Post on 02-Jun-2018
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Spectral Measurements
Case A: Bandwidth exceeds that of
available amplifiers
TA(f)
S(f)
channels
f1
fNreceivers and antennas
detector noise, detect directly or
split frequencies and then detect
before amplification or detection
f1
fN
f1
fN
f
Receivers-G1
1) Extreme bandwidth: use multiple
2) If signal large compared to
3) Use passive frequency splitters
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Spectral Measurements
Case B: Bandwidth permits amplificationf1
fN further frequency splitting
Case C: Bandwidth permits digital spectral analysis
(
fResolution
-~)f(V 2M
2N
)f()( NN
(Permits ~100 more B per cm2 silicon)(Reference: Van Vleck and Middleton, Proc. IEEE, 54, (1966)
)
G2
1) Amplify before either detection or
1) If computer resources permit, compute
N2B
:transformpointNpermultiplysNlogN
2) Or 1-bit (or n-bit) samples)(N
spectraMaverage
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Examples of Passive Multichannel Filters
IN
f1 fnf2
Zo
2. Waveguides
f1 resonant
cavities at ff
virtual
short
/4
fN
RCVRRCVR
f1 f3f2
filters
passive
channel-dropping filters
Zo at f1
f
G3
1. Circuits
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Examples of Passive Multichannel Filters
3. Prism
-bound electron(s)
red blue fo
f
prism
5. Cascaded Dichroics
plane wave
f2 >fn-1(f1>f2>fn)
4. Diffraction grating
(f)
G4
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Digital spectral analysis example: autocorrelation
0 0 f
() (f)[W Hz-1]
analog signals
Possible analog implementation:
1) max lag = max = NT2) sample lag, T sec
3) finite integration time
>> max
)(v
BRF
0f
fRF
0f
fIF
delay line
LO
local oscillator
v(t)
)T(v
)(v
)NT(v
NT = max
BBRF
G5
:onbasedis
T2
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Resolution of autocorrelation analysis
)(W)()( yv =1)
W(
)
0
M = NT-
M
0f
0f
B
v(f) W(f)
~1/2 M Hz
Thus
v(f)
)f(W)f()f( vv
M
=
7777 W(f)
f0
G6
m
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Aliasing in autocorrelation spectrometers
)f(I)f(
)t(i)(
)f(
)(
v
v
v
v
=
=
7777
2)
i(t)
T0t
I(f)
0f(Hz)
-1/T
Aliasing is spectral overlap
-1/T 0 1/T 2/T
B)f(
v
3) Finite averaging time adds noise to )f(,)( vv
G7
1/T 2/T
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Autocorrelation of hard-clipped signals
A/D
( )2
counte
r
delay line
LO
x(t)
vo(t)1
v(t)
)(v
hard clipping
A/D
1
+1
-1
0t
+1 if v(t) > 0
Receivers-I1
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Analysis of 1-bit autocorrelation
( ) ( ) 0-
0xt 212211
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Power spectrum for 1-bit signal
Change
variables
x2
x1
dr
rdr
x1 = r cos x2 = r sin dx
1dx
2= rdr d
( )
=
2
0 0
1
12r
2
2
x 1e
12
12rdd4)( 2
2
(
)( )
= 2
0
21
12
1d4
I3
2sin
21
21
2sin
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Power spectrum for 1-bit signal
(
)( )
=2
0
2
112
1d4
( )
(
)
1sin22
141dsin11
414 1
0x
+
=
=
( )
( )
=
xv 2sin
( ) ( )( )Tx
)Where =
0 b0
p(b)p(a)
b
(see Burns & Yao, Radio Sci., 4(5) p. 431 (1969))
2Let
I4
a
21
2sin
21
-v(tsgnv(t)sgn
exactnotbif
biashas:Note
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Spectral response & sensitivity: autocorrelation receiver
;B
f1
f
T)f( effrms
channel bandwidth
(S. Weinreb empirical result,
MIT EE PhD thesis, 1963)
Apodizing weighting functions:
f
1.099
0.87
0.69
N
s
Nfs
Ns
first
sidelobe
-7 dB
-16 dB
-29 dB
Note trade between spectral resolution, sidelobes in (f) and Trms
=
=
N;NT1f
Ms
uniform
raised cosine
blackman
0
0
0
I5
6.1
f60.0
f13.1
taps#
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Spectral response & sensitivity: autocorrelation receiver
If N delay-line taps, how many spectral samples Ns?
1W()
0 M
Say uniform weighting of ():
Then B = Ns f = Ns /2M) where spectral resolution f 1/2mfor orthogonal channels from boxcar W()
(
)
(
)B2N Ms ====
W(f) for adjacent channel
M
f
In practice: raised cosine widens f by 1/0.6 1.7, so Ns N/1.7
W(f)
I6
(1
taps#Nratenyquistat2B1TBNT2
21
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Types of power
Delivered
AvailableExchangeable
( )
{
}
{ }j tev t R t sin t
=
+
Vg
gZ
LZRg g
+
-
+
-
V
{
}( )
=
gZP
PVIR21P
LDavailable
Dedelivered
Receivers-K1
Receivers Gain and Noise Figure
Ve Re V cos Im V
+ j X
Zifi.e.,,Pmax
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Delivered and Available Power
{
}( )
=
gZP
PVIR21P
LDavailable
Dedelivered
RL
ge ZR-
ge ZR
PD
0Zf ge
Zifi.e.,,Pmax
R:I
optionpowerleexchangeab
ZIm
R:I
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Definition of Gain
gZ LZG1 2
Gpower (= Gp)
12 DDPP
Gavailable
(= GA
) 12 AA
PP
Gtransducer (GT)power
12 AD
PPGexchangeable (=GE)
1
2
E
E
P
P
Ginsertion (= GI)
1
2
D
D
P
Pwithamplifier
without
amplifier
Note: GA, GEdont depend on ZLdo depend on Zg (via PE2)
K3
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Definition: Signal-to-Noise Ratio (SNR)
First define:
N1 =
N2
=
S1 =
S2 =
1zWH
222111 NSSNR;NS
Vg
1 2
gZ
LZ
F
G
gE =
K4
exchangeable noise power spectrum @ Port 1
same, at 2
exchangeable signal power spectrum @ Port 1
same, at 2
SNRDefine
ZfGRecall
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Definition: Noise Figure F
kTNNS
NS
SNR
SNRF oo1
22
11
2
1
[Ref. Proc. IRE, 57(7), p.52 (7/1957); Proc. IEEE, p.436 (3/1963)]
S2 = GES1 (see definition of GE)
N2 = GEN1 + N2T transducer noise
( )N
1NGNGS
NSF
111
11 +=+
= ( )EG
o
R
o
R
1 T
TN1F ==
K5
K290T,where,
GNT2
T2Glet
T2
GkT
GkT
GNexcess noise figure
receiver noise temperature
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Receiver Noise Example
(6FT
F(2FT
)F(1TT1FT
R
R
o
RR
==
===
==+==
Examples:
Excess noise corresponds toreceiver noise temperature TR
+ 1
TR = (F 1) To
290K
noiseless
G, FTA
+1
TA
N2T
TA
TR
K6dB)5.7~FK1500
dB)3K290
dB0K0
F,G
F,G