Spectral Measurements
S(f)
Case A: Bandwidth exceeds that of
available amplifiers
TA(f)
f
channels
f1
1) Extreme bandwidth: use multiple
receivers and antennas
fN
f1
fN
f1
fN
2) If signal large compared to
detector noise, detect directly or
split frequencies and then detect
3) Use passive frequency splitters
before amplification or detection
Receivers-G1
Spectral Measurements
f1
fN
Case B: Bandwidth permits amplification
1) Amplify before either detection or
further frequency splitting
Case C: Bandwidth permits digital spectral analysis
1) If computer resources permit, compute
2
V( f ) N
(~ N log2 N multiplys per N - point transform :
Resolution f 2B N
average M spectra )
2) Or 1-bit (or n-bit) N ( ) N ( f )(N samples)
(Permits ~100 more B per cm2 silicon)
(Reference: Van Vleck and Middleton, Proc. IEEE, 54, (1966)
G2
Examples of Passive Multichannel Filters
1. Circuits
IN
Zo
f2
f1
fn
channel-dropping filters
2. Waveguides
Zo at f1
filters
/4
f
fN
f1
f2
RCVR
f3
RCVR
passive
f1 resonant
cavities at ff
virtual
short
G3
Examples of Passive Multichannel Filters
3. Prism
(f)
bound electron(s)
prism
red
f
blue
fo
4. Diffraction grating
5. Cascaded Dichroics
<f2
<f1
fn
plane wave
>f1
>f2
>fn-1(f1>f2>fn)
G4
Digital spectral analysis example: autocorrelation
(f)
()
[W Hz-1]
analog signals
Possible analog implementation:
BRF
BBRF
v(t)
delay line
fRF
f
LO
local oscillator
v ( ) is based on :
1) max lag = max = NT
2) sample lag, T sec
3) finite integration time >> max
fIF
NT = max
v (T ) v (2T ) v (NT )
G5
Resolution of autocorrelation analysis
W()
1) v ( ) = y ( ) W ( )
7 < M 7
7
7
v (f ) = v (f ) W (f )
-M
M = NT
0
W(f)
0
Thus
v(f) W(f)
v(f)
~1/2 M Hz
0
G6
Aliasing in autocorrelation spectrometers
i(t)
2) v ( ) =
7
v (f ) =
v ( )
7
i( t )
7
v ( f ) I( f )
I(f)
v (f )
B
-1/T 0
-1/T
1/T
1/T 2/T
f(Hz)
2/T
Aliasing is spectral overlap
v (f )
3) Finite averaging time adds noise to v ( ),
G7
Autocorrelation of hard-clipped signals
v(t)
()
delay line
A/D
x(t)
LO
1
hard clipping
vo(t)
c
o
u
n
t
e
r
v ( )
+1 if v(t) > 0
+1
A/D 1
0
-1
Receivers-I1
Analysis of 1-bit autocorrelation
+ 1 x 0
where x1, x 2 are JGRVZM
Let x (t1) x1, x (t 2 ) x 2, sgn x
-1 x < 0
x ( ) = E[sgn x1 sgn x 2 ] =
x12 2x1x 2 + x 22
2
1
2
1
dx dx
e
sgn x1 sgn x 2
1 2
12
(
)
2
1
( )
where ( ) x1x 2 v ( ), = t 2 t1
x ( ) = 2 [p(x1, x 2 )]dx1dx 2 2
= 4 p(x1, x 2 )dx1dx 2 1
0
p(x1, x2 )dx1dx 2
Note : 2 + 2 = 1
0
0
I2
Power spectrum for 1-bit signal
Change
variables
x2
x1 = r cos
x2 = r sin
dx1dx2 = rdr d
rd
r
dr
x1
( )
r
r2
1
e
x ( ) = 4 d d
12
2
2
0
0 2 1
2
= 4 d
0
(1 )
2 12
2(1 sin 2 )
1 sin 2
2
1 2
I3
Power spectrum for 1-bit signal
(1 )
2 12
= 4 d
0
2(1 sin 2 )
Let 2
(
1 )1 2
x ( ) = 4
1
1
d 1 = 4 + sin1 1
1 sin
2 2
0
( ) = sin ( )
v
x
2
Where x ( ) = (sgn v(t) )(sgn v(t - )) T
a
Note : has bias
if b not exact
p(a)
p(b)
0
b0
(see Burns & Yao, Radio Sci., 4(5) p. 431 (1969))
I4
Spectral response & sensitivity: autocorrelation receiver
( f )rms
Teff
f
1 ; 1 .6
B
f
channel bandwidth
Apodizing weighting functions:
(S. Weinreb empirical result,
MIT EE PhD thesis, 1963)
first
sidelobe
1.099
0.60 fs
N
-7 dB
0
raised cosine
0.87
fs N
-16 dB
0.69
uniform
1 N
fs =
;
N
#
taps
=
T M
1.13 fs N
-29 dB
blackman
Note trade between spectral resolution, sidelobes in (f) and Trms
I5
Spectral response & sensitivity: autocorrelation receiver
If N delay-line taps, how many spectral samples Ns?
