Receivers, Antennas,
and Signals
Professor David H. Staelin
6.661 Fall 2001
Slide 1
A1
Subject Content
A
Human
Processor
Radio
Optical, Infrared
Acoustic, other
C
Human
Transducer
Electromagnetic
Environment
B
Processor
Transducer
Communications: A → C (radio, optical)
Active Sensing:
A → C (radar, lidar, sonar)
Passive Sensing: B → C (systems and devices:
environmental, medical, industrial,
consumer, and radio astronomy)
6.661 Fall 2001
Slide 2
A2
Subject Offers
• Physical concepts
• Mathematical methods, system analysis
and design
• Applications examples
• Motivation and integration of prior learning
6.661 Fall 2001
Slide 3
A3
Subject Outline
• Review of signals and probability
• Noise in detectors and systems; physics of detectors
• Receivers and spectrometers; radio, optical, infrared
• Radiation, propagation, and antennas
• Signal modulation, coding, processing and detection
• Communications, radar, radio astronomy, and remote
sensing
• Parameter estimation
6.661 Fall 2001
Slide 4
A4
Review of Signals
Signal Types to be Reviewed:
• Pulses (finite energy)
• Periodic signals
(finite energy per period)
• Random signals
(finite power, infinite energy)
6.661 Fall 2001
Slide 5
A5
Pulses v(t)
∞
−∞
Have Finite Energy : ∫
2
v( t ) dt < ∞
Define Fourier Transform:
∆
V( f ) =
∆
v( t ) =
∞
∫ v( t )e
− j2
πft
N
ω
dt [volts/Hz = volt sec ]
−∞
∞
+ j2πft
V
(
f
)
e
df [volts ]
∫
−∞
Define notation " ↔" for Fourier Transform :
e.g. v(t) ↔ V( f )
Dimensions must only be self consistent;
e.g. v(t) can be dimensionl ess, volts, meters, newtons, etc.
6.661 Fall 2001
Slide 6
B1
Energy Spectral Density S(f)
∆
V( f ) =
S( f ) ∆ V(f)
2
∞
∫ v( t )e
− j2πft
ω
−∞
S(f) can have dimensions of :
sec2
if t is time and v is dimensionless
m2
if t is distance and v is dimensionless
(volts/Hz)2
if t is time and v(t) is volts
(
)
where (v/Hz)2 = v 2 /sec (Hz ) = (v / sec )2
Joules/Hz
if S(f) is dissipated in a 1-ohm resistor
2
by v(t) volts, where Joules = volts • sec/ohm
Et cetera
6.661 Fall 2001
Slide 7
B2
Autocorrelation Function
∞
v( t )v ∗ ( t − τ)dt
−∞
R( τ) ∆ ∫
[v 2 sec ]or [J], etc.
Claim: R( τ) ↔ S( f )
∞
∞
− j2πfτ
S( f ) ? ∫ R( τ)e
dτ = ∫
−∞
−∞
=
{∫
} • {∫
∞
v(t)e− j2πft dt
−∞
{
}
∞
∗
− j2f τ
v
(
t
)
v
(
t
)
dt
e
dτ
−
τ
∫−∞
∆ t′
}
t − t′− dt′
∞
v ∗ (t′)e+ j2πft′ dt′
−∞
Reverses sign of - dt′ for t = constant
2
S( f ) = V( f ) • V ∗ ( f ) = V( f ) Q.E.D.
