This preview shows page 1. Sign up to view the full content.
Unformatted text preview: EE603 Class Notes 09/04/09 John Stensby Chapter 9: Commonly Used Models: NarrowBand Gaussian Noise and Shot Noise
Narrowband, widesensestationary (WSS) Gaussian noise η (t) is used often as a noise
model in communication systems. For example, η(t) might be the noise component in the output
of a radio receiver intermediate frequency (IF) filter/amplifier. In these applications, sample
functions of η (t) are expressed as η(t) = ηc (t)cos ω c t − ηs (t)sin ω c t , (91) where ωc is termed the center frequency (for example, ωc could be the actual center frequency of
the abovementioned IF filter). The quantities ηc(t) and ηs(t) are termed the quadrature components (sometimes, ηc(t) is known as the inphase component and ηs(t) is termed the
quadrature component), and they are assumed to be realvalued. Narrowband noise η (t) can be represented in terms of its envelope R(t) and phase φ(t).
This representation is given as η(t) = R(t) cos( ωc t + φ (t)) , (92) where
2
R(t) ≡ ηc (t) + ηs2(t) (93) −1 φ (t) ≡ tan (ηs (t) / ηc (t)).
Normally, it is assumed that R(t) ≥ 0 and –π < φ (t) ≤ π for all time.
Note the initial assumptions placed on η(t). The assumptions of Gaussian and WSS
behavior are easily understood. The narrowband attribute of η(t) means that ηc(t), ηs(t), R(t)
and φ(t) are lowpass processes; these lowpass processes vary slowly compared to cosωct; they Updates at http://www.ece.uah.edu/courses/ee420500/ 91 EE603 Class Notes 09/04/09 John Stensby watts/Hz Sη(ω) ωc ωc ω
Rad/Sec Fig. 91: Example spectrum of narrowband noise. are on a vastly different time scale from cosωct. Many periods of cosωct occur before there is
notable change in ηc(t), ηs(t), R(t) or φ(t).
A second interpretation can be given for the term narrowband. This is accomplished in
terms of the power spectrum of η(t), denoted as Sη(ω). By the WienerKhinchine theorem,
Sη(ω) is the Fourier transform of Rη(τ), the autocorrelation function for WSS η(t). Since η(t) is real valued, the spectral density Sη(ω) satisfies Sη ( ω) ≥ 0
(94)
Sη ( ω) = Sη ( −ω). Figure 91 depicts an example spectrum of a narrowband process. The narrowband attribute
means that Sη(ω) is zero except for a narrow band of frequencies around ± ωc ; process η(t) has a
bandwidth (however it might be defined) that is small compared to the center frequency ωc.
Power spectrum Sη(ω) may, or may not, have ±ωc as axes of local symmetry. If ωc is
an axis of local symmetry, then Sη (ω + ω c ) = Sη ( − ω + ω c ) (95) for 0 < ω < ωc, and the process is said to be a symmetrical bandpass process (Fig. 91 depicts a
symmetrical bandpass process). It must be emphasized that the symmetry stated by the second
of (94) is always true (i.e., the power spectrum is even); however, the symmetry stated by (95) Updates at http://www.ece.uah.edu/courses/ee420500/ 92 EE603 Class Notes 09/04/09 John Stensby may, or may not, be true. As will be shown in what follows, the analysis of narrowband noise is
simplified if (95) is true.
To avoid confusion when reviewing the engineering literature on narrowband noise, the
reader should remember that different authors use slightly different definitions for the crosscorrelation of jointlystationary, realvalued random processes x(t) and y(t). As used here, the
crosscorrelation of x and y is defined as Rxy(τ ) ≡ E[x(t+τ )y(t)]. However, when defining Rxy,
some authors shift (by τ ) the time variable of the function y instead of the function x.
Fortunately, this possible discrepancy is accounted for easily when comparing the work of
different authors.
η(t) has Zero Mean
The mean of η(t) must be zero. This conclusion follows directly from E[η(t)] = E[ηc (t)]cos ωc t − E[ηs (t)]sin ωc t . (96) The WSS assumption means that E[η(t)] must be time invariant (constant). Inspection of (96)
leads to the conclusion that E[ηc] = E[ηs] = 0 so that E[η] = 0.
ˆ
Quadrature Components In Terms of η and η Let the Hilbert transform of WSS noise η (t) be denoted in the usual way by the use of a
circumflex; that is, η(t) denotes the Hilbert transform of η(t) (see Appendix 9A for a discussion
of the Hilbert transform). The Hilbert transform is a linear, timeinvariant filtering operation
applied to η(t); hence, from the results developed in Chapter 7, η(t) is WSS.
In what follows, some simple properties are needed of the cross correlation of η (t) and
η(t) . Recall that η(t) is the output of a linear, timeinvariant system that is driven by η(t). Also recall that techniques are given in Chapter 7 for expressing the cross correlation between a
system input and output. Using this approach, it can be shown easily that Updates at http://www.ece.uah.edu/courses/ee420500/ 93 EE603 Class Notes 09/04/09 John Stensby ˆ
ˆ
R ηη ( τ) ≡ E[η(t + τ)η(t)] = − R η ( τ)
ˆ
ˆ
ˆ
R ηη ( τ) ≡ E[η(t + τ)η(t)] = R η ( τ)
ˆ (97) R ηη (0) = R ηη (0) = 0
ˆ
ˆ
R η ( τ) = R η ( τ) .
ˆ Equation (91) can be used to express η(t) . The Hilbert transform of the noise signal can
be expressed as η(t) = ηc (t) cos ω c t − ηs (t) sin ω c t = ηc (t) cos ω c t − ηs (t) sin ω c t (98) = ηc (t) sin ω c t + ηs (t) cos ω c t . This result follows from the fact that ωc is much higher than any frequency component in η c or
η s so that the Hilbert transform is only applied to the highfrequency sinusoidal functions (see
Appendix 9A).
The quadrature components can be expressed in terms of η and η . This can be done by
solving (91) and (98) for
ηc (t) = η(t)cos ω c t + η(t)sin ω c t (99) ηs (t) = η(t) cos ω c t − η(t)sin ω c t . These equations express the quadrature components as a linear combination of Gaussian η.
Hence, the components ηc and ηs are Gaussian. In what follows, Equation (99) will be used to
calculate the autocorrelation and crosscorrelation functions of the quadrature components. It
will be shown that the quadrature components are WSS and that ηc and ηs are jointly WSS.
Furthermore, WSS process η(t) is a symmetrical bandpass process if, and only if, ηc and ηs are Updates at http://www.ece.uah.edu/courses/ee420500/ 94 EE603 Class Notes 09/04/09 John Stensby uncorrelated for all time shifts.
Relationships Between Autocorrelation Functions Rη , Rηc and Rηs It is easy to compute, in terms of Rη, the autocorrelation of the quadrature components.
Use (99) and compute the autocorrelation
R ηc ( τ ) = E[ηc (t)ηc (t + τ )]
= E[η(t)η(t + τ )]cos ω c t cos ω c (t + τ ) + E[η(t)η(t + τ )]sin ω c t cos ω c (t + τ ) (910) + E[η(t)η(t + τ )]cos ω c t sin ω c (t + τ ) + E[η(t)η(t + τ )]sin ω c t sin ω c (t + τ ) . This last result can be simplified by using (97) to obtain
R ηc ( τ ) = R η ( τ )[cos ω c t cos ω c (t + τ ) + sin ω c t sin ω c (t + τ )]
+ R η ( τ )[cos ω c t sin ω c (t + τ ) − sin ω c t cos ω c (t + τ )] , a result that can be expressed as R ηc ( τ ) = R η ( τ ) cos ω c τ + R η ( τ )sin ω c τ . (911) The same procedure can be used to compute an identical result for R ηs ; this leads to the conclusion that R ηc ( τ) ≡ R ηs ( τ) (912) for all τ.
