6.02 Fall 2010
Lecture #16
• Reminders
• FD sharing
• Fourier Series and Pictures
• Mod and Demod
• New issue, sine mod -> cosine demod
• Channel Delay
Frequency Domain Sharing – Big Picture
x1 [n]
Mod
x2 [n]
..
.
Mod
xP [n]
Mod
+
X
Channel
Y
DeMod
y1 [n]
DeMod
y2 [n]
DeMod
yP [n]
• Questions
–
–
–
–
–
What
What
What
What
What
is Modulation (Mod in figure)?
do typical X’s look like?
should the Demodulator be?
is the relation between xi [n] and yi [n]
happens if there is channel delay (NEW)
..
.
Modulation by Cosine Multiplication
x1 [n]
..
.
cos -1n
xP [n]
cos -P n
x[n] =
PP
i=1 xi [n] cos -i n
What does a modulated X look like?
Can Fourier Series Help us understand
Modulation?
Periodic Assumption (with period N)
One Period
• Periodic with Period N (N even) means
– Only certain frequency complex exponentials
– x[n] can be represented with a Fourier series
Discrete-Time Fourier Series
OR
Key DTFS Modulation Identity
~
Key DTFS Modulation Identity
/ M Q .lJ.. ",.::h9/l P: re: 't '"e. i\ C'7
Et
1<
x[n]
[
"!<;J.
'1
COS
Q[n] =
<
Om
•..
L
•...
L
K
tJ
J 5l n l4t
.s
iLk
X[k]ejcnk+n.n)n
[{\J
-=
R
r
f.
• •..•
N"
+
L "Z(bJ
1
2
L
K
Q(V1
6.02 Spring 2010
.'
.'
·'tn
J
Y\Jt c,....r..-"-) p;'
J! -::,-:~~
Ffe~telE
X[klej(nk-r~'rr1,)Tl,
k=....-l<
e.-l .(Lt<- ()
i
\
t:~Q.., f\f'
.J. e,- s:t",Il'
j
z,
~ ~~"'LCK]~--\~K+.Q~)(\ -t ~
K-JS
n,
.
- Z 11.
- NM
K::.-R
-
'.
"-i
L-11 k
=:
/ .•
';'n4
•.•••
+e.. ~
k:::::::.-K
1<[(\
1·· ('l
... n.~.,,,,,
s:
1
n+ -«:»
')
?
_. eJ
k~-K
N
1
2
X[k]eJ" /(In k
.......••..
/
b~J~&.Q
k:::...,]S
Lecture 14, Slide #3
Frequency Axes Alternatives
Fourier Coefficient versus Radial Frequency
Careless about
end case
Fourier Coefficient versus Coefficient Index
Fourier Coefficient versus Sampling Frequency
DTFS of x (versus 4Mhz sampling frequency)
DTFS of modulation using
400khz and 800khz cosine multiplication
Mod with 400khz and 800khz cosines
Demod with a 400khz cosine
Demodulation using cosine multiplication
and a Low-pass Filter (Ideal Channel)
Low Pass Filter
Mod by multiplication with nearby cosines
(400khz, 520khz)
Demod of nearby cosine case (at 400khz)
Modulation/Demodulation using
cosine multiplication and I/O LPF’s
Low Pass Filter
Low Pass Filter
x1 [n]
x1 [n]
..
.
cos -1n
Low Pass Filter
..
.
cos -1n
Ideal Channel
xP [n]
Low Pass Filter
xP [n]
cos -P n
cos -P n
LPF’d before cosine multiplication case
LPF then Modulation by
Cosine Multiplication
After Demodulation by
Cosine Multiplication
Mod by LPF then Sine Multiplication,
Demod by Cosine Multiplication then LPF
Low Pass Filter
Low Pass Filter
x1 [n]
x1 [n]
Low Pass Filter
..
.
..
.
cos -1n
Ideal Channel
xP [n]
Low Pass Filter
xP [n]
cos -P n
DTFS of Mod by sine multiplication
DTFS of Demod by cos multiplication,
From Mod by sin multiplication
Mod by Cos Multiplication with Channel Delay
Low Pass Filter
Low Pass Filter
x1 [n]
x1 [n]
..
.
cos -1n
Low Pass Filter
D-sample
Delay
Channel
..
.
???
Low Pass Filter
xP [n]
xP [n]
cos -P n
???
With what shall I demodulation it dear Liza, dear Liza?
I,
}')
\\
I
\
(I, J -."
I ,
~
I..
Modv
of
X
x C~-
[f\-'().
-e..-
1'l
t,
./'
£.
--
\
I
,A
~
J
I
L..
