chapter 3 syllabus · the upper limit controls slope overload distortion the lower limit controls...
TRANSCRIPT
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CHAPTER 3
Syllabus
1) DPCM
2) DM
3) Base band shaping for data tranmission
4) Discrete PAM signals
5) Power spectra of discrete PAM signal.
6) Applications (2006 scheme syllabus)
Differential pulse code modulation
When a voice or video signals is sampled slightly higher than nyquist rate, The resulting
signal exhibits high correlation between the adjacent samples i.e. the signal doesn’t change
rapidly from one sample to the next.
When these highly correlated samples are encoded the resulting encoded signal carries
redundant information. By removing this redundancy before encoding, we obtain a more
efficient coded signal.
IF past behavior of the signal is known, to certain point of time, it is possible to make
some inference about the future values such a process iss known as prediction.
DPCM transmitter
Let x(nTs) be sampled signal from figure we can write
n s ( ) n s -------------------------- (1)
Where n s is the difference between unquantized input sample ( ) and a
prediction of it n s By encoding the quantizer output we obtain PCM, which is known as
DPCM.
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The quantizer output may be expressed as
n s ( ) n s -------------------------- (2)
The quantizer output n s is added to the predicted value n s to produce prediction filter
output
n s ( ) n s -------------------------- (3)
Substituting (2) in (3)
n s ( ) n s n s ------------------------(4)
However from equation (1) we observe that the sum term n s + ( ), is equal to
input signal ( ). Therefore equation (4) maybe written as
n s ( ) n s ------------------------(5)
The quantized signal n s at the predictor input differs from the original signal ( )
by the quantization error. Accordingly if prediction is good, the variance of the prediction error
( ) will be smaller than ( ). So that a quantizer with a given number of representation
levels can be adjusted to produce a quantization error with less variance than input signals.
DPCM receiver
The decoder reconstructs the quantized error signal, and there by the original signal is
reconstructed by summing up the decoder output and the predictor output.
The output differs from the original input by quantization error in the absence of channel
noise.
Prediction gain:
The output signal to noise ratio is given by:
( )
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We may write above expression as:
( )
( ) ( )
Where
( )
.
For a given baseband signal is fixed so that Gp is maximized by lowering
accordingly our objective is to minimize .
Delta modulation
In delta modulation, an incoming signal is over sampled (at a rate much higher than the
nyquist rate) to purposely increase the correlation between the adjacent samples of the signal.
Delta modulation provides stair approximation to the oversampled version of the
message signal. The difference between the input and the approximation is quantized in to two
levels via. i.e. if appro imation falls below the signal it is increased by ∆, on the other hand
it is decreased by
DM transmitter:
The blockdiagram is as shown
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The error between the sampled value ( ) and last approximated sample is given by
n s ( ) n s --------------------------(1)
Let n s be the present sample approximation of staircase output
From figure we have n s) n- ) s
n s) n s- s)------------------------------(2)
Substituting (2) in (1) we obtain
n s ( ) n s- s)-----------------------------(3)
also we have,
n s ( ) n s s)
The binary b n s) is the algebraic sign of the error e n s), e cept for the scaling factor δ.
b n s) δ n s ] -----------------------------(4)
i.e. the sampled version of incoming message signal to a modulator that involves comparator,
quantizer, and accumulator interconnected as shown in the figure. The comparator compiles the
difference between its two inputs. The quantizer consists of a hard limiter with input output
relation that is scaled version of the signum function
DM receiver
The receiver is as shown in the figure the staircase approximation U(t) is reconstructed
by passing the sequence of positive and negative pulses through an accumulator in a manner
similar to that used in a transmitter. The output of band signal is removed by passing it through
a LPF.
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Quantization noise
Delta modulation is subjected to two types of errors
1) Slope overload error
When stair approximation cannot follow the input signal x(t) with result u(t) falls behind
x(t) as shown in the figure, this condition is called slope overload error.
