Ans. Given P(1,3,5)

Cylindrical :-

Similarly

Spherical :-

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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

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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.

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]]>**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 .

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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).

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]]>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.

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i.e, Delta Modulation (DM) is a Modulation scheme in which an incoming message signal is over sampled (i.e, at a rate much higher than the Nyquist rate ) to purposely increase the correlation between adjacent samples of the signal. Over sampling is done to permit the use of a sample Quantizing strategy for constructing the encoded signal.

Signaling rate and Transmission Band Width are quite large in PCM. DM is used to overcome these problems in PCM .

DM transmits one bit per sample.

The process of approximation in Delta Modulation is as follows:-

The difference between the input () and the approximation () is quantized into only two levels corresponding to Positive and negative differences.

i.e, If the approximation () falls below the signal ()at any sampling epoch(the beginning of a period)output signal level is increased by .

On the other hand the approximation () lies above the signal () , output signal level is diminished by provided that the input signal does not change too rapidly from sample to sample.

it is observed that the change in stair case approximation lies with in .

This process can be illustrated in the following figure

**Delta Modulated System:- **The DM system consists of Delta Modulator and Delta Demodulator.

**Delta Modulator:- **

Mathematical equations involved in DM Transmitter are

error signal:

Present sample of the (input) sampled signal:

last sample approximation of stair case signal:

Quantized error signal( output of one-bit Quantizer):

if .

and .

encoding has to be done on the after Quantization that is when the output level is increased by from its previous quantized level, bit ‘1’ is transmitted .

similarly when output is diminished by from the previous level a ‘0’ is transmitted.

from the accumulator

where is the Quantization error.

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The alternative representation if a set of complex exponentials are used,

The resulting representations are known as Fourier Series in Continuous-Time [Fourier Transform in the case of Non-Periodic signal]. Here we focus on representation of Continuous-Time and Discrete-Time periodic signals in terms of basic signals as Fourier Series and extend the analysis to the Fourier Transform representation of broad classes of aperiodic, finite energy signals.

These Fourier Series & Fourier Transform representations are most powerful tools used

- In the analyzation of signals and LTI systems.
- Designing of Signals & Systems.
- Gives insight to S&S.

The development of Fourier series analysis has a long history involving a great many individuals and the investigation of many different physical phenomena.

The concept of using “Trigonometric Sums”, that is sum of harmonically related sines and cosines (or) periodic complex exponentials are used to predict astronomical events.

Similarly, if we consider the vertical deflection of the string at time t and at a distance x along the string then for any fixed instant of time, the normal modes are harmonically related sinusoidal functions of x.

The scientist Fourier’s work, which motivated him physically was the phenomenon of heat propagation and diffusion. So he found that the temperature distribution through a body can be represented by using harmonically related sinusoidal signals.

In addition to this he said that any periodic signal could be represented by such a series.

Fourier obtained a representation for aperiodic (or) non-periodic signals not as weighted sum of harmonically related sinusoidals but as weighted integrals of Sinusoids that are not harmonically related, which is known as Fourier Integral (or) Fourier Transform.

In mathematics, we use the analysis of Fourier Series and Integrals in

- The theory of Integration.
- Point-set topology.
- and in the eigen function expansions.

In addition to the original studies of vibration and heat diffusion, there are numerous other problems in science and Engineering in which sinusoidal signals arise naturally, and therefore Fourier Series and Fourier T/F’s plays an important role.

for example, Sine signals arise naturally in describing the motion of the planets and the periodic behavior of the earth’s climate.

A.C current sources generate sinusoidal signals as voltages and currents. As we will see the tools of Fourier analysis enable us to analyze the response of an LTI system such as a circuit to such Sine inputs.

Waves in the ocean consists of the linear combination of sine waves with different spatial periods (or) wave lengths.

Signals transmitted by radio and T.V stations are sinusoidal in nature as well.

The problems of mathematical physics focus on phenomena in Continuous Time, the tools of Fourier analysis for DT signals and systems have their own distinct historical roots and equally rich set of applications.

In particular, DT concepts and methods are fundamental to the discipline of numerical analysis , formulas for the processing of discrete sets of data points to produce numerical approximations for interpolation and differentiation were being investigated.

FFT known as Fast Fourier Transform algorithm was developed, which suited perfectly for efficient digital implementation and it reduced the time required to compute transform by orders of magnitude (which utilizes the DTFS and DTFT practically).

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Our basic aim is to understand the various modules and sub systems in the system. If we are trying to understand the design and various features of DCS , it is plus imperative that we have to understand how we should design a transmitter and we must understand how to design a very good quality Receiver. Therefore one must know the features of the channel to design a good Transmitter as well as receiver that is the channel and it’s contribution will come repeatedly in digital Communications.

**Source:-** the primary block (or) the starting point of a DCS is an information source, it may be an analog/digital source , for example the signal considered is analog in nature, then the signal generated by the source is some kind of electrical signal which is random in nature. if the signal is a speech signal (not an electrical signal) that has to be converted into electrical signal by means of a Transducer, which can be considered as a part of source itself.

**Sampling & Quantization:-** the secondary block involves the conversion of analog to discrete signal

this involves the following steps

**Sampling:- **it is the process that involves in the conversion of Continuous Amplitude Continuous Time (CACT) signal into Continuous Amplitude Discrete Time (CADT) signal.