A somewhat nonintuitive result can be obtained from (911) and (912). Set τ = 0 in the
last two equations to conclude that Updates at http://www.ece.uah.edu/courses/ee420500/ 95 EE603 Class Notes 09/04/09 John Stensby R η (0) = R ηc (0) = R ηs (0) , (913) an observation that leads to
2
E[η2 (t)] = E[ηc (t)] = E[ηs2(t)] (914) Avg Pwr in η(t) = Avg Pwr in ηc (t) = Avg Pwr in ηs (t). The frequency domain counterpart of (911) relates the spectrums Sη , Sηc and Sηs .
Take the Fourier transform of (911) to obtain Sηc( ω) = Sηs( ω) = 1
( Sη (ω + ωc ) + Sη (ω − ωc ) )
2
− (915) 1
(sgn(ω − ωc )Sη (ω − ωc ) − sgn(ω + ωc )Sη (ω + ωc ) ) .
2 Since ηc and ηs are lowpass processes, Equation (915) can be simplified to produce
Sηc( ω) = Sηs( ω) = Sη ( ω + ωc ) + Sη (ω − ωc ), −ωc ≤ ω ≤ ωc
= 0, (916) otherwise, a relationship that is easier to grasp and remember than is (911).
Equation (916) provides an easy method for obtaining Sηc and/or Sηs given only Sη .
First, make two copies of Sη(ω). Shift the first copy to the left by ωc, and shift the second copy
to the right by ωc. Add together both shifted copies, and truncate the sum to the interval −ωc ≤ ω
≤ ωc to get Sηc . This “shift and add” procedure for creating Sηc is illustrated by Fig. 92. Given only Sη(ω), it is always possible to determine Sηc (which is equal to Sηs ) in this manner.
The converse is not true; given only Sηc , it is not always possible to create Sη(ω) (Why? Think Updates at http://www.ece.uah.edu/courses/ee420500/ 96 EE603 Class Notes 09/04/09 John Stensby Sη(ω) −ωc ωc
Sη(ω+ωc) , ⎮ω⎮ < ωc −ωc ωc
Sη(ω−ωc) , ⎮ω⎮ < ωc −ωc ωc
Sη (ω) = Sη(ω+ωc) + Sη(ω−ωc) , ⎮ω⎮ < ωc
c −ωc ωc Fig. 92: Creation of Sηc from shifting and adding copies of Sη .
about the fact that Sηc(ω ) must be even, but Sη(ω) may not satisfy (95)). The Crosscorrelation R ηc ηs
It is easy to compute the crosscorrelation of the quadrature components. From (99) it
follows that Updates at http://www.ece.uah.edu/courses/ee420500/ 97 EE603 Class Notes 09/04/09 John Stensby R ηc ηs ( τ ) = E[ηc (t + τ )ηs ( t )]
= E[η(t + τ )η(t)]cos ω c (t + τ )cos ω c t − E[η(t + τ )η(t)]cos ω c (t + τ )sin ω c t (917) + E[η(t + τ )η(t)]sin ω c (t + τ )cos ω c t − E[η(t + τ )η(t)]sin ω c (t + τ )sin ω c t . By using (97), Equation (917) can be simplified to obtain
R ηc ηs ( τ ) = R η ( τ )[− sin ω c t cos ω c (t + τ ) + cos ω c t sin ω c (t + τ )] − R η ( τ )[cos ω c t cos ω c (t + τ ) + sin ω c t sin ω c (t + τ )] , a result that can be written as R ηc ηs ( τ ) = R η ( τ )sin ω cτ − R η ( τ )cos ω cτ . (918) The crosscorrelation of the quadrature components is an odd function of τ. This follows
directly from inspection of (918) and the fact that an even function has an odd Hilbert
transform. Finally, the fact that this crosscorrelation is odd implies that R ηc ηs (0) = 0; taken at
the same time, the samples of ηc and ηs are uncorrelated and independent. However, as discussed below, the quadrature components ηc(t1) and ηs(t2) may be correlated for t1 ≠ t2.
The autocorrelation Rη of the narrowband noise can be expressed in terms of the
autocorrelation and crosscorrelation of the quadrature components ηc and ηs . This important
result follows from using (911) and (918) in
R ηc ( τ )cos ω c τ + R ηc ηs ( τ )sin ω cτ = R η ( τ )cos ω cτ + R η ( τ )sin ω cτ cos ω cτ (919) + R η ( τ )sin ω c τ − R η ( τ ) cos ω c τ sin ω c τ . Updates at http://www.ece.uah.edu/courses/ee420500/ 98 EE603 Class Notes 09/04/09 John Stensby However, Rη results from simplification of the right hand side of (919), and the desired
relationship R η ( τ ) = R ηc ( τ ) cos ω c τ + R ηc ηs ( τ )sin ω cτ (920) follows.
Comparison of (916) with the Fourier transform of (920) reveals an “unsymmetrical”
aspect in the relationship between Sη , Sηc and Sηs . In all cases, both Sηc and Sηs can be
obtained by simple translations of Sη as is shown by (916). However, in general, Sη cannot be
expressed in terms of a similar, simple translation of Sηc (or Sηs ), a conclusion reached by
inspection of the Fourier transform of (920). But, as shown next, there is an important special
case where R ηc ηs( τ) is identically zero for all τ, and Sη can be expresses as simple translations
of Sηc . Symmetrical Bandpass Processes
Narrowband process η(t) is said to be a symmetrical bandpass process if Sη (ω + ω c ) = Sη ( − ω + ω c ) (921) for 0 < ω < ωc. Such a bandpass process has its center frequency ωc as an axis of local
symmetry. In nature, symmetry usually leads to simplifications, and this is true of Gaussian
narrowband noise. In what follows, we show that the local symmetry stated by (921) is
equivalent to the condition R ηc ηs ( τ ) = 0 for all τ (not just at τ = 0).
The desired result follows from inspecting the Fourier transform of (918); this transform
is the cross spectrum of the quadrature components, and it vanishes when the narrowband
process has spectral symmetry as defined by (921). To compute this cross spectrum, first note
the Fourier transform pairs Updates at http://www.ece.uah.edu/courses/ee420500/ 99 EE603 Class Notes 09/04/09 John Stensby R η ( τ ) ↔ Sη ( ω )
(922)
R η ( τ ) ↔ − jSgn( ω )Sη ( ω ) , where R +1

Sgn( ω ) ≡ S
 −1
T for ω > 0
(923)
for ω < 0 is the commonly used “sign” function. Now, use Equation (922) and the Frequency Shifting
Theorem to obtain the Fourier transform pairs R η ( τ) sin ω cτ ↔ 1
S η ( ω − ω c ) − Sη ( ω + ω c )
2j (924)
R η ( τ) cos ω cτ ↔ 1
Sgn(ω − ω c ) Sη (ω − ω c ) + Sgn(ω + ω c ) Sη (ω + ω c ) .
