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DSP Based Corrections of Analog Components in
Digital Receivers
DSP Based Corrections of DSP Based Corrections of Analog Components in Analog Components in
Digital ReceiversDigital Receivers
fred harrisfred harris
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 PartsIt is Fastened together by 2.5 Million Rivets
The world manufactures more transistors than it grows grains of rice.
We each own a Billion Transistors
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
Analog IF
I-QBalance
DCCancel
AnalogSynthesizer
DDS DDS
Sampling Clock
2 2 2
ADC DAC
4-to-1 Filter
1-to-8 Filter
LineCancel
VariableBW Filter
Equalizersin(x)(x)
Comp
DC CancelI-Q Balance
Digital TranslationLine Cancel
Variable BW FilterEqualizer
DAC sin(x)/x Compensation
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
LNA
FrequencySynthesizer
Low Pass Filter
Low Pass Filter
ParasiticCoupling
f f f
DC Offset: Self MixingDesired Component: MixingSpectral Image: I-Q Imbalance
DC From A-to-D Converter
x x
x- x-
x x
X(n) X(n)
xq xq
xq xq
xq(n) Xq(n)ADC ADC
Rounding Quantizer Truncating Quantizer
CLK CLK
Truncating QuantizerHighest Allowable Output Value
Not Greater than Measured Value
Nq
(n+1)q
(N-1)q
Sampled Signal Values
Sampled Signal Values
Quantized Sample ValuesQuantized Sample Values
Quantization Errors
Quantization Errors
Amplitude
Sample Time
2’s Complement Number Representation
0 +Nmax-Nmin
b-bitsb-bits
(b-2)-bits(b-2)-bits ErrorError
Positive numbers:Measure displacement from reference. Reference = 0
Negative numbers:Measure displacement from reference. Reference = -Nmin
QuantizedNumber Line
Errors Due to finite Arithmetic
x1 x
x2 hQ
hQh
qA qMqC
s = x + x +q1 2 A p = x h +qQ M.
Quantize Addition Quantize Coefficient Quantize Multiplication
Finite Arithmetic in Radix-2 FFT Algorithm
+
+
+
-
SCL
(2 , 20 -1)e
-j 2πN k
N
N N
NN
N
NN N
2 22
2
22 2
-Pnt
-Pnt
DFT
DFT
k1
n1
n2
k
k+k2
Time Time Freq Freq
n
Radix-2 FFT Signal Flow Diagram
000
011
110
001
100
111
010
101
-
-- -
-
- -
-
- - -
-
w0
w0
w0
w2
w2
w2
w3
w1
Algorithm Noise Due to Finite Arithmetic, Coefficient Noise
Algorithm Noise due to Finite Arithmetic, Scaling Noise
DC Canceller, DC Notch Filter
α
z-1-
x(z)
f(z)
y(z)
)1(1)(α−−
−=
ZZZH
+
1-α
Spectral Response
DC Canceller with Embedded Sigma-Delta
Converter
Z
Z
-1
-1
-
- μ
R1
R2
x(n)y (n)2
Qq_loop
q_out
Suppress Bit Growth withSigma-Delta Converter inFeedback Path
DC Canceller Time Series
Spectra Input and Output of Canceller
Tunable Notch, Spin the Delay Line
α
z-1-
x(z)
f(z)
y(z)φje
φ
φ
α j
j
eZeZZH
)1()(
−−−
=
+
1-α
φ
Spectral Response
Tuning With LP-to-BP Transformation
2α
z-1-
x(z)
f(z)
y(z)
c
cZcZ−
−−
1
)cos(:)21()1(2
12)( 2
2
φαα
=−+−−
+−= c
ZcZcZZZH
++
1-α
1-α
φ
Implementing LP-to-BP Transformation
-
-z z-1 -1
c=cos( )θ mu
x y
b1 b1
Z Z
Z ZZ
-1 -1
-1
-1
-1 -1-1
- -
x(z) x(z)
y(z) y(z)Y(z) zY(z) z
1-cZZ-c
Spectral Response
Self Tuning:Reference Canceling
y(n)x(n)
r(n)
Z-1
w(n) w(n+1)
μ
**
y(n)x(n)
ej nθ e
j nθ
Z-1
w(n) w(n+1)
μ
**
Filters have Same Transfer Function
y(n)x(n)
ej nθ e
j nθ
Z-1
w(n) w(n+1)
μ
**
α
z-1-
x(z)
f(z)
y(z)φje
I-Q Imbalance
Ideal I-Q Mixing
Shape
Shape
Match
Match
CHANNEL
cos( t)ω0 cos( t)ω0
-sin( t)ω0 -sin( t)ω0
n(t)
I(t) I(t)
Q(t)Q(t)
^
^
Real Sinusoids, Time and Frequency
Complex Sinusoids-I
Complex Sinusoids-II
Imbalance: New Spectral Terms
f f
f f
f f
RealReal
RealReal
RealReal
ImagImag