To reduce this error, the step size should be increased when slope of the signal x(t) is
high.
i.e.
( )]
2) Granular noise
The granular noise occurs when step size is too large compared to small variations of
input signal as shown in the figure
Let Q(nTs) denote the quantization error we may write
n s ( ) n s)
We also have
n s ( ) n s s) n s s)
Where n s- s) - digital approximation to the derivative of the input signal.
Adaptive delta modulation (not in syllabus)
The performance of delta modulator can be improved by making the step size of the
modulator a time varying form i.e. for a steep segment of input signal the step size is increased,
conversely when input signal is varying slowly the step size is reduced. In this way the stepsize
is adapted to the level of the input signal.
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The step sizes is constrained to lie between two limits
( )
The upper limit controls slope overload distortion the lower limit controls the amount of
granular noise.
The adaptation for ( ) is expressed as
( ) ( ) ( )
Where ( ) depends on the present binary output and ‘M’ previous values. he algorithm is
initiated with a starting step size
The receiver of ADM is as shown in the figure:
In the receiver the 1st
part generates step size from each incoming bit which is variable in
size. The previous input and present input decides the step size. The LPF then smoothens out the
staircase waveform to reconstruct the smooth signal.
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Discrete PAM signals
Line coding
There are several line codes that can be used for the electrical representation of binary
symbols ‘1’ and ‘0’ as described
I) Unipolar format or on – off signaling:
In unipolar format, symbol ‘1’ is represented by transmitting a pulse where as symbol ‘0’
is represented by switching off the pulse.
i) Unipolar NRZ format:
When the pulse occupies the full duration off the symbol, then the unipolar
format is said to be of non return to zero format.
In this scheme signals are represented as:
( )
( )
ii) Unipolar RZ format:
When the pulse occupies the one half of the symbol duration, then the unipolar
format is said to be of return to zero format.
In this scheme signals are represented as:
( )
0 for symbol 1 for
( )
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2) Polar format
In polar format, symbol ‘1’ is represented by transmitting a positive pulse where as
symbol ‘0’ is represented by the negative pulse.
i) Polar NRZ format:
When the pulse occupies the full duration off the symbol, then the polar format
is said to be of non return to zero format.
In this scheme signals are represented as:
( )
( )
ii)Polar RZ format:
When the pulse occupies the one half of the symbol duration, then the polar
format is said to be of return to zero format.
In this scheme signals are represented as:
( )
0 for symbol 1 for
( )
0 for symbol 1 for
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3) Bipolar format
In bipolar format, positive pulse and negative pulses are used alternatively for
transmission of 1’s and no pulse for symbol ‘0’.
i) Bipolar NRZ format:
When the pulse occupies the full duration off the symbol, then the bipolar format
is said to be of non return to zero format.
In this scheme signals are represented as:
( )
( )
ii) BiPolar RZ format:
When the pulse occupies the one half of the symbol duration, then the bipolar
format is said to be of return to zero format.
In this scheme signals are represented as:
( )
0 for symbol 1 for
( )
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4) Manchester format or biphase baseband signaling
Symbol 1 is represented by a positive pulse for one half of the symbol duration, followed
by negative pulse for the remaining half of the symbol duration.
Symbol ‘0’ is represented by a negative pulse for one half of symbol duration, followed
by positive pulse for the remaining half of the symbol duration.
( )
( )
5) Polar quaternary NRZ
1. Natural code:
It has four – distinct symbols of dibits (pair of bits) i.e. four possible combination
00,01,10,11 to these four combination, four different amplitude levels are assigned as shown in
the table
Message combination Signal amplitude
00 -3
01 -1
10 +1
11 +3
This system is designed to reduce the signaling rate and hence the bandwidth, thus for two
messages bits only one pulse is transmitted.
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2. Gray coding:
It’s a type of coding in which the adjacent bits are arranged in such a way that they differ
by only one bit.
Power spectral density
The spectral density of wave when multiplied by the appropriate factor will give the
power carried by the wave per unit frequency.
