Digital Communications Archives

Amplitude Shift Keying (ASK) (or) On Off Keying (OOK) is the simplest Digital Modulation technique.

In this method, carrier amplitude is switched between two voltages ON and OFF levels depending up on the input binary sequence.

The carrier signal is a continuous wave (or) sinusoidal wave form

$S(t)=A \cos 2\pi f_{c}t$ .

The normalized power is $P=\frac{A^{2}}{2}$

$A=\sqrt{2P_{s}}$ .

The carrier signal can be expresses in terms of power as $S(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t$ .

if energy per bit is $E_{b}$ and the bit interval as $T_{b}$ then the carrier signal is $S(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t$ .

Now according to ASK Binary ‘1’ is represented with carrier voltage and Binary ‘0’ is represented with zero voltage.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$

in terms of Energy and bit duration ASK signal can be written as

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

ASK Transmitter:-

The figure shows the ASK generator (or) ASK Transmitter

It is a simple product Modulator, which modulates the incoming binary sequence (in the form of a signal) with the carrier signal S(t)

i.e, $S_{ASK}(t)=b(t).S(t)$

b(t) represents the binary sequence in the form of a signal.

when the input bit (or) symbol is Binary ‘1’ product Modulator passes the carrier signal and for Binary’0′, A zero output is given which blocks the carrier signal.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

Coherent ASK Detector:-

The figure shows the Block Diagram of coherent ASK/BASK Detector. The ASK signal $S_{ASK}(t)$ is applied to the correlator ( The Block product Modulator followed up by the Integrator).

$S_{ASK}(t)$ is multiplied by local carrier C(t) this carrier C(t) is phase locked with that of the carrier used in the Transmitter. As this is coherent reception.

The product $S_{ASK}(t).C(t)$ is applied to the Integrator. The Integrator integrates the input over one bit interval $T_{b}$ and the output is given to a threshold device. If the threshold voltage is set to 0 V.

the output of threshold device v(t) (or) v is either ‘1’ (or) ‘0’ based on the following condition.

$v\leq 0\rightarrow \ a \ symbol \ '1' \ is \ detected.$

$'0' \ is \ detected otherwise.$

Note:- The input to demodulator is not $S_{ASK}(t)$ always most of the times it is interfered with noise n(t) in the channel.

in coherent detection input to the demodulator is simply $S_{ASK}(t)$ signal where as in Non-coherent detection the input is noisy ASK signal.

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Synchronization methods

Synchronization Methods:-

For optimum demodulation of ASK, FSK and PSK waveforms, timing information is needed at the receiver. In particular, the integrate and dump operation in correlation receivers and the sampling operation in other types of receivers must be carefully controlled and synchronized with the incoming signal for optimum performance.

Three general methods are used for synchronization in digital modulation schemes. These methods are

Use of primary (or) Secondary time standard.
Utilization of a separate synchronization signal.
Extraction of clock information from the modulated waveform itself, referred to as self-synchronization.

In the first method, the receiver and transmitter are solved to a precise master timing source. This method is often used in large data communication networks. In point to point data communication this method is very seldom used because of its cost.

separate synchronization signals in the form of pilot tones are widely used to in point to point data communication systems. In this method, a special synchronization signal (or) sinusoidal signal of known frequency is transmitted along with the information carrying modulation wave form the synchronization signal is sent along with the modulation wave form using one of the following methods.

By frequency division Multiplexing, where in the frequency of the pilot tone is chosen to coincide with a null in the power spectral density of the signalling wave form.
By time division Multiplexing where the modulated wave form is interrupted for a short period of time during which the synchronizing signal is transmitted.
By additional modulation such as the one shown in figure.

In all of the above methods, the synchronization signal is isolated at the receiver and the zero crossings of the synchronization signal control the sampling operations at the Receiver.

All three methods discussed above and over head(or additional requirements) to the system in terms of an increase in power and Band Width requirements (or) a reduction in the data rate in addition to increasing the equipment complexity.

Self-synchronization methods extract a local carrier reference as well as timing information from the received wave forms. The Block Diagram of a system that derives a coherent local carrier from a PSK wave form is shown in figure(1) similar systems can be used to extract such a reference signal for other types of digital modulation schemes.

A feedback version of the squaring synchronizer is shown figure(2) . This version makes use of a PLL for extracting the correct phase and the frequency of the carrier wave form.

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Problem1-Cyclic Codes

pb 1. for a (7,4) Cyclic code if V={0101110} is the received code word . Then find the Syndrome if the generating polynomial is g(x) = x³ + x + 1

Ans. The syndrome polynomial S(x) can be find by dividing V(x) by g(x).

As Syndrome S(x) is zero, the received code word is a valid code word and is nothing but the transmitted code word.

Binary Cyclic Codes

Binary Cyclic Codes:-

Binary Cyclic Codes form a sub class of linear block codes.

Binary Cyclic Codes have two advantages

Encoding and syndrome calculations can be easily implemented using simple shift registers with feedback connections.
These codes have a fair amount of mathematical structure that makes it possible to design codes with useful error correcting properties.

we know that a linear block code can be represented in Matrix form. Here, we will develop a polynomial representation for cyclic codes and this representation is used to derive procedures for encoding and syndrome calculations.

A special sub classes of cyclic codes that are suitable for correcting burst type of errors in a system.

Algebraic structure of Cyclic Codes:-

An (n , k) linear block code C is also called as a cyclic code if it has the following property

if an n-tuple code vector of C is V then the n-tuple

Obtained by shifting V Cyclically one place to the right is also a code vector of ‘C’.

This we can illustrate by using an example

This Cyclical shifting property, allows us to treat the elements of a code word as the coefficients of a polynomial of degree(n-1).

for example is an n-bit code word can be represented as a polynomial of degree (n-1) as

where v_o , v₁ , v₂ , ……..v_n-1are elements of code word are represented as coefficients of polynomial.

The elements are binary 1’s and 0’s in general. Hence these codes are called Binary Cyclic Codes . Then binary addition and multiplication of these elements is as follows

if V is shifted i positions to the right then

we can obtain from Vⁱ(x) from V(x) as the remainder resulting xⁱ V(x) by dividing with xⁿ⁺¹ when

i.e,

Probability of error of Matched Filter

Probability of error of Matched Filter:-

we know that $P_{e}$ of an optimum filter is $P_{e} = \frac{1}{2}\ erfc(\frac{x_{o1}(t)-x_{o2}(t)}{2\sqrt{2}\sigma })$

Let us consider

now chose the ratio $\rho _{max}^{2} =(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}$

from the Matched filter concept $(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}=\frac{2}{N_{o}}\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df-----EQN(1)$

from Parsevel’s relation $\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df =\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt----EQN(2)$

from equations (1) and (2)

$\rho _{max}^{2}=\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt$

Here x(t) is the signal $x(t) =x_{1}(t)-x_{2}(t)$ .

$x(t) =\sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

over a duration of $[0, T_{b}]$ the symbols are transmitted

$\int_{0}^{T_{b}}\left | x(t) \right |^{2}dt=2P_{s}\int_{0}^{T_{b}}\cos^{2} 2\pi f_{c}t \ dt$ .

$=P_{s}T_{b}$ .

from equation(1) $\rho _{max}^{2} = \frac{2}{N_{o}} P_{s}T_{b}$ .

$\rho _{max} = \sqrt{\frac{2P_{s}T_{b}}{N_{o}}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{2P_{s}T_{b}}{N_{o}}}).\frac{1}{2\sqrt{2}})$

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{P_{s}T_{b}}{4N_{o}}})$ .

we know that carrier signal power is $P_{s} = \frac{A^{2}}{2}$ .

$A=\sqrt{2P_{s}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{A^{2}T_{b}}{8N_{o}}})$ .

$\therefore P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ . since $A^{2}T_{b} = P_{s}T_{b} =E_{b}$ .

probability of error of coherent ASK is $P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ (or) in terms of Q function as $P_{e}= Q(\sqrt{\frac{E_{b}}{2N_{o}}})$ .

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Optimum filter

The function of a receiver in a binary Communication system is to distinguish between two transmitted signals $x_{1}(t)\ and \ x_{2}(t)$ (or) ( $s_{1}(t)\ and \ s_{2}(t)$ ) in the presence of noise.

The performance of Receiver is usually measured in terms of the probability of error P_e an the receiver is said to be optimum if it yields the minimum probability of error.

i.e, optimum receiver is the one with minimum probability of error P_e .

optimum receiver takes the form of Matched filter when the noise at the receiver input is white noise.

optimum receiver (or) optimum filter:-

The block diagram of optimum receiver is as shown in the figure below

the decision boundary is set to $\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

Probability of error of optimum filter:-

The probability of error can be obtained as similar to Integrate and dump receiver. Here we will consider noise as Gaussian Noise.

The output of optimum filter is $y(t) = x_{o1}(t)+n_{o}(t)$ .

