## Phase Locked Loop (PLL)

Demodulation of an FM signal using PLL:-

Let the input to PLL is an FM signal

let

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

## Frequency domain representation of a Wide Band FM

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        implies

as      and

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

 n value wide Band FM signal 0 1 -1 … ….

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.     (No Ratings Yet) Loading...

## Matched Filter, impulse response h(t)

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 .     (2 votes, average: 5.00 out of 5) Loading...

## Mutual Information I(X ; Y) Properties

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).     (No Ratings Yet) Loading...

## Example Problems in Electro Magnetic Theory Wave propagation

1. A medium like Copper conductor which is characterized by the parameters  and  uniform plane wave of frequency 50 Hz. Find   and .

Ans.  Given   ,         and

and

Find the Loss tangent

So given medium is a Conductor (Copper)

then

, .

.

.

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.

which implies

.

Ω.     (1 votes, average: 5.00 out of 5) Loading...