Digital Communication 1 Chapter 4
Download
Report
Transcript Digital Communication 1 Chapter 4
KyungHee
University
Chapter 4:
Bandpass Modulation and
Demodulation/Detection
Digital Communication 1
-0-
Chapter 4
4.1 Why Modulate?
Digital Communication 1
KyungHee
University
-1-
Chapter 4
4.1 Why Modulate?
KyungHee
University
Digital modulation :
digital symbol : waveform compatible with the characteristic
of the channel
Why use carrier?
ⓐ reduce size of antenna (=3108m/fc)
e.g.) fc = 3kHz : antenna span : /4 = 25km
fc = 900 MHz : antenna diameter : /4 = 9cm
ⓑ frequency-division multiplexing
ⓒ minimize the effect of interference : spread spectrum
ⓓ place a signal in a frequency band where design
requirements are met (e.g.)RF->IF
Digital Communication 1
-2-
Chapter 4
4.1 Why Modulate?
Digital Communication 1
KyungHee
University
-3-
Chapter 4
4.2 Digital Bandpass Modulation Technique
KyungHee
University
General form of a carrier wave
s(t ) A(t ) cos (t )
(t ) 0t (t )
s(t ) A(t ) cos[0t (t )]
4.2.1 Phasor Representation of a Sinusoid
complex notation of a sinusoidal carrier wave
e j0t cos0t j sin 0t
Digital Communication 1
-4-
Chapter 4
4.2 Digital Bandpass Modulation Technique
Analytical form of transmitted waveform
j0t
e jmt e jmt
s (t ) Re e 1
2
2
KyungHee
University
(cosmt , m 0 ) ( AM )
Analytical representation of narrowband FM(NFM)
s(t) Re e j0t 1 e jmt e jmt
2
2
Digital Communication 1
-5-
Chapter 4
4.2 Digital Bandpass Modulation Technique
KyungHee
University
4.2.2 Phase Shift Keying
si (t )
i (t )
2E
cos0t i (t )
T
2i
M
0 t T
i
1
,...,
M
i 1,...,M
4.2.3 Frequency Shift Keying
si (t )
2E
cos(i t )
T
0 t T
i
1
,...,
M
4.2.4 Amplitude Shift Keying
si (t )
Digital Communication 1
2Ei (t )
cos(0t )
T
-6-
0 t T
i 1,...,M
Chapter 4
4.2 Digital Bandpass Modulation Technique
Digital Communication 1
-7-
KyungHee
University
Chapter 4
4.3 Detection of signals in Gaussian Noise
KyungHee
University
4.3.1 Decision Regions
Two-dimensional signal space (M=2)
Detector decides which of the signals s1 or s2 was transmitted,
after receiving r
=>Minimum-error decision rule chooses the signal class s.t.
d r siis minimized
distance
Decision region
Decision rule
r Region1 s1 sent
r Region2 s2 sent
Digital Communication 1
-8-
Chapter 4
4.3 Detection of signals in Gaussian Noise
KyungHee
University
4.3.2 Correlation Receiver
Received signal
Detection process
r (t ) si (t ) n(t )
0 t T,
i 1,...,M
Step 1 : Transform the waveform r(t) into a single random variable(R.V.)
