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7/29/2019 MODEM Implementation.ppt
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3. Digital Implementation of Mo/Demodulators
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General Structure of a Mo/Demodulator
MOD
)(td
)(tx
amp
CF
)(FD
F
)(FX
FCF
)(FX
FCF
DSB
SSB
ampDEM
)( td
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SSB Re{.}
)(td )(tx
)2( tFj Ce
)(FD
F
F
)(td
)(FX
FCF
)(FD
MOD
Single Side Band (SSB) Modulator
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SSB
)(td )(tx
tFC2cos)(tdR
)(tdI
tFC2sin
where
)(Im)(
)(Re)(
tdtd
tdtd
I
R
Implementation using Real Components
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)(td )(tx
tFC2cos
)(FD
F
)(FX
FCF
DEM
LPF
Single Side Band (SSB) Demodulator
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Digital UpConverter
DUC
M
][nd ][ns
ZOH
)(txAnalog
MOD
)(ts
DISCRETE TIME CONTINUOUS TIME
IFF~ IFC FF ~
sF sMF
)(td
Single Side Band (SSB) Modulator in Discrete Time
sF
Modulator Implemented in two stages:
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Demodulator Implemented in two stages:
Digital Down
Converter
DDC
M
][nd][ns)(ty
Analog
DEM
)(ts
sMF
DISCRETE TIMECONTINUOUS TIME
IFF~ IFC FF ~
sFZOH
)(td
Single Side Band (SSB) Demodulator in Discrete Time
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DUCM
][nd ][ns
IFF~
sF sMF
( )D f
f21
( )S f
fIFf 2
121
DDC
M
][nd ][ns
sMFsF
Digital Down (DDC) and UP (DUC) Converters
F2
sMFF2
sF
kHz for voice
MHz for data
RFBaseband MHz for voice
GHz for data
000,1~MOrder of magnitude of resampling:
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if M is large, very small transition region high complexity filter
][nd ][ns
M LPF
2sF
sFFD
2
sMF
2sMF
LPF
sF
B
B
BFsM
bMF
BF
s
sf 212
Problem with Large Upsampling Factor
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M
Fs2
sFFS
2sF
2sMF
LPF
M
Fs
B
B
BM
Fs
bf MF BMF ss 212/
][nd][nsMLPF
Problem with Large Downsampling Factor
if M is large, very small transition region high complexity filter
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In order to make it more efficient we upsample inL stages
1M )(1 zH LM
][][0 ndnx ][ 22 mx ][][ mymx LL
sFF 0 1F LF
2M )(2 zH
][ 11 mx
)(zHL2F
Ls FMF
LMMMM ...21
Solution: Upsample in Stages
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][ ii mx11 iF
FiX
iFF
iX
1iF)(FHi
iF
2iF
iFB
B
BFi 1
iM ( )iH z
][ 11 ii mx
i
i
F
BF
if21
i-th Stage of Upsampling
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96 )(zH][nd
kHzF 120
)(kHzF
0FFD
4
sec/107.755
656288
6
2250
2881
152,1812
opsFN
N
f
s
MHzF 152.13
][my
This is not only a filter with high complexity, but also it is
computed at a high sampling rate.
Example: Upsample in One Stage
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2 )(1 zH 12][nd
][ 22 mx ][][ 33 mymx
kHzF 120 kHzF 241
4 )(2 zH
][ 11 mx
)(3 zH
kHzF 962 MHzF 152.13
)(kHzF
0FFD
4
3
11
2250
1
61
24812
1
10336
146
sFN
N
f
6
22
2250
2
61
96824
2
1034.1
146
sFN
N
f
6
33
11144
2250
3
14411
1152896
3
105.34
30
sFN
N
f
Total Number of operations/sec=610176.36
a 95% savings!!!!
