Demodulation of an FM signal using PLL:-
Let the input to PLL is an FM signal
Now the signal at the output of VCO is FM signal (another FM signal, which is different from input FM signal) Since Voltage Controlled Oscillator is an FM generator.
the corresponding phase
It is observed that S(t) and b(t) are out of phase by . Now these signals are applied to a phase detector , which is basically a multiplier
the error signal
on further simplification , the product yields a higher frequency term (Sum) and a lower frequency term (difference)
This product e(t) is given to a loop filter , Since the loop filter is a LPF it allows the difference and term and rejects the higher frequency term.
the over all output of a loop filter is
To obtain the frequency-domain representation of Wide Band FM signal for the condition one must express the FM signal in complex representation (or) Phasor Notation (or) in the exponential form
i.e, Single-tone FM signal is
Now by expressing the above signal in terms of Phasor notation ( , None of the terms can be neglected)
Let is the complex envelope of FM signal.
is a periodic function with period . This can be expressed in it’s Complex Fourier Series expansion.
i.e, this approximation is valid over . Now the Fourier Coefficient
let as order Bessel Function of first kind then .
Continuous Fourier Series expansion of
Now substituting this in the Equation (I)
The Frequency spectrum can be obtained by taking Fourier Transform
||wide Band FM signal
From the above Equation it is clear that
- FM signal has infinite number of side bands at frequencies for n values changing from to .
- The relative amplitudes of all the side bands depends on the value of .
- The number of significant side bands depends on the modulation index .
- The average power of FM wave is Watts.
Matched Filter can be considered as a special case of Optimum Filter. Optimum Filter can be treated as Matched Filter when the noise at the input of the receiver is White Gaussian Noise.
Transfer Function of Matched Filter:-
Transfer Function of Optimum filter is
if input noise is white noise , its Power spectral density (Psd) is .
then H(f) becomes
From the properties of Fourier Transforms , by Conjugate Symmetry property
Equation (I) becomes
From Time-shifting property of Fourier Transforms
From Time-Reversal Property
By Shifting the signal by T Seconds in positive direction(time) ,the Fourier Transform is given by
Now the inverse Fourier Transform of the signal from the Equation(II) is
Let the constant is set to 1, then the impulse response of Matched Filter will become .
Property 1:- Mutual Information is Non-Negative
Mutual Information is given by equation
we know that
Substitute Equation (II) in Equation (I)
The above Equation can be written as
we knew that
This result can be applied to Mutual Information , If and be , Both and are two probability distributions on same alphabet , then Equation (III) becomes
i.e, , Which implies that Mutual Information is always Non-negative (Positive).
- A medium like Copper conductor which is characterized by the parameters and uniform plane wave of frequency 50 Hz. Find and .
Ans. Given , and
Find the Loss tangent
So given medium is a Conductor (Copper)
2. If for a medium in which a wave with a frequency of is propagating . Determine the propagation constant and intrinsic impedance of the medium when
Ans: Given , , and .
Since , the given medium is a lossless Di-electric.