Z (T ) or R.V .' Zi (T ) (i 1,...,M )
Matched filter (Correlator) maximizes SNR
T
Z i (T )
r (t ) s
i
(t ) dt
0
Step 2 : Choose waveform si(t) that has the largest correlation with r(t)
Choose the si(t) whose index corresponds to the max Zi(T)
Another detection approach (Fig.4.7.(b)) Any signal set
expressed in terms of some set of basis functions
can be
si (t ) (i 1,...,M ) j (t ) ( j 1,..., N ) where N M
Digital Communication 1
-9-
Chapter 4
4.3 Detection of signals in Gaussian Noise
KyungHee
University
Signal N symbol M
Signal N< symbol M
Ex) M-ary PSK
N=2
Digital Communication 1
- 10 -
Chapter 4
4.3 Detection of signals in Gaussian Noise
KyungHee
University
4.3.2.1 Binary Detection Threshold
Decision stage : choose the signal best matched to the
coefficients aij (with the set of output Zj(T))
Digital Communication 1
- 11 -
Chapter 4
4.3 Detection of signals in Gaussian Noise
KyungHee
University
Two conditional pdfs : likelihood of s1(s2)
p( z )
Digital Communication 1
1
p( z | s1 ) p( z | s2 )
2
- 12 -
Chapter 4
4.3 Detection of signals in Gaussian Noise
KyungHee
University
Minimum error criterion for equally likely binary signals
corrupted by Gaussian noise
For antipodal signals,
or decide s1 (t ) if z1 (T ) z2 (T )
s2 (t ) otherwise
Digital Communication 1
- 13 -
Chapter 4
4.4 Coherent Detection
KyungHee
University
4.4.1Coherent Detection of PSK(BPSK)
Coherent detector
BPSK example
s1 (t )
2E
cos(0 t ) 0 t T
T
s2 (t )
2E
cos(0 t )
T
2E
cos(0 t ) 0 t T
T
( E : signalenergy per sym bol)
Orthonormal basis function
2
cos(0t ) 0 t T
T
si (t ) ai1 1 (t )
1 (t )
s1 (t ) a11 1 (t )
E 1 (t )
s2 (t ) a21 1 (t ) E 1 (t )
Digital Communication 1
- 14 -
Chapter 4
4.4 Coherent Detection
KyungHee
University
4.4.1Coherent Detection of PSK(BPSK)
When s1(t) is transmitted, the expected values of product integrator
T
E z1 | s1 E E 12 (t ) n(t ) 1 (t ) dt
0
T
2
E z2 | s1 E E 12 (t ) n(t ) 1 (t ) dt
then, 1 (t ) cos 0t
T
0
T 2
E z1 | s1 E
E cos 2 0t n(t ) cos 0t ) dt E
0 T
T 2
E z2 | s1 E
E cos 2 0t n(t ) cos 0t )dt E
0 T
Decision stage
Choose the signal with largest value of zi(T)
Digital Communication 1
- 15 -
Chapter 4
4.4.2 Sampled Matched Filter
KyungHee
University
Example 4.1 Sampled Matched Filter Consider the BPSK waveform
set
s1 (t ) cost and s2 (t ) cost
Illustrate how a sampled matched filter or correlator can be used to detect a
received signal, say s1(t), from the BPSK Waveform set, in the absence of
noise.
(e.g. 21000, Ts 0.25m sec, T 1m sec)
Sampled MF (N samples per symbol)
Digital Communication 1
- 16 -
Chapter 4
4.4 Coherent Detection
KyungHee
University
Ex) Sampled MF (4 samples per symbol)
sampled s1
z1 (k 3) 2
s1 (T t )
sampled s2
3
z2 [k 3] s1[3 n]c2 [n]
z2 (k 3) 2
n 0
Digital Communication 1
- 17 -
Chapter 4
4.4 Coherent Detection
KyungHee
University
4.4.3 Coherent Detection of Multiple Phase Shift Keying
Signal space for QPSK(quadri-phase shift keying), M=4 (N=2)
For typical coherent MPSK system,
si (t )
Orthonormal basis function
2E
2i
cos(0t
) 0 t T , i 1,...,M
T
M
Digital Communication 1
- 18 -
1 (t )
2
cos0t , 2 (t )
T
2
sin 0t
T
Chapter 4
4.4.3 Coherent Detection of Multiple PSK
Signal can be written as
Received signal
si (t ) ai1 1 (t ) ai 2 2 (t )
2i
2i
) 1 (t ) E sin(
) 2 (t )
M
M
(0 t T , i 1,...,M )
E cos(
Demodulator
i 1
T
1 (t )
i7
Digital Communication 1
T
0
i 8
i5
r (t ) si (t ) n(t )
upper correlator : X r (t ) 1 (t )dt
2 (t )
M 8
KyungHee
University
lower correlator : Y r (t ) 2 (t )dt
0
ˆ arct an(Y X ) decision
- 19 -
i
Chapter 4
Demodulator of multiple-PSK
KyungHee