2j Finally, use this last equation and (918) to compute the cross spectrum Sηc ηs (ω ) = F [ R ηc ηs ( τ )]
1
=
Sη (ω − ω c )[1 − Sgn(ω − ω c )] − Sη ( ω + ω c )[1 + Sgn( ω + ω c )] .
2j (925) Figure 93 depicts example plots useful for visualizing important properties of (925).
From parts b) and c) of this plot, note that the products on the righthand side of (925) are low
pass processes. Then it is easily seen that Updates at http://www.ece.uah.edu/courses/ee420500/ 910 EE603 Class Notes 09/04/09 a) John Stensby Sη(ω) UL
2ωc LU
ωc ωc 2ω c Sη(ωωc) b)
1Sgn(ωωc)
2ωc ωc UL ωc LU
2ω c Sη(ω+ωc) c) 1+Sgn(ω+ωc)
UL
2ωc LU
ωc ωc 2ω c Figure 93: Symmetrical bandpass processes have ηc(t1) and ηs(t2)
uncorrelated for all t1 and t2.
0
,
⎧
⎪
⎪
Sηcηs ( ω) = ⎨ − j[ Sη ( ω − ωc ) − Sη (ω + ωc )],
⎪
⎪
0
,
⎩ ω > ωc
−ωc < ω < ωc (926) ω < −ωc . Finally, note that Sηc ηs ( ω ) = 0 is equivalent to the narrowband process η satisfying the
symmetry condition (921). Since the cross spectrum is the Fourier transform of the crosscorrelation, this last statement implies that, for all t1 and t2 (not just t1 = t2), ηc(t1) and ηs(t2) are
uncorrelated if and only if (921) holds. On Fig. 93, symmetry implies that the spectral
components labeled with U can be obtained from those labeled with L by a simple folding
operation.
System analysis is simplified greatly if the noise encountered has a symmetrical Updates at http://www.ece.uah.edu/courses/ee420500/ 911 EE603 Class Notes 09/04/09 John Stensby spectrum. Under these conditions, the quadrature components are uncorrelated, and (920)
simplifies to R η ( τ ) = R ηc ( τ ) cos ω c τ . (927) Also, the spectrum Sη of the noise is obtained easily by scaling and translating Sηc ≡ F [R ηc ]
as shown by 1
Sη (ω ) = [Sηc ( ω − ω c ) + Sηc ( ω + ω c )] .
2 (928) This result follows directly by taking the Fourier transform of (927). Hence, when the process
is symmetrical, it is possible to express Sη in terms of a simple translations of Sηc (see the
comment after (920)). Finally, for a symmetrical bandpass process, Equation (916) simplifies
to
Sηc(ω ) = Sηs(ω ) = 2 Sη (ω + ω c ), −ω c ≤ ω ≤ ω c
= 0, . (929) otherwise Example 91: Figure 94 depicts a simple RLC bandpass filter that is driven by white Gaussian
noise with a double sided spectral density of N0/2 watts/Hz. The spectral density of the output is
L C
+ S(ω) = N0/2 +
watts/Hz
(WGN) R η Figure 94: A simple bandpass filter driven by white Gaussian
noise (WGN).
Updates at http://www.ece.uah.edu/courses/ee420500/ 912 EE603 Class Notes 09/04/09 John Stensby given by Sη (ω ) = 2 N0
2α 0 ( jω )
2N
H bp ( jω ) = 0
,
2
2
2 ( α 0 + jω )2 + ω c (930) where α0 = R/2L, ωc = (ωn2  α02)1/2 and ωn = 1/(LC)1/2. In this result, frequency can be
normalized, and (930) can be written as Sη (ω ′ ) = 2 N0
2α ′ ( jω ′ )
0
,
2 ( α ′ + jω ′ ) 2 + 1
0 (931) ′
where α0′ = α0/ωc and ω′ = ω/ωc. Figure 95 illustrates a plot of the output spectrum for α o = .5;
′
note that the output process is not symmetrical. Figure 96 depicts the spectrum for α o = .1 (a
′
′
much “sharper” filter than the α o = .5 case). As the circuit Q becomes large (i.e., α o becomes
small), the filter approximates a symmetrical filter, and the output process approximates a
symmetrical bandpass process. Envelope and Phase of NarrowBand Noise
Zeromean quadrature components ηc(t) and ηs(t) are jointly Gaussian, and they have the
Sη(ω) 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 2 1 0 1
2
ω′ (radians/second) ′
Figure 95: Output Spectrum for α o = .5
Updates at http://www.ece.uah.edu/courses/ee420500/ Sη(ω) 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 2 1 0 1
2
ω′ (radians/second) ′
Figure 96: Output Spectrum for α o = .1
913 EE603 Class Notes 09/04/09 same variance σ2 = R η ( 0) = R ηc ( 0) = R ηs ( 0) . John Stensby Also, taken at the same time t, they are independent. Hence, taken at the same time, processes ηc(t) and ηs(t) are described by the joint
density LM− ηc2 + ηs2 OP .
f ( ηc , ηs ) =
exp
2πσ 2
NM 2σ2 QP
1 (932) We are guilty of a common abuse of notation. Here, symbols ηc and ηs are used to denote
random processes, and sometimes they are used as algebraic variables, as in (932). However,
always, it should be clear from context the intended use of ηc and ηs.
The narrowband noise signal can be represented as
η(t) = ηc (t) cos ωc t − ηs (t) sin ωc t
= Γ1 (t) cos(ωc t + ϕ1 (t)) (933) where 2
2
Γ1 (t) = ηc (t) + ηs (t) ⎛ η (t) ⎞
ϕ1 (t) = Tan −1 ⎜ s ⎟ , π<ϕ1 ≤ π ,
⎝ ηc (t) ⎠ (934) are the envelope and phase, respectively. Note that (934) describes a transformation of ηc(t)
and ηs(t). The inverse is given by
ηc = Γ1 cos(ϕ1 )
ηs = Γ1 sin(ϕ1 ) Updates at http://www.ece.uah.edu/courses/ee420500/ (935) 914 EE603 Class Notes 09/04/09 John Stensby The joint density of Γ1 and ϕ1 can be found by using standard techniques. Since (935) is
the inverse of (933) and (934), we can write f (Γ1, ϕ1 ) = f (ηc , ηs ) det ∂ (ηc , ηs )
∂ (Γ1, ϕ1 ) ηc =Γ1 cos ϕ1 (936) ηs =Γ1 sinϕ1 ∂ (ηc , ηs ) ⎡cos ϕ1
=
⎣
∂ (Γ1, ϕ1 ) ⎢ sinϕ1 −Γ1sinϕ1 ⎤
Γ1 cos ϕ1 ⎥
⎦ (again, the notation is abusive). Finally, substitute (932) into (936) to obtain f ( Γ1, ϕ1 ) = = Γ1 ⎡1
⎤
exp ⎢ − 2 Γ12 (sin 2 ϕ1 + cos2 ϕ1 ) ⎥
2πσ2
⎣ 2σ
⎦ . (937) Γ1 ⎡1
⎤
exp ⎢ − 2 Γ12 ⎥ ,
2
2πσ
⎣ 2σ
⎦ for Γ1 ≥ 0 and π < ϕ1 ≤ π. Finally, note that (937) can be represented as f ( Γ1, ϕ1 ) = f ( Γ1 )f(ϕ1 ) , (938) where LM
N OP
Q 1
Γ
f ( Γ1 ) = 1 exp − 2 Γ12 U( Γ1 )
2σ
σ2 (939) describes a Rayleigh distributed envelope, and f(ϕ1 ) = 1
,  π < ϕ1 ≤ π
2π Updates at http://www.ece.uah.edu/courses/ee420500/ (940) 915 EE603 Class Notes 09/04/09 John Stensby Fig. 97: A hypothetical sample function of narrowband Gaussian noise. The envelope is
Rayleigh and the phase is uniform.
describes a uniformly distributed phase. Finally, note that the envelope and phase are independent. Figure 97 depicts a hypothetical sample function of narrowband Gaussian noise. Envelope and Phase of a Sinusoidal Signal Plus Noise  the Rice Density Function
Many communication problems involve deterministic signals embedded in random noise.