**Quantization:- **it is the process that involves in the conversion of Continuous Amplitude Discrete Time (CADT) signal into Discrete Amplitude Discrete Time (DADT) signal.

**Source Encoder:- ** An important problem in Digital Communications is the efficient representation of data generated by a Discrete Source, this is accomplished by source encoder.

” The process of representation of incoming data from a Discrete source into a more suitable form required for Transmission is known as source encoding”

Note:-The blocks Sampler, Quantizer followed by an Encoder constructs ADC (Analog to Digital Converter).

∴ the output of Source encoder is a Digital Signal, the advantages of Source coding are

- It reduces the Redundancy.
- Minimizes the average bit rate.

**Channel encoder:-**Channel coding is also known as error control coding. Channel coding is a technique which reduces the probability of error by reducing Signal to Noise Ratio at the expense of Transmission Band Width.The device that performs the channel coding is known as Channel encoder.

Channel encoding increases the redundancy of incoming data , this also involves error detection and error correction along with the channel decoder at the receiver.

**Spreading Techniques:-** Spread Spectrum techniques are the methods by which a signal generated with a particular Band Width is deliberately spread in the frequency domain, resulting in a signal with a wider Band width.

There are two types of spreading techniques available

- Direct Sequence Spread Spectrum Techniques.
- Frequency Hopping Spread Spectrum Techniques.

The output of a spreaded signal is very much larger than incoming sequence. Spreading increases the BW required for transmission, which is a disadvantage even though spreading is done for high security of data.

SS techniques are used in Military applications.

**Modulator:-** Spreaded sequence is modulated by using digital modulation schemes like ASK, PSK, FSK etc depending up on the requirement, now the transmitting antenna transmits the modulated data into the channel.

**Receiver:-** Once you understood the process involved in transmitter Block. One should perform reverse operations in the receiver block.

i.e the input of the demodulator is demodulated after that de-spreaded and then the channel decoder removes the redundancy added by the channel encoder ,the output of channel decoder is then source decoded and is given to Destination.

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]]>Communication is the process of establishing Connection (or) link between two points (which are separated by some distance) and transporting information between those two points. The electronic equipment used for communication purpose is called Communication equipment. The equipment when assembled together forms a communication system.

Examples of different types of communications are

- Line Telephony & Telegraphy.
- Radio Broadcasting.
- Point-to-Point Communication.
- Mobile Communication.
- TV Broadcasting.
- Radar and Satellite Communications etc.

**Why Digital?**

A General Communication system has two devices and a medium (channel) connecting those two devices. This can be understood that a Transmitter and Receiver are separated by a medium called as Communication channel. To transport an information-bearing signal from one point to another point over a communication channel either Analog or digital modulation techniques are used.

Now Coming to the point, Why Digital communication is preferred over analog Communication?

Why are communication systems, military and commercial alike, going digital?

- There are many reasons; the primary advantage is the ease with which digital signals compared with analog signals are generated. That is the generation of digital signals is much easier compared to analog signals.
**Propagation of Digital pulse through a Transmission line:-**

When an ideal binary digital pulse propagating along a Transmission line. The shape of the waveform is affected by two mechanisms

- Distortion caused on the ideal pulse because all Transmission lines and Circuits have some Non-ideal frequency Transfer function.
- Unwanted electrical noise (or) other interference further distorts the pulse wave form.

Both of these mechanisms cause the pulse shape to degrade as a function of line length. During the time that the transmitted pulse can still be reliably identified (i.e. before it is degraded to an ambiguous state). The pulse is amplified by a digital amplifier that recovers its original ideal shape. The pulse is thus “re-born” (or) regenerated.

Circuits that perform this function at regular intervals along Transmission system are called “regenerative repeaters’. This is one of the reasons why Digital is preferred over Analog.

**3.Digital Circuits Vs Analog Circuits:-**

Digital Circuits are less subject to distortion and Interference than are analog circuits because binary digital circuits operate in one of two states FULLY ON (or) FULLY OFF to be meaningful, a disturbance must be large enough to change the circuit operating point from one state to another. Such two state operation facilitates signal representation and thus prevents noise and other disturbances from accumulating in transmission.

However, analog signals are not two-state signals, they can take an infinite variety of shapes with analog circuits and even a small disturbance can render the reproduced wave form unacceptably distorted. Once the analog signal is distorted, the distortion cannot be removed by amplification because accumulated noise is irrecoverably bound to analog signals, they cannot be perfectly generated.

4. With digital techniques, extremely low error rates and high signal fidelity is possible through error detection and correction but similar procedures are not available with analog techniques.

5. Digital circuits are more reliable and can be produced at a lower cost than analog circuits also; digital hardware lends itself to more flexible implementation than analog hardware.

Ex: – Microprocessors, Digital switching and large scale Integrated circuits.

6. The combining of Digital signals using Time Division Multiplexing (TDM) is simpler than the combining of analog signals using Frequency Division Multiplexing (FDM).

7. Digital techniques lend themselves naturally to signal processing functions that protect against interference and jamming (or) that provide encryption and privacy and also much data communication is from computer to computer (or) from digital instruments (or) terminal to computer, such digital terminations are normally best served by Digital Communication links.

8. Digital systems tend to be very signal-processing intensive compared with analog systems.

Apart from pros there exists a con in Digital Communications that is non-graceful degradation when the SNR drops below a certain threshold, the quality of service can change suddenly from very good to very poor. In contrast most analog Communication Systems degrade more gracefully.

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