Say uniform weighting of ():
W()
1
0
Then B = Ns f = Ns (1/2M) where spectral resolution f 1/2m
for orthogonal channels from boxcar W()
W(f)
1 2 M
W(f) for adjacent channel
f
Ns = 2MB = 2 NT B (T = 1 2B at nyquist rate ) = N(# taps )
In practice: raised cosine widens f by 1/0.6 1.7, so Ns N/1.7
I6
Receivers Gain and Noise Figure
Types of power
Delivered
Available
Exchangeable
v ( t ) Re Ve jt
Zg
Vg
+
V
Rg + j Xg
ZL
= Re {V} cos t + Im {V} sin t
{ }
R e VI ( PD )
Pdelivered
2
Pavailable max PD , i.e., if ZL = Zg
Receivers-K1
Delivered and Available Power
{ }
1
Pdelivered Re VI ( PD )
2
Pavailable max PD, i.e., if ZL = Z
PD
If : R e Z g > 0
Im Z g = 0
PA
- Re Zg
0
Re Zg
RL
PD
If : R e Z g < 0
Re Zg
RL
- Re Zg
Pexchangeab le PD
( finite - power option )
ZL = Z g
K2
Definition of Gain
1
Zg
Gpower (= Gp)
Gavailable (= GA)
Gtransducer (GT)
power
PD
PA
PD
PD
PA
PA
Note: GA, GE
1
1
ZL
Ginsertion (= GI)
PD
PD
with
amplifier
without
1 amplifier
Gexchangeable (=GE)
PE
PE
dont depend on ZL
do depend on Zg (via PE2)
K3
Definition: Signal-to-Noise Ratio (SNR)
First define:
WH z1
N1
N2
S1
S2
=
=
=
=
exchangeable noise power spectrum @ Port 1
same, at 2
exchangeable signal power spectrum @ Port 1
same, at 2
( )
Recall GE = f Zg
Zg
F
ZL
Vg
G
1
Define SNR1 S1 N1 ; SNR 2 S2 N2
K4
Definition: Noise Figure F
SNR1 S1 N1
F
, where N1 kTo , To 290 K
SNR 2 S2 N2
[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
S1 N1
N2T
F =
= 1+
GS1 (GN1 + N2T )
N1G
N2T kTRG TR
F 1 =
=
N1G kToG To
(let G GE )
receiver noise temperature
excess noise figure
K5
Receiver Noise Example
TA
G, F 1
TA
G, F
noiseless
TR
TA
TR = (F 1) To
G, F 1
290K
Excess noise corresponds to
receiver noise temperature TR
N2T
Examples:
TR
TR
TR
TR
= 0K
F = 1+
= 1 (F = 0 dB)
To
= 290K F = 2
(F = 3 dB)
= 1500K F 6
(F ~= 7.5 dB)
K6