The simplest such combination of signal and noise is that of a constant frequency sinusoid added
to narrowband Gaussian noise. In the 1940s, Steven Rice analyzed this combination and
published his results in the paper Statistical Properties of a Sinewave Plus Random Noise, Bell
System Technical Journal, 27, pp. 109157, January 1948. His work is outlined in this section.
Consider the sinusoid s(t) = A0 cos( ωc t + θ0 ) = A0 cos θ0 cos ωc t − A0 sin θ0 sin ωc t , (941) where A0, ωc, and θ0 are known constants. To signal s(t) we add noise η(t) given by (91), a
zeromean WSS bandpass process with power σ2 = E[η2] = E[ηc2] = E[ηs2]. This sum of signal
and noise can be written as
s(t) + η(t) = [A0 cos θ0 + ηc (t)]cos ωc t − [A0 sin θ0 + ηs (t)]sin ωc t
= Γ2 (t) cos[ωc t + ϕ 2 ] , Updates at http://www.ece.uah.edu/courses/ee420500/ (942) 916 EE603 Class Notes 09/04/09 John Stensby where Γ 2 (t) = [A 0 cos θ0 + ηc (t)]2 + [A 0 sin θ0 + ηs (t)]2 ϕ 2 (t) = tan −1 ⎡ A 0 sin θ0 + ηs (t) ⎤
⎢
⎥,
A 0 cos θ0 + ηc (t) ⎦
⎣ (943)
−π < ϕ 2 ≤ π , are the envelope and phase, respectively, of the signal+noise process. Note that the quantity
(A0 / 2 )2 / σ2 is the signaltonoise ratio, a ratio of powers.
Equation (943) represents a transformation from the components ηc and ηs into the
envelope Γ2 and phase ϕ2. The inverse of this transformation is given by
ηc (t) = Γ2 (t) cos ϕ 2 (t) − A0 cos θ0
ηs (t) = Γ 2 (t)sin ϕ 2 (t) − A0 sin θ0 . (944) Note that constants A0cosθ0 and A0sinθ0 only influence the mean of ηc and ηs. In the remainder
of this section, we describe the statistical properties of envelope Γ2 and phase ϕ2.
At the same time t, processes ηc(t) and ηs(t) are statistically independent (however, for τ
≠ 0, ηc(t) and ηs(t+τ) may be dependent). Hence, for ηc(t) and ηs(t) we can write the joint
density f ( ηc , ηs ) = exp[−( ηc2 + ηs2 ) / 2σ 2 ]
2πσ 2 (945) (we choose to abuse notation for our convenience: ηc and ηs are used to denote both random
processes and, as in (945), algebraic variables).
The joint density f(Γ2, ϕ2) can be found by transforming (945). To accomplish this, the
Jacobian Updates at http://www.ece.uah.edu/courses/ee420500/ 917 EE603 Class Notes 09/04/09 ∂ ( ηc , ηs ) ⎡cos ϕ 2
=
⎣
∂ ( Γ 2 , ϕ 2 ) ⎢ sinϕ 2 John Stensby −Γ2sinϕ 2 ⎤
Γ2 cos ϕ 2 ⎥
⎦ (946) can be used to write the joint density f (Γ 2 , ϕ 2 ) = f (ηc , ηs ) det ∂ (ηc , ηs )
∂ (Γ 2 , ϕ 2 ) ηc =Γ 2 cos ϕ 2 − A0 cos θ0 (947) ηs =Γ 2 sinϕ 2 − A 0 sin θ0 Γ2 f (Γ 2 , ϕ 2 ) = 2πσ 2 { } exp − 1 2 [Γ 22 − 2A 0Γ 2 cos(ϕ 2 − θ0 ) + A 02 ] U(Γ 2 ) .
2σ Now, the marginal density f(Γ2) can be found by integrating out the ϕ 2 variable to obtain
f ( Γ2 ) = ∫ 2π 0 f ( Γ 2 , ϕ 2 ) dϕ 2 { Γ2 } A0Γ 2 2π (948) = 2 exp − 1 2 [ Γ22 + A02 ] U( Γ 2 ) 21 ∫ exp{ 2 cos(ϕ 2 − θ0 )}dϕ 2 .
π0
2σ
σ
σ This result can be written by using the tabulated function z 2π
I0 (β ) ≡ 21
exp{β cos(θ )}dθ ,
π (949) 0 the modified Bessel function of order zero. Now, use definition (949) in (948) to write FH IK R
S
T U
V
W Γ
ΓA
f ( Γ2 ) = 2 I0 2 2 0 exp − 1 2 [Γ22 + A 02 ] U( Γ2 ) ,
2
σ
2σ
σ (950) a result known as the Rice probability density. As expected, θ0 does not enter into f(Γ2).
Equation (950) is an important result. Updates at http://www.ece.uah.edu/courses/ee420500/ It is the density function that statistically 918 EE603 Class Notes 09/04/09 John Stensby describes the envelope Γ2 at time t; for various values of A0/σ, the function σ f( Γ2) is plotted on
Figure 98 (the quantity (A0 / 2)2 / σ2 is the signaltonoise ratio). For A0/σ = 0, the case of no
sinusoid, only noise, the density is Rayleigh. For large A0/σ the density becomes Gaussian. To
observe this asymptotic behavior, note that for large β the approximation I 0 (β ) ≈ eβ
, β >> 1,
2πβ (951) becomes valid. Hence, for large Γ2A0/σ2 Equation (950) can be approximated by f ( Γ2 ) ≈ Γ2
2πA 0σ 2 R
S
T U
V
W exp − 1 2 [Γ2 − A 0 ]2 U ( Γ2 ) .
2σ (952) 0.8 σ f (Γ2) 0.7 A 0 /σ = 0 0.6 A 0 /σ = 1
A 0 /σ = 2 0.5 A0/σ = 3 A0/σ = 4 0.4
0.3
0.2
0.1
0.0 0 1 2 3 4 5 6 Γ2/σ Figure 98: Rice density function for sinusoid plus noise. Plots
are given for several values of A0/σ. Note that f is approximately
Rayleigh for small, positive A0/σ; density f is approximately
Gaussian for large A0/σ.
Updates at http://www.ece.uah.edu/courses/ee420500/ 919 EE603 Class Notes 09/04/09 John Stensby For A0 >> σ, this function has a very sharp peak at Γ2 = A0, and it falls off rapidly from its peak
value. Under these conditions, the approximation
1 f ( Γ2 ) ≈ 2πσ 2 R
S
T exp − 1 2 [Γ2 − A 0 ]2
2σ U
V
W (953) holds for values of Γ2 near A0 (i.e., Γ2 ≈ A0) where f(Γ2) is significant. Hence, for large A0/σ,
envelope Γ2 is approximately Gaussian distributed.