The output of sampler is $y(T) = \left\{\begin{matrix} x_{o1}(T)+n_{o}(T) \ for \ binary \ i/p \ '1'\\ x_{o2}(T)+n_{o}(T) \ for \ binary \ i/p \ '0' \end{matrix}\right.$

suppose if Binary ‘1’ is transmitted then the input is $x(t) = x_{1}(t)$ , to find the probability of error this transmitted ‘1’ should be received as ‘0’.

this is possible when the condition $\left | y(T) \right | <\frac{x_{o1}(T)+x_{o2}(T)}{2}$ is true.

1 will be received as 0 $\Rightarrow x_{o1}(T)+n_{o}(T) <\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T) <\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

similarly a Binary ‘0’ will be received as ‘1’ if and only if

$\Rightarrow x_{o2}(T)+n_{o}(T) >\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T) >\frac{x_{o1}(T)-x_{o2}(T)}{2}$ .

the conditions are summarized in the table

Noe the Probability Distribution Function of Gaussian noise with zero mean and standard deviation $\sigma$ is given by

$f(n_{o}(T)) = \frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{n_{o}^{2}(T)}{2}}$ .

Probability of error= probability ‘1’ will be received as ‘0’ =probability ‘0’ will be received as ‘1’.

$\therefore P_{e} =$ area under the curve $n_{o}(T) >\frac{x_{o1}(T)-x_{o2}(T)}{2}$ (or) area under the curve $n_{o}(T) <\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

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Amplitude Shift Keying(ASK)

Amplitude Shift Keying (ASK) (or) On Off Keying (OOK) is the simplest Digital Modulation technique.

In this method, carrier amplitude is switched between two voltages ON and OFF levels depending up on the input binary sequence.

The carrier signal is a continuous wave (or) sinusoidal wave form

$S(t)=A \cos 2\pi f_{c}t$ .

The normalized power is $P=\frac{A^{2}}{2}$

$A=\sqrt{2P_{s}}$ .

The carrier signal can be expresses in terms of power as $S(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t$ .

if energy per bit is $E_{b}$ and the bit interval as $T_{b}$ then the carrier signal is $S(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t$ .

Now according to ASK Binary ‘1’ is represented with carrier voltage and Binary ‘0’ is represented with zero voltage.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$

in terms of Energy and bit duration ASK signal can be written as

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

ASK Transmitter:-

The figure shows the ASK generator (or) ASK Transmitter

It is a simple product Modulator, which modulates the incoming binary sequence (in the form of a signal) with the carrier signal S(t)

i.e, $S_{ASK}(t)=b(t).S(t)$

b(t) represents the binary sequence in the form of a signal.

when the input bit (or) symbol is Binary ‘1’ product Modulator passes the carrier signal and for Binary’0′, A zero output is given which blocks the carrier signal.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

Coherent ASK Detector:-

The figure shows the Block Diagram of coherent ASK/BASK Detector. The ASK signal $S_{ASK}(t)$ is applied to the correlator ( The Block product Modulator followed up by the Integrator).

$S_{ASK}(t)$ is multiplied by local carrier C(t) this carrier C(t) is phase locked with that of the carrier used in the Transmitter. As this is coherent reception.

the output of threshold device v(t) (or) v is either ‘1’ (or) ‘0’ based on the following condition.

$v\leq 0\rightarrow \ a \ symbol \ '1' \ is \ detected.$

Note:- The input to demodulator is not $S_{ASK}(t)$ always most of the times it is interfered with noise n(t) in the channel.

in coherent detection input to the demodulator is simply $S_{ASK}(t)$ signal where as in Non-coherent detection the input is noisy ASK signal.

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Burst and Random error correcting codes

Burst and Random error correcting codes:-

To correct specific, well-defined classes of error patterns a variety of codes are designed but the problems of correcting random errors and burst errors are treated separately.

But in practical systems errors occur neither independently nor in well-defined bursts. Errors may occur in a fashion, that as a mixture of random and burst errors.

therefore either random error correcting codes (or) single-burst error correcting codes are insufficient and inefficient to correct errors which involves as a mixture of both random and burst errors.

So for channels in which both types of errors occur, there is a need for special types of codes, which will correct both random and burst errors simultaneously.

The most effective method uses the interlacing technique.

Interlace code:-

for an (n, k) cyclic code a $(\lambda n,\lambda k)$ interlaced code can be constructed by simply arranging $\lambda$ code vectors of the original code into rows of a rectangular array and transmitting them by column by column the resulting code is called interlaced code with an interlacing degree $\lambda$ .

In an interlaced code, a burst of length $\lambda$ (or) less will affect not more than one digit in each row .

If the original code can correct single errors, then the interlaced code can correct single bursts of length $\lambda$ (or) less.

If the original code can correct‘t’ errors (t>1) then the interlaced code can correct any combination of t bursts of length $\lambda$ (or) less.

consider for example a (15,7) BCH code is generated by $g(x)=x^{8}+x^{4}+x^{2}+x+1$ which will have $d_{min}=5$ (minimum distance) it is able to correct $\leq \left \Rightarrow \leq 2$ .

it may correct 2 errors. so for this code we can construct a (75, 35) interlaced code with as (75, 35) with a burst error correcting capability of 10.

Now the message block length is 35 bits, this 35 bit message block is divided into 5 ,’7’ bit blocks as

and each 7 bit message block is converted into a 15 bit code word by using $g(x)$ .

These code words are arranged as five rows of a 5 X 15 matrix. The column of the matrix is transmitted in the order indicated as a 75 bit long code vector.

Error correcting capabilities:-

To illustrate the burst and random error correcting capabilities of this code.

assume that errors havee occures in bit positions 5,37 through 43 and 69.

at the decoder, the decoder operates on the rows of the table that each row has a maximum of 2 errors and from (15,7) BCCH code , we know that the code is able to correct up to ‘2’ errors per row.

therefore the error pattern occurred in the table can be corrected. The errors in bit positions 5 and 69 as random errors and from 37 to 43 as burst error.

while operating on the rows of the code array may be an obvious way to encode and decode an interlaced code which is not the simplest implementation.

The simplest implementation results from the property that if the original code is cyclic, then the interlaced code is also cyclic.

The polynomial in interlaced code is $g(x^{\lambda })$ if the original polynomial is $g(x)$ .

Thus encoding and decoding can be accomplished by using shift registers. The decoder for the interlaced code can be derived from the decoder for original code by replacing each shift register stage of the original decoder by $\lambda$ stages without changing the other connections.

each shift register stage by $\lambda$ stages without changing other connections. This allows the decoder to look at successive rows of the code array on successive decoder cycles.

If the decoder for the original code was simple then the decoder for interlaced code will also be simple.

Therefore The interlacing technique is an effective tool for deriving long powerful codes from short optimal codes.

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QPSK equation, wave forms and Signal space diagram

QPSK equation:-

The meaning of QPSK is that the carrier signal takes on different phases Π/4, 3Π/4, 5Π/4 and 7Π/4 based on incoming di-bit combination or symbol.

$\large S_{QPSK}(t)= \sqrt{\frac{2E_{s}}{T_{s}}}cos(2\pi f_{c}t +(2i-1)\frac{\pi}{4}), 0\leq t\leq T_{s}$

= 0, elsewhere, where i = 1,2,3,4.

E_b and T_b are the bit energy and bit-interval , E_s and T_s are the energy per symbol and symbol duration. T_s= 2 T_b .

The carrier frequency f_c = n_c /T_s. where n_c is a fixed integer.

each possible value of phase corresponds to a unique di-bit. then the foregoing phase values to represent the gray encoded set of di-bits 11,01,10 and 00, where only a single bit is changed from one di-bit to the next.

QPSK equation can be represented in another format as follows

$\large S_{QPSK}(t) = \sqrt{\frac{2E_{s}}{T_{s}}}cos (2\pi f_{c}t+(2i+1)\frac{\pi }{4} ), 0\leq t\leq T_{s}$

= 0, elsewhere ,where i=0,1,2,3.

The above two equations are same, there is a change in i values. alternately the equation can be represented as follows.

$S_{QPSK}(t)= \sqrt{\frac{2E_{s}}{T_{s}}}cos (2i-1)\frac{\pi }{4}cos2\pi f_{c}t - \sqrt{\frac{2E_{s}}{T_{s}}}sin (2i-1)\frac{\pi }{4}sin2\pi f_{c}t$ , where i= 1,2,3,4.

