Calculation of receiver sensitivity
vo
where ∂ v o ∂TA
rms
∆
∆Trms (°K )
∂ v o ∂TA calibrates voltage as temperature
Φ o ( f )DC ⇒ v o
Φ o ( f )AC ⇒ v o
rms
v o ∝ (TA + TR )
vo(t) compressed time
vrms → ∆Trms
vo
TA + TR
0
t
Approach:
v d(t) ↔
?
↓
φd ( t ) ↔ Φ d ( f ) ⇒ Φ o ( f )
Receivers-B1
Calculation of Φd(f), Power spectrum of vi2(t), vi gaussian
φd ( τ) = E[ v d ( t ) • v d ( t − τ)] = E[ v i2 ( t )v i2 ( t − τ)]
not gaussian
gaussian vi
It can be shown that:
E[wxyz] = E[wx]E[yz] + E[wy]E[xz] + E[wz]E[xy]
if w,x,y,z are jointly gaussian random variables [JGRV]
with zero mean [JGRVZM]
v
u,v are JGRVZM
u
v
u,v are NOT JGRV,
but are each GRVZM
u
B2
Calculation of Φd(f), Power spectrum of vi2(t), vi gaussian
φd ( τ) = E[ v d ( t ) • v d ( t − τ)] = E[ v i2 ( t )v i2 ( t − τ)]
not gaussian
gaussian vi
It can be shown that:
E[wxyz] = E[wx]E[yz] + E[wy]E[xz] + E[wz]E[xy]
if w,x,y,z are jointly gaussian random variables [JGRV]
with zero mean [JGRVZM]
2
2
2
∴ φd ( τ) = v i ( t )v i ( t − τ) + 2v i ( t )v i ( t − τ) = φi2 (0) + 2φi2 ( τ) [Ergodic]
7
Φ d(f ) =
7
7
φi2 (0)δ( f ) + 2Φi ( f ) ∗ Φ∗i ( f )
B2
Evaluation of Φd(f)
Φ d ( f ) = φi2 (0)δ( f ) + 2Φi ( f ) ∗ Φ∗i ( f )
1) φi (0) = v i2 ( t ) =
∞
∆T +T )
Φ
(
f
)
df
(
=
kT
B
T
∫ i
eff
eff
A
R
−∞
Φi(f)
where
kTeff/2
B
-fo
2)
Φi ( f ) ∗ Φ∗i ( f )
0
fo
f
(kTeff 2)2 • 2B
=
2B
-2fo ↑
-B 0
B
f
2B
↑ +2fo
B3
Evaluation of Φd(f)
Φ d ( f ) = φi2 (0)δ( f ) + 2Φi ( f ) ∗ Φ∗i ( f )
1) φi (0) = v i2 ( t ) =
2)
Φi ( f ) ∗ Φ∗i ( f )
∞
∆T +T )
Φ
(
f
)
df
(
=
kT
B
T
∫ i
eff
eff
A
R
−∞
(kTeff 2)2 • 2B
=
2B
-2fo ↑
3) Therefore:
(kTeff B)
2
-B 0
f
2B
↑ +2fo
Φd(f)
δ( f )
-2fo
B
(kTeff )2B
-B 0
B
+2fo
f
B4
Filtered output power density spectrum Φo(f)
B
PA = kTA
B
Φo(f)
( )2
+
0 fo
PR = kTR
display
h(t)
vo(t)
vd(t)
v o ( t ) = v d ( t ) ∗ h( t )
Φ o ( f ) = Φ d ( f ) • H( f )
2
AC+DC terms
Say:h(t)
H( f )
Integrator impulse response
A
0
(τ 2) delay
• e− j2πf τ 2
τ/2
τ
t
1/τ
0
f
B5
Filtered output power density spectrum Φo(f)
v o ( t ) = v d ( t ) ∗ h( t )
Φ o ( f ) = Φ d ( f ) • H( f )
2
AC+DC terms
Say:h(t)
A
0
(τ 2) delay
H( f )
Integrator impulse response
• e− j2πf τ 2
τ
τ/2
Then : H( f = 0) =
t
1/τ
f
0
∞
− j2π( f = 0 )t
h
(
t
)
e
dt
∫
H( f )
2
−∞
= Aτ
0
f
1/τ
(Typically 1/τ << B)
B6
Filtered output power density spectrum Φo(f)
H( f )
2
f
0
1/τ
(Typically 1/τ << B)
Thus : Φ o ( f ) = (kTeff B )2 (Aτ )2 δ( f ) = DC power
DC
Po
AC
=
∞
∫
−∞
Φo
AC
( f )df ≅ (kTeff )2 B •
∞
2
∫ H( f ) df
−∞
Note that if 1/τ << B, only the value of Φd(f = 0) is important,
so this integral is trivial.
