fred harris
DSP Based Corrections of
Analog Components in
Digital Receivers
IEEE Communications, Signal Processing, and
Vehicular Technology Chapters
Coastal Los Angeles Section
24-April 2008
It’s all done with Computer Chips
We each own a
BillionTransistors
We have an amazing wealth of resources
at our disposal! Just How big is a Billion?
A stack of a billion dollar notes would be
76.2 kilometers High.
A billion seconds ago was 32.5 years ago.
We each own a
Billion Transistors
For Comparison, the Eiffel Tower Contains 18,084 Parts
It is Fastened together by 2.5 Million Rivets
We each own a
Billion Transistors
The world manufactures more
transistors than it grows grains of rice.
How big is a billion grains of rice?
•
•
•
•
•
•
•
•
8mm x 2mm x 2mm (Long Grain)
1-billion grains of rice
8 Meters x 2 Meters x 2 Meters
Or 32 Cubic Meters
Or a cube 3.2 Meters on a side
It weighs 24,000 kgr (26.6 tons)
Market price, $1,000/ton; $26,600
CLS-350 Mercedes Benz weighs 2,200 kgr
(gross wt)
Digital Translator
Sampling
Clock
Variable
BW Filter
Analog
IF
1-to-8
Filter
ADC
2
DAC
DC
Cancel
2
I-Q
Balance
DDS
Analog
Synthesizer
4-to-1
Filter
Line
Cancel
Equalizer
DC Cancel
I-Q Balance
Digital Translation
Line Cancel
Variable BW Filter
Equalizer
DAC sin(x)/x Compensation
2
DDS
(x)
sin(x)
Comp
From Where Did DC Come?
• Self Mixing in Analog Down Converter
• ADC Injects DC
• 2’s Complement Arithmetic Injects DC
Self Mixing in
Down Converter
Parasitic
Coupling
Frequency
Synthesizer
Low Pass
Filter
LNA
Low Pass
Filter
f
f
f
DC Offset:
Self Mixing
Desired Component: Mixing
Spectral Image:
I-Q Imbalance
DC From A-to-D Converter
CLK
X(n)
ADC
xq
xq
CLK
xq(n)
X(n)
ADC
Xq(n)
x
x
x- xq
x- xq
x
Rounding Quantizer
x
Truncating Quantizer
Truncating Quantizer
Highest Allowable Output Value
Not Greater than Measured Value
Amplitude
Sampled Signal Values
(n+1)q
Sampled Signal Values
Quantization Errors
Quantization Errors
Nq
Quantized Sample Values
Quantized Sample Values
(N-1)q
Sample Time
2’s Complement
Number Representation
b-bits
(b-2)-bits
b-bits
Error
-Nmin
Negative numbers:
Measure displacement from reference.
Reference = -Nmin
(b-2)-bits
0
Error
Quantized
Number Line
+Nmax
Positive numbers:
Measure displacement from reference.
Reference = 0
Errors Due to finite
Arithmetic
x1
x2
qA
qC
h
s = x1+ x2+qA
Quantize Addition
x
hQ
hQ
Quantize Coefficient
qM
p = x.hQ +qM
Quantize Multiplication
Finite Arithmetic in
Radix-2 FFT Algorithm
Time
Time
n
N
2
N
N
2
Freq
n1
n2
N -Pnt
2
DFT
N -Pnt
2
DFT
+
k1
Freq
+
N
2
k+ N2
+
k2
N
2
k
-j 2π k
e N
SCL
0
-1)
(2 , 2
N
Radix-2 FFT
Signal Flow Diagram
000
001
-
010
0
w
2
w
011
-
-
100
0
w
1
w
101
-
110
0
2
w
111
-
2
-
3
-
w
w
-
-
w
-
Algorithm Noise Due to Finite
Arithmetic, Coefficient Noise
Algorithm Noise due to Finite
Arithmetic, Scaling Noise
DC Canceller,
DC Notch Filter
x(z)
y(z)
-
f(z)
z -1
α
1-α
H (Z ) =
Z −1
Z − (1 − α )
+