The above equation can be expanded cos(A+B). There are two orthogonal functions Φ₁(t) and Φ₂(t) where

$\Phi _{1}(t)=\sqrt{\frac{2}{T_{s}}}cos 2\pi f_{c}t, 0\leq t\leq T_{s}, \Phi _{2}(t)=\sqrt{\frac{2}{T_{s}}}sin 2\pi f_{c}t, 0\leq t\leq T_{s}$

$S_{QPSK}(t)=\sqrt{E_{s}}cos(2i-1)\frac{\pi }{4} * \Phi _{1}(t) - \sqrt{E_{s}}sin(2i-1)\frac{\pi }{4} * \Phi _{2}(t)$

Let $b_{o}(t)= \sqrt{E_{s}}cos(2i-1)\frac{\pi }{4}$ and $b_{e}(t)= -\sqrt{E_{s}}sin(2i-1)\frac{\pi }{4}$

then the resultant equation is: $S_{QPSK}(t)= b_{o}(t) * \Phi _{1}(t) + b_{e}(t) * \Phi _{2}(t)$ .

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Drawbacks in Delta Modulation (or) errors in Delta Modulation

Adaptive Delta Modulation(ADM)

Adaptive Delta Modulation, a modification of Linear Delta Modulation (LDM) is a scheme that circumvents the deficiency of DM. In ADM step size Δ of the Quantizer is not a constant but varies with time , we shall express Δ as $\Delta (n)= 2 \delta (n)$ .

where $\delta (n)$ increases during a steep segment of input and decreases for a slowly varying segment of input.

The adaptive step size control which forms the basis of an ADM scheme can be classified in various ways such as

Discrete or Continuous.
instantaneous (or) syllabic (fairly gradual change).
forward (or) backward.

we shall describe an adaptation scheme that is backward, instantaneous and discrete in practical implementation , the step size $\delta (n)$ is constrained in between some pre-determined minimum and maximum values.

$\delta _{min} \leq \delta (n)\leq \delta _{max}$

The upper limit $\delta _{max}$ controls the amount of Slope over load distortion and the lower limit $\delta _{min}$ controls the granular noise (or) Idle noise.

The adaptive rule for $\delta (n)$ can be expressed in the general form $\delta (n) = g(n) \delta (n-1)$

where the time varying gain g(n) depends on the present binary output b(n) and M previous values b(n-1),b(n-2) ……….b(n-M). The algorithm is usually initiated with $\delta _{min}$ .

when M=1, b(n) and b(n-1) are compared to detect probable slope over load {b(n) = b(n-1)} (or) probable granularity {b(n) ≠ b(n-1)} then g(n) is

$g(n) = P$ if $b(n) =b(n-1)$ .
$g(n)=\frac{1}{P}$ if $b(n) \neq b(n-1)$ .

when $b(n) =b(n-1)$ Slope overload distortion is detected and when $b(n) \neq b(n-1)$ Idle noise is detected.

where $P\geq 1$ , note that $P=1$ represents LDM. $P_{optimum}=1.5$ minimizes the Quantization noise for speech signal, where $1< P< 2$ is for broad class of signals.

Adaptive Delta Modulation System:-

ADM Transmitter:-

The figure shows ADM Transmitter and ADM Receiver which includes the logic for step size control both in Transmitter and receiver.

The logic for step size control is added in the diagram the step size increases (or) decreases according to specified rule depending up on one bit Quantizer output.

for example if one bit Quantizer output is high then the step size may be doubled for next sample.

if one bit Quantizer output is low, the step size may be reduced by one step.

ADM Receiver:-

In the Receiver of Adaptive Delta Modulator there are two parts logic for step size and accumulator.

The first block produces the step size from each incoming bit, which uses the same principle used in the Transmitter.

The previous input and present input decides the step size. It is then applied to an accumulator, which builds up stair case wave form. The LPF then smoothens out the stair case wave form to reconstruct the original signal.

Advantages of ADM:-

ADM has certain advantages over DM

The signal to Noise Ratio becomes better than ordinary Delta modulation because of the reduction in slope over load distortion and Idle noise.
Because of the variable step size the dynamic range of ADM is wider than simple DM.
Utilization of Band Width is better than DM.

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Pulse Code Modulation(PCM)

To describe Pulse Code Modulation (PCM) the idea about Sampling and Quantization are required in hand.

PCM is the most basic form of digital pulse modulation

PCM :- In PCM a message signal is represented by a sequence of coded pulses , which is accomplished by representing the signal in discrete form in both time and Amplitude.

PCM Block Diagram:-

The Block Diagram of a PCM system is given in the following figure. This Block Diagram consists of the basic elements of a PCM system.

Transmitter.
Transmission path.
Receiver.

PCM Transmitter:-

the basic operations performed in the transmitter are Sampling, Quantization and Coding -Analog to Digital Conversion(ADC).

LPF:- The Low Pass Filter to sampler is inclined to prevent aliasing of the message signal.

Sampler:- Converts the analog band limited signal to discrete signal.

ADC:- It’s a device which performs two operations known as Quantization and Encoding.

Transmission path:-

Regenerative repeaters:- PCM encoded output is transmitted through the transmission path to the receiver, during transmission the PCM output get distorted in the channel by noise which can be regenerated by the device called regenerative repeater.

PCM Receiver:-

The essential operations in the receiver are decoding, regeneration of impaired PCM output .

the device which performs decoding and demodulation at the receiver is known as (DAC) Digital-to-Analog Converter.

Few points about PCM:-

we know that PAM, PWM and PPM are analog pulse modulation techniques, where as PCM is a pure Digital Modulation scheme.
It should be understood that the PCM is not a modulation technique in the conventional sense.
The output of PCM is in the coded digital form that is in the form of code words. A PCM system consists of PCM encoder as well as decoder.

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Differential Pulse Code Modulation (DPCM)

Differential Pulse Code Modulation (DPCM):-

When a voice signal (or) video signal is sampled at a rate higher than Nyquist rate , which is usually done in PCM.

The resulting sampled signal is found to exhibit a high degree of correlation between adjacent samples.

The meaning of this high correlation is that the signal does not change rapidly from one sample to the next.

As a result ,

The variance of adjacent samples < variance of the signal x(t) (or) m(t).

When these correlated samples are encoded, as in the standard PCM system, the resulting encoded signal contains redundant information.

As a result of encoding process, redundant samples that are not essential in the transmission of information are generated.

By removing the redundancy before encoding, we obtain a more efficient coded signal which is the basic idea behind DPCM.

If we know the past behavior of a signal x(t) up to certain point of time, we may use prediction to make an estimate of a future value of the signal.

The figure shows the DPCM transmitter

The fact that motivates the scheme DPCM is the possibility of prediction of future value of the signal x(t).

i.e, the DPCM works on the principle of prediction the value of the present sample is predicted from the past samples the prediction may not be exact but it is very close to that actual sample value.

Equations in DPCM:-

$e[n]=x[n]-\widehat{x[n]}$ .

Where $x[n]$ – is the input sample.

$\widehat{x[n]}$ -Prediction of input sample.

$e[n]$ – is the difference between un Quantized input sample and prediction of it.

$\widehat{x[n]}$ – The predicted value is produced by using a linear prediction filter the input to the prediction filter is a quantized version of input sample $x_{q}[n]$ .

$\therefore$ The difference signal $e[n]$ is the prediction error.

Prediction error:- it is the amount by which the prediction filter fails to predict the input exactly.

The Quantizer output may be expressed as $e_{q}[n] = e[n]+q[n]$ .

$e_{q}[n]$ – Quantized version of $e[n]$

$e[n]$ – is the prediction error

$q[n]$ – Quantization error.

$x_{q}[n]=e_{q}[n]+\widehat{x[n]}$ .

$e_{q}[n]$ – Output of Quantizer.

$\widehat{x[n]}$ – Prediction of $x[n]$ .

form equation(2)

$x_{q}[n]=\widehat{x[n]}+e[n]+q[n]$

$\therefore \widehat{x[n]} +e[n] =x[n]$ from equation (1)

$\therefore x_{q}[n]=x[n]+q[n]$ .

By encoding the Quantizer output, we obtain a variant of PCM known as differential pulse code Modulation (DPCM).

DPCM Receiver :-

The figure shows the block diagram of DPCM receiver

The decoder first reconstructs the Quantized error signal from incoming binary signal.

The prediction filter output and the quantized error signals are summed up to give the quantized version of the original signal.

Same prediction filter is used at the receiver as that of transmitter.

Thus the signal at the receiver differ from actual signal $x[n]$ by Quantization error $q[n]$ , which is introduced permanently in the reconstructed signal.

Comparison of DPCM with DM and PCM:-

DPCM includes M as a special case.

DPCM and DM are basically similar except for two important differences.

DM is the ‘1’ bit version of DPCM.

Note that unlike a standard PCM system, the transmitters of both the DPCM and DM involve the use of feedback.

Like DM , in DPCM also slope overload distortion exists whenever input signal changes too rapidly for the prediction filter to track it.

Like PCM,DPCM suffers Quantization error.

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Phase Shift Keying (PSK)

It is the most efficient of the 3 digital modulation techniques ASK,PSK and FSK. It is used for high bit rates.

Definition:-

In BPSK, the Phase of the carrier is shifted/ Changed according to the incoming Binary data sequence.

