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## Resistors Lecture Notes

• Resistors Lecture Notes: A resistor is a passive component with a specified resistance value. Resistors are the most commonly used components in electronic equipments. As its name states, a resistor is a device that resists the flow of current passed through it. The resistance of any material related to its dimensions and the resistivity of the material by using the formula:

R = (ρ*l)/A

The above equation shows that the resistance of a material is directly proportional to:
1.The length of the material used.
2.The specific resistance (The nature of the taken material)

And inversely proportional to:

1.The area of cross section.

Now we are going to discuss about the specifications of resistors:

Specifications of a resistor:

The important specifications of a resistor are:
1.Resistance value.
2.Tolerance   and
3.Wattage rating.

In addition to these specifications, specifications like voltage coefficient of resistance, voltage rating and temperature coefficient are also mentioned for different applications.

The value of a resistor is the resistance value. The unit of resistance is expressed in ohms. The resistance value can be either written on its body or it can use color codes. The percentage deviation from the rated value is called as the ‘tolerance’ of a resistor.

The maximum power that the resistor can dissipate safely is the ’wattage rating or power rating’. The temperature rise should be very high beyond this rating that the resistor gets damaged. The physical size of of a resistor gives its wattage (power) rating.

As we know the resistivity of a material is generally temperature dependent, so the resistance value changes with change in temperature

Colour Coding of Resistors:

Carbon resistors are very small size resistors. The colour code of the resistors will indicate the resistance values. The Electronic Industries Association (EIA) standardized the colour coding of resistors. The given table shows the colour coding schemes of resistors.

 Colour Digit Multiplier Tolerance Black 0 10^0 - Brown 1 10^1 (+- )1% Red 2 10^2 (+-)2% Orange 3 10^3 - Yellow 4 10^4 - Green 5 10^5 - Blue 6 10^6 - Violet 7 10^7 - Gray 8 10^8 - White 9 10^9 - Gold 10 10^-1 (+-)5% Silver 11 10^-2 (+-)10% NO colour 12 - (+-)20%

The colour bands are printed at one end of the re3sistor. The colour bands are read from left to right. The first colour indicates the first digit, the second colour the second digit and third colour gives the multiplier (the number of zeros to be added after the first two digits). The tolerance value is given by the fourth colour. The fifth band (it is an optional band) indicates the reliability level (failure rate) for which the colour code is given by:

Brown = 1%
Red      =  0.1%
Orange = 0.01%
Yellow = 0.001%

Consider an example for the colour code: A resistor with colour code Blue, Green, Red, Gold will give a value of (652+-5%) Ω.

Consider another example:
If the colour code is Red, Gray, Brown, Silver then the resistance value is (281+-10%)
Another type of resistance code is basically based on B.S1825 code. Here the letters R,K,M denote the value of the resistor in ohms, kilo ohms and mega ohms.

The tolerance code is given in the table:

 Letter Tolerance F +-1% G +-2% J +-5% K +-10% M +-20%

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## Region of Convergence of Z Transform Properties

• The z-transform exists when the infinite sum converges.

The sum may not converge for all values of ‘z’. The value of ‘z’ for which the sum converges is called Region of Convergence (ROC).

PROPERTIES OF REGION OF CONVERGENCE

1. The ROC is a concentric ring or a circle in the z-plane centered at the origin.

2. The ROC cannot contain any poles.

3. If x(n) is a finite duration causal sequence, the ROC is entire z-plane except at z=0.

If x(n) is a finite duration anti causal sequence, then the ROC is the entire z-plane       except at z=∞.
If x(n) is a finite duration 2-sided sequence, then the ROC will the entire z-plane except at z=0 and z=∞.

4.If x(n) is a right sided sequence and if the circle |z|=r0 is in the ROC, then all finite values of ‘z’ for which |z|>ro will also be in ROC.

5. If x(n) is a left sided sequence and if the circle |z|=r0 is in the ROC, then all values of z for which 0<|z|<ro will be in ROC.

6. If x(n) is a 2-sided sequence and if the circle |z|=r0 is in the ROC, then the ROC will consists of a ring in the z-plane that includes the circle |z|=r0

7. If the z-Transform X(z) of x(n) is rational, then its ROC is bounded by poles or extends to ‘∞’.

8. If the z-Transform X(z) of x(n) is rational and if x(n) is right sided, then ROC is the region in the z-plane outside the outermost pole. In other words, Outside the radius of circle = the largest magnitude of pole of x(z). If x(n) is causal then ROC also includes Z= ∞.