Spectral Response
DC Canceller with
Embedded Sigma-Delta
Converter
y2 (n)
x(n)
-
q_loop
Q
q_out
R1
Z
-1
μ
Z
-1
R2
Suppress Bit Growth with
Sigma-Delta Converter in
Feedback Path
DC Canceller Time Series
Spectra
Input and Output of Canceller
Tunable Notch,
Spin the Delay Line
x(z)
y(z)
e jφ
f(z)
1-α
z -1
φ
α
Z − e jφ
H (Z ) =
Z − (1 − α )e jφ
+
Spectral Response
Tuning With LP-to-BP
Transformation
y(z)
x(z)
c
f(z)
−
1 − cZ
Z −c
1-α
z -1
+
φ
2α
1-α
Z 2 − 2cZ + 1
H (Z ) = 2
: c = cos(φ )
Z − 2c(1 − α ) Z + (1 − 2α )
+
Implementing LP-to-BP
Transformation
x(z)
Z
-
x(z)
-1
Z
-
b1
-1
Z
y(z)
-1
Z
-1
b1
-1
Z
-1
Y(z) z
y(z)
Y(z) z-1
1-cZ
Z-c
y
x
z
-1
z
-1
c=cos(θ)
-
-
mu
Spectral Response
Self Tuning:
Reference Canceling
y(n)
x(n)
w(n)
w(n+1)
*
*
-1
Z
μ
r(n)
y(n)
x(n)
*
w(n+1)
w(n)
*
-1
Z
μ
jθn
e
j θn
e
Filters have Same
Transfer Function
y(n)
x(n)
*
w(n+1)
w(n)
*
-1
Z
μ
jθn
j θn
e
e
x(z)
y(z)
e jφ
f(z)
z -1
α
I-Q Imbalance
Ideal I-Q Mixing
cos(ω0t)
cos(ω0t)
I(t)
Shape
Match
^
I(t)
CHANNEL
Q(t)
n(t)
Shape
-sin(ω0t)
Match
-sin(ω0t)
^
Q(t)
Real Sinusoids,
Time and Frequency
Complex Sinusoids-I
Complex Sinusoids-II
Imbalance: New Spectral Terms
_A
2
_A
Real
2
_A
-f 0
X={ A cos(2πf 0t) }
X=F{ A cos(2πf0t) }
_A
-f 0
2
Imag
f
_A α
2
-
_A
f0
i Y=F{i (1+ ε) A sin(2πf0t+α) }
-f 0
Imag
f
2
X+i Y={ A exp(j 2πf 0t) }
A
_A α
-f 0
2
Imag
f0
f
_A α
2
Real
f
Real
2
i Y={ i A sin(2 πf 0t) } -f 0
Imag
f0
_A (1+ε)
Real
2
A
2
Imag
f0
_A
Real
Real
f0
f
_A (1+ε)
2
X+i Y=F{sum}
-f 0
Imag
f0
_A α _
A
2
ε
2
f
Arbitrary Signals,
Time and Frequency
Complex Baseband &
Complex Band-Centered
Complex Baseband &
Real Band-Centered
Complex Down Conversion
Gain and Phase Imbalance in
I-Q Mixers
Balanced Mixers
Gain Imbalance
Phase Imbalance
Gain and Phase Imbalance
Filter Imbalance
Effect of Gain Imbalance
Effect of Phase Imbalance
Description and Model of
Baseband Down Conversion
Effect of I-Q Imbalance on
16-QAM Constellation
I-Q S A M P LE S OF 16-QA M S IGNA L W ITH I-Q GA IN A ND P HA S E M IS M A TCH
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Input Signal Conditioning
and Correction
Analog Signals
Analog I/Q
Down Convert
Synthesizer
Analog
Low Pass
Filters
Digital Signals
A-to-D
Converter
DC
Cancel
Phase
Balance
Gain
Balance
Fixed
Equalizer
DC-Cancel and I-Q
Imbalance Correction
DCI
I
I’ I’
1+ε
DC
Cancel
^I
α
Q
Q’ Q’
DCQ
DC
Cancel
-
^
1-ε
α
ε
estimate
^
α estimate
^
Q
I-Q Imbalance
Correction Algorithms
Gain and Phase Parameter
Trajectories
PHASE PARAMETER
0.5
AMPLITUDE
0.4
0.3
0.2
0.1
0
0
1000
2000
3000
4000
5000
6000
7000
8000
3000
4000
5000
TIME (in SAMPLES)
6000
7000
8000
GAIN PARAMETER
1.05
AMPLITUDE
1
0.95
0.9
0.85
0.8
0
1000
2000
Constellation after Correcting
Gain and Phase Imbalance
CONSTELLATION OF I-Q SAMPLES AFTER PHASE AND GAIN ADAPTION
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
• That was the Easy part
• The Easy part is over
• Consider a channelizer!