'Qr
is
~
/~
~(~\
'7
/
)~
_
,....
Ii"-
7
--~ -
i
r~
----~--
.i->:
,...f'"
1
.
.1'
...,.---
.-
/'\
<"
/
i
~
;
jll.
..
)
I
I
I
~
""
0-
J~T
-~ ?
A
.•..•.
"'-~c.
,
.
J
-
.
\ ..
J
-
-. -
J
("JS ..HJI
/\...;- \.r r-
\
·h
I r(
\
---L Pt=
J
/
~(
f) ;~MjDr~ ~y ,
"-
t\.
"
(.....\J ~..,\(...,
/'
J LM \~c.rJ
JL
.~
7\
II
--'..,---
~ 1\
.-
Cf\\
X
-
--
X Cf\l
~vrv\1f\c.
-,
(~
I,,'
C- 05.1[.j)
-
!'Ul-Vj
2-
,
rr-:
~'
7\
J
,.
~
~
j
.-",
'\.;
I
I
~
J
i. '
J I
.
-~I~
...........
/>::
7
~\.-
I
.eli'.
~ 'I
f
(1
A
'X [~ -
~
..,
..
0
"'"
V\
11\ fi
\1 r•••••.••.•v
USA
~
_
Ls-,
--
....,
Example: Consider modulation and demodulation using
sine and cosine multiplication. The signal, x[n], is
multiplied by a cosine, then transmitted through a channel
with a D-sample delay, then multiplie d by a sine and a
cosine.
For this example, we will assume periodicity, with a N=1000
sample period. Then, the only frequencies that are periodic
with period 1000 are multiples of
As a specific case, let x[n] be given by
and be modulated by
.
Note that if the channel has a five-sample delay, then
The plots on the following slides show the impact of a fivesample channel delay on modulation and demodulation for
four different modulation frequencies (but always a fivesample delay). The different modulation frequencies will
change the phase shift introduced by the delay, and generate
different outputs from sine and cosine demodulation. In
particular, the modulation frequencies considered are 25, 50,
75 and 100 times
.
Plots of x[n] versus n, Real and Imaginary
parts of X[k] versus
Note: 0.3 for coefficient 0 and
½*0.3 for coefficients -5, -4,-3
and 3, 4, 5
Modulation using multiplication by
Cos25 Omega_1 n
Note: ½*0.3 for coefficient
25 and -25, 1/4*0.3 for
coefficients -30, -29,-28,
22,-21, -20 and 30, 29,
28, 22, 21, 20
Plots of x after modulation without and
with 5 sample delay (note periodicity!)
Modulation with Cos25 Omega_1 n
multiplication followed by 5 sample delay
Note impact of pi/4 phase
shift (peaks are 0.3*½
*sqrt(2)/2 for real and
imaginary part).
Demod by Cos25 Omega_1 n multiplication
after delay
Demod by Sin 25 Omega_1 n multiplication
after delay
Modulation by Cos50 Omega_1 n multiplication
Note: ½*0.3 for coefficient 50
and -50, 1/4*0.3 for coefficients
-55, -54, -53, -47,-46, -45 and
55, 54, 53,47,46, 45
Modulation by Cos50 Omega_1 n
multiplication followed by 5 sample delay
Note impact of pi/2 phase shift
(peaks are 0.3*½ but are now
in the imaginary part).
Demod by Cos50 Omega_1 n multiplication
after delay
Demod by Sin 50 Omega_1 n multiplication
after delay
Modulation by Cos75 Omega_1 n
multiplication
Note: ½*0.3 for coefficient
75 and -75 and 1/4*0.3 for
coefficients -80, -79,-78,
-72,-71,-70, 80, 79, 78,
72,71, 70
Modulation by Cos 75 Omega_1 n
multiplication followed by 5 sample delay
Note impact of approximately 3pi/4 phase
shift (peaks are 0.3*½*sqrt(2)/2 for real
and imaginary part, and opposite in sign
from 25 Omega_1 case.
Demod by Cos75 Omega_1 n
multiplication after delay
Demod by Sin 75 Omega_1 multiplication
after delay
Modulation by Cos100 Omega_1 n
multiplication
Note: ½*0.3 for coefficient 100
and -100, and 1/4*0.3 for
coefficients -105, -104, -103,
-97, -96, -95, 105, 104, 103,
97, 96, 95
Mod by Cos 100 Omega_1 n multiplication
followed by 5 sample delay
Note impact of pi
phase shift (peaks are
0.3* ½ and real, but
flipped in sign).
Demod by Cos100 Omega_1 n multiplication
after delay
Demod by Sin 100 Omega_1 n multiplication
after delay