BPSK was developed during the early days of the deep space programs. PSK is now widely used in both Military and commercial communication systems.

we represent data sequence $\left \{ b_{k} \right \}$ in Bipolar-NRZ scheme.

Expressions of PSK:-

The carrier signal is a continuous wave (or) sinusoidal wave form

$S(t)=A \cos 2\pi f_{c}t$ .

The normalized power is $P=\frac{A^{2}}{2}$

$A=\sqrt{2P_{s}}$ .

The carrier signal can be expresses in terms of power as $S(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t$ .

if energy per bit is $E_{b}$ and the bit interval as $T_{b}$ then the carrier signal is $S(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t$ .

Now according to PSK Binary ‘1’ is represented with carrier phase $0^{o}$ and Binary ‘0’ is represented with a phase $180^{o}$ .

$\left\{\begin{matrix} S_{PSK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =\sqrt{2P_{s}} \cos (2\pi f_{c}t +\pi )\ \rightarrow \ Binary\ '0' \end{matrix}\right.$

in terms of Energy and bit duration ASK signal can be written as

$\left\{\begin{matrix} S_{PSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ \ \ =\sqrt{\frac{2E_{b}}{T_{b}}} \cos (2\pi f_{c}t+\pi )\ \rightarrow \ Binary\ '0' \end{matrix}\right.$

i.e,

$\left\{\begin{matrix} S_{PSK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ \ \ \ =- \sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

The wave forms of PSK modulation scheme are shown in the figure , where b(t) represents the polar NRZ representation of binary sequence $\left \{ b_{k} \right \}$

i.e, $b(t) = \left\{\begin{matrix} \ +v \ (or) \ 1\ volt \ when \left \{ b_{k} \right \}=1\\ \ -v \ (or) \ -1\ volt \ when \left \{ b_{k} \right \}=0 \end{matrix}\right.$ .

PSK Transmitter:-

The figure shows the PSK generator (or) PSK Transmitter

The incoming binary sequence $\left \{ b_{k} \right \}$ (in the form of a signal) into $\pm 1 \ volt$ by using NRZ level encoder.

$b(t)= \left\{\begin{matrix} +1V \ for \ symbol \1\\ -1V \ for \ symbol \0 \end{matrix}\right.$

Now b(t) and the carrier S(t) are applied to product modulator, to get the PSK modulated signal at the output.

i.e, $S_{PSK}(t)=b(t).S(t)$

finally,

Coherent PSK Detector:-

The figure shows the Block Diagram of coherent PSK/BPSK Detector. The PSK signal $S_{PSK}(t)$ is applied to the correlator (The Block product Modulator followed up by the Integrator).

$S_{PSK}(t)$ is multiplied by local carrier C(t) this carrier C(t) is phase locked with that of the carrier used in the Transmitter. As this is coherent reception.

The product $S_{PSK}(t).C(t)$ is applied to the Integrator. The integrator eliminates the noise.

The Integrator integrates the input over one bit interval $T_{b}$ and the output is given to a threshold device. If the threshold voltage is set to 0 V.

the output of threshold device v(t) (or) v is either ‘1’ (or) ‘0’ based on the following condition.

$v\leq 0\rightarrow \ a \ symbol \ '1' \ is \ detected.$

Note:- The input to demodulator is not $S_{PSK}(t)$ always most of the times it is interfered with noise n(t) in the channel.

in coherent detection input to the demodulator is simply $S_{PSK}(t)$ signal where as in Non-coherent detection the input is noisy PSK signal.

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Mutual Information I(X ; Y) – Properties

Property 1:- Mutual Information is Non-Negative

Mutual Information is given by equation $I(X ; Y) =\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2} \frac{P(\frac{x_{i}}{y_{j}})}{P(x_{i})}---------Equation(I)$

we know that $P(\frac{x_{i}}{y_{j}})=\frac{P(x_{i}, y_{j})}{P(y_{j})}-------Equation(II)$

Substitute Equation (II) in Equation (I)

$I(X ; Y) =\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2}\frac{P(x_{i}, y_{j})}{P(x_{i})P(y_{j})}$

The above Equation can be written as

$I(X ; Y) =-\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2}\frac{P(x_{i})P(y_{j})}{P(x_{i}, y_{j})}$

$-I(X ; Y) =\sum_{i=1}^{m}\sum_{j=1}^{n}P(x_{i}, y_{j})\log _{2}\frac{P(x_{i})P(y_{j})}{P(x_{i}, y_{j})}------Equation(III)$

we knew that $\sum_{k=1}^{m} p_{k}\log _{2}(\frac{q_{k}}{p_{k}})\leq 0---Equation(IV)$

This result can be applied to Mutual Information $I(X ; Y)$ , If $p_{k} = P(x_{i}, y_{j})$ and $q_{k}$ be $P(x_{i}) P( y_{j})$ , Both $p_{k}$ and $q_{k}$ are two probability distributions on same alphabet , then Equation (III) becomes

$-I(X ; Y) \leq 0$

i.e, $I(X ; Y) \geq 0$ , Which implies that Mutual Information is always Non-negative (Positive).

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Correlation Receiver (Special case of Optimum Receiver)

we know that one form of Optimum filter is Matched filter, we will now derive another form of Optimum filter that is different from Matched filter Let the input to the Optimum filter is $v(t)$ which is a noisy input that is

$v(t)=x(t)+n(t)$

from the figure output of the filter after sampling at $t=T_{b}$ seconds is $v_{o}(T_{b})$

$v_{o}(t)=v(t)*h(t)$

$v_{o}(t) = \int_{-\infty }^{T_{b}}v(\tau ) h(t-\tau )d\tau$

at $t= T_{b}$ output becomes $v_{o}(T_{b}) = \int_{-\infty }^{T_{b}}v(\tau ) h(T_{b}-\tau )d\tau -----Equation(1)$

Now by substituting $h(\tau )=x_{2}(T_{b}-\tau )-x_{1}(T_{b}-\tau )$

$h(T_{b}-\tau )=x_{2}(T_{b}-T_{b}+\tau )-x_{1}(T_{b}-T_{b}+\tau )$

$h(T_{b}-\tau )=x_{2}(\tau )-x_{1}(\tau )------Equation(2)$

by substituting the Equation(2) in Equation(1) over the limits $[0,T_{b}]$

$v_{o}(T_{b}) = \int_{0 }^{T_{b}}v(\tau ) (x_{2}(\tau )-x_{1}(\tau ))d\tau$

$v_{o}(T_{b}) = \int_{0 }^{T_{b}}v(\tau ) x_{2}(\tau ) d\tau-\int_{0}^{T_{b}}v(\tau )x_{1}(\tau )d\tau$

Now by replacing $\tau$ with $t$ the above equation becomes

$v_{o}(T_{b}) = \int_{0 }^{T_{b}}v(t ) x_{2}(t ) dt-\int_{0}^{T_{b}}v(t )x_{1}(t )dt-----Equation(3)$

The equation (3) suggests that the Optimum Receiver can be implemented as shown in the figure, this form of the Receiver is called as correlation Receiver. This receiver requires the integration operation be ideal with zero initial conditions. Correlation Receiver performs coherent-detection.

in general Correlation Receiver can be approximated with Integrate and dump filter.

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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 $H(f)=\frac{k X^{*}(f)e^{-j2\pi fT}}{S_{ni}(f)}$

if input noise is white noise , its Power spectral density (Psd) is $S_{ni}(f)=\frac{N_{o}}{2}$ .

then H(f) becomes $H(f)=\frac{k X^{*}(f)e^{-j2\pi fT}}{\frac{N_{o}}{2}}$

$H(f)=\frac{2k}{N_{o}}X^{*}(f)e^{-j2\pi fT}-----Equation(I)$

From the properties of Fourier Transforms , by Conjugate Symmetry property $X^{*}(f) = X(-f)$

Equation (I) becomes

$H(f)=\frac{2k}{N_{o}}X(-f)e^{-j2\pi fT}------Equation(II)$

From Time-shifting property of Fourier Transforms

$x(t)\leftrightarrow X(f)$

From Time-Reversal Property $x(-t)\leftrightarrow X(-f)$

By Shifting the signal $x(-t)$ by T Seconds in positive direction(time) ,the Fourier Transform is given by $x(T-t)\leftrightarrow X(-f)e^{-j2\pi ft}$

Now the inverse Fourier Transform of the signal from the Equation(II) is

$F^{-1}=F^{-1}$

$h(t)=\frac{2k}{N_{o}}x(T-t)$

Let the constant $\frac{2k}{N_{o}}$ is set to 1, then the impulse response of Matched Filter will become $h(t) = x(T-t)$ .

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Condition required to eliminate Slope Overload Distortion in Delta Modulation

As we have seen that Slope Overload distortion in Delta Modulation if slope of the input signal is too high.