9. If the z-Transform X(z) of x(n) is rational and if x(n) is left sided, then ROC is the region in the z-plane inside the outermost ‘non zero pole’.

In other words, inside the circle of radius = the smallest magnitude of pole of x(z) other than at z=0 and extending inwards to and possibly including z=0. If x(n) is anti-causal, ROC includes z=0.

10.If x(n) is a finite duration 2-sided sequence, then ROC will consists of a circular ring in the z-plane bounded on the interior and exterior by a pole and not containing any pole.

11. The ROC of an LTI (Linear Time Invariant) system contains the unit circle.

12. ROC must be a connected region.

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## Inverse Z-Transform with Examples

• represents the discrete time fourier transform of (r^-n*x (n)).

The Fourier inverse of X (r*e^ (j*ω)) is

x (n)*(r^-n) = (1/(2*π))* X (r*e^ (j*ω))* e^ (j*ω*n) dω

Multiplying by (r^n) on both sides, we get,

x (n)= (1/(2*π))* X (r*e^ (j*ω))* (r*e^ (j*ω))^n dω

Changing the variable of integration from ω to z

r*e^ (j*ω) = z

Differentiating the above equation, we get

r*j*( e^ (j*ω)) dω = dz

dω = (dz/ r*j*( e^ (j*ω)))

That is,   dω = (dz/ j*(r*( e^ (j*ω))))

dω = (dz/(z*j))

Therefore,

x (n) = (1/(2*π))* X (z)*(z^n)/ (j*z) dz =(1/(2*π*j))* X (z)*(z^(n-1)) dz

Since the integration over a(2*π) interval in ‘ω’, in terms of ‘z’ corresponds to one transfers around a circle |z|=r

x (n) =(1/(2*π*j))* X (z)*(z^(n-1)) dz

Which gives the inverse z-transform of x(z)

X (z) =(b0+(b1*z^(-1))+(b2*z^(-2))+………………..+(bm*z^(-m)))/(1+(a1*z^(-1)) +
(a2*z^(-2))+…………………..+(an*z^(-n)));

Where (n>m).

The roots of the numerator polynomial are those values of ‘z’ for which x(z)=0 and are referred as zeroes of X (z).Zeros are represented by ‘0’ in the z-plane.

The roots of the denominator polynomial are those values of ‘z’ for which X (z) = ∞ and referred as poles of X (z).poles are represented by ‘x’ in the z-plane.

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## Z-Transform Basics with Formulas

• The Z-Transform is the discrete time counter part of Laplace transform. Z-transform allows us to perform transform analysis of unstable systems and to develop additional insights and tools for LTI (linear Time Invariant)system analysis. The Z-transform transforms difference equation into algebraic equations and hence the discrete time system analysis is specified. Z Transform Basics with Z transform formulas are explained below,

The Z-Transform of a discrete time signal x(n) is defined as:

Where ‘z’ is a complex variable and z=r*e^ (j*ω)

Where ‘r’ is the radius of the circle.

If the sequence x(n) exists for ‘n’ in the range( -∞ to ∞), then,

represents a bilateral or two sided Z transform.

If the sequence x(n) exists only for n>=0,then

which is called one sided or unilateral Z-Transform.

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## Hamming Code with Example

• For a hamming code,

n = (2^m)-1
k = (2^m)-1-m
n-k = m,      where m>=3.
If this condition is satisfied then the code is called hamming code.

Hamming Distance:

Hamming distance is actually defined as the difference in number of elements in the respective locations.
Consider an example:
1101       1111

Hamming Weight:

Each code vector will have a hamming weight. Hamming weight is the number of non zero elements in the code vector.

Ci, t                                                                           Cj, t

T = the number of bits up to which the linear block code can correct.

In other words, it can correct the number of bits with in the circle.
If Ci and Cj are together taken, then the correct bits will be:

D (Ci, Cj) >= (2*t)+1

(t+t+1)

t – Ci , t – Cj .

i.e.  d (Ci, Cj)  >=  (2*t)+1

That is d min >= (2*t)+1
d min-1 >= (2*t)
(2*t) <= d min-1
t <= ½*(d min– 1)
d min
The minimum distance (d min) between two codes is the minimum code vector of the linear block codes.