Same Example in Three Stages
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0 ( )H z
][][0 ndnx ][ 22 mx
][][ mymx LL
sFF 0 1F
][ 11 mx
sL
FF
M
1M 1( )H z2
F2M 1( )LH z
1LF LM
0F
LMMMM ...21
Downsample in Stages
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][ ii mx
11 iF
F
iX
1( )iH F
iF
1
2iF
B
B
iF B
1( )iH z][ 11 ii mx
1
2i
i
F B
i Ff
iM
1
2iF1
2iF
i
Fi
F
X
Bi
FiF
1iF
noise
keep aliased noiseaway from signal
i-th Stage of Downsampling
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200)(zH][nd
1 12F kHz
)(kHzF
0FFD
4
12 8 12400 600
5022
9
0
600 1, 364
3.273 10 / sec
f
N
N F ops
0 2.4F MHz
][my
Example: Downsample in One Stage
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40 ( )H z][nd
3
12F kHz
)(kHzF
0FFD
4
0 2.4F MHz
][my
51( )H z 102 ( )H z
1 600F kHz 2 120F kHz
600 8 10 2400 4.05
500 22
6
0 0
4.05 10
24 10
f
N
N F
120 8 11 600 5.36
501 22
6
1 1
5.36 13
7.8 10
f
N
N F
12 8 12 120 30
502 22
6
2 2
30 68
8.16 10
f
N
N F
Total Number of operations/sec =639.96 10
a savings of almost 99% !!!
Same Example in Three Stages
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1M )(1 zH LM
][nd ][my
sFF 0 LF
1LM )(1 zHL )(zHL
Ls FMF
0 ( )H z
][nd
sFF 0 1F sL
FF
M
1M 1( )H z 2M 1( )LH z
LM0F
][my
highest rates
the highest sampling rates are close to carrier frequencies, thus very
high;
properly choose intermediate frequencies to have simple filters at
highest rates
1LF
Stages at the Highest Rates
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11 LFF
LX
1LF
wide region
LM][my
LF)(zHL
Ls FMF
BFL 1B
][1 nxL
Last Stage in UpSampling
1LF
LL FFB 12
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0 ( )H z
][nd
sFF 01M
0F
][1 mx
11 FFX
1F
wide region BF 1B
First Stage in DownSampling
BFF 210
1F
V i l L P Filt th C b I t t C d
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Very simple Low Pass Filter: the Comb Integrator Cascade
(CIC)
][][]1[][ Nnxnxnyny
these two are the same!
1
0][][
N
nxny
Notice: no multiplications!
11
1z
Nz1][ny
Comb Integrator
)1(1 ...1 Nzz][nx
][nx
same!!!
1
0
][][N
nxny
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Frequency Response of the Comb Filter
fNje
eeee
fNj
fNjfNjfNjfNj
sin2
1 2
like a comb!
fjez
Nz2
1
fN1
N2
N3
N2
N1
fNje 21
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Impulse Response of the CIC
11
1
z
z N][n ][0 mc
N
1
0
0 ][][N
mmc
][m][n][0 mc
0 1N
interpolating sequence
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The CIC in the Time Domain
11
1
z
z N][nx ][my
N
][nx ][ms
][my
][][][ Nmxms
][][][ 0 Nmcxmy
like a discrete time ZOH!
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Two Important Identities: The Noble Identities
N][nx
][][ kNmNxmy kNz
][ kNnx
N][nx
])[(][ Nkmxmy kz
][mNx Same !!!
As a consequence we have one of two Noble Identities:
N
][nx
NzH][my
N
][nx
zH][my
Same!!!
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N][nx
kz][ knx
As a consequence we have the other of the two Noble Identities:
N
][nx
NzH
][my
N
][nx
zH][my
N][nx ][1 my
kNz
][ kNmy
n
nNkNmnxkNmy ][][][1
n
nNmknxmy ][][][2
Same !!!
Other Noble Identity
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N][my][nx
11 z 111
z
1z
N1z
][nx][my
Use Noble Identity:
Very simple implementation (no multiplications):
111
z
][nx ][myN Nz1
Efficient Implementation of Upsampling CIC
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N][my][nx
11 z111
z
1z
N1z
][nx][my
Use Noble Identity:
Very simple implementation (no multiplications):
111 z
][nx ][my
NNz1
Efficient Implementation of Downsampling CIC
Frequency Response of the CIC
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Frequency Response of the CIC
Not a very good Low Pass Filter. We want a better attenuation in the
stopband!