To eliminate Slope Overload error we require a condition that must be satisfied is

slope of the Delta Modulator $\geq$ Maximum slope of input signal x(t).

i.e, $\frac{\Delta }{T_{s}}\geq max value of \left | \frac{dx(t)}{dt} \right |$ .

where $\frac{\Delta }{T_{s}}$ is the slope of approximated signal $\widehat{x(t)}$ .

slope = $\frac{\Delta }{T_{s}}$ .

when x(t) is a single-tone:-

if x(t) is a single-tone then $x(t) = A_{m}\sin 2\pi f_{m}t$

now slope of $x(t)\ is \ \frac{dx(t)}{dt}= 2 \pi f_{m}\ A_{m}\cos 2\pi f_{m}t$ .

$max\ value \ of \ \frac{dx(t)}{dt}= 2 \pi f_{m}\ A_{m}$ .

$\therefore$ to eliminate slope overload error in the single-tone case

$\frac{\Delta }{T_{s}}\geq max value of \left | \frac{dx(t)}{dt} \right |$ .

$\frac{\Delta }{T_{s}}\geq 2 \pi f_{m}\ A_{m}$

Non uniform Quantization in PCM

Quantization has been divided into two types

Uniform Quantization.
Non-Uniform Quantization.

Uniform Quantization

Non-Uniform Quantization

$\Delta$ is constant through out Quantization process.

$\Delta$ is variable that is the step size is variable.

characteristics of the Quantizer are linear.

ex:- Mid-tread type Quantizer. Mid-rise type Quantizer.

characteristics of the Quantizer are Non-linear.

we know that the sampled version of message signal is Quantized. The Quantization process may follow a Quantization law , which is uniform in general.

But in tele – communication applications like PCM telephony however it is preferable to use a variable separation between the representation levels.

i.e, Non-Uniform Quantization law is to be followed.

for example the range of voltages covered by voice signals from the peaks of loud talk to the weak passages of weak talk is on the order of 1000 to 1.

weak passages (weak talk) which need more protection can be Quantized with smaller step size.

and loud talk will be Quantized with large step size.

Need for Non-Uniform Quantization for speech signal (or) Need for companding in PCM:-

as said above weak talk can be protected more than that of loud talk of a speech signal.

Since at the receiving end of a telephone system a loud talk can be received with tolerable reduced voltage but a weak talk can not be received if the voltage level is reduced during Quantization process.

we know that speech and music signals are characterized by large crest factor.

for a signal with large crest factor, the signal to noise ratio is given by

$SQR=\frac{P}{N_{q}}=\frac{Signal Power}{Noise Power}$ .

if we keep $\Delta$ as constant that is Uniform Quantization is applied then noise power $N_{q} = \frac{\Delta ^{2}}{12}$ is almost constant.

$SQR\propto P$ .

SQR is proportional to signal power .

if signal power is low then SQR also becomes low which is inevitable at the receiver.

if signal power is high then SQR also becomes high which is desirable at the receiver.

to maintain constant SQR at the receiving end companding is preferred in a PCM system (or) for the transmission of voice / speech / Music signals.

i.e, for low power levels of input signal $\Delta$ is reduced

$\therefore N_{q} =\frac{\Delta ^{2}}{12}$ becomes less and $\because \frac{S}{N_{q}} =\frac{P}{N_{q}}$ is improved.

similarly for high power level $\Delta$ is increased ,

P is more Noise power increases and SQR is reduced.

In this way , SQR is maintained uniform throughout the Quantization process.

This Uniform Quantization can be achieved through companding.

companding in PCM :-

is required to improve SQR of weak signals.
is also known as Non-Uniform Quantization.

In practical it is difficult to implement the Non-Uniform Quantization because it is not known in advance hoe the signal level is varying?

we follow a particular method as mentioned below

Before the application of the signal to a Uniform Quantizer in the Transmitter weak signals are being amplified and strong signals are being attenuated.
then uniform Quantization is used on the modified signal.
At the receiver a reverse process is to be done.

in general compander is a combination of two devices a compressor(at the Transmitter) and an expander (at the receiver).

the use of Non-Uniform Quantizer is equivalent to passing the base band signal through a compressor passing through a compressor and then applying the compressed signal to a uniform Quantizer.

a particular compression law is used in practice the so called -law and A-law .

to restore the signal samples to their correct relative values we must use a device in the receiver with a characteristic complementary to the compressor such a device is called as an expander.

ideally, the compression and expansion laws are exactly inverse.

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Compression laws (A-law and u-law)

The laws used in compressor of a non-uniform Quantizer are known as compression laws.

two laws are available

$\mu$ – law.
$A$ -law.

$\mu$ -law:- A particular form of compression law that is used in practice is the so called $\mu$ – law which is defined by

$\left | v \right |= sgn(x){\frac{\ln (1+\mu \left | x \right |)}{(1+\mu )}}, for\ 0\leq \left | x \right |\leq 1$ .

where $\left | v \right |$ – Normalized compressed output voltage.

$\left | x \right |$ – Normalized input voltage to the compressor.

and $\mu$ is a positive constant and its value decides the curvature of $\left | v \right |$ .

$\mu =0$ corresponds to no compression, which is the case of uniform quanization, the curve is almost linear as the value of $\mu$ increases signal compression increases.

$\mu =255$ is the north american standard for PCM voice telephony.

for a given value of $\mu$ , the reciprocal slope of the compression curve, which defines the quantum steps is given by the derivative of $\left | x \right |$ with respect to $\left | v \right |$

$\frac{dx}{dv} = \frac{\ln (1+\mu )}{\mu } (1+\mu \left | x \right |)$

we see that therefore $\mu$ law is neither strictly linear nor strictly logarithmic.

i.e,

It is approximately linear at low levels of input corresponding to $\mu \left | x \right |<<1$ . and is approximately logarithmic at high input levels corresponding to .

typical value of $\mu =255$

A-law :- another compression law that is used in practice is the so called A- law defined by

$\left | v \right | =\left\{\begin{matrix} sgn(x) (\frac{A\left | x \right |}{1+\ln A}), 0\leq \left | x \right |\leq \frac{1}{A}.\\ sgn(x) (\frac{1+ \ln A\left | x \right |}{1+\ln A}),\frac{1}{A} \leq \left | x \right |\leq 1. \end{matrix}\right.$

A=1 corresponds to the case of uniform Quantization. A-law has been plotted for three different values of A. A=1, A=2, A=87.56.

typical value of A is 87.56 in European Commercial PCM standard which is being followed in India.

for both the $\mu$ -law and A-law, the dynamic range capability of the compander improves with increasing $\mu$ and A respectively. The SNR for low-level signals increases at the expense of the SNR for high level signals.

to accommodate these two conflicting requirements (i.e, a reasonable SNR for both low and high-level signals), a compromise is usually made in choosing the value of parameter $\mu$ for the $\mu$ -law and parameter A for the A-law. The typical values used in practice are $\mu=255$ and $A=87.56$ .

It is also interested to note that in actual PCM systems, the companding circuitry does not produce an exact replica of the non-linear compression curves shown in the compression law characteristics.

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Probability of error of PSK

Probability of error of PSK using coherent Detection:-

The equation of PSK from it’s basic definition

$S_{ASK}(t)=\left\{\begin{matrix} +\sqrt{2P_{s}}\cos 2\pi f_{c}t\ \rightarrow \ symbol\ '1'\\ -\sqrt{2P_{s}}\cos 2\pi f_{c}t\ \rightarrow \ symbol\ '0' \end{matrix}\right.$ .

The inputs to the optimum filter are $x_{1}(t) \ and \ x_{2}(t)$ when the symbols 1 and 0 are being transmitted at the transmitter.

$\therefore x_{1}(t) = \sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

$x_{2}(t)=-\sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

we know that $P_{e}$ of an optimum filter is $P_{e} = \frac{1}{2}\ erfc(\frac{x_{o1}(t)-x_{o2}(t)}{2\sqrt{2}\sigma })$

now chose the ratio $\rho _{max}^{2} =(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}$

from the Matched filter concept $(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}=\frac{2}{N_{o}}\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df-----EQN(1)$

from Parsevel’s relation $\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df =\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt----EQN(2)$

from equations (1) and (2)

$\rho _{max}^{2}=\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt$

Here x(t) is the signal $x(t) =x_{1}(t)-x_{2}(t)$ .

$x(t) =2\sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

over a duration of $[0, T_{b}]$ the symbols are transmitted

$\int_{0}^{T_{b}}\left | x(t) \right |^{2}dt=8P_{s}\int_{0}^{T_{b}}\cos^{2} 2\pi f_{c}t \ dt$ .

$=4P_{s}T_{b} =4E_{b}$ .

from equation(1) $\rho _{max}^{2} = \frac{2}{N_{o}} 4 P_{s}T_{b}$ .

