北邮通信工程专业大三课程——《移动通信》课件

∑ [ a k – jb k ] p ( t – kT ) .
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We can pictorially view different modulation formats by plotting the different values a k – jb k can take on.
School of Telecommunications Engineering, BUPT
Complex Representations of Signals and Noise
* Examples bk ak bk bk
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BPSK: a k ∈ { ± A }, b k = 0 bk
QPSK: a k ∈ { ± A }, b k ∈ { ± A }
Example: QAM s m ( t ) = A m cos ( ω c t + θ ) + B m sin ( ω c t + θ ) . Choose: φ 1 ( t ) = 2 ---- cos ( ω c t + θ ) and φ 2 ( t ) = Ts 2 ---- sin ( ω c t + θ ) . Ts
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School of Telecommunications Engineering, BUPT
Wireless Communication Systems - Course Notes .
A Geometric View of Common Modulation Formats
Example: M-ary PSK s m ( t ) = A cos ( ω c t + θ m ) i = 0, 1, 2, …, M – 1 . = A cos ( θ m ) cos ( ω c t ) – A sin ( θ m ) sin ( ω c t ) Choose: φ 1 ( t ) = 2 ---- cos ( ω c t ) and φ 2 ( t ) = Ts ⇒ sm ( t ) = sm = 2 ---- sin ( ω c t ) . Ts
stellation. t 2 Q ( x ) = ∫ ---------- exp – --- dt 2 x 2π
School of Telecommunications Engineering, BUPT
Detection of Common Modulation Formats
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bk
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Biblioteka Baidu
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ak
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A Geometric View of Common Modulation Formats
* Consider a modulation format which sends one of M signals { s 1 ( t ), …, s M ( t ) } every T s seconds. * Define a set of D orthonormal basis functions for the space spanned by the transmitted signals. That is choose: { φ 1 ( t ), …, φ D ( t ) } such that 1) 2)
k k
⇔ gs ( t ) =
∑ [ a k p ( t – kT ) – jb k q ( t – kT ) ] .
k
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If we assume p ( t ) = q ( t ) (same pulse shape used on the I and Q channel) then gs ( t ) =
where No ------ = PSD of white Gaussian noise, 2 d min = minimum Euclidean distance between points in the signal constellation, Nd
min
= average number of points at a distance d min from a point in the con∞ 1
∫0 ∫0
Ts Ts
φ i ( t )φ j* ( t ) dt = 0 for all i ≠ j , φ i ( t ) 2 dt = 1 for all i ,
D
3)
sm ( t ) =
∑
i=1
s m, i φ ( t ) ↔ s m, i =
i
∫0
Ts
s m ( t )φ i* ( t ) dt
2 E b Q --------- for BPSK N o Pe = 2Eb π 2Q 2 - for all others --------- log 2 ( M ) sin --- M N o
Example: MPSK π 2 d min = 4 E s sin 2 --- M π = 4 E b log 2 ( M ) sin 2 --- M Nd 1 = 2 M = 2 M>2
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School of Telecommunications Engineering, BUPT
E s ( cos ( θ m )φ 1 ( t ) – sin ( θ m ) φ 2 ( t ) )
A 2 Ts - = energy per symbol. E s ( cos ( θ m ), – sin ( θ m ) ) where E s = ----------2
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BPSK d min = 2 E s = 2 E b
(Generalized Fourier Series)
School of Telecommunications Engineering, BUPT
A Geometric View of Common Modulation Formats
* Each possible transmitted symbol can be viewed as a point in a D-dimensional signal space. s m ( t ) ↔ s m = ( s m, 1, s m, 2, …, s m, D ) . * This is similar to the visual representation used in the previous slides, but allows for signals to occupy more than two dimensions. The set of M such points is referred to as a signal constellation and gives us much insight into the characteristics of a format. o o Bandwidth is roughly proportional to the number of dimensions. Bit error rate is dependent on the distance between nearest neighbors in the signal constellation.
X r(t) φ1 ( t )
∫ ( ) dt
d1 Compute dm = r – sm d2 Choose smallest dM
m = 1, 2, …, M X φD ( t )
∫ ( ) dt
rD
School of Telecommunications Engineering, BUPT
Detection of Arbitrary Modulation Formats
Complex Representations of Signals and Noise
* Now consider a general modulation format of the form s(t) =
∑ ak p ( t – kT ) cos ( ωc t ) + ∑ bk q ( t – kT ) sin ( ω c t )
8-PSK: a k + jb k ∈ exp ( j π m ⁄ 4 )
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16QAM: { a k, b k } ∈ { ± A, ± 3 A }
School of Telecommunications Engineering, BUPT
* The probability of error of the optimum receiver can generally be well approximated by
2 d min P e = Pr ( error ) ≈ N d Q ----------- min 2 N o
Wireless Communication Systems - Course Notes - .
Complex Representations of Signals and Noise
* The lowpass complex equivalent model is more than just a mathematical convenience. Many receivers being built today work by first converting the received signal down to complex baseband and then performing all the required signal processing at complex baseband.
Ts T s -, B m ---- . Note: E s = E [ s m 2 ] . s m = A m ---2 2 d min = 2 A 2 T s Es = 5 A2 Ts 2 -E d min = -5 s 8 - E = 1.265 E b = -5 b
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M=16
School of Telecommunications Engineering, BUPT
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Detection of Arbitrary Modulation Formats
* The optimum receiver (for an additive white Gaussian noise channel) will take the received signal, r ( t ) , and project it onto the signal space to produce a received vector, r . The receiver then chooses in favor of the signal vector closest to r . r1
d min =
QPSK 2 Es = 2 Eb
8PSK d min = 0.765 E s = 1.326 E b
School of Telecommunications Engineering, BUPT
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A Geometric View of Common Modulation Formats
X r(t) 2 cos ( ω c t ) X – 2 sin ( ω c t )
LPF gr ( t ) A/D DSP data out
LPF
Analog Front End
School of Telecommunications Engineering, BUPT
Wireless Communication Systems - Course Notes - .