Analog Block Conversion
Digital Second Conversion
Description and Model of
Block Down Conversion
Constellations of channel +k and -k
Crosstalk Between Channels k and –k
Due to gain and Phase Imbalance
Constellation after Gradient Descent
Correction of Gain and Phase Imbalance
Crosstalk Between Channels
k and Empty Channel–k
Constellation after Gradient Descent
Correction of Gain and Phase Imbalance
Estimating Correcting
Gain Terms
The Plot Thickens
Analog Block Down Conversion
With a Frequency Offset
Removed by Digital
Down Conversion
Analog Block Conversion
with Frequency Offset
Digital Down Conversion and
Removal of Frequency Offset
Description of Model and Block Down
Conversion With frequency Offset
Estimating Correcting Gain
Terms with Frequency Offset
Constellations with I-Q Imbalance with
and without Frequency Offset
Constellations, With I-Q Imbalance
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Constellations, With I-Q Imbalance and Frequency Offset
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Constellation Trajectories
During Adaptive Convergence
Constellations, With I-Q Imbalance and Frequency Offset
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
0.5
1
Constellations, Compensated for I-Q Imbalance
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
0
Processing Task
Variable BW
Variable Length
FIR Filter
Narrower Bandwidth Filters Have
Narrower Transition Bandwidths
Hence Are Longer Filters
Spectrum and Group Delay
IIR Filter and Equalizer
Spectrum and Group Delay
2- BW IIR Filters and Equalizers
Pole-Zero Diagrams for
IIR Filter and Equalizer
Filter Coefficient Variation
with Filter Bandwidth
Z 2 + b1Z + 1
H (Z ) = 2
Z + a1Z + a2
Equalizer Coefficient Variation
with Filter Bandwidth
a2 Z 4 + a1Z 2 + 1
E (Z ) = 4
Z + a1Z 2 + a2
Recursive Filter for Variable
Bandwidth with Companion AllPass Equalizer
x(n)
bw
Recursive Filter
Recursive Equalizer
an(bw)
cn(bw)
bn(bw)
dn(bw)
Coefficient Polynomials
y(n)
Compare FIR Filter and
Equalized IIR Filter
Narrower Bandwidth-> Longer Filter,
Longer Filter -> More Taps
True for Fixed Sample Rate
Narrower Bandwidth-> Longer Filter,
Longer Filter -> Same Taps
at Lower Sample Rate
Continuously Variable Bandwidth
FIR Filter
with Fixed Number of Taps
P-to-Q
Arbitrary
Interpolator
Fixed BW
Fixed Length
FIR Filter
Q-to-P
Arbitrary
Interpolator
Spectra at Input and Output of the
Three Processing blocks of the Variable
Bandwidth Filter
Impulse Response of
Variable BW FIR Filter
Power Amplifier Linearization
1-to-8
Filter
IF
Filter
Pre-Distort
DAC
2
NL
Model
Gain
Correct
Table
LP
Filter
(x)
sin(x)
Comp
Analog
Synthesizer
DDS
IF
Filter
2
ADC
PA Linearization
Peak-to-Average Ratio Control
PA
Non Linear Amplifier and
Precompensating Gain
Nonlinear Transfer Function of Amplifier
3
Output Magnitude
2.5
2
1-dB Compression Point
1.5
1
0.5
0
0
0.5
1
1.5
2
Input Magnitude
Compensating Gain of Transfer Function
2.5
3
0
0.5
1
2.5
3
1.3
Compensating Gain
1.25
1.2
1.15
1.1
1.05
1
0.95
1.5
Input Magnitude
2
Transition Diagram Input and
Output of Amplifier and input
and output of Precompensator
Transition Diagram: Input and Compressed Amplifier
Transition Diagram: Input and Precompensated
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2
-1
0
1
2
-2
-2
-1
0
1
2
Spectra: Input and Output of Amplifier and Output
of Precompensator and Precompensated Amplifier
To Clip or Not to Clip;
That is the Question!
s1(t)
+LCLIP
s2(t)
LCLIP
s3
s2
s1
-LCLIP
LCLIP
-LCLIP
s3(t)=s 2(t)-s 1(t)
-LCLIP
s1
-LCLIP
LCLIP
Band Limited
Subtractive Clipping
a1
L
L
S1 (n)
k2
-
k1
-L
a1
k1
C(k2)
a2
k2
k2
k1
-L
a2
C(k1)
S3 (n)
Band-Limited Clipping
LCLIP
s1(t)
s2(t)
CLIP-1
-
s1(t)
s2(t)
s3(t)
CLIP-2
s3(t)
LCLIP
s1(t)
s2(t)
Delay
CLIP-2
LCLIP
s3(t)
Filter
s4(t)
Spectra: Input Signal, Clipping
Component and Clipped Signal
Magnitude of Input Signal
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
1400
1600
1800
2000
Magnitude of Input Signal Exceeding Top
0.4
0.3
0.2
0.1
0
0
200
400
600
800
1000
1200
Magnitude of Input Signal - Level Exceeding Top
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Spectra: Input Signal, Band Limited
Clipping Component and Clipped Signal
Magnitude of Input Signal
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
1800
2000
Magnitude of Input Signal - Level Exceeding Top and Filtered Version
0.4
0.3
0.2
0.1
0
0
200
400
600
800
1000
1200
1400
1600
Magnitude of Input Signal - Filtered Level Exceeding Top
2.5
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Spectra: Input, Clip Component, Band
Limited Clip, and Band Limited Clip
Input Spectra to P A R Control
S pectra; S ignal Exceeding Top
0
20*log (mag), (dB)
-20
-40
10
10
20*log (mag), (dB)
0
-60
-80
-4
-2
0
Norm alized S pectra
2
-20
-40
-60
-80
-4
4
Spectra; B andlim ited V ersion E xceeding Top
2
4
0
20*log (mag), (dB)
-20
-40
10
10
20*log (mag), (dB)
0
Norm alized S pectra
S pectra; S ignal with Bandlim iting Top
0
-60
-80
-4
-2
-2
0
Norm alized S pectra
2
4
-20
-40
-60
-80
-4
-2
0
Norm alized S pectra
2
4
Professor harris, may I be excused?
My brain is full.
SOFTWARE
DEFINED
RADIO
MAN
Is Open For Questions