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## Lempel Ziv Algorithm with Example

• Coding for sources with memory:

A drawback of Huffman’s code is that it requires simple probabilities. For real time applications, Huffman encoding becomes impractical as a source statistics are not always known apriori. A code that might be more efficient to use statistical interdependence of the letters in the alphabet along with their individual probabilities of occurrences is the Lempel-ziv algorithm. This algorithm belongs to the class of universal source coding algorithms.

The logic behind Lempel-ziv universal coding is as follows:

* The compression of an arbitrary binary sequence is possible by coding a series of zeros and ones as some previous such string (prefix string) + one new bit.

* The new string formed by such parsing becomes a potential prefix, string for future strings.
These variables are called phrases (sub sequences). The phrases are listed in a dictionary or code book which stores the existing phrases and their locations. In encoding a new phrase be specified the location of the existing phrase in the code book and append the new letter.

Consider an example:

1.     Determine the Lempel-ziv code for the given sequence:
000101110010100101…………..  ?
The given sequence is:
00, 01, 011, 10, 010, 100, 101
Depending on this a table can be formed as :

 Numeric position 1 2 3 4 5 6 7 8 9 Sub Sequence 0 1 00 01 011 10 010 100 101 Numerical representation - - 11 12 42 21 41 61 62 Binary Coding Sequence - - 0010 0011 1001 0100 1000 1100 1101

Maximum number in numerical representation = 6
6=110
No of bits = 3

i.e,.
0 = 000
1 = 001
2 = 0101
4 = 100
5 = 101
6 = 110

The last symbol of each sub sequence in the code book is an innovation sequence corresponding the last bit of each uniform blocks of bits in the binary encoded representation of the data stream represents innovation symbol for the particular subsequence under consideration. The remaining bits provide the equal binary representation of the pointer in the root subsequence that matches the one in question except for the innovation number

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## Satellite Communication Lecture Notes

• SATELLITE COMMUNICATION

A satellite is an object that revolves around another object. For example, earth is a satellite of The Sun, and moon is a satellite of earth etc. All planets in the solar system can be considered as satellites of sun. An object in the solar system can have more than one satellite.

A communication satellite is a microwave repeater station in a space that is used for telecommunication, radio and television signals. A communication satellite process the data coming from one earth station and it converts the data into another form and send it to the second earth station.

HOW A SATELLITE WORKS

Two stations on earth want to communicate through radio broadcast but are too far away to use conventional means. The two stations can use a relay station for their communication.
One earth station transmits the signal to the satellite. Uplink frequency is the frequency at which ground station is communicating with satellite.
The satellite transponder converts the signal and sends it down to the second earth station, this is called Downlink.

1. The Coverage area is very high than that of terrestrial systems.
2. The transmission cost is independent of the coverage area.
3. Higher bandwidths are possible.

1. Launching satellites into orbits is a costly process.
2. The bandwidths are gradually used up.
3. High propagation delay for satellite systems than the conventional terrestrial systems.

PATH TRACED BY A SATELLITE

The path traced by a satellite around the earth is an elliptical path with two focuses.

How Do Satellites Work :

Satellite Communication Basics :

The process of satellite communication begins at an earth station. Here an installation designed to transmit and receive signals from a satellite in orbit around the earth. Earth stations send information to satellites in the form of high powered, high frequency (GHz range) signals. The satellites, which receive and retransmit the signals back to earth where they are received by other earth stations in the coverage area of the satellite. Satellite's footprint is the area which receives a signal of useful strength from the satellite. The transmission system from the earth station to the satellite through a channel is called the uplink. The system from the satellite to the earth station through the channel  is called the downlink. Below Figure shows the basic elements of a satellite communications system.

Satellite Frequency Bands:

The satellite frequency bands which was commonly used for communication are the C-band, Ku-band, and Ka-band. C-band and Ku-band are the commonly used frequency spectrums by today's satellites. It is important to note that there is an inverse relationship between frequency and wavelength i.e. when frequency increases, wavelength decreases this helps to  understand the relationship between antenna diameter and transmission frequency. Larger antennas (satellite dishes) are necessary to gather the signal with increasing wavelength.

