac cos(2π fct βsin(2π fmt)) - Ac cos[2π fcT + 2π f(t/T - 1) + βsin(2π fm(t-T))]
y(t) = Ac [cos(2π fct βsin(2π fmt)) - cos(2π fcT + 2π f(t/T - 1) + βsin(2π fm(t-T)))]
Using trigonometric identity, we get
y(t) = -2Ac sin[2π f((t/T) - 1/2) + πβcos(2π fmt)/(2π fm)]
On envelope detection, we get
a(t) = |-2Ac sin[2π f((t/T) - 1/2) + πβcos(2π fmt)/(2π fm)]|
a(t) = 2Ac |sin[2π f((t/T) - 1/2) + πβcos(2π fmt)/(2π fm)]|
Explanation of the Result
The given frequency demodulation scheme uses a delay line to introduce a phase shift of π/2. This phase shift is used to convert the frequency modulation into amplitude modulation. By subtracting the output of the delay line from the incoming FM signal, we get a signal that is proportional to the frequency deviation. This signal is then envelope detected to obtain the baseband signal. The resulting a(t) is proportional to the frequency deviation and can be used to recover the modulating signal.