EE 445S Real-Time Digital
Signal Processing Lab
Fall 2013
Lab #5.1
Pulse Amplitude Modulation/ BPSK
Outline





Block Diagram & Expressions of Transmitter
Block Diagram & Expressions of Receiver
Inter Symbol Interference & its Nyquist Criteria
Raised Cosine Filter
Digital Interpolation & Pulse Shaping Filter Banks
Examples
2
Block Diagram of M-PAM Transmitter
Bit Rate: Rd bits/sec
Symbol Rate: Rd/J bits/sec
Courtesy: Steven Tretters
Chapter 11 Recitation Slides
Example of
2J
Mapping
M
M
 
1
,....,
0
,....,
li d(2i1) i
2
2
3
Expressions for Transmitter
•An Impulse Modulator is

*
s
(
t)
a

(
t
kT
)

k
k


•Output ofTransmit Filter it
s
(
t)
a
g
(
t
k
T
)

k
k



•Rectangular
pulse shaped BPSK:

s
(
t
)

a
[
u
(
t

kT
)

u
(
t

(
k

1
)
T
)]

k
k


4
Block Diagram of Receiver
•Removes out of
band noise
•Forms perfect
pulse shape with Tx
•Eliminate small
deviations
Courtesy: Steven Tretters
Chapter 11 Recitation Slides
5
Expressions for Receiver
•Let us define g(t) as
g
(
t
)

g
(
t
)
*
c
(
t
)
*
g
(
t
)
T
R
•Output of receive filter is

x
(
t
)

a
g
(
tk

T
)g
(
t
)
*
v
(
t
)

k
R
k



6
Inter Symbol Interference


(
nT
)

a
g
(
nT

kT
)
The received filter output: x

k

(Assuming no additive white
Gaussian noise)
k





g
(
nT

kT
)


x
(
nT
)

g
(
0
)
a

a

n
k
 k
g
(
0
) 





n
 k


We can rewrite this as:

The condition on g(t) that needs to be satisfied for
no ISI is: g(nT
)[n]
7
Inter Symbol Interference (eye pattern)
Superimpose every two symbols on each other for several times
Binary PSK with ISI
Courtesy: http://www.answers.com/topic/intersymbolinterference
Binary PSK without ISI
Courtesy: http://www.answers.com/topic/intersymbolinterference
8
Raised Cosine Filter


s
sin(
t)cos(
 st)
g
(t)
2

s
t
2
2

t
1

4
( )2
T


s
T
for


(
1


)

2


 



T
T

s
s
s
G
(

)

1

sin
(


)
for
(
1


)



(
1


)
 


2
2
 2
2
2
 


0
elsewhere




:excess
bandwidth
factor
Frequency Domain
Time Domain
 [0,1]
Courtesy: http://en.wikipedia.org/wiki/Raised-cosine_filter
9
Square Root Raised Cosine Filter

The system should be designed in such a manner
that the combined effect of Tx filter and Rx filter
should be a Raised Cosine filter.
0
.
5
G
(

)

G
(

)

[
G
(

)]
T
R
10
Digital Interpolation Example
16 bits
44.1 kHz
4 16 bits
176.4
kHz
FIR Filter
Digital 4x Oversampling Filter


Input to Upsampler by
4
28 bits
176.4
kHz
Upsampling by 4 (denoted by 4)
Output input sample followed by 3 zeros
Four times the samples on output as input
Increases sampling rate by factor of 4
n
0
1
2
Output of Upsampler by
4
n’
0 1 2 3 4 5 6 7 8
Output of FIR Filter
FIR filter performs interpolation
0 1 2 3 4 5
Lowpass filter with stopband frequency stopband   / 4
For fsampling = 176.4 kHz, =  / 4 corresponds to 22.05 kHz
n’
6 7 8
13 - 11
Pulse Shaping Filter Bank Example


L = 4 samples per symbol
Pulse shape g[m] lasts for 2 symbols (8 samples)
bits
encoding
s[m] = x[m] * g[m]
No multiplication by zeros
L polyphase filters
…a2a1a0
↑4
s[0] = a0 g[0]
s[1] = a0 g[1]
s[2] = a0 g[2]
s[3] = a0 g[3]
…,s[4],s[0]
{g[0],g[4]}
…,a1,a0
…000a1000a0
x[m]
g[m]
s[m]
s[4] = a0 g[4] + a1 g[0]
s[5] = a0 g[5] + a1 g[1]
s[6] = a0 g[6] + a1 g[2]
s[7] = a0 g[7] + a1 g[3]
m=0
…,s[5],s[1]
{g[1],g[5]}
s[m]
…,s[6],s[2]
{g[2],g[6]}
…,s[7],s[3]
{g[3],g[7]}
Commutator
(Periodic)
Filter
Bank
13 - 12