lecture ii introduction to digital communications following lecture iii next week: 4. … matched...
TRANSCRIPT
![Page 1: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/1.jpg)
Lecture IIIntroduction to DigitalCommunications
Following Lecture III next week:4. …Matched Filtering (…continued from L2) (ch. 2 – part 0 “Notes”)
5. Statistical Decision Theory - Hypothesis Testing(ch. 4 – part 0 “Notes”)
2. Antipodal transmission (a special case of PAM)3. Finite Energy Signal Space representations4. Matched Filtering in AWGN for PAM antipodal links• (ch. 2 – part 0 “Notes”)
V4 –just moved to L2 the slides that we did not have time for
![Page 2: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/2.jpg)
Review (and elaboration) of L1(3 slides)
![Page 3: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/3.jpg)
Target parameters for communication system design optimization
• Max Transmission rate (bit-rate, symbol-rate)• Min Error Probability (bit/symbol/block error rate, outage)• Min Power (PSD, SNR)• Min Bandwidth (max spectral efficiency bps/Hz)• Min Complexity (Cost)• Min Delay• (multi-user) Max # of users• (multi-user) Min Mutual Interference• Other parameters
![Page 4: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/4.jpg)
Pulse Modulation and Pulse Amplitude Modulation (PAM)
tt
PAM
P(t)AK
Figure 1.12:
PULSE MODULATOR
( ) ( )
( ) ( )
( )
0
( )
0
1 1
: ( )
: ( )
: ( )M M
H S t
H S t
H S t
…
…
Index
CODER (MAPPER)
Bitstream ( )( )
k
kS t Tk
( ) ( )kk
S t p tA kT
( )S t
( )( ) ( ) ( )S t A p t
![Page 5: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/5.jpg)
LTI
t
PAM
( )p t
“Single shot” @ t=0 – Isolated Pulse Amp. Mod.
0 ( )A t
0 ( )A p t
t
0 0( )A A
( )p t
0 ( )A p tt
![Page 6: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/6.jpg)
End of Review
![Page 7: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/7.jpg)
General PAM Link Analysis
SlicerAK S(t)CODER
PAM
g(t)
SAMP [T]Bits CH. Filter
b(t)
N(t)
RX Filter
f(t)
r(t) q(t) qK
TX Medium/Channel RX
Figure 1.17:
(“multi-shot” analysis in TA classes…)
( )g t
Channel Imp. Resp.
( )b t
PAMPulseshape
0( ) ~ [0, / 2]N t WGN N
( )f t
RX filter Imp. Resp.
SAMPLER
SLICER /DECISION
( )0A Single-shot:
kAMulti-shot: t kT0t
Single-shot:
![Page 8: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/8.jpg)
Antipodal transmission system analysis
![Page 9: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/9.jpg)
Antipodal digital link
LTI Filter
f(t)
Pulse Modulator
N(t)
Slicer“0” g(t)“1” -g(t)
Bitstream
1010111...
White Gaussian noise one-sided spectral density No
RXTX
Medium
Antipodal modulation: Special case of PAM modulation:modulating symbols equal +/-1
0 0( ) ( ); 1A Ag t g t
![Page 10: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/10.jpg)
A0 , A1 , ... AkBitstreamCoder
"0" -1"1" 1
PAMg(t)
s(t)Ak {-1,1}
g(t)
0
1
T
s(t)
0
1
T 2T 3T 4T
bitstream: ...1 0 1 1 0 1....
symbolstream: 1 -1 1 1 -1 1....
waveform s(t) = g(t) - g(t-T) + g(t-2T)+ g(t-3T) - g(t-4T) + g(t-5T)
Example of antipodalTransmitter – flat pulses
![Page 11: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/11.jpg)
Antipodal digital link analysis
RX Filter
f(t)
Pulse Modulator
N(t)
SlicerBitstream
1010111...
