sampling and baseband modulation

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Formatting & Baseband

ModulationChapter No.2

(Digital Communications Fundamentalsand Applications By Bernard Sklar)

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INTRODUCTION TO DIGITAL COMMUNICATIO

A digital signal is a physical signal that is arepresentation of a sequence of discrete values (aquantified discrete-time signal).

The term digital signal can refer to

a continuous-time waveform signal used in any form of digitalcommunication.

a pulse train signal that switches between a discrete numberof voltage levels or levels of light intensity, also known as a a

line coded signal, for example a signal found in digitalelectronics or in serial communications using digital basebandtransmission in, or a pulse code modulation (PCM)representation of a digitized analog signal.

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Baseband Vs Broadband signals

In a baseband transmission, the entire bandwidth of the cable isconsumed by a single signal. In broadband transmission, signalsare sent on multiple frequencies, allowing multiple signals to besent simultaneously.

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Baseband Signalling: Broadband Signalling:

1)Uses digital signalling 1)Uses analog signalling

2)No frequency-division multiplexing 2) Frequency-division multiplexing ispossible

3)Bi-directional transmission 3) Unidirectional transmission

4)Signal travels over short distances 4)Signal can travel over long distancesbefore being attenuated


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3.8

Baseband transmission using a dedicated medium

Baseband transmission of a digital signalthat preserves the shape of the digital

signal is possible only if we have a low-pass channel with an infinite or very wide

bandwidth.


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3.9

An example of a dedicated channel where the entirebandwidth of the medium is used as one single channel is aLAN. Almost every wired LAN today uses a dedicatedchannel for two stations communicating with each other. In abus topology LAN with multipoint connections, only twostations can communicate with each other at each moment in

time (timesharing); the other stations need to refrain fromsending data. In a star topology LAN, the entire channelbetween each station and the hub is used for communicationbetween these two entities.

Example

In baseband transmission, the required bandwidth is proportionalto the bit rate;if we need to send bits faster, we need more bandwidth.


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Bandwidth of a bandpass channel

If the available channel is a bandpass channel, we cannot sendthe digital signal directly to the channel;

we need to convert the digital signal to an analog signal beforetransmission.

i.e. We do MODULATION


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3.11

Modulation of a digital signal for transmission on a bandpasschannel


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Formatting and transmission of baseband signal

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EncodeTransmit

PulsemodulateSample Quantize

Demodulate/Detect

Channel

Receive

Low-pass

filterDecode

PulsewaveformsBit stream

Format

Format

Digital info.

Textualinfo.

Analog

info.

Textualinfo.

Analoginfo.

Digital info.

source

sink


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Format analog signals

To transform an analog waveform into a form that is compatible

with a digital communication, the following steps are taken:1. Sampling

This is the conversion of analog continuous time signal into discrete timesignal by taking the samples of the signal at discrete time intervals

2. Quantization and encoding Conversion of a discrete time continuous valued signal into discrete

valued discrete time (digital) signal. The value of each signal sample by avalue selected from a finite set of possible values

The difference between unquantized and quantized sample is known asquantization error

Each discrete sample is represented by a b-bit binary sequence

3. Baseband transmission Conversion of binary sequence into digital waveform

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Figure 4.22 Three dif ferent sampling methods for PCM


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Impulse Sampling & Natural Sampling

Impulse Sampling Also known as Ideal Sampling A band limited signal x d (t) is multiplied with periodic train

of unit impulse function x(t) , to get the sampled version x s(t) . Natural Sampling

In natural sampling a band limited signal x(t) is multiplied

by the pulse train or switching waveform x p(t) which haswidth T and amplitude 1/T

It can b viewed as opening and closing of a switch Top of each sampled pulse x p(t) of maintains the original

shape of its analog pulse1/12/2015 16


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In electronics , a sample and hold (' S/H , also "follow-and-hold") circuit isan analog device that samples (captures, grabs) the voltage of a continuouslyvarying analog signal and holds (locks, freezes) its value at a constant levelfor a specified minimal period of time.

A very common, and easily implemented method of sampling of an analogsignal uses the sample-and-hold operation. This produces flat top samples.

