Nyquist criterion for distortion less baseband binary channel

Nyquist Criterion for Distortion-
less Baseband Binary Channel
Done by,
Priyanga.K.R,
AP/ECE

This is the diagram of the binary transmission system
From design point of view – frequency response of the channel and
transmitted pulse shape are specified; the frequency response of the transmit
and receive filters has to be determined so as to reconstruct [bk].
Done by extracting and decoding the corresponding sequence of
coefficients [ak] from the output y(t).

Reconstructing:
• extracting
▫ sampling of the output y(t) at times t = iTb
• decoding
▫ the weighted pulse contribution should be free from
ISI to take a proper logical decision
▫ the weighted pulse contribution is:
( )k b ba p iT kT
1,
( ) (4.49)
0,
b b
i k
p iT kT
i k


  
 
To be free from ISI it
has to meet the
condition:
p(0)=1 by normalization

• if the condition given in (4.49) is satisfied then
the pulse will be free from ISI and its form will
be a perfect pulse (of course not considering
the noise):
• So, the condition (4.49), formulated in the time
domain, ensures perfect symbol recovery if
there is no noise.
• In the next slides we will try to formulate this
condition in the frequency domain.
• y(ti)=µai

In the Frequency Domain:
• If we consider a sequence of samples {p[nTb]},
where n = 0, +/-1, +/-2… in the time domain, we
will have periodicity in the frequency domain
(duality property) (discusses in detail in Ch.3) which
in general can be expressed as:
( ) ( ) (4.50)b b
n
P f R P f nR


 
Rb=1/Tb
bit rate (b/s)
FT of an infinite
sequence of
samples

Pδ(f) in our case is the FT of an infinite sequence
of delta functions with period Tb, weighted by the
sample values of p(t).
So, it can also be expressed by:
( ) [ ( ) ( )]exp( 2 )
(4.51)
b bn
m
P f p mT t mT j ft dt  



  
where if we let m = i – k for i = k we have m = 0 and for i ≠
k we have correspondingly m ≠0

• further on if we impose the conditions (4.49)
on the sample values of p(t):
( ) (0) ( )exp( 2 ) (0)
(4.52)
P f p t j ft dt p  


  
1,
( ) (4.49)
0,
b b
i k
p iT kT
i k


  
  ..and using the
sifting property of
the delta function
(4.51) can be written
as:

As we have the normalizing condition p(0) = 1
substituting (4.52) in (4.50)
we finally get:
( ) (4.53)b b
n
P f nR T


 
( ) ( ) (4.50)b b
n
P f R P f nR


 
which represents the condition for zero ISI in the frequency
domain

Conclusion:
• The case when ISI is equal to zero is known as
distortion-less channel.
• We have derived the condition for distortion-less
channel both in the time (4.49) and frequency
domain (4.53) in the absence of noise
• We can formulate the Nyquist criterion for
distortion-less bandpass transmission in the
absence of noise as follows:

• The frequency function P(f) eliminates ISI for
samples taken at intervals Tb providing that it
satisfies equation (4.53).
• It is important to note that P(f) refers to the
whole system, including the transmission
filter, the channel and the receiver filter in
accordance with equation (4.47):
( ) ( ) ( ) ( ) (4.47)P f G f H f C f 
( ) (4.53)b b
n
P f nR T


 

• As it is very unlikely in real life that a channel
itself will exhibit Nyquist transfer response and
the condition for distortion-less transmission is
incorporated in the design of the filters used.
• This is also known as Nyquist channel filtering
and Nyquist channel reponse.
• Often the Nyquist filtering response needed
for zero ISI is split between Tx and Rx using a
root raised cosine filter pair (which we will
discuss later)
• Next we discuss what is understood by “Ideal
Nyquist channel”

Ideal Nyquist Channel
• If we have to design a
filter to meet the
condition in (4.53) for
recovering pulses free
from ISI
• one possible function that
we can specify for the
frequency function P(f) is
the rectangular function:
( ) (4.53)b b
n
P f nR T


 
1
,
2
( )
0, | |
1
2 2
(4.54)
W f W
W
P f
f W
f
rect
W W

  
 

 
 
  
 
rect function of
unit amplitude at f
= 0
overall
system
bandwidth
1
2 2
b
b
R
W
T
 

Figure 4.8
(a) Ideal magnitude response.
(b) Ideal basic pulse shape.

