SYMBOL TIMING

ESTIMATION

Introduction

Timing recovery: The process of extracting clock signal at the receiver is known as timing recovery or symbol synchronization.

Need -To perform the periodic sampling at time instants = + , we require clock signal at the receiver.

Timing phase: The choice of sampling instant within the symbol interval of duration T is called the timing phase.

Need- To perform sampling receiver must know not only the frequency (1/T) at which the outputs of the demodulator are sampled , but also where to take to

take samples within each symbol interval.

Two ways to achieve symbol synchronization:

1. The transmitter and receiver are synchronized to a master clock, which provides a very precise timing signal. Used in Radio communication system.

2. The transmitter simultaneously transmit the clock frequency 1/T or multiple of 1/T along with the information signal. Used in Telephone transmission system.

adv: -simple to implement.

disad:- Some transmitter power must be allocated to Tx of clock signal.

- Fraction of channel bandwidth must be allocated for Tx of clock signal.

The clock signal can also be extracted from the received data signal. It is called self-synchronization.

Methods for self synchronization: The methods to achieve self synchronization are:

1. Decision-Directed methods.

2. Non-decision directed methods.

Maximum-Likelihood Timing Estimation: Let us consider the signal is baseband

PAM wave form, represented as

= ; + ().(1)where

s(; ) = ( )..(2) Here we are estimating the time delay . Also, we are considering the case of decision directed timing estimation, in which

the o/p of the demodulator is treated as a known transmitted sequence.

In this case the log-likelihood function has a form :() = 0

; ..(3)

Substituting eq(2) in eq(3):

() = 0 ( )

= ()where ()= 0

( )

a necessary condition for the estimate of to be the ML estimate is that

d()

=

d

0

( )

= d

() = 0

This equation suggests the implementation of tracking loop as shown in figure below:

Summation in the loop serves as loop filter whose o/p drives the VCC which controls the sampling time for the i/p to the loop.

Since the information seq. is used in estimation of the estimate is decision-directed.

The technique can be extended to carrier modulated signal formats as QAM and PSK.

Digital communication

through band limited

channels

Introduction In reality, due to the properties of the transmission medium, the underlying

transmission channel is bandlimited

So, we consider the problem of signal design when the channel is band limited to some specified bandwidth of W Hz which can be modelled as a linear filter having low pass frequency response C(f) i.e. zero for |f|>W.

Design of signal pulse is dependent upon the characteristics C(f) of the channel

We design the signal pulse g(t) in a linearly modulated (PAM, PSK, or QAM)signal, represented as

v(t)= ( )

that efficiently utilizes the total available channel bandwidth W.

Characterization of bandlimited channels Many communication channels, including telephone channels and some radio

channels, may be characterized as bandlimited linear filters having an equivalent

low pass frequency response characteristics C(f).

Then if a signal of the form = [()2] is transmitted over this channel then the received signal is

= + ()

The signal term in frequency domain is represented as V(f)C(f).

If the channel is bandlimited to W Hz then C(f)=0 for |f|>W. Due to which we limit the bandwidth of transmitted signal to W Hz.

Within the bandwidth of channel, the freq response is = |()|() .

The envelop delay characteristic is define as = 1

2

.

A channel is non-distorting or ideal within the bandwidth W if,

|C f | = , and = . ,

Where and are constants. On the other hand if channel is not following the above conditions we say that the channel distorts the V(f) in amplitude and delay.

Due to the amplitude and delay distortion caused by non-ideal channel, a succession of pulses transmitted through the channel at a rate comparable to the

bandwidth W are smeared.

Individual pulses might not be distinguishable at the receiver and we have inter-symbol interference (ISI).

For eg. The effect of delay distortion on transmitted pulse, figure a illustrates a bandlimited pulse having zeros periodically at times ,2, .

figure:

Note the received pulse for a non-ideal channel does not have zero crossings at T , 2T ,

and so on.

It is possible to compensate for the non-idealfrequency response characteristic of the channel

by use of a filter or equalizer at the receiver

The extent of ISI on telephone channel can be seen by observing the frequency response characteristics of the channel.

The corresponding impulse response duration is about 10ms.the transmittedsymbol rates on such a channel is of orderof 2500 pulses or symbols per seconds. Hence the ISI might extend over

20 -30 symbols. Figure: The channel is subjected to other impairments viz. Nonlinear distortion-due to amp & cmpander., small & very difficult to correct. Frequency offset-due to carrier equip. , removed by use of carrier recovery loop. Phase jitter-a low-index freq. modulation, can be compensated at demodulator. Impulse noise-additive disturbance, arises from switching equipment in system. Thermal noise-present at levels of 30 dB or more below the signal. The degree to which they affect the system depends on Transmission rates over

the modulation technique.