The marginal density f(ϕ2) can be found by integrating Γ 2 out of (947). Before integrating, complete the square in Γ 2 , and express (947) as f ( Γ2 , ϕ 2 ) = Γ2
2πσ S
}R
T { U
V
W 2 exp − 1 2 [Γ2 − A 0 cos(ϕ 2 − θ0 )]2 exp − A 02 sin 2 (ϕ 2 − θ0 ) U( Γ2 ) .
2
2σ
2σ (954) Now, integrate Γ 2 out of (954) to obtain
∞ f (ϕ 2 ) = ∫ f ( Γ 2 , ϕ 2 )dΓ2
0 { } 2
⎧
⎫ ∞ Γ2
exp − 12 [Γ2 − A0 cos(ϕ 2 − θ0 )]2 dΓ 2 .
= exp ⎨ − A0 sin 2 (ϕ 2 − θ0 ) ⎬
2
0 2 πσ2
2σ
⎩ 2σ
⎭ ∫ (955) On the righthandside of (955), the integral can be expressed as the two integrals z ∞ Γ2
2πσ { } 2 exp − 1 2 [Γ2 − A 0 cos(ϕ 2 − θ0 )]2 dΓ2
2σ = 0 z ∞ 2{Γ2 0 − A 0 cos(ϕ 2 − θ0 )}
4πσ
+ 2 { z A 0 cos(ϕ 2 − θ0 ) ∞
2πσ } exp − 1 2 [Γ2 − A 0 cos(ϕ 2 − θ0 )]2 dΓ2
2σ 2 Updates at http://www.ece.uah.edu/courses/ee420500/ 0 { (956) } exp − 1 2 [Γ2 − A 0 cos(ϕ 2 − θ0 )]2 dΓ2
2σ 920 EE603 Class Notes 09/04/09 John Stensby After a change of variable ν = [Γ2  A0cos(ϕ2  θ0)]2 , the first integral on the righthandside of
(956) can be expressed as
− A0 cos(ϕ 2 − θ0 )} ∞ 2{Γ 2 ∫0 4πσ 2 { } exp − 1 2 [ Γ 2 − A0 cos(ϕ 2 − θ0 )]2 dΓ 2
2σ = ∞
exp[ − υ2 ]dν
2 A 2 cos2 (ϕ 2 − θ0 )
2σ
4πσ
0 = ⎡ A2 cos2 (ϕ 2 − θ0 ) ⎤
1
exp ⎢ − 0
⎥.
2π
2 σ2
⎣
⎦ 1 ∫ (957) After a change of variable ν = [Γ2  A0cos(ϕ2  θ0)]/σ, the second integral on the righthandside
of (956) can be expressed as { } ∞
1
2
1
∫0 exp − 2σ2 [Γ2 − A0 cos(ϕ 2 − θ0 )] dΓ2
2 πσ { } 1∞
2
∫ −(A0 / σ) cos[ϕ2 − θ0 ] exp − 1 ν dν
2
2π = { (958) } 1 −(A0 / σ) cos[ϕ 2 − θ0 ]
exp − 1 ν2 dν
∫−∞
2
2π = 1− A
= F( σ0 cos[ϕ 2 − θ0 ]), where z x 2 exp − ν dν
2
2 π −∞ F( x ) ≡ 1 is the distribution function for a zeromean, unit variance Gaussian random variable (the identity
F(x) = 1  F(x) was used to obtain (958)). Updates at http://www.ece.uah.edu/courses/ee420500/ 921 EE603 Class Notes 09/04/09 John Stensby Finally, we are in a position to write f(ϕ 2), the density function for the instantaneous
phase. This density can be written by using (957) and (958) in (955) to write f (ϕ 2 ) = ⎡ A2 ⎤
1
exp ⎢ − 0 ⎥
2π
⎣ 2 σ2 ⎦ , (959) 2
A cos(ϕ 2 − θ0 )
⎧
⎫A
exp ⎨ − A0 sin 2 (ϕ 2 − θ0 ) ⎬ F( σ0 cos[ϕ 2 − θ0 ])
+0
2 πσ
⎩ 2 σ2
⎭ the density function for the phase of a sinusoid embedded in narrowband noise. For various
values of SNR and for θ0 = 0, density f(ϕ2) is plotted on Fig. 99. For a SNR of zero (i.e., A0 =
0), the phase is uniform. As SNR A02/σ2 increases, the density becomes more sharply peaked (in
general, the density will peak at θ0, the phase of the sinusoid). As SNR A02/σ2 approaches
infinity, the density of the phase approaches a delta function at θ0. 2.0
1.8
1.6
f (ϕ2) 1.4
1.2 A0/σ = 4 1.0
0.8
0.6 A0/σ = 2 0.4 A0/σ = 1 A0/ σ = 0 0.2
0.0 π π/2 0 π/2 π Phase Angle ϕ2 Figure 99: Density function for phase of signal plus noise A0cos(ω0t+θ0) +
{ηc(t)cos(ω0t)  ηs(t)sin(ω0t)} for the case θ0 = 0.
Updates at http://www.ece.uah.edu/courses/ee420500/ 922 EE603 Class Notes 09/04/09 John Stensby Shot Noise
Shot noise results from filtering a large number of independent and randomlyoccuring intime impulses. For example, in a temperaturelimited vacuum diode, independent electrons
reach the anode at independent times to produce a shot noise process in the diode output circuit.
A similar phenomenon occurs in diffusionlimited pn junctions. To understand shot noise, you
must first understand Poisson point processes and Poisson impulses.
Recall the definition and properties of the Poissson point process that was discussed in
Chapters 2 and 7 (also, see Appendix 9B). The Poisson points occur at times ti with an average
density of λd points per unit length. In an interval of length τ, the number of points is distributed
with a Poisson density with parameter λdτ.
Use this Poisson process to form a sequence of Poisson Impulses, a sequence of impulses
located at the Poisson points and expressed as
z(t) = ∑ δ(t − t i ) , (960) i where the ti are the Poisson points. Note that z(t) is a generalized random process; like the delta
function, it can only be characterized by its behavior under an integral sign. When z(t) is
integrated, the result is the Poisson random process
⎧ n(0, t), t > 0
⎪
t
⎪
x(t) = ∫ z(τ)dτ = ⎨ 0,
t=0
0
⎪
⎪n(0, t) t < 0,
⎩ (961) where n(t1,t2) is the number of Poisson points in the interval (t1,t2]. Likewise, by passing the
Poisson process x(t) through a generalized differentiator (as illustrated by Fig. 910), it is
possible to obtain z(t). Updates at http://www.ece.uah.edu/courses/ee420500/ 923 EE603 Class Notes 09/04/09 x(t) ... × John Stensby
z(t) × × × ... d/dt Poisson Process ... ... Poisson Impulses Figure 910: Differentiate the Poisson Process to get Poisson impulses.