University
T
upper correlator : X r (t ) 1 (t )dt
0
T
lower correlator : Y r (t ) 2 (t )dt
0
ˆ arct an(Y X )
Digital Communication 1
- 20 -
Chapter 4
4.4.4 Coherent Detection of FSK
KyungHee
University
Typical set of FSK signal waveforms
si (t )
2E
cos(i t ) 0 t T
T
where
i 1,..., M
Orthonormal set
j (t )
E
cos j t ( j 1,...,N )
T
T
aij (t )
0
2E
2
cosi t
cos j tdt
T
T
aij (t )
E for i j
Distance between any two prototype
signal vectors is constant
d ( si , s j ) si s j
2 E for i j
0 otherwise
The ith prptotype signal vector is located on the ith coordinated axis a displacement
from origin
E
Digital Communication 1
- 21 -
Chapter 4
KyungHee
University
4.4 Coherent Detection
Example:3-ary FSK signal
i 2i M
Digital Communication 1
- 22 -
Chapter 4
4.5 Non-coherent Detection
KyungHee
University
4.5.1 Detection of Differential PSK
Non-coherent detection : actual value of the phase
of the incoming signal is not required
Tx signal: si (t )
2E
cos[0t i (t )]
T
Rx signal: r (t )
2E
cos[0t i (t ) ] n(t ) (0 t T , i 1,...,M )
T
• For coherent detection, MF is used
• For non-coherent detection, this is not possible because MF
output is a function of unknown angle α
Digital Communication 1
- 23 -
Chapter 4
4.5.1 Detection of Differential PSK
KyungHee
University
Differential encoding : information is carried by
the difference in phase between two successive
waveforms. To sent the i-th message (i=0,…,M),
the present signal must have its phase advanced by
over the previous signal
i 2i
M
Differential coherent detection : non-coherent because
it does not require a reference in phase with received
carrier Assuming that αvaries slowly relative to 2T,
phase difference is independent of α as
[k (T2 ) ] [ j (T1 ) ] k (T2 ) j (T1 ) i (T2 )
Digital Communication 1
- 24 -
Chapter 4
4.5.1 Detection of Differential PSK
KyungHee
University
DPSK Vs. PSK
DPSK : 3dB worse than PSK
PSK compares signal with clean reference
DPSK compares two noisy signals,
reducing complexity
Digital Communication 1
- 25 -
Chapter 4
4.5.2 Binary Differential PSK Example
KyungHee
University
c(k ) c(k 1) m(k ) or
c(k ) c(k 1) m(k ) (used here)
Sample index k
Original message
encoder
Differential message
Correspondng phase
1 Arbitrary setting
Digital Communication 1
decoder
- 26 -
Chapter 4
4.5 .3 Non-coherent Detection of Binary Differential FSK
KyungHee
University
• Just an energy detector without phase measurement
• Twice as many channel branches
• Quadrature receiver
Digital Communication 1
- 27 -
Chapter 4
4.5 .3 Non-coherent Detection of Binary Differential FSK
KyungHee
University
Three different cases :
1) r (t ) cos1t n(t )
2) r (t ) sin 1t n(t )
3) r (t ) cos(1t ) n(t )
Another implementation for
non-coherent FSK detection
Envelop detector :
rectifier and LPF
Looks simpler, but (analog)
filter require more complexity
Digital Communication 1
- 28 -
Chapter 4
4.5.4 Required Tone Spacing for Non-coherent
Orthogonal FSK Signaling
KyungHee
University
In order for the signal set to be orthogonal, any
pair of adjacent tones must have a frequency
separation of a multiple of 1/T[Hz] cf) Nyquist filter
si (t ) (cos 2 f i t )rect ( t )
T
1 for T t T
2
2
where rect ( t )
T
0 for t T
2
Fourier transform
F{si (t )} Tsinc( f f i )T
Digital Communication 1
- 29 -
Minimum tone separation:1/T[Hz]
Chapter 4
4.5 Non-coherent Detection : Example 4.3
cos(2 f1t ) cos 2 f 2t
⊙ Non-coherent FSK signal :
T
0
cos(2 f1t ) cos 2 f 2tdt 0 for orthogonality
KyungHee
University
where f1 f 2
f1 f 2
1
T
e.g. two tones f1 10, 000 Hz and f 2 11, 000 Hz orthogonal ?
