ENSC327

Communication Systems

26: Raised Cosine Pulse

and Eye Diagram

1

Jie Liang

School of Engineering Science

Simon Fraser University

Outline

� 6.4 Raised cosine pulse spectrum

� 6.6 Eye Diagram

2

Nyquist Pulse Shaping Condition

f

ETb

B0=1/(2Tb)-B0

� Nyquist channel and Nyquist bandwidth:

� If the bit duration is fixed at Tb, or bit rate Rb is

1/Tb, then the transmission bandwidth is

.)( ETT

kfP

b

k b

=+∑∞

−∞=

3

t

t=Tb=1/(2B0)p(t)

1/Tb, then the transmission bandwidth is

so the minimal bandwidth is half of the bit rate.

� On the other hand, if the transmission bandwidth

is fixed at BT, then

� the maximal transmitted bits per second (bit rate)

is twice of the bandwidth.

b

b

TR

TBB

2

1

2

1

0==≥

TbBR 2≤

Examples of zero ISI spectrum

� If BW of P(f) is greater than 1/(2Tb), there are

infinite possible solutions to satisfy

.)( ETT

kfP

b

k b

=+∑∞

−∞=

4

� In particular,

then the summation only involves two terms at each

frequency, which can be easily satisfied:

,/1 than less is )( ofBW theif TbfP

bT2

1

bT

1f

k b−∞=

Raised Cosine Spectrum

� For example, the previous case can be achieved by

using raised cosine function.

� If Tb = 2. Let

( )ffP π2cos12/1)( +=

11

5

( )ffP π2cos12/1)( +=1/2 1/2

Many other functions also satisfy the requirement, for example,

P(f) can be a triangle. But the cosine function has some nice properties.

Raised Cosine Spectrum

� The previous example can be generalized.

� Raised Cosine Spectrum: Let 2B0 = 1/Tb, or0

2B

EET

b=

6

frequency. off-cut :1f

Matlab functions:

rcosfir( ): FIR

rcosine( ): FIR/IIR

Example: Raised Cosine Spectrum

)(20 fPE

BRC

2B0

1

� Bandwidth is 2B0-f1

Roll-off region

7

f2B0

B0f1 2B0 –f1

0

( ):

)-(210

1

fB

ff −π

� Constant for f < f1, raised cosine for f in [f1, 2B0 – f1].

scales [f1, 2B0 – f1] to [ 0, π].

( ):

)-(2cos1

10

1

−+

fB

ffπMapped to 1 + cos(x), x in [ 0, π].

Example: Raised Cosine Spectrum

Roll-off factor, or excess-bandwidth factor (over Nyquist bw B0):

Cut-off frequency:

8

� f1 and α can be adjusted to control the trade-off between

the bandwidth and length of the impulse response.

� α > 0: more bandwidth than Nyquist, but filter is shorter.

Bandwidth:

Example: Raised Cosine Spectrum

A nice thing about raised cosine window is that its impulse response

has closed-form expression:

01)1( Bf α−=

9

t).2sinc(p(t) :00

BEa =→=

� If rolloff factor α increases

� f1 decreases

� Bandwidth increases

� But p(t) is shorter �More robust to timing error.

This is a scaled sinc function:

Root Raised Cosine Pulse

� If the raised cosine pulse is used, we have

)()()()( =∗∗= tqthtgtp

� One way to achieve this is:

Raised cosine function

10

� Given the channel H(f), use the first equation to find

transmitter filter G(f).

� Received filter is the root raised cosine.

Example: Bandwidth of T1 system

� T1 system: multiplexing 24 telephone inputs.

� Bit duration:

�� Bit rate:

� The Nyquist bandwidth is (minimal required BW):

sTb

µ647.0=

Mb/s 544.1/1 ==bbTR

11

� The Nyquist bandwidth is (minimal required BW):

�A more realistic choice is to use α = 1:

�Note: the unit of bandwidth is Hz,

and the unit of bit rate is bits/sec (b/s).

Outline

� 6.4 Raised cosine pulse spectrum

� 6.6 Eye Diagram

12

6.6 Eye Diagram

� An effective way to observe ISI

13

� Extract one or more symbol periods

� Superimpose all possible results

� Can be easily obtained by oscilloscope

cont …

14

cont …

� Easy to show on a scope

http://www.highfrequencyelectronics.com/Archives/Nov05/HFE1105_Tutorial.pdf

15

Example: sinc pulse

� If the composite filter p(t) is a sinc pulse:

t

Tbp(t)

� If only interferences from the immediate neighboring

pulses are considered: ==> 8 possibilities

� {1, 1, 1}, {1, 1, -1}, {-1, 1, 1}, {-1, 1, -1}

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� {1, 1, 1}, {1, 1, -1}, {-1, 1, 1}, {-1, 1, -1}

� {-1, -1, -1}, {-1, -1, 1}, {1, -1, -1}, {1, -1, 1}

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Example: sinc pulse

1

1.5

Details within [-Tb, Tb]

1

1.5

Superimposed results

17

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

Example: sinc pulse

Details within [-Tb/2, Tb/2]

1

1.5

If there is no noise and no

timing error, the data 1 or

18

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-1.5

-1

-0.5

0

0.5 timing error, the data 1 or

-1 can be perfected

detected at time 0.

Eye Diagram Summary

� Practical eye diagrams have some errors:

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