White Gaussian noise one-sidedspectral density No
" ":
" "
0
1 :
( )
( )
h t
h t
RX Filter
f(t)Medium
b(t)SlicerPAM MOD
1 q t 0q
0N t ~WGN 0,N 2
t 0
g(t)g(t)
h(t)
N(t) n t 0nf(t)
F t 0signal analysis:
noise analysis:
RXMediumTX
TX+ Medium RX
δ(t) p t
0 0q (0)p p h(t) f(t)h(t)
00 0pq n
![Page 12: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/12.jpg)
Signal propagation through the receive filter
done on the board
![Page 13: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/13.jpg)
Noise propagation through the receive filter
N(t) n t 0nf(t)
F t 02~ [0, ]nN
![Page 14: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/14.jpg)
RX Filter
f(t)
Pulse Modulator
N(t)
SlicerBitstream
1010111...
White Gaussian noise one-sidedspectral density No
" ":
" "
0
1 :
( )
( )
h t
h t
Effective gaussian channel
00 0pq n
0q -axis
0p0p 0
![Page 15: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/15.jpg)
RX Filter
f(t)
Pulse Modulator
N(t)
SlicerBitstream
1010111...
White Gaussian noise one-sidedspectral density No
" ":
" "
0
1 :
( )
( )
h t
h t
The statistics of decision
00 0pq n
-4 -2 2 4
0.1
0.2
0.3
0.4
0p0p 0
)0(0( | )p Hq )1(
0( | )p Hq
DECIDE ( )0H DECIDE ( )1H
0q0n
![Page 16: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/16.jpg)
Effective gaussian scalar channel
Self-read
![Page 17: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/17.jpg)
The gaussian-Q function
![Page 18: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/18.jpg)
The Q-function is the complementary cdf of a normalized gaussian r.v.
22
( )2
zeP z
0
Area Q t
t
Figure 1.36:
![Page 19: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/19.jpg)
The gaussian Q-function^
Gaussian integral functionor Q-function
=Prob. of “upper tail” of normalized gaussian r.v.
22
( )2
zeP z
0
Area Q t
t-4 -2 2 4
0.2
0.4
0.6
0.8
1
5 70.001 3.1{0.5, , 100. 0.022, , , 2 0 }16 .93 1
{ [0], , , [3],[1 [2 [5] ,] }][4]Q QQQ QQ
[ ]Q t
t
![Page 20: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/20.jpg)
Q(t) and some of its upper bounds
t
2z /2
tQ(t) 1/ 2π e dz
2t /2Q(t) 1/ 2π t e
2t /2Q(t) 1/2 e
Figure 1.38:
![Page 21: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/21.jpg)
The dog wagging the (gaussian) tail vs. the tail wagging the dog
22
( )2
zeP z
0
t-t
Q t Q -t 1
~ [0,1]N
![Page 22: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/22.jpg)
^
Deviation oft from the meanmeasured in units of the standard deviation
^ ^
^
0,1
Calculating cdf-s of gaussian variables with the Q-function
^ ^
done on the board
Self-study
![Page 23: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/23.jpg)
Error Probability calculation for the antipodal link
![Page 24: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/24.jpg)
Probability of error (conditioned on the “0” hypothesis)
0q-4 -2 2 4
0.1
0.2
0.3
0.4
)0(0( | )p Hq
0p
0p00n
0 ~ [0,1]Nnnoise indep.