Therefore:
∞
R( τ) =
S( f )e + j2πfτ df
−∞
∞ 2
∞
R(0) =
v ( t ) dt =
S( f ) df Parseval' s Theorem
−∞
−∞
∫
∫
6.661 Fall 2001
Slide 8
∫
B3
Compact Transform Notation
[v ]
v( t ) ↔ V ( f )
↓
↓
↓
2
R( τ) ↔ V( f ) ∆ S( f )
↔ [v / Hz]
↓
[v 2 sec ] ↔ [v / Hz]2
[Joules] ↔ [J / Hz]
6.661 Fall 2001
Slide 9
If power is
dissipated in a
1-ohm resistor
B4
Define “Unit Impulse”
ε
uo ( t ) ∆ δ( t ) where lim
uo ( t ) dt = 1, uo ( t ) = 0 for t > 0
−
ε
ε →0
∫
t
un −1( t ) ∆
u ( t ) dt
−∞ n
∫
Ramp
Step
uo ( t ) Impulse
t
0
6.661 Fall 2001
Slide 10
u−1( t )
0
t
u− 2 ( t )
0
Slope = +1
t
B5
Define “Convolution”
∞
a( t ) b(t − τ) dt
−∞
a( t ) ∗ b( t ) ∆ ∫
a(t)
= c( τ)
a( t ) ∗ b( t ) = c( τ)
b(t)
*
t
0
6.661 Fall 2001
Slide 11
τ
t
0
0
B6
Useful Transformation Pairs for Pulses
∞
a( t ) e − j2πft dt
−∞
a( t )
↔ A (f )
A( f )∆ ∫
uo ( t ) = δ( t )
↔ 1
1
↔ uo (f)
Have ∞ energy
(treated as special pulses)
a′( t )
↔ jωA (f )
un ( t )
↔ ( jω)n
a( t ) e jωo t
↔ A (f − fo )
a( t − t o )
↔ A ( f ) e − j ωt o
u−1( t ) e − αt
↔ 1/ ( jω + α )
6.661 Fall 2001
Slide 12
ω∆2πf
∞
A( f )e j2πft df
−∞
a( t ) = ∫
ωo ≡ 2πfo
C1
Transforms: Even/Odd Functions
ae ( t )
↔ R e {A (f )}
where:
so:
0
6.661 Fall 2001
Slide 13
EVEN
ao (t) ∆ [a( t ) − a(− t )] / 2 = ao (− t )
ODD
a(t) = ae (t) + ao (t)
a( t )
e.g.
ao ( t )
ae ( t ) ∆ [a( t ) + a(− t )] / 2 = ae (− t )
t
=
ae ( t )
0
t
+ ao ( t )
0
t
↔ j Im{A( f )}
C2
Transforms: Operators and Gaussians
a1( t ) • a2 ( t ) ↔ A1( f ) ∗ A 2 ( f )
a1( t ) ∗ a2 ( t ) ↔ A1( f ) • A 2 ( f )
v(t)
σ
0
6.661 Fall 2001
Slide 14
A
−( t / σ )2 / 2
−( σω)2 / 2
v( t ) ∆
e
↔ Ae
t
σ 2π
All
Gaussians
↓
↓
A 2 −( τ / σ 2 )2 2
2 −( σω)2
e
↔ A e
2σ π
C3
Linear Systems
x(t)
h(t)
y(t)
Characterized by:
h(t) = " Impulse Response, " where
∞
x( τ)h( t − τ)dτ ← " superposition integral"
−∞
y(t) ∆ x(t) ∗ h( t ) ∆ ∫
Test : If x(t) = δ(t), then y(t) = h(t)
If h(t) = δ(t), then y(t) = x(t)
Note:
A ∗ (B + C) = (A ∗ B) + (A ∗ C)
A ∗B
=B∗A
A ∗ (B ∗ C) = (A ∗ B) ∗ C
6.661 Fall 2001
Slide 15
“Distributive”
“Commutative”
“Associative”
C4
Periodic Signals
∞ 2
T 2
Although ∫ v ( t )dt = ∞, ∫ v ( t )dt < ∞ where Period ∆ T
o
−∞
v(t)
–T
m =1
m=0
T
0
Fourier Series:
1 T/2
v( t )e − jm( 2π / T )t dt
Vm ∆ ∫
T −T / 2
ω = 2πf
o
v(t) =
Infinite energy,
finite power
2T
3T
where m = 0, ± 1, ± 2,...
o
∞
jm 2πfo t
∆ 1/T )
(
V
e
f
∑ m
o
m = −∞
6.661 Fall 2001
Slide 16
E1
Autocorrelation, Power Spectrum
Autocorrelation Function:
∞
1 T/2
2
∗
v( t )v (t − τ )dt = ∑ V m e jm2πfo τ
R( τ) ∆ ∫
T −T / 2
m = −∞
Power Spectrum:
Φm ∆ V m
6.661 Fall 2001
Slide 17
2
1 T/2
= ∫
R( τ)e − jm 2πfo τ dt
T −T / 2
E2
Compact Notation
↔ Vm
↓
2
↔ V m ∆ Φm ↔ Φ(f)
v( t )
↓
R( τ)
Typical dimensions:
[volts]
↓
[ volts2 ]
[volts]
↔
↓
↔
[ volts2 ]
In 1-ohm resistor:
[ watts]
6.661 Fall 2001
Slide 18
↔
[W ]
E3
Transforms of Impulse Trains
a
− 2T
a/T
v( t )
0
2T
t
0
v( t )
(a / T )
2T
τ
−3/T
Vm
0
R( τ)
ah
area = a2 / T
e.g.