Sample and Hold Operations/ flat-top Sampling
http://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Analog_circuithttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Analog_signalhttp://en.wikipedia.org/wiki/Analog_signalhttp://en.wikipedia.org/wiki/Analog_signalhttp://en.wikipedia.org/wiki/Analog_signalhttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Analog_circuithttp://en.wikipedia.org/wiki/Analog_circuithttp://en.wikipedia.org/wiki/Analog_circuithttp://en.wikipedia.org/wiki/Electronics


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Sampling

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Time domain Frequency domain

)()()( t xt xt x s )()()( f X f X f X s |)(| f X

)(t x

|)(| f X

|)(| f X s)(t x s

)(t x


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AliasingIn signal processing and related disciplines, aliasing refers toan effect that causes different signals to become

indistinguishable (or aliases of one another) when sampled . Italso refers to the distortion or artifact that results when thesignal reconstructed from samples is different from the originalcontinuous signal. OR
http://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Sampling_(signal_processing)http://en.wikipedia.org/wiki/Distortionhttp://en.wikipedia.org/wiki/Artifact_(error)http://en.wikipedia.org/wiki/Artifact_(error)http://en.wikipedia.org/wiki/Distortionhttp://en.wikipedia.org/wiki/Sampling_(signal_processing)http://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processing


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Aliasing

When a digital image is viewed, a reconstruction isperformed by a display or printer device, and by theeyes and the brain. If the image data is not properlyprocessed during sampling or reconstruction, thereconstructed image will differ from the original image,and an alias is seen.
http://en.wikipedia.org/wiki/Signal_reconstructionhttp://en.wikipedia.org/wiki/Signal_reconstruction


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Aliasing effect

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LP filter

Nyquist rate

aliasing

Aliased spectral components represents the ambiguousdata that appears in frequency band (f s-f m) and f m

f s-f m and f m


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Sampling theorem

Sampling theorem: A band limited signal

with no spectral components beyond , canbe uniquely determined by values sampledat uniform intervals of

The sampling rate, is calledNyquist rate .

SamplingprocessAnalogsignalPulse amplitude

modulated (PAM) signal

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A di t th N i t th th li t t b t l t 2


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Figure 4.23 Nyquist sampli ng r ate for low-pass and bandpass signals

According to the Nyquist theorem, the sampling rate must be at least 2times the highest frequency contained in the signal.


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F or an intuitive example of the Nyquist theorem, let us sample asimple sine wave at th ree sampling rates: f

s = 4f (2 times the Nyqui st

rate), f s = 2f (Nyquist rate), andf s = f (one-half the Nyquist r ate). F igur e 4.24 shows the sampli ngand the subsequent recovery of the signal .

I t can be seen that sampling at the Nyquist r ate can create a goodapproximation of the ori ginal sine wave (par t a). Oversampling inpart b can also create the same approximation, but i t i s redundantand unnecessary. Sampling below the Nyquist r ate (par t c) does not

produce a signal that looks l ike the original sine wave.

Example 4.6


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Figure 4.24 Recovery of a sampled sine wave for different sampli ng r ates


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A complex low-pass signal has a bandwidth of 200 kH z. What i s theminimum sampling rate for thi s signal?

SolutionThe bandwidth of a low-pass signal is between 0 and f, where f isthe maximum frequency in the signal. Therefore, we can samplethis signal at 2 times the highest frequency (200 kHz). The

sampling rate is therefore 400,000 samples per second.

Example 4.10

Example 4.11A complex bandpass signal has a bandwidth of 200 kH z. What i sthe minimum sampling rate for thi s signal?

SolutionWe cannot find the minimum sampling rate in this case because wedo not know where the bandwidth starts or ends. We do not knowthe maximum frequency in the signal.


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Figure 4.21 Components of PCM encoder


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Quantization The samples from the analog signal are not yet in the digital

form (bit stream) because these samples have infiniteamplitudes.

To make the amplitudes have finite levels (i.e. belong to afinite range of amplitude levels), different amplitude levelsare created known as the quantized level.

Each sample is going to be mapped into one of these levels(i.e. the nearest level for that sample).

More levels mean lower error of mapping known as thequantized error but at the expense of increasing BW.

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Q i i


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Quantization

Amplitude quantizing: Mapping samples of a continuous amplitude waveformto a finite set of amplitudes.

In

Out

Q u a n

t i z e

d

v a l u

e s

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Vpp=peak-to-peak voltage= V p -(-Vp )= 2 Vp Volts The step size between quantization levels is known as

quantile interval, denoted by q volts Each sample value of analog pulse is approximated with

a quantized pulse: the approximation results in error nolonger than q/2 in either direction

Q i i


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QuantizationAverage quantization noise power

2

=

+

=+

1/

Signal peak power

V p2 = /2 2 =( ) 2 = 2

Signal power to average quantization noise power

q=/4

/


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Encoding (PCM)

A uniform linear quantizer is called Pulse Code Modulation(PCM).

Pulse code modulation (PCM): Encoding the quantized signalsinto a digital word ( PCM word or codeword).

Each quantized sample is digitally encoded into an l bits codewordwhere L in the number of quantization levels and

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Figure 4.26 Quanti zation and encoding of a sampled signal


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What is the SNR dB in the example of F igure 4.26?

SolutionWe can use the formula to find the quantization. We have eight

levels and 3 bits per sample, so SNR dB = 6.02(3) + 1.76 = 19.82 dB

Increasing the number of levels increases the SNR.