∞
By Parseval’s theorem:
2
∫ H( f ) df =
−∞
∞
2
2
h
(
t
)
dt
=
A
τ
∫
−∞
B7
Total-Power Radiometer Sensitivity ∆Trms
TA + TR
∆Trms =
(∂
(kTeff )2 B • A 2τ kTeff A Bτ
[°K ] =
=
(∂[kTeff BAτ] ∂TA ) kA Bτ
∂TA )
PAC
PDC
TA + TR
for total - power radiometer
∴ ∆Trms =
Bτ
B8
Effect of different integrator impulse response
Suppose h(t)
1
e−t τ
Recall Φ o ( f ) = Φ d ( f ) • H( f ) 2
We need to compute
H( f = 0) =
∞
∞
−∞
−∞
∫ h( t )dt = τ
kTeff Bτ 2
Then ∆Trms =
kBτ
τ
0
2
∫ H( f ) df =
∆Trms =
t
∞
2
h
∫ ( t )dt = τ 2
−∞
(TA + TR )
2Bτ
Greater sensitivity, but at the expense of a longer memory
B9
Example: Radio telescope receiver
Possible : TA + TR = 30°K, B = 100 MHz
then : ∆Trms = 30
108 • 1sec = 0.003°K ⇒ 300 µK for 100 s
Example: Voice radio, AM
If : TA + TR = 10,000K, B = 10kHz, τ = 10 ­ 4 sec
then : ∆Trms = 10 4
10 4 • 10 − 4 = 10 4 K = TA + TR
B10
Receiver sensitivity derivation: sampled signals
Sampling-Theorem approach for the total-power radiometer
TA
vi
+
0
B
( )2
f
TR
vd
h(t)
vo(t)
t
τ
0 T
i(t)
i(t)
T < 1/2B ⇒ pulse correlation
T > 1/2B ⇒ lost information
0 T 2T
T = 1/2B is the
Nyquist rate
t
T
Nyquist sampling: e.g.
t
Highest f (=B here) T = 1/2B
Receivers-C1
Computation of ∆Trms for a sampled system
vi(t)
Boxcar having τ/T = 2Bτ samples
h(t)
t
0 T
2Bτ samples
1
0 T
vd(t)
τ
t
2Bτ
t
0 T 2T
∆Trms =
vo
rms AC
⎡∂ vo ⎤
⎢
⎥
T
∂
⎣ A ⎦
∑ v d ⇒ Gaussian if 2Bτ = τ T >> 1
boxcar
(central limit theorem)
= (output fluctuation/scale calibration)
Variance of v o = 2Bτ • σ2d
# samples
variance of vd
C2
∆Trms =
vo
rms AC
⎡∂ vo ⎤
⎢
⎥
T
⎣∂ A ⎦
= (output fluctuation/scale calibration)
Variance of v o = 2Bτ • σ2d
# samples
σ2d
( ) (where v = JGRVZM)
− 2(v ) + (v ) = v − (v )
∆ (v − v )2 = v 2 − v 2 2
d
d
i
i
=
v i4
variance of vd
2
i
2
2
i
2
4
i
i
2
i
2
Recall : xn = 1• 3 • 5 • •(n − 1), if even; xn = 0, if n odd
(where x = JGRVZM)
Let : x 2 ≡ 1 here and v i2 = Teff • a • x 2 (this equation defines " a" )
2⎤
⎡
2
2 2
2
4
2
2 2 ⎢ 4 ⎛⎜ 2 ⎞⎟ ⎥
Thus : σd = v i − v i = Teff a xN − ⎜ xN ⎟ = 2Teff
a
⎢3 ⎝ 1⎠ ⎥
⎦
and the variance of v = 2Bτ • 2T 2 a2 ⎣
( )
o
eff
C3
Computation of ∆Trms for a sampled system
2 2
variance of v o = 2Bτ • 2Teff
a
v o = 2Bτ • v i2 = 2Bτ • Teff a
variance of v o Teff a 4Bτ
=
∴ ∆Trms =
2Bτa
∂ v o ∂TA
∆Trms = Teff
Bτ
Note: # samples = 2Bτ,
as before
variance ∝ 1 Bτ
C4
Gain fluctuations in total-power radiometers
Teff g(t) = (TA + TR)g(t)
g( t ) ≅ 1 + m( t ), m << 1
2
rms = m2Teff
t
0
∆Trms ≅
2
(∆Tthermal )2 + m2Teff
= Teff
1
+ m2
Bτ
(Note: 0.1% gain fluct. @ Teff = 2000K ⇒ 2K!)