if rate 1, 000 symvols / s, then orthogonal
if rate 1, 000 symvols / s, then not orthogonal
⊙ Coherent FSK signal :
T
0
0
f1 f 2
1
2T
cos(2 f1t ) cos 2 f 2tdt
T
T
0
0
cos cos 2 f1t cos 2 f 2tdt sin sin 2 f1t cos 2 f 2tdt
cos
sin 2 ( f1 f 2 )T
cos 2 ( f1 f 2 )T 1
sin
2 ( f1 f 2 )
2 ( f1 f 2 )
sin 2 ( f1 f 2 )T cos 2 ( f1 f 2 )T
0
2 ( f1 f 2 )
2 ( f1 f 2 )
since
Digital Communication 1
Non-coherent이면
둘 다 0이어야 함
- 30 -
f1 f 2 0
Chapter 4
4.7 Error Performance for Binary Systems
KyungHee
University
4.7.1Probability of Bit Error for Coherently Detected BPSK
Antipodal signals
s1 (t )
2E
cos(0t )
T
s2 (t )
2E
cos(0t )
T
s (t ) a11 1 (t ) E 1 (t )
1
s2 (t ) a21 1 (t ) E 1 (t )
2E
cos(0t ) 0 t T
T
Basis function
1 (t )
Decision rule is
2
cos0t for 0 t T
T
Digital Communication 1
- 31 -
s1 (t ) if z (T ) 0 0
s2 (t ) otherwise
Chapter 4
4.7 Error Performnace for Binary Systems
1
PB P( H 2 | s1 ) P( s1 ) P( H1 | s2 ) P( s2 ) P( s1 ) P ( s2 )
2
1
1
PB P( H 2 | s1 ) P( H1 | s2 )
2
2
The same
PB P( H 2 | s1 ) P( H1 | s2 ) sym m etry of pdf
priori
0
a
probability
P( z | s2 )dz
PB
KyungHee
University
( a1 a2 )
2
2
1
1 z a2
dz
exp
2 0
( a1 a2 ) 0 2
u
u
a1
a2
2
1
Q( X )
2
Digital Communication 1
- 32 -
u2
z a2
x exp 2 du, u 0
Chapter 4
4.7 Error Performnace for Binary Systems
u
PB
u
a1 a2
u2
1
exp du Q
2
2
2 0
a1
Eb , a2 Eb , Eb : signal energy per binary sym bol
( a1 a2 )
2
Since 02
N0
2
(n(t ) : white noise with PSD
PB
KyungHee
University
2 Eb N 0
N0
2
Rn ( )
2 Eb
u2
1
exp du Q
2
2
N0
Digital Communication 1
- 33 -
N0
2
( ))
Chapter 4
4.7 Error Performance for Binary Systems
KyungHee
University
Another approach (1)
Digital Communication 1
- 34 -
Chapter 4
4.7 Error Performance for Binary Systems
KyungHee
University
Another approach (2)
2
BPSK
s2
Ed 4Eb
Eb
BFSK
s2
s1
Eb
Eb
Digital Communication 1
Ed ( 2Eb ) 2
s1
Eb
- 35 -
1
Chapter 4
4.7 Error Performance for Binary Systems
KyungHee
University
Probability of bit error for several types of binary systems
TABLE 4.1 Probability of Error for Selected
Binary Modulation Schemes
Modulation
PSK(coherent)
DPSK
(dfferentially coherent)
Orthogonal FSK
(coherent)
Orthogonal FSK
(noncoherent)
Digital Communication 1
PB
2 Eb
Q
N0
E
1
exp b
2
N0
Q
Eb
N0
E
1
exp b
2
2 N0
- 36 -
Chapter 4
4.8 M-ary Signaling and Performance
KyungHee