0 0.1 0.2 0.3 0.4 0.5-25
-20
-15
-10
-5
0
5
f=F/Fs
dB
PASSf STOPf
only 13 dB attenuation
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Put M Stages together
M
MfNj
fj
fNj
Mf
fNe
e
efC
sin
sin
1
1)( )1(
2
2
1
1
1
1
MNz
z
][nx ][myN
1
1
1
MNz
z
][nx ][myN
Frequency Response:
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0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
f=F/Fs
dB
Resampling Factor N=10
2M
3M
4M
5M
WithM=4 or 5 we already get a very good attenuation.
Improved Frequency Response of CIC Filter
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0 0.1 0.2 0.3 0.4 0.5-80
-70
-60
-50
-40
-30
-20
-10
0
f=F/Fs
dB
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
-8
-6
-4
-2
0
f=F/Fs
dB
Example: M=4 Stages
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Use Noble Identity:
N][my][nx
Mz 11 M
z
11
1
1z
1z
N1z 1z
][nx ][my
1
1
1
M
z
][nx ][myN 1
MNz
Implementation of M Stage CIC Filter: Upsampling
I l i f M S CIC Fil D li
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N
][ny][nx][nxNM
N
z
z
11
1
Use Noble Identity:
N][ny][nx
Mz 11 M
z
11
1
1z
1z
N
][nx ][ny
1z 1z
Implementation of M Stage CIC Filter: Downsampling
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N][ny][nx
Mz 11 M
z
11
1
1z
1z
N
][nx ][ny
1z 1z
Now we have to be careful: the output of the integrator will easily go to
infinity
Problem: DownSampling CIC is Unstable
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CIC Implementation.
N
][ny][nx[ ]
Mx n
1
0
MN
k
k
z
]1[...]1[][][ 111 Nnxnxnxnx pppp
This implies: |][|max|][|max 1 nxNnx pp
N][ny
][nx[ ]Mx n1
0
Nk
k
z
1
0
Nk
k
z
1
0
Nk
k
z
1
[ ]x n 2[ ]x n 1[ ]px n [ ]px n
At thep stage:
and |][|max|][|max nxNnx MM
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If we use Q bits for the integrators then we need to guarantee
1max | [ ] | 2Q
M
x n
1 1max | [ ] | max | [ ] | 2 2M M L QMx n N x n N
Let the input data use L bits:
1max | [ ] | 2Lx n
][nx
Then:
NMLQ 2log
input bitsnumber of stages
decimation factor
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Application: Software Defined Radio
Definitions:
Software Defined Radio: modulation, bandwidth allocation all in software
Field Programmable Gate Array (FPGA): reprogrammable logic device which is
able to perform a number of operations in parallel. They can process data at a rate
of several 100s of MHz
DSP Chip: optimized for DSP operations by some hardwired ops (such asmultiplies).
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An HF SSB Software Defined Radio
by Dick Benson, The Mathworks,
Rec/Tr
DAC
64MHz
RF IQ
Rec.
RFIQ
Trans.
FPGA
AUDIO
AUDIO
DSP Chip
Rec.
Trans.
15.6kHz 7.8kHzsF
Transmitter:
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Transmitter:
( )x t
7.8125 kHz
2 FIR
DSP Chip
Q
AUDIO
I
2 FIR
I
Q
8 FIR 8 FIR 64 CIC
8 FIR 8 FIR 64 CIC
64SF MHz
RF
FPGA
Xilinx Library Modules
SSB
nfC2cos
nfC2sin
Receiver:
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RF CIC
CIC
64
64
FIR
FIR
8
8
FIR
FIR
8
8
I
Q
Receiver:
Xilinx Library Modules
FPGA
Q
IFIR
FIR
2
2
DSP Chip
AUDIO
nfC2sin
nfC2cos