• in the time domain
this corresponds to:
• the special rate Rb is
the well known
Nyquist rate
• the W = Rb/2 is the
Nyquist bandwidth
sin(2 )
( )
2
sin (2 )
(4.56)
Wt
p t
Wt
c Wt





Conclusions:
• The ideal baseband pulse transmission system (channel)
which satisfies
▫ eq. (4.54) in the frequency domain,
▫ eq. (4.56) in the time domain
• is known as the Ideal Nyquist Channel.
• The function p(t) is regarded as the impulse response
of that channel. It is in fact the impulse response
of an ideal low-pass filter with pass-band
magnitude response 1/2W and bandwidth W.
• It has its peak at origin, and goes through zero at integer
multiples of the bit duration Tb.
• If such a waveform is sampled at t = 0, +/-Tb, +/-2Tb..
the pulses defined by µp(t-iTb) with amplitude µ will not
interfere with each other.

Figure 4.9
A series of sinc pulses corresponding to the sequence 1011010.

Unfortunately,
• This is difficult to realize in practice because:
▫ requires flat magnitude characteristic P(f) from –
W to +W, and 0 elsewhere
▫ there is no margin of error for the sampling in the
receiver (p(t) decreases slowly and decays for large
t)
• A practical solution is the raised cosine filter
mentioned before.

Raised Cosine Spectrum
• We extend the min value of W = Rb/2 to an
adjustable value between W and 2W.
• We specify a condition for the overall frequency
response P(f) .
• Specifically in the equation for the ideal
frequency response (4.53) we consider only
three terms (three harmonics) and restrict the
bandwidth to (-W, + W).

• Then (4.43) reduces to the following expression:
1
( ) ( 2 ) ( 2 ) ,
2
(4.59)
P f P f W P f W W f W
W
       
• There are several possible band-limited
functions to satisfy this equation.
• Of great practical interest is the raised cosine
spectrum whose frequency domain
characteristic is given on the next slide:

1
1 1
1
1
1
, 0 | |
2
1 (| | )
( ) 1 sin , | | 2
4 2 2
0, | | 2
(4.60)
f f
W
f W
P f f f W f
W W f
f W f


 

    
       
   
  


1
1
f
W
  
where f1 and the
band-width W are
related by
rolloff factor
indicates the
excess bandwidth
over the ideal
solution W
BT=2W-f1
=W(1+α)

Figure 4.10
Responses for different
roll-off factors.
(a) Frequency response.
(b) Time response.

Remarks:
• In Figure 4.10a we have the normalized P(f) .
Increasing the roll off factor we see that it
gradually cuts off compared to Ideal Niquist
Channel (α = 0)
• In Figure 4.10b we have p(t) which is
obtained from P(f) (4.60) using the FT.
2 2 2
cos(2 )
( ) (sin (2 )) (4.62)
1 16
Wt
p t c Wt
W t


 
  
 

2 2 2
cos(2 )
( ) (sin (2 )) (4.62)
1 16
Wt
p t c Wt
W t


 
  
 
factor characterizing the
Ideal Nyquist Channel
factor decreasing with
time,
proportional to 1/ |t2|
ensures zero crossings of p(t)
at the desired time instants
t = iT
reduces the tails of the
pulse considerably
below that of the INC

Conclusions:
• For α = 1 we have the most gradual cut offs and also the
smallest amplitude of the tails in the time domain.
• This may be interpreted as the intersymbol interference
resulting from timing error decreasing as the roll off
factor α is increased from 0 to 1.
• The special case of α = 1 is known as the full-cosine
roll off characteristic
• Its frequency response is given as:
1
1 cos , 0 | | 2
( ) (4.63)4 2
0, | | 2
f
f W
P f W W
f W
   
         
 

• correspondingly in the
time domain given as:
2 2
sin(4 )
( )
1 16
(4.64)
Wt
p t
W t


The time response has two interesting properties:
At t = ±Tb/2 = ±1/4W p(t) = 0.5 – the pulse width
measured
at half amplitude is equal to the bit duration
There are zero crossings at t = ±3Tb/2, ±5Tb/2, … in
addition to the usual zero crossings at t = ±Tb, ±2Tb….
Note: These two properties are extremely important when
extracting
a timing signal from the received signal for synchronization.
Price: Double bandwidth compared to the INC.

Nyquist criterion for distortion less baseband binary channel

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