nyquist pulse shaping.pdf

7/27/2019 nyquist pulse shaping.pdf

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2008/10/26 ISI & Nyquist's Criterion 1

Baseband Pulse TransmissionBaseband Pulse TransmissionIntersymbol Interference andIntersymbol Interference and NyquistNyquistss CriterionCriterion

ChengCheng ChiaChia--HsinHsin

Department of Information & Communication EngineeringDepartment of Information & Communication Engineering

ChaoyangChaoyang University of TechnologyUniversity of Technology


Intersymbol InterferenceIntersymbol Interference

Motivations

z The next source of bit errors in a baseband-pulse transmission systemthat we wish to study is intersymbol interference (ISI).

z In practical, the communication channel isbandlimited () and

dispersive (

).z Hence, the ISI often is arisen in the practical communication systems.

z The baseband transmission of digital data is assumed to be discretepulse-amplitude-modulated (PAM) in this material.


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Baseband binary PAM ()z Suppose the incoming binary sequence {bk} consists of symbols 1 and 0,

each ofduration Tb.

z Polar form11,

01,

k

k

k

if symbol b isa

if symbol b is

+=

( )s

x t



z The signal at the source output is

z The transmitted signal is given by

z The channel introduces additional filtering, which is imposed by a filterwith transfer function H(f) and additive Gaussian noise represented byw(t).

( ) ( )s kk

bt a t k T

=

=

( ) ( ) ( )( ) k b k bk k

a t kT g t a g t k s t T

= =

= =

Bit duration

Time-domian Signal


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z Thus the channel output, or received input, is given by

z The receiver consists ofa filter with transfer function C(f) and impulse

response ofc(t), a sampler, anda decision device.

z The comparison thresholdcomparison thresholdcan be set at zero because of the symmetry of

the data and noise amplitude distributions about zero.

z The output of the receiver filter, denoted by y(t), is given by (the

derivation is summarized in the next slider)

0 0( )( ( ) ( ) ( ) ( )) x t w t where x t s t x h tt = + =

( ) ( ) ( )bk ky t p t kTa n t= +



[ ]

[ ]0( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )

s

k bk

k bk

k bk

y t x t c t x t w t c t

s t h t w t c t

x t g t h t c t w t c t

a t kT g t h t c t w t c t

a t kT p t w t c t

a p t kT n t

= = +

= +

= +

= + = +

= +

( ) ( ) ( )

( ) ( ) ( ) ( )

n t w t c t where

p t g t h t c t

=

=

is a scale factor chosen such thatp(0) = 1, n (t) is the noise component at thereceiver filter output, andp(t) is the pulse shape at the receiver filter output.

Receiver input


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z The receiver output y(t) is sampled at time ti = iTb (with i taking oninteger values), yielding

[ ]

[ ]

( ) ( ) ( )

( ) ( )

i k b i

k

i k b i

kk i

y t a p i k T n t

a a p i k T n t

=

=

= +

= + +

The contribution of the ithtransmitted bit.

1. The residual effect ofall other transmitted bits onthe detection of the ith bit.

2. This residual effect due to the occurrence of pulsebefore and after the sampling instant ti is calledintersymbol interference (ISI)

AWGN noise sample

i.e., the desired signal is a



( ) ( )k bk

s t a g t kT = 0 ( ) ( ) ( )x t s t h t=

0( ) ( ) ( )x t x t w t= +

( ) ( ) ( )k bky t a p t kT n t= +

[ ]( ) ( ) ( )i k b ikk

i

i

y t a p i k T n ta

=

= + +

( ) ( ) ( ) ( )p t g t h t c t=

144444424444443

1442443

644474448

ISIdesired signal

[ ]1,

( )0,

b

i kp i k T

i k

= =

No ISI

. ., (0) 1i e p =


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z In the absence of both ISI and noise, we observe from the previous resultthat

which shows that, under these ideal conditions, the ith transmitted bit isdecoded correctly.

z The unavoidable presence of ISI and noise in the system, however,introduces errors in the decision device at the receiver output.

z Therefore, in the design of the transmit and receive filters, the objectiveis to minimize the effects of noise and ISI and thereby deliver the digitaldata to their destination with the smallest error rate possible.

z In the following sections, we wish to determine the pulse waveformp(t)for which the ISI is completely eliminated.

