Last time we talked about: Receiver structure Impact of AWGN and ISI on the transmitted signal Optimum filter to maximize SNR Matched filter receiver and Correlator receiver... Re
Trang 1Digital Communications I: Modulation and Coding Course
Period 3 - 2007 Catharina Logothetis
Lecture 4
Trang 2Last time we talked about:
Receiver structure
Impact of AWGN and ISI on the transmitted signal
Optimum filter to maximize SNR
Matched filter receiver and Correlator receiver
Trang 3Receiver job
Demodulation and sampling:
Waveform recovery and preparing the received signal for detection:
Improving the signal power to the noise power (SNR) using matched filter
Reducing ISI using equalizer
Sampling the recovered waveform
Detection:
Estimate the transmitted symbol based on the
received sample
Trang 4Receiver structure
Frequency down-conversion
Receiving filter
Equalizing filter
Threshold comparison
channel induced ISI
Baseband pulse
(test statistic) Baseband pulse
Trang 5Implementation of matched filter receiver
1 T
z
) (
*
1 T t
) (
*
t T
Bank of M matched filters
)(
)(t s T t r
z i = ∗ ∗i −
M
i =1, ,
), ,
,())(), ,
(),((z1 T z2 T z M T = z1 z2 z M
=
z
Trang 6Implementation of correlator receiver
dt t s t r
1 T z
), ,
,())(), ,
(),((z1 T z2 T z M T = z1 z2 z M
=
z
M
i =1, ,
Trang 7Today, we are going to talk about:
Detection:
Estimate the transmitted symbol based on the
received sample
Signal space used for detection
Orthogonal N-dimensional space
Signal to waveform transformation and vice versa
Trang 8Signal space
What is a signal space?
Vector representations of signals in an N-dimensional
orthogonal space
Why do we need a signal space?
It is a means to convert signals to vectors and vice versa.
It is a means to calculate signals energy and Euclidean
distances between signals.
Why are we interested in Euclidean distances between signals?
For detection purposes: The received signal is transformed to
a received vectors The signal which has the minimum
distance to the received signal is estimated as the transmitted signal.
Trang 9Schematic example of a signal space
) ,
( )
( )
( )
(
) , ( )
( )
( )
(
22 21 2
2 22 1
21 2
12 11 1
2 12 1
11 1
a a t
a t
a t
s
a a t
a t
a t
s
=
⇔ +
=
=
⇔ +
=
=
⇔ +
=
s s
ψ ψ
ψ ψ
ψ ψ
)(
1 t
ψ
)(
2 t
ψ
) ,
( 11 12
1 = a a
s
),
( 21 22
2 = a a
s
) ,
3 = a a
s
),(z1 z2
=
z
Transmitted signal
alternatives
Trang 10Signal space
To form a signal space, first we need to know the inner product between two signals
(functions):
Inner (scalar) product:
Properties of inner product:
Trang 11 Norm between two signals:
We refer to the norm between two signals as the
x E dt
t x t
x t x t
2
)()
(),()
(
)()
(t a x t
)()
(
, x t y t
= “ length ” of x(t)
Trang 12Example of distances in signal space
)(
1 t
ψ
)(
2 t
ψ
) ,
( 11 12
1 = a a
s
),
( 21 22
2 = a a
s
) ,
3 = a a
s
),(z1 z2
=
z
z s
d ,
1
z s
d ,
2
z s
d ,
3
The Euclidean distance between signals z(t) and s(t):
) (
) (
) ( )
Trang 13Orthogonal signal space
N-dimensional orthogonal signal space is characterized by
N linearly independent functions called basis
functions The basis functions must satisfy the orthogonalitycondition
ψ
ji i j
T i j
N i
j i
Trang 14Example of an orthonormal bases
Example: 2-dimensional orthonormal signal space
Example: 1-dimensional orthonornal signal space
1 )
( )
(
0 )
( ) ( )
( ), (
0 )
/ 2 sin(
2 )
(
0 )
/ 2 cos(
2 )
(
2 1
2 0
1 2
1 2 1
dt t t
t t
T t T
t T
t
T t T
t T
t
T
ψ ψ
ψ ψ
ψ ψ
π ψ
π ψ
) (
1 t
ψ
) (
2 t
ψ
0
1 ) (
ψ
) (
1 t
ψ
0
Trang 15Signal space …
Any arbitrary finite set of waveforms
where each member of the set is of duration T, can be
expressed as a linear combination of N orthonogal
s
1
)()
( ψ i =1, ,M
M
N ≤
dt t t
s K
t t
s K
a
T
j i
j
j i
Trang 16i t a t
s
1
) ( )
) (
Trang 17Example of projecting signals to an
orthonormal signal space
) ,
( )
( )
( )
(
) , ( )
( )
( )
(
22 21 2
2 22 1
21 2
12 11 1
2 12 1
11 1
a a t
a t
a t
s
a a t
a t
a t
s
=
⇔ +
=
=
⇔ +
=
s
s
ψ ψ
ψ ψ
)(
1 t
ψ
)(
2 t
ψ
) ,
( 11 12
1 = a a
s
),
( 21 22
2 = a a
s
) ,
Trang 18Signal space – cont’d
To find an orthonormal basis functions for a given
set of signals, Gram-Schmidt procedure can be
If , do not assign any basis function.
1 Renumber the basis functions such that basis is
This is only necessary if for any i in step 2
ψ
) ( / ) ( /
) ( )
( ), ( )
( )
(
i
j
j j
i i
0 ) (t ≠
d i ψi(t)= d i(t)/ d i(t)
0 ) (t =
d i
{ψ1(t), ψ2(t), , ψN(t)}
0 ) (t =
d i M
N ≤
Trang 19Example of Gram-Schmidt procedure
Find the basis functions and plot the signal space for the following
transmitted signals:
Using Gram-Schmidt procedure:
) (
1 t s
) (
2 t s
) (
) (
) ( )
(
) ( )
(
2 1
1 2
1 1
A A
t A
t s
t A
t s
ψ ψ
T A
( ) ( )
( ), (
/ ) ( /
) ( )
(
) (
1 2
1 2
1 1
1 1
0
2 2
1 1
t t
s t
t
s
A t
s E
t s t
A dt
t s E
T
T
ψ ψ
ψ
1
2
Trang 20Implementation of matched filter receiver
1 T −t
∗
ψ
) (T t
−
∗
), ,
,(z1 z2 z N
s
1
)()
Trang 21Implementation of correlator receiver
), ,
,(z1 z2 z N
s
1
)()
( ψ i =1, ,M
M
Trang 22Example of matched filter receivers using basic functions
Number of matched filters (or correlators) is reduced by 1 compared to using
2 t s
) (
Trang 23White noise in orthonormal signal space
AWGN n(t) can be expressed as
) (
~ )
( ˆ )
( t n t n t
Noise projected on the signal space
which impacts the detection process.
Noise outside on the signal space
>
=< n(t), (t)
0)
(),
(
~ >=
< n t ψ j t
)()
(
ˆ
1
t n
(n1 n2 n N
=
n
) (
ˆ t
n
independent zero-mean Gaussain random variables with variance
j j
n
1
=
2 / )
var(n j = N0