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
M
(~ N log2 N multiplys per N - point transform :
Resolution ∆f ≥ 2B N
average M spectra )
2) Or 1-bit (or n-bit) φN ( τ) ↔ ΦN ( 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
λ/4
≠f
fN
f1
f2
RCVR
f3
RCVR
passive
f1 resonant
cavities at ff
virtual
short
G3
Examples of Passive Multichannel Filters
3. Prism
ε(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
Φ(f)
φ(τ)
[W Hz-1]
τ
0
f
0
analog signals
Possible analog implementation:
BRF
B≤BRF
v(t)
delay line
×
0
fRF
f
LO
“local oscillator”
φˆ v ( τ) is based on :
1) max lag = τmax = NT
2) sample lag, T sec
3) finite integration time τ >> τmax
0
fIF
f
×
×
×
∫τ
∫τ
NT = τmax
∫τ
φˆ v (T ) φˆ v (2T ) φˆ v (NT )
G5
Resolution of autocorrelation analysis
W(τ)
1) φˆ v ( τ) = φ y ( τ) • W ( τ)
7 τ < τM 7
7
7
ˆ v (f ) = Φ v (f ) ∗ W (f )
∴Φ
-τM
τ
τM = NT
0
W(f)
½ τm
f
0
Thus
Φv(f) ∗ W(f)
Φv(f)
~1/2 τM Hz
0
f
0
B
f
G6
Aliasing in autocorrelation spectrometers
i(t)
2) φˆ v ( τ) =
7
ˆ v (f ) =
Φ
φ v ( τ)
7
• i( t )
7
7
Φ v ( f ) ∗ I( f )
0
T
t
I(f)
ˆ v (f )
Φ
B
-1/T 0
-1/T
0
1/T
1/T 2/T
f(Hz)
2/T
“Aliasing” is spectral overlap
ˆ v (f )
3) Finite averaging time τ adds noise to φˆ v ( τ), Φ
G7
Autocorrelation of hard-clipped signals
×
v(t)
()
2
delay line
A/D
x(t)
×
×
×
LO
∫
±1
“hard clipping”
vo(t)
c
o
u
n
t
e
r
φˆ v ( τ)
+1 if v(t) > 0
+1
A/D ⇒ ±1
0
t
-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
φ x ( τ) = E[sgn x1 sgn x 2 ] =
⎡
x12 −2ρx1x 2 + x 22 ⎤
−
∞
⎥
⎢
2
1
−
ρ
2
1
⎥ dx dx
e
∫∫ sgn x1 sgn x 2 ⎢⎢
1 2
12
⎥
(
)
2
1
π
−
ρ
−∞
⎥
⎢
⎦
⎣
( )
where ρ( τ) ∆ x1x 2 ≡ φv ( τ), τ = t 2 − t1
∞
0 ∞
0
−∞ 0
φ x (τ ) = 2 ∫ ∫ [p(x1, x 2 )]dx1dx 2 − 2
∞
= 4 ∫ ∫ p(x1, x 2 )dx1dx 2 − 1
0
∫ ∫ p(x1, x2 )dx1dx 2
∞
0 ∞ ⎫
⎧⎪
⎪
⎨Note : 2 ∫ ∫ + 2 ∫ ∫ = 1⎬
⎪⎩
−∞ 0 ⎪
0
⎭
I2
Power spectrum for 1-bit signal
Change
variables
x2
x1 = r cos θ
x2 = r sin θ
dx1dx2 = rdr dθ
rdθ
r
dr
θ
x1
( )
−r
⎛ r2 ⎞
1
e
φ x ( τ) = 4 ∫ dθ ∫ d⎜ ⎟
12
⎜2⎟
2
0
0 ⎝ ⎠ 2π 1 − ρ
π2
∞
(
π2
= 4 ∫ dθ
0
(1− ρ )
)
2 12
2π(1 − ρ sin 2θ )
2
⎛ 1− ρ sin 2θ ⎞
⎟
2 ⎜
⎜ 1− ρ 2 ⎟
⎠
⎝
−1
−1
I3
Power spectrum for 1-bit signal
(1− ρ )
2 12
π2
= 4 ∫ dθ
0
2π(1 − ρ sin 2θ )
Let φ ∆ 2θ
−1
π
(
1 − ρ )1 2
φ x (τ ) = 4
∫
4π
1
⎧ 1 ⎛π
⎞⎫
dφ − 1 = 4⎨ ⎜ + sin−1 ρ ⎟⎬ − 1
1 − ρ sin φ
⎠⎭
⎩ 2π ⎝ 2
0
ˆφ (τ ) ≡ ρˆ = sin⎛⎜ π φˆ (τ )⎞⎟
v
x
⎝2
⎠
Where φˆ x (τ ) = (sgn v(t) )(sgn v(t - τ)) T
a
Note : ρ̂ has bias
if b not exact
p(a)
p(b)
0
b0
b
(see Burns & Yao, Radio Sci., 4(5) p. 431 (1969))
I4
Spectral response & sensitivity: autocorrelation receiver
σ( f )rms ≅
αβ Teff
∆f
1 − ; β ≅ 1 .6
B
τ∆f
channel bandwidth
“Apodizing” weighting functions:
(S. Weinreb empirical result,
MIT EE PhD thesis, 1963)
first
sidelobe
α
∆f
τ
1.099
0.60 fs
N
-7 dB
τ
0
raised cosine
0.87
fs N
-16 dB
τ
0.69
0
0
uniform
⎛
⎞
1 N
⎜ fs ∆ =
⎟
;
N
#
taps
=
⎜
⎟
T τM
⎝
⎠
1.13 fs N
-29 dB
blackman
Note trade between spectral resolution, sidelobes in Φ(f) and ∆Trms
I5
Spectral response & sensitivity: autocorrelation receiver
If N delay-line taps, how many spectral samples Ns?
Say uniform weighting of φ(τ):
W(τ)
1
0
τM
τ
Then B = Ns • ∆f = Ns • (1/2τM) where spectral resolution ∆f ≅ 1/2τm
for orthogonal channels from boxcar W(τ)
W(f)
1 2 τM
W(f) for adjacent channel
f
∴ Ns = 2τMB = 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 jωt
+
Zg
Vg
+
V
-
Rg + j Xg
ZL
-
= Re {V} cos ωt + Im {V} sin ωt
{ }
1
∆
R e VI∗ (∆ PD )
Pdelivered
2
Pavailable ∆ max PD , i.e., if ZL = Z∗g
Receivers-K1
Delivered and Available Power
{ }
1
Pdelivered ∆ Re VI∗ (∆ PD )
2
Pavailable ∆ max PD, i.e., if ZL = Z∗
g
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
2
G
2
PD
2
PA
2
PA
Note: GA, GE
1
1
1
ZL
Ginsertion (= GI)
∆
PD
2
PD
with
amplifier
without
1 amplifier
Gexchangeable (=GE) ∆
PE
2
PE
1
don’t depend on ZL
do depend on Zg (via PE2)
K3
Definition: Signal-to-Noise Ratio (SNR)
First define:
WH z−1
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
2
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
= 0°K
⇒ F = 1+
= 1 (F = 0 dB)
To
= 290°K ⇒ F = 2
(F = 3 dB)
= 1500°K ⇒ F ≅ 6
(F ~= 7.5 dB)
K6