Nyquist criterion for zero ISI

Nyquist Criterion

1. EC8501 Digital Communication
Mr.P.Gunaseakran
Assistant Professor
Ramco Institute of Technology
Rajapalayam

2. Nyquist’s Criterion for Distortion less Baseband
Binary Transmission
Is it possible to completely eliminate ISI (in principle) by
selecting a proper g(t) ?

3. Nyquist’s Criterion for Distortion less Baseband
Binary Transmission
Let P(f) = G(f)H(f)C(f).
Sample p(t) with sampling period T
b to produce P
δ(f).
From Slide 3-4, we get:



Also from Slide 3-4, we have:
∑
∞
−∞=








−=
n bb T
n
fP
T
fP
1
)(
δ
( ) 12exp)()( =−=∑
∞
−∞=n
bb
fnTjnTpfPπ
δ

4. Nyquist’s Criterion for Distortion less Baseband
Binary Transmission
This concludes that the condition for zero ISI is:



This is named the Nyquist criterion.
The overall system frequency function P (f) suffers no
ISI for samples taken at interval T
b if it satisfies the
above equation.
Notably, P(f) represents the overall accumulative effect
of transmit filter, channel response, and receive filter.
b
n b
T
T
n
fP =








−∑
∞
−∞=

5. Ideal Nyquist Channel
The simplest P(f) that satisfies Nyquist criterion is the
rectangular function:
.)()( and
2
1
||,0
2
1
||,
)(
b
b
b
b
TWPWP
T
Wf
T
WfT
fP =+−







=>
=<
=
)2(sinc
2
)2sin(
)( Wt
Wt
Wt
tp ==⇒
π
π

8. Infeasibility of Ideal Nyquist Channel
Rectangular P(f) is infeasible because:
p(t) extends to negative infinity, which means that each a
k
has already been transmitted at t = – ∞!
A system response being flat from –W to W, and zero
elsewhere is physically unrealizable.
The error margin is quite small, as a slight (erroneous)
shift in sampling time (such as, iT
b+ε), will cause a very
large ISI.
Note that p( t) decays to zero at a very slow rate of 1/|t|.

9. Infeasibility of Ideal Nyquist Channel
Examination of timing error margin
Let ∆t be the sampling time difference between
transmitter and receiver.


For simplicity, set i = 0.
∑
∞
−∞=
∆+−=∆+
k
bkb
tTkipatiTy ))(()(
∑
∑
∞
−∞=
∞
−∞=
−∆
−∆
=
−∆=∆
k b
b
k
k
bk kTtW
kTtW
a
kTtpaty
)(2
)](2sin[
)()(π
π

10. ∑
∑
∑
∞
≠
−∞=
∞
−∞=
∞
−∞=
−∆
−∆
+
∆
∆
=
−∆
∆−
=
−∆
−∆
=∆
0
0
2
)1(]2sin[
2
]2sin[
2
]2sin[)1(
2
]2sin[
)(
k
k
k
k
k
k
k
k
k
ktW
atW
tW
tW
a
ktW
tW
a
ktW
ktW
atyπ
π
π
π
ππ
π
ππ
ππ
.0 small fixedany for
2
)1(
that such }{ exists There
0
>∆∞=
−∆
−∑
∞
≠
−∞=
t
ktW
a
a
k
k
k
k
k

11. Raised Cosine Spectrum
exists.
)(
then,)( if
),( function enonnegativa For
k
k
k
f
fP
dttpt
tpδ
δ
∞<∫
∞
∞−
α
α
α

12. Raised Cosine Spectrum
We extend the bandwidth of p(t) from W to 2W, and require
that

So, the price to pay is a larger bandwidth.
One of the P (f) that satisfies the above condition is the
raised cosine spectrum.
.||for
2
1
)2()2()( Wf
W
WfPWfPfP <=++−+
Wf
WfW
W
Wf
W
Wf
W
fP
)1(||,0
)1(||)1(,
2
))1(|(|
cos1
4
1
)1(||0,
2
1
)(α
αα
α
απ
α+≥
+<≤−











 −−
+
−<≤
=

13. Raised Cosine Spectrum
The transmission bandwidth of the raised cosine spectrum
is equal to:

where
α is the rolloff factor, which is the excess bandwidth
over the ideal solution.
)1(2α+=WB
T
α
α
α

14. large || as
||
1
~
161
)2cos(
)2(sinc)(
3
222
t
t
tW
Wt
Wttp 





−
=
α
πα

15. Raised Cosine Spectrum
 consists of two terms:

The first term ensures the desired zero crossing of p( t).
The second term provides the necessary tail
convergence rate of p(t).

The special case of
α = 1 is known as the full- cosine rolloff
characteristic.







−
=
222
161
)2cos(
)2(sinc)(
tW
Wt
Wttpα
πα
22
161
)4(sinc
)(
tW
Wt
tp
−
=

16. Raised Cosine Spectrum
Useful property of full-cosine spectrum.




We have more “zero-crossing” at ± 3T
b/2, ±5T
b/2, ±7T
b/2,…
in addition to the desired ±T
b, ±2T
b, ±3T
b…
This is useful in synchronization. (Think of when
“synchronized,” the quantity should be small both at ±3T
b/2,
±5T
b/2, ±7T
b/2,… and at ±T
b, ±2T
b, ±3T
b…)
However, the price to pay for this excessive synchronization
information is to “double the bandwidth.”





≥
=
=
=





±
2,0
1,
2
1
0,1
2
i
i
i
iT
p
b

17. Reference
1.S.Haykin, “Digital Communications”, John Wiley, 2005