Receivers-E1
One solution to gain variations: “Synchronous detection”
Dicke radiometer
TR
TA
vo(t)
receiver
TCAL
Zo
vo(t)
+
∫
-
vSD(t) ∝ TA - TCAL
integrated by upper integrator
∝(TR + TA)
∝(TR + TC)
∫
~VSD
t
0
integrated by lower integrator
Note: gain is irrelevant
when TA = TC
= unchanged (looking at same signal all the time)
rms
1
former value (we view TA half the time)
but ∂v SD ∂Teff =
2
v SD
∴ ∆TrmsDicke = 2Teff
Bτ (at null only)
E2
Asymmetric Dicke radiometer
vo(t)
β
α
TR + TA
vo(t)
∫τ A
RCVR
TR + TC
0
VSD
γ∫
γ∆β α
+
τC
-
t
Want: τA << desired-signal fluctuation time constant
τc > τA (typically τc ≥ γτA)
Integration times τA and τc should be shorter than the fluctuation
times of the desired signal and system gain, respectively.
E3
Filtered Dicke radiometer
TA
receiver
TCAL
vo(t)
Narrowband
filter at
Dicke frequency
v’o(t)
Zo
vo(t)
∫τ
+
vSD
∫τ
-
v’o(t)
t
∆Trms =
t
π
T
2 eff
narrowband Dicke π
Bτ
2 = 2.22
E4
Correlation radiometer
Typical circuit:
power/2
S(t)
+
na(t)
TR
°K
TA
S( t )
volts
2
0
B
f va(t)
h(t)
×
TR
vo(t)
0
nb(t)
τ t
vb(t)
+
0
B f
vm(t)
Uses:
1) To reduce radiometric gain modulation effects
(similar to Dicke receiver)
2) As a correlator
Correlator power density spectrum, pre-integrator:
φm ( τ) = E[v a ( t )v b ( t )v a ( t − τ)v b ( t − τ)]
E5
Correlator power density spectrum, pre-integrator
φm ( τ) = E[v a ( t )v b ( t )v a ( t − τ)v b ( t − τ)]
⎡⎛ S1
⎞⎛ S1
⎞⎛ S2
⎞⎛ S2
⎞⎤
= E ⎢⎜
+ na ⎟⎜
+ nb ⎟⎜
+ na ⎟⎜
+ nb ⎟⎥
1 ⎠⎝ 2
1 ⎠⎝ 2
2 ⎠⎝ 2
2 ⎠⎦
⎣⎝ 2
Where S1 ∆ S(t), n1 ∆ n(t), S2 ∆ S(t - τ), n2 ∆ n(t - τ)
All JGRVZM, so : ABCD = AB • CD + AC • BD + AD • BC
φm ( τ) = 1 φs2 (0) + 1 φ2s ( τ) + φs ( τ)φn ( τ) + φn2 ( τ)
4
2 N
n×n
S×S terms
S×n
1
1 2
Φm ( f ) = φs (0)δ( f ) + Φ s ( f ) ∗ Φ s ( f ) + Φ s ( f ) ∗ Φn ( f ) + Φn ( f ) ∗ Φn ( f )
4 2 S×S
S×n
n×n
E6
Sensitivity of correlation radiometer
1 2
1
Φm ( f ) = φs (0)δ( f ) + Φ s ( f ) ∗ Φ s ( f ) + Φ s ( f ) ∗ Φn ( f ) + Φn ( f ) ∗ Φn ( f )
4 2 S×n
S×S
n×n
1 2
Pdc follows from φs (0)δ( f ), and Pac from the other terms
4
Pac
Teff
∆Trms =
=
∂ Pdc ∂TA
Bτ
2
where Teff
= TA2 + 2TA TR + 2TR2
∆Trms ≅
2TR
Bτ
∆Trms ≅ TA
Bτ
for the weak-signal case (TA << TR )
for the strong-signal case (TA >> TR )
E7
Summary – radiometer sensitivity
Radiometer type
∆Trms
Total power
Teff
Bτ
Correlation
Teff
Bτ
Dicke
2Teff
Dicke narrowband
post detector
Teff2
π
T
2 eff
Bτ
Bτ
(TA + TR )2
(TA + TR )2 + TR2
(TA + TR )2 at null
(TA + TR )2
Relative sensitivity
to small fluctuations
1
2
2 at null
2.22
E8