通信系统ch1-2-Random Processes

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Representations of Band-Pass Signals (Appendix 2)
Bandwidth
If x(f) = 0, when |f|>B, call B as bandwidth Not strictly limited in frequency In practice, how to defined bandwidth? Positive frequency? Negative frequency
1. 2. If the input process to a stable linear filter is Gaussian, then the output process is also Gaussian. The set of random variables obtained by sampling a Gaussian random process at different times are jointly Gaussian. (Can be used as a definition of Gaussian process)
Thermal Noise（热噪声） （热噪声）
Arising due to random motion of electrons in a conductor Usually modeled using the Thévenin equivalent circuit or the Norton equivalent circuit Available noise power is kT∆f watts. ∆
2 T w '(t ) = ∫0 w(t ) cos(2π fct ) dt T E[ w '(t )] = 0
2 T T σ = ∫ ∫ RW (t1 , t2 ) cos(2π f c t1 ) cos(2π f c t2 ) dt1dt2 T 0 0 N0 2 T T = ∫0 ∫0 δ (t1 − t2 ) cos(2π fct1 ) cos(2π fct2 ) dt1dt2 2 T N0 2 T N0 2 = ∫0 cos (2π fct ) dt = 2 2 T
1 f X ( t1 ), L, X ( tn ) ( x1 , L , xn ) = exp − (x − µ)T ∑ −1 (x − µ) (2π ) n / 2 ∆1/ 2 2 1
Determinant of ∑ Mean vector Covariance matrix
B
Figure 1.17 Characteristics of low-pass filtered white noise. (a) Power spectral density. (b) Autocorrelation function. 8
Example 1.11 Correlation of White Noise with a Sinusoidal Wave
Yi are normalized version of Xi
Yi = (Xi - µx) / σX, i = 1, 2, …
The central limit theorem:
1 VN = N
∑Y
i =1
N
i
approaches Ν (0,1) as N → ∞
2
Properties of a Gaussian Process
∫
)1/ 2
| g (t ) |2 df
Trms *Wrms
1 ≥ 4π
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Only gaussian pulse can get equality sign
Noise equivalent bandwidth
noise-equivalent bandwidth for a low-pass filter
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Bandwidth
Metric
Null-to-null bandwidth: Main lobe sinc(t) 3-dB bandwidth: ½ power Root-mean square (rms) bandwidth:
Wrms
∫ =(
f | G ( f ) | df
2
2
∫ | G( f ) |
∫ B=
∞
0
| H ( f ) | df H (0)
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2
2
Representations of Band-Pass Signals (Appendix 2.3, 2.4) Hilbert Transform: different perspective
Fourier transform: frequency-selective Hilbert Transform: phase selective 90 degree phase shifting
) df
1/2
2
11
Time-bandwidth product
(duration * bandwidth) = constant A signal cannot be strictly limited in both time and frequency
Trms = (
∫
t 2 | g (t ) |2 df
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Hilbert Transform
以信号与系统的观点看
传输函数为
h (t) =
1
ˆ (t ) = f (t ) ∗ 1 f πt
πt
时域
2011-12-16
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Hilbert Transform
频率表现形式：求H(w) 频率表现形式：
比较相似 傅立叶变换的对偶定理
1 h(t ) = πt
1 t>0 2 sgn(t ) = 0 t = 0 ⇔ jω −1 t < 0
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1.9 Noise
Shot noise（射击噪声;散弹噪声 散粒噪声 起伏噪声 ） （射击噪声 散弹噪声 散粒噪声;起伏噪声 散弹噪声;散粒噪声
Arising due to the discrete nature of current flow in some electronic devices Number of arrivers in a pre-defined interval follows Poisson distribution
g(τ ) 1 ˆ g(t) = ∫ dτ = ⊗ g(t) π −∞ t −τ πt 1
∞
ˆ g (t ) 1 ˆ g (τ ) = − ∫ dτ = − ⊗ g (t ) −∞ t − τ π πt 1
∞
ˆ G ( f ) = − j sgn( f )G ( f )
ቤተ መጻሕፍቲ ባይዱ
1 ⇔ − j sgn(ω ) πt
N 0 = kTe
Equivalent noise temperature
Figure 1.16 Characteristics of white noise. (a) Power spectral density. (b) Autocorrelation function.
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White Noise (Cont’d)
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White Noise
An idealized form of noise for noise analysis of communication systems white：all frequency ：
N0 SW ( f ) = 2
Boltzmann’s comstant
N0 RW (τ ) = δ (τ ) 2
( y − µY ) 2 fY ( y ) = exp − 2 2σ Y 2πσ Y 1
pdf of normalized Gaussian distribution Y ∼ N(0, 1):
y2 1 fY ( y ) = exp − 2π 2
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Central Limit Theorem
Independently and identically distributed (i.i.d.) random variables Xi, i = 1, 2, …
The Xi are statistically independent The Xi have the same probability distribution
Samples at different times on a white noise are uncorrelated If the white noise is Gaussian (called white Gaussian noise), the samples are also statistically independent (the ultimate randomness) As long as the bandwidth of a noise process at the input of a system is appreciably larger than that of the system itself, we may model the noise process as white noise.
Infinite average power, not phisysically
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Example 1.10 Ideal Low-Pass Filtered White Noise
N0 , −B< f <B SN ( f ) = 2 0, f >B
N0 RN (τ ) = ∫ exp( j 2π f τ ) df −B 2 = N 0 B sinc(2 Bτ )
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Properties of a Gaussian Process (Cont’d)