$\rho _{max} = \sqrt{\frac{8P_{s}T_{b}}{N_{o}}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc (\sqrt{\frac{8P_{s}T_{b}}{N_{o}}}).(\frac{1}{2\sqrt{2}})$

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{P_{s}T_{b}}{N_{o}}})$ .

we know that carrier signal power is $P_{s} = \frac{A^{2}}{2}$ .

$\therefore P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{N_{o}}})$ . since $A^{2}T_{b} = P_{s}T_{b} =E_{b}$ .

probability of error of coherent PSK is $P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{N_{o}}})$ (or) in terms of Q function as $P_{e}= Q(\sqrt{\frac{2E_{b}}{N_{o}}})$ .

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Probability of error of ASK

Probability of error of ASK using coherent Detection:-

The equation of ASK from it’s basic definition

$S_{ASK}(t)=\left\{\begin{matrix} \sqrt{2P_{s}}\cos 2\pi f_{c}t\ \rightarrow \ symbol\ '1'\\ 0\ \rightarrow \ symbol\ '0' \end{matrix}\right.$ .

The inputs to the optimum filter are $x_{1}(t) \ and \ x_{2}(t)$ when the symbols 1 and 0 are being transmitted at the transmitter.

$\therefore x_{1}(t) = \sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

$x_{2}(t)=0$ .

we know that $P_{e}$ of an optimum filter is $P_{e} = \frac{1}{2}\ erfc(\frac{x_{o1}(t)-x_{o2}(t)}{2\sqrt{2}\sigma })$

now chose the ratio $\rho _{max}^{2} =(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}$

from the Matched filter concept $(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}=\frac{2}{N_{o}}\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df-----EQN(1)$

from Parsevel’s relation $\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df =\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt----EQN(2)$

from equations (1) and (2)

$\rho _{max}^{2}=\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt$

Here x(t) is the signal $x(t) =x_{1}(t)-x_{2}(t)$ .

$x(t) =\sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

over a duration of $[0, T_{b}]$ the symbols are transmitted

$\int_{0}^{T_{b}}\left | x(t) \right |^{2}dt=2P_{s}\int_{0}^{T_{b}}\cos^{2} 2\pi f_{c}t \ dt$ .

$=P_{s}T_{b}$ .

from equation(1) $\rho _{max}^{2} = \frac{2}{N_{o}} P_{s}T_{b}$ .

$\rho _{max} = \sqrt{\frac{2P_{s}T_{b}}{N_{o}}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{2P_{s}T_{b}}{N_{o}}}).\frac{1}{2\sqrt{2}})$

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{P_{s}T_{b}}{4N_{o}}})$ .

we know that carrier signal power is $P_{s} = \frac{A^{2}}{2}$ .

$A=\sqrt{2P_{s}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{A^{2}T_{b}}{8N_{o}}})$ .

$\therefore P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ . since $A^{2}T_{b} = P_{s}T_{b} =E_{b}$ .

probability of error of coherent ASK is $P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ (or) in terms of Q function as $P_{e}= Q(\sqrt{\frac{E_{b}}{2N_{o}}})$ .

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SNR in PCM (or) Signal-to-Quantization Noise Ratio in PCM system

we know that signal-to-Noise Ratio is defined as

$SNR=\frac{S}{N}=\frac{Normalized\ signal\ power}{Normalized\ Noise \ power}$ .

let $x(t)$ is a given signal with which is an arbitrary signal with Normalized signal power P watts.

and this $x(t)$ is some arbitrary signal oscillating between $+x_{max}$ and $-x_{max}$ .

then the step size used in the Quantization process is $\Delta =\frac{2 . x_{max}}{L}$ .

and $L=2^{n}$ .

where L- is the number of Quantization levels.

n- no.of bits required to encode each Quantization level.

$\therefore \Delta =\frac{2 . x_{max}}{2^{n}} ----EQN(1)$ .

Quantization Noise Power $\bg_black N \ (or ) \ N_{Q}$ :-

if uniform (or) linear Quantization is used in PCM system, during the approximation process of $x(t)$ with $x_{q}(t) \ (or) \ \widehat{x}(t)$ there exists some error between these two signals . This error is called as Quantization error (or) noise.

$x(t) \approx x_{q}(t)$

In discrete time domain $e(nT_{S}) = x_{q}(nT_{S})-x(nT_{S})$ .

Quantization error = Quantized signal- original signal.

we know that step size is $\Delta =\frac{2 . x_{max}}{L}$ .

Now to find out Quantization noise , assume it is uniformly distributed random variable .

now the Probability density function of this uniformly distributed random variable is $f_{\epsilon }(\epsilon )$

$f_{\epsilon }(\epsilon ) = \left\{\begin{matrix} \frac{1}{\Delta } , \frac{-\Delta }{2}\leq0\leq \frac{\Delta }{2}.\\ 0,otherwise. \end{matrix}\right.$

Mean square value of this random variable is with zero mean

$E(\epsilon ^{2}) = \int_{-\infty }^{\infty } \epsilon ^{2}f_{\epsilon }(\epsilon )d\epsilon$ .

$E(\epsilon ^{2}) = \int_{\frac{-\Delta }{2} }^{\frac{\Delta }{2} } \frac{1}{\Delta }d\epsilon$ .

simplification gives $E(\epsilon ^{2}) = \frac{\Delta ^{2}}{12}$ .

Mean Square value= Quantization Noise Power.

$\therefore N_{q} = \frac{\Delta ^{2}}{12}----EQN(2)$ .

by substituting $\Delta$ in equation (2) ,

$N_{q} = \frac{(\frac{2 . x_{max}}{2^{n}})^{2}}{12}$ .

$N_{q} = \frac{x_{max}^{2}}{3X2^{2n}}$ .

$\therefore SQR \ in \ PCM system=\frac{Signal \ power}{Noise \ power}$ .

$SQR = \frac{P}{\frac{x_{max}^{2}}{3(2^{2n})}}$ .

$\frac{S}{N} = \frac{S}{N_q} = SQR=\frac{3P2^{2n}}{x_{max}^{2}}$ .

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SNR in DM (or) Signal-to-Quantization Noise Ratio in Delta Modulation system

we know that signal-to-Noise Ratio is defined as

$SNR=\frac{S}{N}=\frac{Normalized\ signal\ power}{Normalized\ Noise \ power}$ .

let $x(t)$ is a given signal which is a single tone signal $x(t) =A_{m} \cos 2\pi f_{m}t \ where \ \omega _{m} = 2\pi f_{m}$ .

The maximum value of RMS signal power is $P_{rms} = \frac{A_{m}^{2}}{2R}$ .

Normalized signal power $P=\frac{A_{m}^{2}}{2}$ with R=1.

we know that slope overload distortion can be eliminated if and only if $A_{m}\leq \frac{\Delta}{2\pi f_{m}T_{s}}$ .

let $A_{m}= \frac{\Delta}{2\pi f_{m}T_{s}}$

By substituting $A_{m}$ value in P then the power results to be $P= \frac{(\frac{\Delta}{2\pi f_{m}T_{s}})^{2}}{2}$ .

$P= \frac{\Delta^{2}}{8\pi^{2} f_{m}^{2}T_{s}^{2}}----EQN(1)$

Quantization Noise Power $\bg_black N \ (or ) \ N_{Q}$ :-

if uniform (or) linear Quantization is used in DM system, during the approximation process of $x(t)$ with $x_{q}(t) \ (or) \ \widehat{x}(t)$ there exists some error between these two signals . This error is called as Quantization error (or) noise.

$x(t) \approx x_{q}(t)$ (approximation process)

In discrete time domain $e(nT_{S}) = x_{q}(nT_{S})-x(nT_{S})$ .

Quantization error = Quantized signal- original signal.

Now to find out Quantization noise , assume it is uniformly distributed random variable $(+\Delta ,-\Delta )$

now the Probability density function of this uniformly distributed random variable is $f_{\epsilon }(\epsilon )$

$f_{\epsilon }(\epsilon ) = \left\{\begin{matrix} \frac{1}{2\Delta } , -\Delta \leq0\leq \Delta .\\ 0,otherwise. \end{matrix}\right.$

Mean square value of this random variable is with zero mean

$E(\epsilon ^{2}) = \int_{-\infty }^{\infty } \epsilon ^{2}f_{\epsilon }(\epsilon )d\epsilon$ .

$E(\epsilon ^{2}) = \int_{\Delta }^{\Delta }\epsilon ^{2} \frac{1}{2\Delta }d\epsilon$ .

simplification gives $E(\epsilon ^{2}) = \frac{\Delta ^{2}}{3}$ .

Mean Square value= Quantization Noise Power.

$\therefore N_{q} = \frac{\Delta ^{2}}{3}----EQN(2)$ .