The mean of z(t) is simply the derivative of the mean value of x(t). Since E[x(t)]=λdt, we
can write ηz = E[z(t)] = d
E[x(t)] = λ d .
dt (962) This formal result needs a physical interpretation. One possible interpretation is to view ηz as
t/2
ηz = limit 1 ∫
z( τ)dτ = limit 1 ( λ d t + random fluctuation with increasing t ) = λ d .
t −t / 2
t
t →∞ t →∞ (963) For large t, the integral in (963) fluctuates around mean λdt with a variance of λdt (both the
mean and variance of the number of Poisson points in (t/2, t/2] is λdt). But, the integral is
multiplied by 1/t; the product has a mean of λd and a variance like λd/t. Hence, as t becomes
large, the random temporal fluctuations become insignificant compared to λd, the infinitetimeinterval average ηz.
Important correlations involving z(t) can be calculated easily. Because Rx(t1,t2) = λ d2 t1t2
+ λdmin(t1,t2) (see Chapter 7), we obtain R xz (t1, t 2 ) =
R z (t1, t 2 ) = ∂
2
R x (t1, t 2 ) = λ d t1 + λ d U(t1 − t 2 )
∂t 2 (964) ∂
2
R xz (t1, t 2 ) = λ d + λ d δ(t1 − t 2 ) .
∂t1 Updates at http://www.ece.uah.edu/courses/ee420500/ 924 EE603 Class Notes 09/04/09 John Stensby The Fourier transform of Rz(τ) yields
2
Sz ( ω) = λ d + 2π λ d δ(ω) , (965) the power spectrum of the Poisson impulse process.
Let h(t) be a realvalued function of time and define
s(t) = ∑ h(t − t i ) , (966) i a sum known as shot noise. The basic idea here is illustrated by Fig. 911. A sequence of δ
functions described by (960) (i.e., process z(t)) is input to system h(t) to form output shot noise
process s(t). The idea is simple: process s(t) is the output of a system activated by a sequence of
impulses (that model electrons arriving at an anode, for example) that occur at the random
Poisson points ti.
Determined easily are the elementary properties of shot noise s(t). Using the method
discussed in Chapter 7, we obtain the mean
∞ ηs = E [s(t) ] = E [z(t) ∗ h(t)] = h(t) ∗ E [z(t)] = λ d ∫ h(t)dt = λ d H(0) . (967) 0 Shot noise s(t) has the power spectrum
2 2 2 2
2
Ss ( ω) = H(ω) Sz ( ω) = 2πλ d H 2 (0)δ(ω) + λ d H(ω) = 2πηs δ(ω) + λ d H(ω) . (968) Finally, the autocorrelation is
λ∞
2
2
2
R s ( τ) = F 1 [ Ss ( ω)] = λ d H 2 (0) + d ∫ H(ω) e jωτdω = λ d H 2 (0) + λ d ρ( τ) ,
2π −∞
Updates at http://www.ece.uah.edu/courses/ee420500/ (969) 925 EE603 Class Notes 09/04/09 John Stensby h(t) s(t) = h(t)*z(t) z(t) ... ...
ti1 ti ... h(t) ti+1 ti+2 ...
ti1 ti Poisson Impulses ti+1 ti+2 Shot Noise Figure 911: Converting Poisson impulses z(t) into shot noise s(t)
where ρ(τ) = ∞
1∞
2 jωτ
∫−∞ H(ω) e dω = ∫−∞ h(t)h(t + τ)dt .
2π (970) From (967) and (969), shot noise has a mean and variance of
ηs = λ d H(0)
σs2 2
= [λ d H 2 (0) + λ d ρ(0)]  [λ d H(0)]2 λ∞
2
= λ d ρ(0) = d ∫ H(ω) dω ,
2π −∞ (971) respectively (Equation (971) is known as Campbell’s Theorem). Example: Let h(t) = eβtU(t) so that H(ω) = 1/(β + jω), ρ(t) = e λ
ηs = E[s(t)] = d
β λ −β τ ⎡ λ d ⎤
R s ( τ) = d e
+⎢ ⎥
2β
⎣β⎦ −β t / 2β and 2 (972)
2λ
σs = d
2β 2 ⎡λ ⎤
λ
Ss (ω) = 2π ⎢ d ⎥ δ(ω) + 2 d 2
β +ω
⎣β⎦ FirstOrder Density Function for Shot Noise
In general, the firstorder density function fs(x;t) that describes shot noise s(t) cannot be
calculated easily. Before tackling the difficult general case, we first consider a simpler special
Updates at http://www.ece.uah.edu/courses/ee420500/ 926 EE603 Class Notes 09/04/09 John Stensby case where it is assumed that h(t) is of finite duration T. That is, we assume initially that h(t) = 0, t < 0 and t ≥ T . (973) Because of (973), shot noise s at time t depends only on the Poisson impulses in the
interval (t  T, t]. To see this, note that s(t) = ∫ ∞ −∞ h(t − τ)∑ δ( τ − t i )dτ =
i ∑ t −T < t i ≤ t h(t − t i ) , (974) so that only the impulses in (t  T, t] influence the output at time t. Let random variable nT
denote the number of Poisson impulses during (t  T, t]. From Chapter 1, we know that P[n T = k] = e −λ d T (λ dT) k
.
k! (975) Now, the Law of Total Probability (see Ch. 1 and Ch. 2 of these notes) can be applied to write
the firstorder density function of the shot noise process s(t) as fs (x) = ∞ ∑ fs (x⎮n k =0 T = k)P[n T = k] = ∞ ∑ fs (x⎮n k =0 T = k)e −λ d T (λ d T) k
k! (976) (note that fs(x) is independent of absolute time t). We must find fs(x⎮nT = k), the density of shot
noise s(t) conditioned on there being exactly k Poisson impulses in the interval (t  T, t].
For each fixed value of k that is used on the righthandside of (976), conditional density
fs(x⎮nT = k) describes the filter output due to an input of exactly k impulses on (t  T, t]. That is,
we have conditioned on there being exactly k impulses in (t  T, t]. As a result of the conditioning, the k impulse locations can be modeled as k independent, identically distributed Updates at http://www.ece.uah.edu/courses/ee420500/ 927 EE603 Class Notes 09/04/09 John Stensby (iid) random variables (all locations ti, 1 ≤ i ≤ k, are uniform on the interval).
For the case k = 1, at any fixed time t, fs(x⎮nT = 1) is actually equal to the density g1(x)
of the random variable x1 (t) ≡ h(t − t1 ) , (977) where random variable t1 is uniformly distributed on (t  T, t). That is, g1(x) ≡ fs(x⎮nT = 1)
describes the result that is obtained by transforming a uniform density (used to describe t1) by the
transformation h(t  t1).
Convince yourself that density g1(x) = fs(x⎮nT = 1) does not depend on time. Note that
for any given time t, random variable t1 is uniform on (tT, t), and x1(t) ≡ h(tt1) is assigned
values in the set {h(α) : 0 < α < T}, the assignment not depending on t. Hence, density g1(x) ≡
fs(x⎮nT = 1) does not depend on t.
The density fs(x⎮nT = 2) can be found in a similar manner. Let t1 and t2 denote
independent random variables, each of which is uniformly distributed on (t  T, t), and define x 2 (t) ≡ h(t − t1 ) + h(t − t 2 ) . (978) At fixed time t, the random variable x2(t) is described by the density fs(x⎮nT = 2) = g1∗g1 (i.e.,
the convolution of g1 with itself) since h(t  t1) and h(t  t2) are independent and identically
distributed with density g1.