of signal
![Page 25: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/25.jpg)
Probability of error (conditioned on the “1” hypothesis)
Self-read
![Page 26: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/26.jpg)
The two conditional Error Probabilities
(graphical representation)
0p 0p
(0)0
( )q H
p q(1)
0
( )q H
p q
Decide (0)H Decide (1)H
0
(0) (0)0Area Pr q 0 H Pr e H
(1) (1)0< Area Pr q 0 H Pr e H
0
n
Qp
![Page 27: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/27.jpg)
Total error probability (I)
on the board
![Page 28: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/28.jpg)
Total error probability (II)
on the board
![Page 29: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/29.jpg)
Total error probability (III)
This completes the error prob. eval. for the antipodal linkNext: optimize it, i.e. design the system to reduce BER
SNR=
0
neP Q
p
antipodal system
![Page 30: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/30.jpg)
Signal Spaces
![Page 31: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/31.jpg)
Signal Spaces (vector spaces of functions of time)
denotes a vector
The key vector property:Vector
Examples:(0) Arrows(i) D-tuples(ii) functions(iii) r.v.-s
![Page 32: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/32.jpg)
VECTOR SPACES Self-study
![Page 33: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/33.jpg)
Gram-Schmidt
Self-study
![Page 34: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/34.jpg)
Inner product spaces (I)
![Page 35: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/35.jpg)
Inner product examples
D
![Page 36: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/36.jpg)
Norm, energy, distanceThe norm is the “length” of a vectoror the root of its energy
![Page 37: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/37.jpg)
Norm, energy – examples
![Page 38: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/38.jpg)
Cauchy-Schwartz (C-S) inequality
Example: geometric vectors:
![Page 39: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/39.jpg)
C-S inequality – more examples
![Page 40: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/40.jpg)
Correlation coefficient
![Page 41: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/41.jpg)
Correlation coefficient - examples
![Page 42: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/42.jpg)
…back to the error probability of the antipodal link
RX Filter
f(t)
Pulse Modulator
N(t)
SlicerBitstream
1010111...
White Gaussian noise one-sidedspectral density No
" ":
" "
0
1 :
( )
( )
h t
h t
00 0pq n
![Page 43: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/43.jpg)
Total error probability (III) - revisit
Rewrite in terms of vector notation for functions:
| |
![Page 44: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/44.jpg)
Error probability optimization – antipodal transmissionFor given h(t) maximize the s-factor by selecting the receive filter f(t):
Use C-S ineq.:
C-S ineq. becomes C-S eq. (i.e. s-factor is maximized) when:
![Page 45: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/45.jpg)
The effect of scaling the receive and transmit filters
(root) SNR does not change when both signal and noise are scaledby the same factor
Constant gain (in RX) does not matter
…but transmitting more power (making h(t) larger is beneficial – though we run into limits…
s
![Page 46: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/46.jpg)
Matched filters
![Page 47: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/47.jpg)
h(-t) 0p
0t
h(t)
Filter matched to h(t)
Matched filter f(t)=C h(-t) minimizes the error probability
receive filter f(t)
C
![Page 48: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/48.jpg)
Optimal receiver for antipodal transmission is based on a matched receive filter
h(-t)
h(t)Filter matched to h(t)
h(t)
TX
g(t)
Medium
b(t)
Optimum PAM RX
Figure 1.48:
![Page 49: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/49.jpg)
Matched filter f(t)=C h(-t)
What is the value of the optimum (min.) error probability?It corresponds to the max s-factor:
…by maximizing the SNR (or s-factor):
max0
filterEnergySNR
/ 2 NoiseSpectralDensity(2s)h
N
minimizes the antipodal error probability…
![Page 50: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/50.jpg)
Optimal Error Probability for antipodal linkas a function of SNR
Pe
0 0b/N or P/ N W or SNRε
0 0
2 PbPr(e) SN N W
εQ Q Q
![Page 51: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/51.jpg)
P is the average received power (energy per unit time)
W is the bandwidth of the received signal (and the receive filter)
A bound for the optimal probability of error in terms of bandwidth and received power
Symbol rate cannot exceed twice the bandwidthEquality achieved for the so-called Nyquist pulses (see TA class)
Self-study
![Page 52: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/52.jpg)
Antipodal transmission operational point:For 10^-5 Error Probability, SNR must be 9.6 dB
0b N2EQ
Pe
= P
roba
bili
ty o
f bi
t err
or
SNR (dB) NE 0b
9.6 dB
Figure 1.41:
![Page 53: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/53.jpg)
Performance of antipodal receiver using a mismatched filter
f(-t)h(t)
Figure 1.49:
<1
Performancedegraded
Proof:
![Page 54: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/54.jpg)
Causal Matched Filter
f(t) =h(T-t)
t Th(t)
Filter matched to h(t)at time T0
p(t) p(T0)
When the input signal h(t) is causal, the impulse response h(-t) of the matched filter is non-causal.Sufficiently delaying this non-causal response turns it causal.Given the time-invariance, we must also delay our sampling instant
h(-t) 0p
t 0h(t)
Filter matched to h(t)(at time zero)
Use this receiver front-end for optimal antipodal detection
![Page 55: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/55.jpg)
Correlators vs. matched filters
![Page 56: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/56.jpg)
A correlator <h(t)| > (I) (multiply & integrate implementation)
.