The M signal is passed through a reconstruction Low pass Filter at the output of a DM Receiver . The Band width of this filter is $f_{M}$ in such a way that $f_{M}\geq f_{m} \ and \ f_{M}< < f_{s}$ .

where $f_{s}$ is the sampling frequency of the signal.

now assume that Quantization noise is distributed over a frequency band up to $f_{s}$ and is given by $\frac{\Delta ^{2}}{3}$ .

then the noise power $N_{q}^{'}$ distributed over $f_{M}$ will be

$N_{q}^{'} = \frac{f_{M}}{f_{s}}\frac{\Delta ^{2}}{3}---EQN(3)$ .

$\therefore SQR \ in \ DM system=\frac{Signal \ power}{Noise \ power}$ .

$\frac{S}{N} = \frac{S}{N_q^{'}} = SQR=\frac{\frac{\Delta^{2}}{8\pi^{2} f_{m}^{2}T_{s}^{2}}}{\frac{f_{M}}{f_{s}}\frac{\Delta ^{2}}{3}}$ .

$SQR_{DM} = \frac{3}{8\pi ^{2}f_{m}^{2}T_{s}^{3}f_{M}} \ where \ T_{s}=\frac{1}{f_{s}}$ .

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Optimum filter

The function of a receiver in a binary Communication system is to distinguish between two transmitted signals $x_{1}(t)\ and \ x_{2}(t)$ (or) ( $s_{1}(t)\ and \ s_{2}(t)$ ) in the presence of noise.

The performance of Receiver is usually measured in terms of the probability of error P_e and the receiver is said to be optimum if it yields the minimum probability of error.

i.e, optimum receiver is the one with minimum probability of error P_e .

optimum receiver takes the form of Matched filter when the noise at the receiver input is white noise.

optimum receiver (or) optimum filter: –

The block diagram of optimum receiver is as shown in the figure below

the decision boundary is set to $\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

Probability of error of optimum filter:-

The probability of error can be obtained as similar to Integrate and dump receiver. Here we will consider noise as Gaussian Noise.

The output of optimum filter is $y(t) = x_{o1}(t)+n_{o}(t)$ .

The output of sampler is $y(T) = \left\{\begin{matrix} x_{o1}(T)+n_{o}(T) \ for \ binary \ i/p \ '1'\\ x_{o2}(T)+n_{o}(T) \ for \ binary \ i/p \ '0' \end{matrix}\right.$

suppose if Binary ‘1’ is transmitted then the input is $x(t) = x_{1}(t)$ , to find the probability of error this transmitted ‘1’ should be received as ‘0’.

this is possible when the condition $\left | y(T) \right | <\frac{x_{o1}(T)+x_{o2}(T)}{2}$ is true.

1 will be received as 0 $\Rightarrow x_{o1}(T)+n_{o}(T) <\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T) <\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

similarly a Binary ‘0’ will be received as ‘1’ if and only if

is true.

1 will be received as 0 .

the conditions are summarized in the table

Noe the Probability Distribution Function of Gaussian noise with zero mean and standard deviation $\sigma$ is given by

$f(n_{o}(T)) = \frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{n_{o}^{2}(T)}{2}}$ .

Probability of error= probability ‘1’ will be received as ‘0’ =probability ‘0’ will be received as ‘1’.

$\therefore P_{e} =$ area under the curve

(or) area under the curve $n_{o}(T) <\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

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Amplitude Shift Keying(ASK)

Amplitude Shift Keying (ASK) (or) On Off Keying (OOK) is the simplest Digital Modulation technique.

In this method, carrier amplitude is switched between two voltages ON and OFF levels depending up on the input binary sequence.

The carrier signal is a continuous wave (or) sinusoidal wave form

$S(t)=A \cos 2\pi f_{c}t$ .

The normalized power is $P=\frac{A^{2}}{2}$

$A=\sqrt{2P_{s}}$ .

The carrier signal can be expresses in terms of power as $S(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t$ .

if energy per bit is $E_{b}$ and the bit interval as $T_{b}$ then the carrier signal is $S(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t$ .

Now according to ASK Binary ‘1’ is represented with carrier voltage and Binary ‘0’ is represented with zero voltage.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$

in terms of Energy and bit duration ASK signal can be written as

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

ASK Transmitter:-

The figure shows the ASK generator (or) ASK Transmitter

It is a simple product Modulator, which modulates the incoming binary sequence (in the form of a signal) with the carrier signal S(t)

i.e, $S_{ASK}(t)=b(t).S(t)$

b(t) represents the binary sequence in the form of a signal.

when the input bit (or) symbol is Binary ‘1’ product Modulator passes the carrier signal and for Binary’0′, A zero output is given which blocks the carrier signal.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

Coherent ASK Detector:-

The figure shows the Block Diagram of coherent ASK/BASK Detector. The ASK signal $S_{ASK}(t)$ is applied to the correlator ( The Block product Modulator followed up by the Integrator).

$S_{ASK}(t)$ is multiplied by local carrier C(t) this carrier C(t) is phase locked with that of the carrier used in the Transmitter. As this is coherent reception.

the output of threshold device v(t) (or) v is either ‘1’ (or) ‘0’ based on the following condition.

$v> 0\rightarrow \ a \ symbol \ '0' \ is \ detected.$

$v\leq 0\rightarrow \ a \ symbol \ '1' \ is \ detected.$

Note:- The input to demodulator is not $S_{ASK}(t)$ always most of the times it is interfered with noise n(t) in the channel.

in coherent detection input to the demodulator is simply $S_{ASK}(t)$ signal where as in Non-coherent detection the input is noisy ASK signal.

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Compression laws (A-law and u-law)

The laws used in compressor of a non-uniform Quantizer are known as compression laws.

two laws are available

$\mu$ – law.
$A$ -law.

$\mu$ -law:- A particular form of compression law that is used in practice is the so called $\mu$ – law which is defined by

$\left | v \right |= sgn(x){\frac{\ln (1+\mu \left | x \right |)}{(1+\mu )}}, for\ 0\leq \left | x \right |\leq 1$ .

where $\left | v \right |$ – Normalized compressed output voltage.

$\left | x \right |$ – Normalized input voltage to the compressor.

and $\mu$ is a positive constant and its value decides the curvature of $\left | v \right |$ .

$\mu =0$ corresponds to no compression, which is the case of uniform quanization, the curve is almost linear as the value of $\mu$ increases signal compression increases.

$\mu =255$ is the north american standard for PCM voice telephony.

for a given value of $\mu$ , the reciprocal slope of the compression curve, which defines the quantum steps is given by the derivative of $\left | x \right |$ with respect to $\left | v \right |$

$\frac{dx}{dv} = \frac{\ln (1+\mu )}{\mu } (1+\mu \left | x \right |)$

we see that therefore $\mu$ law is neither strictly linear nor strictly logarithmic.

i.e,

It is approximately linear at low levels of input corresponding to $\mu \left | x \right |<<1$ . and is approximately logarithmic at high input levels corresponding to $\mu \left | x \right |>>1$ .

typical value of $\mu =255$

A-law :- another compression law that is used in practice is the so called A- law defined by

A=1 corresponds to the case of uniform Quantization. A-law has been plotted for three different values of A. A=1, A=2, A=87.56.

typical value of A is 87.56 in European Commercial PCM standard which is being followed in India.

It is also interested to note that in actual PCM systems, the companding circuitry does not produce an exact replica of the non-linear compression curves shown in the compression law characteristics.

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Quadrature Phase Shift Keying (QPSK) Transmitter and Receiver

Quadrature Phase Shift keying:-

The designing of digital communication system requires two important goals to achieve
1. To achieve low probability of error P_e.
2. To utilize Channel Band width efficiently.
QPSK is A Band width conserving modulation scheme, which is an example of Quadrature Carrier Multiplexing.
The modulation schemes such as ASK, PSK & FSK does not meet the Band width requirements of data Communication systems since the Bit rate and Baud rate are same in these schemes. Since the channel band width depends up on the bit rate (or) signalling rate of the modulation scheme. If two (or) more bits are combined into a symbol, then the signalling rate is reduced. Therefore the frequency of the carrier is also reduced, this reduces the transmission channel band width. Thus grouping of bits into symbols reduces Channel Band width.

Meaning of QPSK:-

In Quadri Phase Shift Keying as with Binary PSK information carried by the transmitted signal is contained in the phase of the carrier. The phase of the carrier Φ_c takes on one of four equally spaced values such as π/4, 3π/4, 5π/4 and 7π/4 that is in QPSK two successive bits are combined into a di-bit or symbol and each possible value of the phase corresponds to a unique di-bit.
for example the foregoing set of phase values are chosen to represent the gray encoded set of di-bits 10, 00, 01 and 11 , where only a single bit is changed from one di-bit to the next.

