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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}
16
� {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:
19