The general case fs(x⎮nT = k) is similar. At fixed time t, the density that describes x k (t) ≡ h(t − t1 ) + h(t − t 2 ) + + h(t − t k ) (979) is Updates at http://www.ece.uah.edu/courses/ee420500/ 928 EE603 Class Notes 09/04/09 g k (x) ≡ fs (x⎮nT = k) = g1(x) ∗ g1(x) ∗ ∗ g1(x) , John Stensby
(980) k convolutions the density g1 convolved with itself k times.
The desired density can be expressed in terms of results given above. Simply substitute
(980) into (976) and obtain fs (x) = e −λ d T ∞ ∑ gk (x) k =0 (λ d T) k
.
k! (981) When nT = 0, there are no Poisson points in (t  T, t], and we have g 0 (x) ≡ fs (x⎮nT = 0) = δ(x) (982) since the output is zero. Convergence is fast, and (981) is useful for computing the density fs
when λdT is small (the case for low density shot noise), say on the order of 1, so that, on the
average, there are only a few Poisson impulses in the interval (t  T, t]. For the case of low
density shot noise, (981) cannot be approximated by a Gaussian density. fs(x) For An Infinite Duration h(t)
The firstorder density function fs(x) is much more difficult to calculate for the general
case where h(t) is of infinite duration (not subject to the restriction (973)). We show that shot
noise is approximately Gaussian distributed when λd is large compared to the time interval over
which h(t) is significant (so that, on the average, many Poisson impulses are filtered to form s(t)). To establish this fact, consider first a finite duration interval (T/2, T/2), and let random
variable nT, described by (975), denote the number of Poisson impulses that are contained in the
interval. Also, define the timelimited shot noise Updates at http://www.ece.uah.edu/courses/ee420500/ 929 EE603 Class Notes
sT (t) ≡ 09/04/09 nT ∑ h(t − t k ), −T / 2 < t < T / 2 , John Stensby
(983) k =1 where the random variables ti denote the times at which the Poisson impulses occur in the
interval. Shot noise s(t) is the limit of sT(t) as T approaches infinity.
In our analysis of s(t), we first consider the characteristic function
Φs (ω) = E ⎡e jωs ⎤ = limit E ⎡e jω sT ⎤ .
⎣
⎦ T →∞ ⎣
⎦ (984) Now, write the characteristic function of sT as E ⎡ e jω sT ⎤ =
⎣
⎦ ∞ ∑ E ⎡e jωsT⎮nT = k ⎤ P [nT = k ] ,
⎣
⎦ (985) k =0 where P[nT = k] is given by (975). In the conditional expectation used in (985), output sT
results from filtering exactly k impulses (this is different from the sT that appears on the lefthandside of the equation). Due to the conditioning, we can model the impulse locations as k
independent, identically distributed (iid – they are uniform on (T/2, T/2)) random variables. As
a result, the terms h(t  ti) in sT(t) are independent so that ( E ⎡e jω sT ⎮nT = k ⎤ = E ⎡ e jω sT ⎮nT = 1⎤
⎣
⎦
⎣
⎦ ),
k (986) where
1 T / 2 jωh(t − x)
E ⎡ e jω sT ⎮nT = 1⎤ = ∫
e
dx,
⎣
⎦ T −T / 2 Updates at http://www.ece.uah.edu/courses/ee420500/ −T / 2 < t < T / 2 , (987) 930 EE603 Class Notes 09/04/09 John Stensby since each ti is uniformly distributed on (T/2, T/2). Finally, by using (984) through (987), we
can write Φ s ( ω) = limit E ⎡e jω sT ⎤ = limit
⎦ T →∞
T →∞ ⎣ ∞ ⎡
⎤
∑ E ⎣e jωsT⎮nT = k ⎦ P [nT = k ] k =0 ∞ ⎛ 1 T / 2 jωh(t − x) ⎞ k −λ d T (λ dT)
e
dx ⎟ e
= limit ∑ ⎜ ∫
k!
T →∞ k = 0 ⎝ T − T / 2
⎠ k (988) k ⎛ λ T / 2 e jωh(t − x)dx ⎞
⎜ d∫
⎟
−λ d T
⎠.
= limit e
∑ ⎝ −T / 2
T →∞
k!
k =0
∞ Recalling the Taylor series of the exponential function, we can write (988) as ( ) T / 2 jωh(t − x) ⎞
∞
Φ s ( ω) = limit exp{−λ d T}exp ⎛ λ d ∫
e
dx ⎟ = exp ⎡λ d ∫
e jωh(t − x) − 1 dx ⎤ ,
⎜
⎢
⎥
−T / 2
−∞
⎝
⎠
⎣
⎦
T →∞ (989) a general formula for the characteristic function of the shot noise process.
In general, Equation (989) is impossible to evaluate in closed form. However, this
formula can be used to show that shot noise is approximately Gaussian distributed when λd is
large compared to the time constants in h(t) (i.e., compared to the time duration where h(t) is
significant). First, this task will be made simpler if we standardize s(t) to s(t) ≡ s(t)λ d H(0)
,
λd (990) so that Updates at http://www.ece.uah.edu/courses/ee420500/ 931 EE603 Class Notes 09/04/09 John Stensby E [ s] = 0
R s (τ) = ρ(τ) = ∫ ∞ −∞ . (991) h(t)h(t + τ)dt (see (967) and (969)). The characteristic functions of s and s are related by
⎡
⎡ s  λ d H(0) ⎤ ⎤
Φ s ( ω) = E ⎡e jωs ⎤ = E ⎢ exp ⎢ jω
⎥ ⎥ = exp ⎡ − jω λ d H(0) ⎤ Φ s (ω
⎣
⎦
⎣
⎦
λd ⎥ ⎥
⎢
⎢
⎣
⎦⎦
⎣ λd ) . (992) Use (989) in (992) to write
∞⎧
⎡
⎫⎤
jω
jω
Φ s ( ω) = exp ⎢λ d ∫ ⎨exp ⎡
h(t − x) ⎤ − 1 −
h(t − x) ⎬ dx ⎥ .
⎣ λd
⎦
−∞ ⎩
λd
⎭⎦
⎣ (993) Now, in the integrand, expand the exponential in a power series, and cancel out the zero and
firstorder terms to obtain ⎡
∞ ∞ ( jω) k
Φ s ( ω) = exp ⎢λ d ∫ ∑ k!
−∞
⎢
k =2
⎣ k
k
⎤
⎡
⎤
∞
k
⎧ h(t − x) ⎫
⎧
⎫
⎪
⎪
⎥ = exp ⎢λ ∑ ( jω) ∞ ⎪ h(x) ⎪ dx ⎥ .
dx
⎨
⎬
k! ∫−∞ ⎨ λ ⎬
⎥
⎢d
⎥
⎪ λd ⎪
⎪ d⎪
⎩
⎭
⎩
⎭
k =2
⎦
⎣
⎦ (994) Finally, assume that λd is large compared to the time duration during which h(t) is significant.