Generation of QPSK/ QPSK transmitter:-

Consider the generation and detection of QPSK signals. The figure shows a Block diagram of a typical QPSK Transmitter.The incoming binary sequence is first transmitted into polar form by a Non-Return to zero level encoder. Thus symbols 1 and 0 are represented by
√ E_s and –√ E_s
This binary wave is next divided by means of a de-multiplexer into two separate binary waves. Consisting of the odd and even numbered input bits {b_e(t)} and {b_o(t)} represents those two binary waves.
The two bit streams b_e(t) and b_o(t) are modulated by two ortho-normal basis functions Φ₁(t) and Φ₂(t).finally, the two binary PSK signals are added to produce the desired QPSK signal.
i.e, S_QPSK(t) = S_e(t) + S_o(t).
S_oPSK(t)= b_o(t)* √(2/T_s)* cos 2πf_c t
S_ePSK(t)= b_e(t)* √(2/T_s)* sin 2πf_c t

S_QPSK(t)= b_o(t)* √(2/T_s)* cos 2πf_c t + b_e(t)* √(2/T_s)* sin 2πf_c t.

QPSK Receiver:-

The QPSK Receiver consists of a pair of correlators called as In-phase channel and Quadrature phase channel with a common input. The input x(t) is supplied with a pair of coherent reference signals Φ₁(t) and Φ₂(t). The two correlators produces two signals x₁(t) and x₂(t) in response to the received signal x(t). these signals x₁(t) and x₂(t) are compared with threshold voltage 0V by the decision devices in the two channels.

If x₁ >0, a decision has been made in favor of symbol ‘1’ for the in-phase channel output. but if x₁<0 a decision has been made in favor of ‘0’. similarly for the Q-phase channel,

x₂>0—-> a symbol ‘1’ is decided.

x₂<0—-> a symbol ‘0’ is decided.

finally, these two binary sequences at the I-phase and Q-phase channel outputs are combined in a multiplexer to reproduce the original binary sequence at the Receiver output with the minimum probability of symbol error in the AWGN channel.

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FSK Generator /BFSK generator

we know that the input to the FSK Generator is a binary sequence 1010…etc.

FSK generator uses two product modulators upper-modulator and lower-modulator with carriers

and

A level shifter is there in which the output of the level shifter is

when the input is a binary ‘1’ and ‘0’ volts for the input ‘0’ level shifter.

i.e,

The working of the FSK generator is as follows when the input binary sequence is ‘1’

on the upper modulator 1 has been shifted to a voltage so that the output of product modulator 1 is

and on the lower modulator input ‘1’ is passed through an inverter and if the output of the inverter is ‘0’ then the output of the level shifter will not change it remains at ‘0’ volt itself.

then the product modulator 2 output is

then the overall output

similarly, when the input sequence is a binary ‘0’

Frequency Shift Keying (FSK)/BFSK

In a Binary FSK system, symbols 1 and 0 are distinguished from each other by transmitting one of two sinusoidal waves that differ in frequency by a fixed amount.

(or)

The frequency of the carrier signal shifted to two frequencies for symbols ‘1’ and ‘0’ transmission.

The equation for FSK signal is

\[ S_{FSK}(t)= \sqrt{\frac{2E_{b}}{T_{b}}} \ \ cos (2\pi f_{c_{i}}t),\ 0\leq t\leq T_{b} , \ i=1,2 \]

\[ S_{FSK}(t)= 0 elsewhere \]

i.e,

where

is generally a high frequency.

is a low frequency and vice-versa is also true.

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Types of digital Modulation techniques (or) systems:-

Digital Modulation techniques may be classified into coherent (or) Non-coherent techniques, depending on whether the receiver is equipped with a phase-recovery circuit (or) not.

Coherent digital modulation techniques (or) systems	Non-coherent digital Modulation techniques (or) systems /Envelope detection.
1. In this scheme, the local carrier generated at the receiver is phase locked with the carrier at the transmitter. i.e, phase lock exists between Transmitter and Receiver.	1. There exists no such phase lock between Transmitter and Receiver.
2. This is also called Synchronous detection.	2. This is known as Non-synchronous detection.
3. complexity increases in terms of designing of the receiver.	3. less complexity in terms of designing the receiver.
4. probability of error decreases. Examples:- coherent ASK, PSK, and FSK systems.	4. error probability increases. examples:-Non-coherent ASK, PSK, and FSK systems.

Base band Vs Pass band Transmission

Baseband data transmission	Passband data Transmission
1. The digital data is transmitted over the channel directly, there is no carrier (or) any modulation.	1. The digital data modulates high-frequency sinusoidal carrier. Hence it is also called as digital CW modulation . ∴carrier is required.
2. This is suitable for transmission over short distances. Examples:- Ethernet signals operating over a LAN (Local Area Network) The most common baseband modulation is (PAM) and PCM in local digital computer links.	2. suitable for long distances transmission. Examples:- Microwave links, Satellite Communication links are called Passband communication systems.
3. Baseband transmission sends the information signal as it is without modulation. i.e, without frequency shifting.	3. passband transmission shifts the signal(information) to be transmitted in low frequency to a higher frequency. i.e, Modulation is required.
4. baseband signals are in general low-frequency signals i. human voice(20Hz-5KHz). ii. video signal from a TV camera (0Hz-5.5MHz). Examples:- The telephone systems used for offices and homes (one room to other) transmits baseband signal as it is the system falls into baseband communication systems.	4. whereas long-distance call that is transmitted via microwave (or) satellite links uses modulation which is known as passband communication systems. examples :- Passband Modulation techniques ASK,DPSK,FSK,QPSK,PSK,M-ary PSK etc.

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Drawbacks in Delta Modulation (or) errors in Delta Modulation

Delta Modulation is subject to two types of quantization error

slope Overload Distortion (SOD)/ Slope Overload error
Granular Noise / Granular error.

During the process of digital equivalent integration of x(t) that is approximating x(t) with there exists an error called Quantization error as shown by

if time instance is (n-1)^thinstance.

the Quantization error q[n] is of two types in Delta Modulation.

Slope-Overload Distortion:- if the rate of rise of input signal is so high

i.e, the slope of the signal is so high so that the stair case signal can not approximate to x(t) .

i.e, as is large enough

in this case the step size becomes to small for the stair case approximation to follow a steep segment of the input wave form x(t) with the result that falls behind which can be clearly visible in the figure.

the Qunatization error that exist between x(t) and in this condition is called as slope overload distortion.

Generaaly DM is often referred as a linear Delta Modulator because the step size is fixed during approximation process, and also its maximum (or) minimum slopes occur along straight lines.

To avoid slope Overload Distortion, step size must be increased.

Granular Noise (or) Idle Noise:-

In contrast to slope overload distortion Granular noise occurs when the step size is too large relative to the local slope characteristic of the input wave form x(t)

∴ This large value of causes that the stair case approximation to hunt around a flat segment of the input wave form as shown in the above figure

i.e, oscillates between when. x(t) is almost straight.

∴ The error between and in this condition is called as Granular noise (or) Idle noise.

To eliminate this error is to make the step size small.

Granular noise occurs that for a very small variations in the input signal causes a very large variations in the approximated signal .

Thus we see that there is a need to have a large step size to accomodate a wide dynamic range of input signal.

and a small step size is required to accurate representation of relatively low-level signals.

i.e, large step size is required to reduce slope overload distortion and small step size is required to reduce Granular noise.

∴ It is clear that the choice of the optimum step size that minimizes the Mean Square value of the Quantization error in a Linear Delat Modulator will be the result of a compromise between Slope overload Distortion and Granular Noise.

To satisfy such a requirement , we need to make the Delta modulator “Adaptive” in the sense that the step size is made to vary in accordance with the input signal x(t).

This can be further discussed in the topic called as “Adaptive Delta Modulation ” scheme (ADM).

Block Diagram of Digital Communication system

Block Diagram of Digital Communication System/Elements of DCS:-

A General Communication System can be viewed as a Transmitting unit and a Receiving Unit connected by a medium(Channel). Obviously, Transmitter and Receiver consist of various sub-systems (or) blocks.
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 a receiver that is the channel and its 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, if 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 an electrical signal by means of a Transducer, which can be considered as a part of the 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 avaerage bit rate.

Channel encoder:- Channel coding is also known as error control coding. Channel coding is a technique that 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 the 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 Bandwidth.

There are two types of spreading techniques available

1. Direct Sequence Spread Spectrum Techniques.

2. 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.

Digital Communication systems Vs Analog Communication Systems

Introduction:-
Communication is the process of establishing a 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 purposes is called Communication equipment. The equipment when assembled together forms a communication system.
Examples of different types of communications

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

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 a 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?

1. 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.
2. Propagation of Digital pulse through a Transmission line:-
When an ideal binary digital pulse propagates 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 waveform.
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 the Transmission system are called “regenerative repeaters’. This

is one of the reasons why digital is preferred over
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 waveform 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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Digital Communications tutorial questions

Digital Communications Tutorial-1 Questions on PCM, DM

Digital Communications Assignment-1

Digital Communications Assignment-2