From 4 to 8 GHz frequency range, C-band satellite transmissions occupy. Than the Ku-band or Ka-band, these relatively low frequencies translate to larger wavelengths. These larger wavelengths of the C-band mean that a larger satellite antenna is required to gather the minimum signal strength. Hence the minimum size of an average C-band antenna is approximately 2-3 meters in diameter. It is shown in Figure.

The frequency range from 11 to 17 GHz occupied by Ku-band satellite transmissions. These relatively high frequency transmissions correspond to shorter wavelengths. Thus a smaller antenna can be used to receive the minimum signal strength. Ku-band antennas can be as small as 18 inches in diameter. It can be commonly seen in the RCA DSS and Sony DSS systems. The Ku-band antenna of the Sony DSS system is shown in below figure.

The 20 to 30 GHz frequency range is occupied by the Ka-band satellite transmissions. These very high frequency transmissions means very small wavelengths and therefore very small diameter receiving antennas are used.

Geosynchronous Earth Orbit (GEO) Satellites

The majority of satellites in orbit around the earth are positioned at a point 22,238 miles above the earth's equator in a special type of geosynchronous earth orbit (GSO) known as Geostationary earth orbit (GEO). it is also called as the Clarke orbit. This is in honour of Arthur C. Clarke. He is the man who first suggested in 1945 that satellites in geosynchronous orbits could be used for communications purposes. A satellite can maintain an orbit with a period of rotation around the earth exactly equal to 24 hours at the precise distance of 22,238 miles. satellites appear stationary from the earth’s surface, since the they revolve at the same rotational speed of the earth. Due to this reason, most earth station antennas (satellite dishes) don't need to move once they have been properly aimed at a target satellite in the sky. The mathematical derivation of the Clarke orbit can be obtained as a straight-forward calculus problem.

The Clarke Orbit

Medium Earth Orbit (MEO) Satellites

The technological innovations in space communications during the last few years, have given rise to new orbits and totally new systems designs. Medium earth orbit (MEO) satellite networks will orbit at distances of about 8000 miles from earth's surface. Signals transmitted from a MEO satellite travel a shorter distance. This translates to improved signal strength at the receiving end. This shows that smaller, more lightweight receiving terminals can be used at the receiving end. Also, since the signal is travelling a shorter distance to and from the satellite. Hence there is less transmission delay. Transmission delay can be defined as the time it takes for a signal to travel up to a satellite and back down to a receiving station. For real-time communications, the shorter the transmission delay, better the communication system. As an example, a GEO satellite requires .25 seconds for a round trip. A MEO satellite requires less than .1 seconds to complete a round trip. MEOs operates in the frequency range of 2 GHz and above.

Low Earth Orbit (LEO) Satellites

The LEO satellites are mainly classified into three categories: little LEOs, big LEOs, and Mega-LEOs. LEOs will orbit at a distance of 500 to 1000 miles above the earth's surface. This relatively short distance reduces transmission delay to only .05 seconds. This further reduces the need for sensitive and bulky receiving equipment. Little LEOs will operate in the 800 MHz (.8 GHz) range. Big LEOs will operate in the 2 GHz or above range, and Mega-LEOs operates in the 20-30 GHz range. The higher frequencies associated with Mega-LEOs translates into more information carrying capacity and yields to the capability of real-time, low delay video transmission scheme. Microsoft Corporation and McCaw Cellular (now known as AT&T Wireless Services) have partnered to deploy 840 satellites to form Teledesic. It is a proposed Mega-LEO satellite network.

High Altitude Long Endurance (HALE) Platforms

Experimental HALE platforms are basically highly efficient and lightweight airplanes carrying communications equipments. This will act as very low earth orbit geosynchronous satellites. These crafts will be powered by a combination of battery and solar power or high efficiency turbine engines. HALE platforms will offer transmission delays of less than .001 seconds at an altitude of only 70,000 feet, and even better signal strength for very lightweight hand-held receiving devices.

Orbital Slots

Here there may arise a question that with more than 200 satellites up there in geosynchronous orbit, how do we keep them from running into each other or from attempting to use the same location in space?. To answer this problem, international regulatory bodies like the International Telecommunications Union (ITU) and national government organizations like the Federal Communications Commission (FCC) designate the locations on the geosynchronous orbit where the communications satellites can be located. These locations are specified in degrees of longitude and are called as orbital slots. The FCC and ITU have progressively reduced the required spacing down to only 2 degrees for C-band and Ku-band satellites due to the huge demand for orbital slots.