University
4.8.2 M-ary Signaling(M=2k k:bits, M=# of waveforms)
①
M-ary orthogonal
k↑ BER↑ BW↑
M-ary PSK
k↑ BER↑
Shannon
Limit
-1.6dB
k=∞
same BW
②
(R, Eb/No, BER, BW) : fundamental “trade-off”
Digital Communication 1
- 37 -
Chapter 4
4.8.3 Vectorial View of MPSK Signaling
KyungHee
University
① (M=2k↑, the same Eb/No)
bandwidth efficiency (R/W) ↑, PB ↑
② (M=2k↑, the same PB)
bandwidth efficiency (R/W) ↑, Eb/No ↑
Digital Communication 1
- 38 -
Chapter 4
4.8.4 BPSK and QPSK : the same bit error probability
General relationship
BPSK
Eb S / 2 2W
N0
N R
QPSK
S W
NR
Digital Communication 1
KyungHee
University
QPSK = two orthogonal BPSK
channel (I stream, Q stream)
Magnitude
(A)
I stream
( A/root(2) )
Q stream
( A/root(2) )
Power/bit
Half
Half
Bit rate
Half
Half
- 39 -
Chapter 4
S REb
4.8 M-ary Signaling and Performance
If original QPSK is given by R[bps], S[watt],
each BPSK:
N WN0
KyungHee
University
Eb S 2 W S 1
N0
N 0 R 2 N 0 R
Same BER, BW efficiency : BPSK=1,QPSK=2[bit/s/Hz]
Eb/N0 vs. SNR
Eb N 0 ( Normalized SNR ):the most meaningful way of comparing one digital system with another
Eb
E
Eb
S W
S WT S WT
S 1
b
N0 N R
N 0 N log 2 M N k
N0 N k
log 2 M k
where W : Detection BW , R
: data rate, WT 1(typical )
T
T
Effect of normalized SNR : noise increases as k increases
Digital Communication 1
- 40 -
Chapter 4
4.8 M-ary Signaling and Performance
KyungHee
University
Fig. 4.34 : M-ary orthogonal signaling at PE=10-3 in
dB(decibel, nonlinear), factor(linear)
k=10 (1024-ary symbol), 20SNR(factor)→2SNR per bit(factor);
each bit require 2.
Eb
S
[dB] [dB] 10 log k
N0
N
Eb
N [dB]
0
Eb
N [dB]
0
Eb [dB]
N0
Eb [dB]
N0
: k 1( BPSK)
S
[dB] 3[dB]
: k 2(QPSK)
N
S
[dB] 4.77[dB] : k 3(8 PSK)
N
S
[dB] 10[dB] : k 10
N
S
[dB]
N
Digital Communication 1
- 41 -
Chapter 4
4.9 Symbol Error Performance for M-ary System(M>2)
KyungHee
University
4.9.4 Bit Error Probability vs. Symbol Error Probability for Multiple Phase Signaling
Assume that the symbol(011) is transmitted
If an error occur, (010) or (100) is likely→3bit errors
Gray code : neighboring symbols differ from one another in only one bit position
PE
P
E ( for PE 1), PE sym bol erroer probability
log2 M
k
Note that PE 1 (1 PB )k .
BPSK vs. QPSK
PB
PE PB
for BPSK
PE 2 PB for QPSK
Digital Communication 1
- 42 -
Chapter 4