Correlatorh rh(t) r t
Correlator
h(t)
r(t) rh“Multiply & Integrate”implementation
waveform in… …number out
![Page 57: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/57.jpg)
A correlator <h(t)| > (II)matched filter & sample implementation
.
0Tt
0h T -t
Correlator
r(t) 0q(T ) h rq(t)
Correlatorh rh(t) r t
![Page 58: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/58.jpg)
A correlator <h(t)| > (III).
Proof that “matched filter & sample” works as a correlator:
![Page 59: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/59.jpg)
Correlators and their implementations
Correlatorr(t) rhh(t)
Correlator
h(t)
r(t) rh
0Tt t-Th 0
Correlator
r(t) rh
0t t-h
Correlator
r(t) rh
Implementations:
“Multiply & Integrate”
“Matched-filter & Sample
“Matched-filter & Sample”
Figure 1.52:
Causal implementation
Non-causal implementation
The abstract view:
May use any of thesefor optimal antipodal detection
![Page 60: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/60.jpg)
Correlator optimization for maximum SNR the “Matched correlator” (I)
tf
Choose f(t) for maximum SNR atthe output of the correlator
Nfhfrf N(t)h(t)r(t)
2N0,WGN~ 0“Signal”
f(t) = ?
![Page 61: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/61.jpg)
What is SNR?
![Page 62: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/62.jpg)
tf f NN(t)
2N0,WGN~ 0
White Noise propagation through the Correlator
Var =?
![Page 63: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/63.jpg)
Correlator optimization for maximum SNR the “Matched correlator” (II)
tf
Choose f(t) for maximum SNR atthe output of the correlator
Nfhfrf N(t)h(t)r(t)
2N0,WGN~ 0“Signal”
![Page 64: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/64.jpg)
Correlator optimization for maximum SNR the “Matched correlator” (III)
tf
Choose f(t) for maximum SNR atthe output of the correlator
Nfhfrf N(t)h(t)r(t)
2N0,WGN~ 0“Signal”
Alternative “tricky” view:Maximize the output signal while constraining
the kernel energy to be fixed
But when the kernel energy is fixed so is the noise variance at the output!So signal is maximized, while the noise is fixed -> SNR is maximized
It all happens when f(t) is “matched” to h(t)
(any scale of f(t) will do)
Self-study
![Page 65: Lecture II Introduction to Digital Communications Following Lecture III next week: 4. … Matched Filtering ( … continued from L2) (ch. 2 – part 0 “ Notes](https://reader034.vdocuments.us/reader034/viewer/2022051416/56649e3c5503460f94b2e4ba/html5/thumbnails/65.jpg)
Optimum Antipodal PAM Receiver:The matched correlator view
b(t)δ(t)
tN
g(t) th th
Can use either theMultiply&integrate orMatched filter&sampleimplementations
The SNR is maximizedby the matched correlation receiver,yielding minimum BER
Example: Integrate&dump receiver for flat pulse-shape g(t). Must multiply by a constant and integrate for T seconds, However multiplication is irrelevant (does not affect SNR)So, just integrate for T seconds (before dumping)
RXMediumTX