DIGITAL PAM TRANSMISSION

THROUGH BANDLIMITED CHANNELS

v(t)

Symbol

Timing

estimator

sampler

SIGNAL DESIGN FOR BAND LIMITED

CHANNELS The transmitted signal is represented as

v = ( )

where represents the information bearing sequence of symbols and g(t) is a pulse for which frequency response G(f)=0 for |f|>W.

This signal is transmitted over a bandlimited channel whose freq. response is C(f), also limited |f|

Now the received signal is passed through a filter and then sampled at a rate 1/T samples/s. Then the output of the filter is given as

= =0 ( ) + ()

Where x(t) is pulse due to response of filter to i/p h(t) ;x(t)= h(t)*()and

v(t) is response of filter to noise z(t) =z(t)*()

Now y(t) is sampled at times t=kT+ 0(transmission delay),k=0,1..then

+ 0 = =0 + 0 + + 0

= =0 + , = 0,1,2

The sample values can be expressed as:

= 0( + 1/0 =0

) + , = 0,1,2

0 is the arbitrary scale factor which is set unity. Then

= + =0

+ , = 0,1,2

The amount of ISI and noise can be viewed on oscilloscope. For PAM signals:

y(t) is displayed on vertical input with horizontal sweep rate set at 1/T. The resulting oscilloscope pattern is called eye pattern.

The effect of ISI is to cause eye to close as shown:

Two distortions are caused by ISI:

a) The eye closes => less noise margin.

b) Distort the zero crossing => Synchronization errors.

SIGNAL DESIGN FOR NO ISI

We assume that the band-limited channel has ideal frequency response characteristics i.e., = 1 . Then the pulse x(t) has a spectral characteristics X(f) = |()|2 where,

= ()2

We have to determine the spectral properties of pulse x(t). Since = + =0

+

The condition for no ISI is = = 1 = 0 (1)

0 0It is known as Nyquist pulse shaping criterion or Nyquist condition for zero ISI.

THEOREM: The necessary and sufficient condition for x(t) to satisfy

= 1 = 00 n 0

is that its Fourier transform x(f) must satisfy

+

=

PROOF. We know that x(t) is the inverse Fourier transform of X(f) given as

= ()2

At sampling instants t= , this relation becomes

= ()2

Let us break this integral as

= = (21)/2

(2+1)/2()2

Substituting =/ => =+/ Integration limits, lower limit =/1/2 => =1/2

Upper limit =/+1/2 => =+1/2 Function becomes ()=(+/) And the exponent 2=2(+/)=2, because shifting to

multiples of 2 has no effect. Recalling f as f

= = 1/2

+1/2( +/)2

= 1/2+1/2[ = ( + /)]2

= 1/2+1/2()2 (2)

Where we have defined

B(f)= = ( +/)

as B(f) is a periodic function with period 1/T.

It can be represented using Fourier coefficients {} as B(f)= =

2 (3)

Where

=T 1/2+1/2()2 (4)

Comparing equation (2) and (4) we obtain

=T()Therefore the necessary and sufficient condition for eq. (1) to be satisfied is that

=T n = 00 n 0

Substituting above in eq. (3) gives

B(f)=T

Or = ( +/) =T

This is the condition for zero ISI.

Three important cases

For the ideal band limited channel, we distinguish three cases.

Case I: 2

() consists of non-overlapping replicas of X(f) separated by 1/T as shown below

No choice for X(f) to make sure = (+)=. Therefore, No-way to design

the system for zero ISI.

Case II =1/2 or 1/=2 ,he replications of X(f) separated by 1/T are shown:

There exists only one ()= || ()=(/)/(/)=(/)

Difficulty with sinc:

A) non causal (non-realizable): can be made realizable by using a delayed version of sinc, and (0/)0

Case III >12

() Consists of overlapping replicas separated by 1/. We have numerous choices for X(f) such that B(f)=T.

A pulse spectrum that has desirable spectral properties is raised cosine spectrum whose frequency characteristics is given as where is called roll-off factor, 0 1.

The bandwidth occupied by signal beyondthe nyquist frequency 1/2T is called the Excess bandwidth, expressed as a %age of`1Nyquist frequency

Thank

you