Nyquist Criterion 5n4f6d

4. Nyquist Criterion for Distortionless Baseband Binary Transmission Objective: To design hT (t ) and hd (t ) under the following two conditions: (a). There is no ISI at the sampling instants (Nyquist criterion, this section ). (b). A controlled amount of ISI is allowed (correlative coding, next section)

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Design of Bandlimited Signals for Zero ISI - Nyquist criterion Recall the output of the receiving filter, sampled at t = kT, is given by y (kT ) = µbk + µ bn p( kT − nT ) + no (kT )

∑ n≠ k

Thus, in time domain, a sufficient condition for µp(t) such that it is ISI free is

1 p ( nT ) =  0

n=0 n≠0

(1)

Question. What is the condition for P(f) in order for p(t) to satisfy (1) (Nyquist, 1928)? @G. Gong

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Theorem. (Nyquist) A necessary and and sufficient condition for p(t) to satisfy (1) is that the Fourier transform P(f) satisfies

∑ n

n P( f − ) = T T

(2)

This is known as the Nyquist pulse-shaping criterion or Nyquist condition for zero ISI. Proof.

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Proof. When we sample p (t ) at On the other hand t = kT , k = 0, ± 1, ± 2,L , Pδ ( f ) = p (kT ) F (δ (t − kT )) we have the following pulses k pδ (t ) ≡ p(t ) δ (t − kT ) ( p(kT ) is constant for t.)

∑

∑ k

=

∑ p(kT )δ (t − kT ) k

The Fourier transform of pδ (t ) is given by Pδ ( f ) = F ( pδ (t ))   = F p ( kT )δ (t − kT )     k 

∑

1 = T

∑ k

k P( f − ) T

( 3)

= ∑ p(kT ) exp(− j 2πfkT ) k

= 1 ( from (1) ) ( 4) From (3) and (4), ISI free ⇔ 1 T

∑ k

k P( f − ) = 1 T

which gives the result in (2).

Investigate possible pulses which satisfy the Nyquist criterion Suppose that the channel has a bandwidth of W, then H c ( f ) = 0 for | f |> W

Since

P( f ) = H T ( f ) H c ( f ) H d ( f ) , we have

P ( f ) = 0 for | f |> W We write Z ( f ) = ∑ P ( f − n / T ) n

and distinguish the following three cases:

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Z(f)

-1/T-W

-1/T -1/T+W -W

W

1/T-W 1/T

1/T+W

f

Fig. 4.1 Z(f) for the case T < 1/(2W) Z(f)

W=

-1/T

1 2T

1/T

f

Fig. 4.2 Z(f) for the case T = 1/(2W) Z(f)

-1/T

-W

1/T-W W 1/T

-1/T+W

Fig. 4.3 Z(f) for the case T > 1/(2W)

f

T<

1 2W

T=

1 2W

1 > 2W T

1. , or (i.e., bit rate > 2W, impossible!) No choices for P(f) such that Z(f) = 0. 2.

, i.e.,

W=

1 2T

(the Nyquist rate)

In this case, if we choose T P( f ) =  0

which results in

| f| ≤ W otherwise

i.e.,

 f  P ( f ) = T ⋅ rect   2 W  

t p (t ) = sin c   T 

This means that the smallest value of T for which the transmission with zero ISI is possible is T=

1 2W

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(

R≡

1 = 2W, T

bit rate )

This is called the ideal Nyquist channel. 7

In other words,

Ideal Nyquist channel : 1 R = 2 Bo = T W = Bo ( R = Rb , T = Tb )

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Disadvantages: (a) an ideal LPF is not physically realizable. (a) Note that 1 t  p (t ) = sin c   ∝ T  |t |

Thus, the rate of convergence to zero is slow since the tails of p(t) decay as 1/|t|. Hence, a small mistiming error in sampling the output of the matched filter at the demodulator results in an infinite series of ISI components.

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1 T> 2W

1 < 2W T

3. For , i.e, , in this case, there exists numerous choices for P(f) such that Z(f) = T. The important one is so called the raised cosine spectrum. The raised cosine frequency characteristic is given by  1  2B  0  1 P( f ) =   4 B0 0  

0 ≤ f < (1 − α ) B0  π (| f | −(1 − α ) B0 )  1 + cos   (1 − α ) B0 ≤ f < (1 + α ) B0 2αB0   f ≥ (1 + α ) B0

where α ∈ [0,1] is called the rolloff factor and ( i.e.,

B0 =

@G. Gong

1 2T

) .

B0 =

R 2

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1 2B0 − B0 B0 − (1 + α ) B0 − (1 − α )B0 (1 − α ) B0 (1 + α ) B0

Z(f) = T by the following sum of three at any interval of length 2Bo:

P( f ) + P( f − 2 B0 ) + P ( f + 2 B0 ) = T P( f ) + P( f − 2 B0 ) + P( f + 2 B0 ) = T …

− B0 ≤ f ≤ B0 B0 ≤ f ≤ 3B0

f

The time response p(t), the inverse Fourier transform of P(f), is given by

cos2παB0t p (t ) = sinc 2 B0t 2 2 2 1 − 16α B0 t

(5)

This function has much better convergence property than the ideal Nyquist channel. The first factor in (5) is associated with the ideal filter, and the second factor that decreases as 1/|t|2 for large |t|. Thus

1 p (t ) ∝ 3 t @G. Gong

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Summary:

Nyquist Criteria +∞

1 P ( f − nR) = T , R = ∑ T n = −∞

-Bo

Bo

Ideal Nyquist Channel Raised Cosine Spectrum

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