( )i iy t a=


NyquistNyquistss CriterionCriterion

Designing for zero ISI: Nyquists pulse-shaping criterion

z Since the channel is fixed, the goal is to choose G(f) and C(f) tominimize the combined effects of ISI and noise on the decision process.

z The effects of ISI can be completely negated if it is possible to obtain a

received pulse shape,p

(t), with the property

z Nyquists pulse-shaping criterion: IfP(f) = F[p(t)] satisfies the condition

then

1, 0( )

0, 0b

np nT

n

==

1,

2

b

k b b

kP f T f

T T

=

+ =

1, 0( )

0, 0b

np nT

n

==

( )

For example : 0

1, 2b

b

k

P f T f T

=

=


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z Proof: Break the inverse Fourier transform integral forp(t) up intocontiguous intervals of length 1/Thertz.

z This results in the summation

(2 1) 2

(2 1) 2

2

1 2

1 2

1 2

1 2

( ) ( )

( )exp( 2 )

exp( 2 )

exp( 2 ) sinc( )

1, 0

0,

( )

0

j ft

k

T

t nT

k T

k T

Tk

T

T

kP u

T

p t P f e df

P f j fnT df

j nTu du

T j nTu du n

n

n

=

+

=

=

+

=

=

=

= =

==



Ideal Nyquist channel

z The simplest way of satisfying Nyquists criterion is to specify thefrequency P(f) to be in the form of a rectangular function, as shown by

1 2 ,( ) 0,

1rect

2 2

1

2 2b

b

W f WWP f f W

f

W W

Rwhere W

T

<

=

= =

sin(2 )( )

2

sinc(2 )

Wtp t

Wt

Wt

=

=

W(BW)R

b

BWRb/2


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An example



Practical considerations

z Although the use of the ideal Nyquist channel does indeed achieveeconomy in bandwidth in that it solves the problem of zero intersymbolinterference with the minimum bandwidth possible,

z

there are two practical difficulties that make it an undesirable objectivefor system design:

1. It requires that the magnitude characteristic ofP(f) be flat form Wto Wand zero elsewhere. This is physically unrealizable because of theabrupt transitions at the band edges W.

2. The functionp(t) decreases as 1/|t| forlarge |t|, resulting in a slow rateof decay. This also caused by the discontinuity of P(f) at W.Accordingly, there is practically no margin of error in sampling times inthe receiver.


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The effect of timing error

z Consider the sample ofy(t) at t= t, where tis the timing error.

z To simplify the exposition, the correct sampling time ti is assumed to be

zero.z In the absence of noise

[ ]

0

0

( ) ( )

sin 2 ( )

2 ( )

( 1)sin(2 )sinc(2 )

(2 )

k b

k

b

k

k b

k

k

kk

y t a p kT

W kTa

W t kT

aWa W t

W t k

t

t

t

=

=

= +

( ) ( )k bk

y t a p t kT=



Raised cosine spectrum

z We may overcome the practical difficulties encountered with the idealNyquist channel by extending the bandwidth from the minimum value W=Rb/2 to an adjustable value between Wand 2W.

z

The overall frequency responseP

(f

) to satisfy a condition more elaboratethan that for the ideal Nyquist channel

z A particular form of P(f) that embodies many desirable features isprovided by a raised cosine spectrum ()

( 2 ) ( 2 )1

( ) ,2

P f WP f Wf WP fW

W+ = + +

1

1 1

1

1

1 2 , 0

( )( ) 1 4 1 sin , 22 2

20,

W f f

f WP f W f f W f

W f

f W f

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