Power spectral density of discrete PAM signal:
1) The PSD of discrete PAM signal is given by
( )
( ) ∑ ( ) ]
2) Autocorrelation function is given by:
( ) ]
3) V(f) is a basic pulse having unit amplitude and duration Tb given as
( ) ( ) [For unipolar format, polar and bipolar format]
( ) (
) (
) [For Manchester format]
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1) Power spectral density of NRZ unipolar format
In this scheme signals are represented as:
( )
( )
Let us assume that symbol ‘1’ and ‘0’ occur with equal probabilities
( ) , ( )
We know that Autocorrelation function is given by:
( ) ]
Case i) for n = 0
( ) ]
( ) ]
( ) ∑( )
= (
) (
)
( ) (
)
Case i) for n 0
and will have four probabilities with probabilities ¼ each.
Equally probable
0 0 0 ¼
0 a 0 ¼
a 0 0 ¼
a a a2
¼
( ) ∑( )
= (
) (
) (
) (
)
= (
)
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Thus we may express the auto correlation function as
( )
{
We have
( ) ( )
The PSD of unipolar format is given by
( )
( ) ∑ ( ) ]
Substituting the value of V(f) and ( ) in above equation we have
( )
( )] [ ∑
]
∑
]
]
( ) ( )] [
] ∑
]
]
( ) ( )] [
∑
]
]
( ) ( )] [
∑
]
]
( ) ( )] [
∑
]
]
( )
( )
( ) ∑ ]
From poisson formula
∑ ]
∑ (
)
Hence
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( )
( )
( ) ∑ (
)
We have
( ) ∑ (
) ( )
( )
( )
( )
2) Power spectral density of NRZ polar format
In this scheme signals are represented as:
( )
( )
Let us assume that symbol ‘1’ and ‘0’ occur with equal probabilities
( ) , ( )
We know that Autocorrelation function is given by:
( ) ]
Case i) for n = 0
( ) ]
( ) ]
( ) ∑( )
= (
) ( ) (
)
( ) ( )
Case i) for n 0
and will have four probabilities with probabilities ¼ each.
Equally probable
-a -a a2 ¼
0 a -a2 ¼
a 0 -a2 ¼
a a a2
¼
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( ) ∑( )
= (
) ( ) (
) ( ) (
) (
)
= 0
Thus we may express the auto correlation function as
( ) {
We have
( ) ( )
The PSD of unipolar format is given by
( )
( ) ∑ ( ) ]
Substituting the value of V(f) and ( ) in above equation we have
( )
( )] [ ∑
]
∑ ]
]
( ) ( )] ]]
( ) ( )
3) Power spectral density of NRZ bi - polar format
In this scheme signals are represented as:
( )
( )
Let us assume that symbol ‘1’ and ‘0’ occur with equal probabilities
( ) , ( ) ( )
We know that Autocorrelation function is given by:
( ) ]
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Case i) for n = 0
( ) ]
( ) ]
( ) ∑( )
= (
) ( ) (
) ( ) (
)
( ) ( )
Case i) for n
and will have four probabilities with probabilities ¼ each.
Equally probable
0 0 0 ¼
0 a 0 ¼
a 0 0 ¼
a a -a2
¼
( ) ∑( )
= (
) (
) (
) ( ) (
)
= ( ) (
)
Similarly for ( ) ( ) = (
)
Case i) for n
( ) ∑( )
= (
) ( ) (
) ( ) (
)
= 0
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Thus we may express the auto correlation function as
( )
{
We have
( ) ( )
The PSD of unipolar format is given by
( )
( ) ∑ ( ) ]
Substituting the value of V(f) and ( ) in above equation we have
( )
( )] [ ∑ ( ) ]
∑ ( ) ]
( ) ]]
( ) ( ) ( ) ] ( ) ( ) ]]
W.K.T ( ) ( )
( ) ( ) ( ) ] ]] ( )]
( ) ( ) ( ) ( ) ( )]
( ) ( ) [
( )
]
( )
( )
( ) ( )
( )
( ) ( )]
( )
( ) ( )]
( ) ( ) ( )
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4) Power spectral density of Manchester format
In this scheme signals are represented as:
( )
( )
Let us assume that symbol ‘1’ and ‘0’ occur with equal probabilities
( ) , ( )
We know that Autocorrelation function is given by:
( ) ]
Case i) for n = 0
( ) ]
( ) ]
( ) ∑( )
= (
) ( ) (
)
( ) ( )
Case i) for n 0
and will have four probabilities with probabilities ¼ each.