Example 4.12

Example 4.13A telephone subscriber line must have an SNR dB above 40. What i sthe minimum number of bits per sample?

SolutionWe can calculate the number of bits as

Telephone companies usually assign 7 or 8 bits per sample.


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Figure 4.27 Components of a PCM decoder

Example 4.15

We have a low-pass analog signal of 4 kH z. I f we send the analogsignal, we need a channel with a minimum bandwidth of 4 kH z. I fwe digiti ze the signal and send 8 bits per sample, we need a channelwith a mini mum bandwidth of 8 4 kH z = 32 kH z.


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Probability and random signals

For random signal, we use the statistical andprobabilistic methods to describe a signal sincethere is no definite expression for such randomsignals.

What is a random variable?


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A Random Variable is a function, which assigns unique numerical values to allpossible outcomes of a random experiment under fixed conditions

OR A random variable is not a variable but rather a function that maps events to

numbers

A random variable X is a function that assign a real number, X( ) to each outcome in the sample space S of a random experiment.

Domain of the random variable -- S Range of the random variable -- S x

S

x

X()



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Example 1 : Suppose that a coin is tossed 3 times and the sequence of headsand tails is noted.

Sample space S={HHH,HHT,HTH,HTT,THH,THT,TTH, TTT} X :number of heads in three coin tosses.

Outcome HHH HHT HTH THH HTT THT TTH TTT

X() 3 2 2 2 1 1 1 0

A discrete random variable has a countable number of possible values.

X: number of heads when trying 5 tossing of coins.The values are countable

A continuous random variable takes all values in an interval of numbers.X: the time it takes for a bulb to burn out.The values are not countable

S x ={0,1,2,3}

Two types of random variables


l


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Distribution Functions of a Random Variab le

A random variable is characterized by a distribution. Thedistributions associated with the discrete and continuous RVs arePMF/PDF is equal to or within an interval ; CDF is less than orequal to

Discrete random variable If the random variable is discrete , then the probability mass function (PMF)

is the probability that the discrete assumes the exact value of x; i.e., PMF:P(X = x).

Continuous random variable If the random variable is continuous , then the probability density function

(PDF) is the probability that the random variable assumes a value over aninterval or range; i.e., PDF: P(x 1 < X


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Probability of random variable

Example 2 : The event { X =k } ={k heads in three coin tosses} occurs whenthe outcome of the coin tossing experiment contains k heads.P[ X =0]=P[{TTT}]=1/8P[X=1]=P[{HTH}]+P[{THT}]+P[{TTH}]=3/8P[X=2]=P[{HHT}]+P[{HTH}]+P[{THH}]=3/8P[X=3]=P[{HHH}]=1/8



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The function p x (x) is called the probability mass function (pmf) of the discrete r.v. X .

k

k X

k X

k X

x p

k x x x p

k x p

1)(

,...)2,1(0)(

,...2,11)(0 Properties of p x (x):

If FX(x) (cdf) changes value only in jumps and is constant between jumps, i.e. F X(x)is a staircase function, then X is called a discrete random variable.


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A typical example of cdf

46

x X x pk =P[X=k] F X ( x )

-101234

{TTT}

{TTT,TTH,THT,HTT}{TTT,TTH,THT,HTT,HHT,HTH,THH}

SS

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11

Tossing a coin 3 times and counting the number of heads

Example of cdf for discrete r.v.


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Example of pmf for discrete r.v.


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What does it mean?


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The Issue of Noise

Noise is a non-predictable (i.e. random ), corrupting signal thatadds to the desired signal

For RF receiver, most of it comes from the analog circuits

that amplify and demodulate the input signal An undesired signal is a predictable , corrupting signal which

also adds to the desired signal May be called noise if it is difficult to predict

Experiment to see Statistical Distribution of Noise


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p

Create histograms of samplevalues from trials of

increasing lengths Histogram should converge

to a shape known as aprobability density function

(PDF)

Take on differentsequences fordifferent trials

Think of trials asdifferentmeasurementintervals from thesame experimentalsetup

A Histogram generates data for thenumber of occurrences of a value ina data set.

Samplevalues


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Formalizing Probability


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g y

The probability that random variable x takes on a value in therange of x 1 to x 2 is calculated from the PDF of x as:

Note that probability values are always in the range of 0 to 1 Higher probability values imply greater likelihood that the event

will occur

Example Probability Calculation


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Example Probability Calculation

Verify that overall area is 1:

Probability that x takes on a value between 0.5 and 1.0:

This shape isreferred to as auniform PDF

AWGN


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AWGN

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The noise signal has low amplitudes with high probabilitiesand high amplitudes with low probability. So the noise signal is mostly low amplitude signals.

Properties


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Normal or Gaussian PDF

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sampling and baseband modulation

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