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
*
* *
stellation. t 2 Q ( x ) = ∫ ---------- exp – --- dt 2 x 2π
School of Telecommunications Engineering, BUPT
Detection of Common Modulation Formats
∑ [ a k – jb k ] p ( t – kT ) .
k
*
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
* *
* *
* *
*
*
ak
*
* *
ak
*
*
BPSK: a k ∈ { ± A }, b k = 0 bk
QPSK: a k ∈ { ± A }, b k ∈ { ± A }
∫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
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 rier 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.
* * * * * * * * * *
M=16
School of Telecommunications Engineering, BUPT
*
* *
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
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.
d min =
QPSK 2 Es = 2 Eb
8PSK d min = 0.765 E s = 1.326 E b
School of Telecommunications Engineering, BUPT
* *
* *
*
* *
*
*
A Geometric View of Common Modulation Formats
8-PSK: a k + jb k ∈ exp ( j π m ⁄ 4 )
* * * * * * * * ak * * * * * * * *
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
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
* *
*
*
*
BPSK d min = 2 E s = 2 E b
* * * * * *
* * * * * * * * * * * * * * * * * * 32-Cross
bk
*
*
*
*
*
*
*
*
ak
*
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)
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 - .
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
where No ------ = PSD of white Gaussian noise, 2 d min = minimum Euclidean distance between points in the signal constellation, Nd