1/ h
τ
Area = a is impulse value
6.661 Fall 2001
Slide 19
0
2
a 2 / T R( τ)
− 2T
−3/T
Vm
2/T
2
f
→ Φ( f )
2/T
f
a2h2 ⎛ 1 ⎞
•⎜ ⎟
T
⎝h⎠
= R(0) = a2h / T
τ
− 1 / h 0 1/ h
Let h → ∞ so area a2 / T
becomes impulse value
E4
Receiver Noise Processes
Receivers, Antennas, and Signals
Professor David H. Staelin
6.661 Fall 2001
Slide 20
Random Signals
Random signals generally have finite power, infinite energy,
and are unpredictable
Example:
[WH ]
z
−1
Φ( f )
0 f MAX
f
Since we have infinite information for infinite time and
a finite frequency band, then “V(f)” is not an analytic function
and our approach must be different.
New definitions are required.
6.661 Fall 2001
Slide 21
G1
Expected Value of x
Finite or infinite ensemble of xi (t)
∞
E[x(t )] ∆ ∑ xi (t) p{xi (t )} → ∫ x p(x ) dx
−∞
i
t1
xi ε " ensemble"
t2
xi ( t )
∑ p( x i ) ∆ 1
t
i
p[xi (t )]
t
x j (t)
A “random signal” is drawn from some ensemble
[
]
Autocorrelation Function : φv (t1, t 2 ) ∆ E v (t1)v ∗ (t 2 )
6.661 Fall 2001
Slide 22
G2
Stationarity
v(t) is “wide-sense stationary” if:
φ v (t1, t 2 ) = φ v (t1 + ∆, t 2 + ∆ ) = φ(τ )
where τ ∆ t 2 − t1 for all t1, t 2 , ∆
v(t) is “strict-sense stationary” if:
E[g{v (t1), v (t 2 ),..., v (t n )}] = E[g{v (t1 + ∆ ), v (t 2 + ∆ ),..., v (tn + ∆ )}]
for any function g
v(t) is “Ergodic” if: v(t) is wide-sense stationary and
1 T
∗
v(t)
v
φv ( τ ) = lim
( t − τ ) dt,
∫
T
−
T →∞ 2T
i.e., ensemble average = time average
transitions
Otherwise v(t) is
occur only
“Non-stationary” – e.g.: 0 1 0 1 1 0 0 1 t
at clock ticks
(time-origin sensitive)
0 1 2 3 4 5 6
6.661 Fall 2001
Slide 23
G3
Transform Diagram: Random Signals
v(t)
↔
(?)
↔
↓
Φ(f)
↓
φv(τ)
Typical Sets of Units
[V]
↔
↓
[ V2 ]
↔
(?)
[V]
↓
↓
[ V2/Hz ]
[W]
Power to
1-ohm resistor
6.661 Fall 2001
Slide 24
↔
(?)
↓
↔
[W/Hz]
Power
spectral
density
G4
Power Spectral Density
2⎤
⎡1 T
− j2πft
Φ( f ) = lim E ⎢
v( t )e
dt ⎥
∫
T
−
T → ∞ ⎣ 2T
⎦
Why use E[ ] if v(t) is ergodic?
Because lim σ2T ( f ) ≠ 0! where σ2T ( f ) ∆ E[Φ T ( f ) − Φ( f )]2
Spectral resolution increases with T,
becoming infinite as T → ∞
e.g.
Infinite information in finite
t bandwidth unless ensemble
is averaged
Power Spectral Density Computation:
For a single ergodic waveform, take ensemble average
over successive intervals of width 2T. Use T adequate to
yield desired or meaningful spectral resolution.
6.661 Fall 2001
Slide 25
G5