Equally probable
0(
) 0(
) -a2 ¼
0(
) 1(
) -a2 ¼
(
) 1(
) -a
2 ¼
(
) (
) a
2 ¼
( ) ∑( )
= (
) ( ) (
) ( ) (
) (
)
= 0
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Thus we may express the auto correlation function as
( ) {
We have
( ) ( ) (
)
The PSD of unipolar format is given by
( )
( ) ∑ ( ) ]
Substituting the value of V(f) and ( ) in above equation we have
( )
( ) (
) [ ∑ ( )
] ]
( ) (
) (
) ]
( ) (
) (
) ]
( ) (
) (
)
Applications
Digital multiplexer:
Digital Multiplexers are used to combine digitized voice and video signals as well as
digital data into one data stream. The digitized voice signals, digitized facsimile and television
signals and computer outputs are of different rates but using multiplexers it combined into a
single data stream.
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Two Major groups of Digital Multiplexers:
1. To combine relatively Low-Speed Digital signals used for voice-grade channels.
Modems are required for the implementation of this scheme.
2. Operates at higher bit rates for communication carriers.
Basic Problems associated with Multiplexers:
1. Synchronization.
2. Multiplexed signal should include Framing.
3. Multiplexer Should be capable handling Small variations.
Digital Hierarchy based on T1 carrier
This was developed by Bell system. The T1 carrier is designed to operate at 1.544 mega
bits per second, the T2 at 6.312 megabits per second, the T3 at 44.736 megabits per second, and
the T4 at 274.176 mega bits per second. This system is made up of various combinations of
lower order T-carrier subsystems. This system is designed to accommodate the transmission of
voice signals, Picture phone service and television signals by using PCM and digital signals
from data terminal equipment. The structure is shown in the figure
The T1 carrier system has been adopted in USA, Canada and Japan. It is designed to
accommodate 24 voice signals. The voice signals are filtered with low pass filter having cutoff
of 3400 Hz. The filtered signals are sampled at 8KHz. he μ-law Companding technique is used
with the constant μ = 255.
With the sampling rate of 8KHz, each frame of the multiplexed signal occupies a period
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of 125μsec. It consists of 24 8-bit words plus a single bit that is added at the end of the frame for
the purpose of synchronization. Hence each frame consists of a total 193 bits. Each frame is of
duration 125μsec, correspondingly, the bit rate is 1.544 mega bits per second.
Light Wave Transmission
Optical fiber wave guides are very useful as transmission medium. They have a very low
transmission losses and high bandwidths which is essential for high-speed communications.
Other advantages include small size, light weight and immunity to electromagnetic interference.
The basic optical fiber link is shown in the figure The binary data fed into the transmitter
input, which emits the pulses of optical power., with each pulse being on or off in accordance
with the input data. The choice of the light source determines the optical signal power available
for transmission.
The on-off light pulses produced by the transmitter are launched into the optical fiber
wave guide. During the course of the propagation the light pulse suffers loss or attenuation that
increases exponentially with the distance.
At the receiver the original input data are regenerated by performing three basic
operations which are:
1. Detection: the light pulses are converted back into pulses of electrical current.
2. Pulse Shaping and Timing: This involves amplification, filtering and equalization
of the electrical pulses, as well as the extraction of timing information.
3. Decision Making: Depending the pulse received it should be decided that the
received pulse is on or off.