Equivalently, λd is large compared to all of the filter time constants. This insures that, on the
average and at any give time, shot noise s(t) results from filtering a large number of random
Poisson impulses. For this case, only the first term in the sum is significant; for large λd,
Equation (994) can be approximated as ⎡ ( jω)2 ∞ 2
⎤
22
Φ s (ω) ≈ exp ⎢
∫−∞ h (x)dx ⎥ = exp ⎡− 1 σs ω ⎤ ,
2
⎣2
⎦
⎣
⎦ Updates at http://www.ece.uah.edu/courses/ee420500/ (995) 932 EE603 Class Notes 09/04/09 John Stensby where σs2 = R s (0) (996) is the variance of standardized shot noise s(t) (see (991)). Note that Equation (995) is the
characteristic function of a zeromean, Gaussian random variable with variance (996). Hence,
shot noise is approximately Gaussian distributed when λd is large compared to the time interval
over which h(t) is significant (so that, on the average, a large number of Poisson impulses are filtered to form s(t)).
Example: TemperatureLimited Vacuum Diode
In classical communications system theory, a temperaturelimited vacuum diode is the
quintessential example of a shot noise generator (the phenomenon was first predicted and
analyzed theoretically by Schottky in his 1918 paper: Theory of Shot Effect, Ann. Phys., Vol 57,
Dec. 1918, pp. 541568). In fact, over the years, noise generators (used for testing/aligning
communication receivers, low noise preamplifiers, etc.) based on vacuum diodes (i.e., Sylvania
5722 special purpose noise generator diode) have been offered on a commercial basis.
Vacuum noise generating diodes are operated in a temperaturelimited, or saturated,
mode. Essentially, all of the available electrons are collected by the plate (few return to the
cathode) so that increasing plate voltage does not increase plate current (i.e., the tube is
saturated). The only way to increase plate current is to increase filament/cathode temperature. Under this condition, between electrons, space charge effects can be negligible so that individual
electrons are, more or less, independent of each other.
The basic circuit is illustrated by Figure 912. In a random manner, electrons are emitted
by the cathode, and they flow a distance d to the plate to form the current i(t). If emitted at t = 0,
an independent electron contributes a current h(t), and the aggregate plate current is given by Updates at http://www.ece.uah.edu/courses/ee420500/ 933 EE603 Class Notes 09/04/09 John Stensby i(t) d RL +  + Vf
Filament Vp
Plate Figure 912: Temperaturelimited vacuum
diode used as a shot noise generator. i(t) = ∑ h(t − t k ) , (997) k where tk are the Poissondistributed independent times at which electrons are emitted by the
cathode (see Equation (966)). In what follows, we approximate h(t).
As discussed above, space charge effects are negligible and the electrons are
independent. Since there is no space charge between the cathode and plate, the potential distribution V in this region satisfies Laplace’s equation
∂ 2V
∂2x = 0. (998) The potential must satisfy the boundary conditions V(0) = 0 and V(d ) = Vp. Hence, simple
integration yields V= Vp
d x, 0≤x≤d. (999) As an electron flows from the cathode to the plate, its velocity and energy increase. At
point x between the cathode and plate, the energy increase is given by E n (x) = eV (x) = e Vp
d x, Updates at http://www.ece.uah.edu/courses/ee420500/ (9100) 934 EE603 Class Notes 09/04/09 John Stensby where e is the basic electronic charge.
Power is the rate at which energy changes. Hence, the instantaneous power flowing from
the battery into the tube is Vp dx
dE n dE n dx
=
=e
= Vp h ,
dt
dx dt
d dt (9101) where h(t) is current due to the flow of a single electron (note that d 1dx/dt has units of sec1 so
that (e/d ) dx/dt has units of charge/sec, or current). Equation (9101) can be solved for current to
obtain h= e dx e
= vx ,
d dt d (9102) where vx is the instantaneous velocity of the electron.
Electron velocity can be found by applying Newton’s laws. The force on an electron is
just e(Vp/d), the product of electronic charge and electric field strength. Since force is equal to
the product of electron mass m and acceleration ax, we have ax = e Vp
.
md (9103) As it is emitted by the cathode, an electron has an initial velocity that is Maxwellian distributed.
However, to simplify this example we will assume that the initial velocity is zero. With this
assumption, electron velocity can be obtained by integrating (9103) to obtain vx = e Vp
t.
md Updates at http://www.ece.uah.edu/courses/ee420500/ (9104) 935 EE603 Class Notes 09/04/09 John Stensby h(t)
2e/tT tT
t
Figure 913: Current due to a single electron emitted by
the cathode at t = 0.
Over transition time tT the average velocity is vx = 1 tT
e Vp
d
∫0 v x dt = 2m d tT = tT .
tT (9105) Finally, combine these last two equations to obtain ⎛ 2d ⎞
vx = ⎜
⎟ t, 0 ≤ t ≤ tT .
⎜t2 ⎟
⎝T ⎠ (9106) With the aid of this last relationship, we can determine current as a function of time.
Simply combine (9102) and (9106) to obtain ⎛ 2e ⎞
h(t) = ⎜ ⎟ t, 0 ≤ t ≤ tT ,
⎜t2⎟
⎝T⎠ (9107) the current pulse generated by a single electron as it travels from the cathode to the plate. This
current pulse is depicted by Figure 913.
The bandwidth of shot noise s(t) is of interest. For example, we may use the noise
generator to make relative measurements on a communication receiver, and we may require the Updates at http://www.ece.uah.edu/courses/ee420500/ 936 EE603 Class Notes 09/04/09 John Stensby
10Log{S(ω)/S(0)} Rs(τ) 2 Relative Power (dB) 4e 2
3tT 0
2
4
6
8 tT τ tT Figure 914: Autocorrelation function of
normalized shot noise process. 10 π/tT ω (Rad/Sec) Figure 915: Relative power spectrum of normalized shot noise process. noise spectrum to be “flat” (or “white”) over the receiver bandwidth (the noise spectrum
amplitude is not important since we are making relative measurements). To a certain “flatness”,
we can compute and examine the power spectrum of standardized s(t) described by (990). As
given by (991), the autocorrelation of s(t) is ( )∫ R s ( τ) = 2 ( )( ) e 2 tT −τ
4 e2
τ2
τ
t(t + τ)dt =
1−
1+
, 0 ≤ τ ≤ tT
tT
3 tT
tT
2tT
0
= R s (−τ), −tT ≤ τ ≤ 0 . = 0, otherwise (9108) The power spectrum of s(t) is the Fourier transform of (9108), a result given by
∞ S (ω) = 2∫ R s (τ) cos(ωτ)dτ =
0 4
(ωtT ) 4 ( (ωtT )2 + 2(1 − cos ωtT − ωtT sin ωtT ) ) . (9109) Plots of the autocorrelation and relative power spectrum (plotted in dB relative to peak power at
ω = 0) are given by Figures 914 and 915, respectively.
To within 3dB, the power spectrum is “flat” from DC to a little over ω = π/tT. For the
Sylvania 5722 noise generator diode, the cathodetoplate spacing is .0375 inches and the transit Updates at http://www.ece.uah.edu/courses/ee420500/ 937 EE603 Class Notes 09/04/09 John Stensby time is about 3×1010 seconds. For this diode, the 3dB cutoff would be about 1/2tT = 1600Mhz.
In practical application, where electrode/circuit stray capacitance/inductance limits frequency
range, the Sylvania 5722 has been used in commercial noise generators operating at over
400Mhz. Updates at http://www.ece.uah.edu/courses/ee420500/ 938 ...
View
Full
Document
This note was uploaded on 10/12/2009 for the course EE EE603 taught by Professor Johnstensby during the Spring '09 term at University of Alabama in Huntsville.
 Spring '09
 JohnStensby

Click to edit the document details