ImagImag
ImagImag
A
AA
A
A
A
A
A
A
A
A
A
A
A
A
2
22
2
2
2
2
2
2
2
2
2
2
_
__
_
_
_
_
_
_
_
_
_
_ α
α
(1+ )ε
(1+ )ε
α
α
ε
-
f0
f0
f0
f0
f0
-f0
-f0
-f0 -f0
-f0
-f0
f0
A cos(2 f t)π 0
A cos(2 f t)π 0
(1+ ) A sin(2 f t+ε π α)0
A sin(2 f t)π 0
A exp(j 2 f t)π 0
X={ }
X=F{ }
i Y={ i }i Y=F{i }
X+i Y={ }
X+i Y=F{sum}
Arbitrary Signals,Time and Frequency
Complex Baseband & Complex Band-Centered
Complex Baseband & Real Band-Centered
Complex Down Conversion
Gain and Phase Imbalance inI-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
-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
2.5I-Q SAMPLES OF 16-QAM SIGNAL W ITH I-Q GAIN AND PHASE MISMATCH
Input Signal Conditioning and Correction
Analog I/QDown Convert
AnalogLow Pass Filters
A-to-DConverter
DCCancel
PhaseBalance
GainBalance
FixedEqualizer
Synthesizer
Analog Signals Digital Signals
DC-Cancel and I-Q Imbalance Correction
I I’
Q Q’
α1+ε
DCI
DCQ
II’
QQ’
α
1-ε
- ^
^
^
^ αestimate
εestimate
DCCancel
DCCancel
I-Q Imbalance Correction Algorithms
Gain and Phase Parameter Trajectories
0 1000 2000 3000 4000 5000 6000 7000 80000
0.1
0.2
0.3
0.4
0.5PHASE PARAMETER
AM
PLI
TUD
E
0 1000 2000 3000 4000 5000 6000 7000 80000.8
0.85
0.9
0.95
1
1.05GAIN PARAMETER
TIME (in SAMPLES)
AM
PLI
TUD
E
Constellation after Correcting Gain and Phase Imbalance
-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
2.5CONSTELLATION OF I-Q SAMPLES AFTER PHASE AND GAIN ADAPTION
• 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 –kDue 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 ConversionWith 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
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Constellations, With I-Q Imbalance
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Constellations, With I-Q Imbalance and Frequency Offset
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Constellation Trajectories During Adaptive Convergence
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Constellations, With I-Q Imbalance and Frequency Offset
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Constellations, Compensated for I-Q Imbalance
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
Processing Task
Variable BWVariable Length FIR Filter
Narrower Bandwidth Filters Have Narrower Transition Bandwidths
Hence Are Longer Filters
Spectrum and Group DelayIIR Filter and Equalizer
Spectrum and Group Delay2- BW IIR Filters and Equalizers
Pole-Zero Diagrams for IIR Filter and Equalizer
Filter Coefficient Variation with Filter Bandwidth
21
21 2
1( ) Z b ZH ZZ a Z a
+ +=
+ +
Equalizer Coefficient Variation with Filter Bandwidth
4 22 1
4 21 2
1( ) a Z a ZE ZZ a Z a
+ +=
+ +
Recursive Filter for Variable Bandwidth with Companion All-
Pass Equalizer
a (bw)n
d (bw)n
c (bw)n
b (bw)n
Recursive Filter Recursive Equalizerx(n) y(n)
bwCoefficient Polynomials
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
Fixed BWFixed Length FIR Filter
P-to-Q ArbitraryInterpolator
Q-to-P ArbitraryInterpolator
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
AnalogSynthesizer
DDS
2
2
DAC
ADC
1-to-8 Filter
LPFilter
IFFilter
IFFilter
Pre-Distort
NLModel
GainCorrect Table
sin(x)(x)
PA
Comp
PA LinearizationPeak-to-Average Ratio Control
Non Linear Amplifier and Precompensating Gain
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3Nonlinear Transfer Function of Amplifier
1-dB Compression Point
Input Magnitude
Out
put M
agni
tude
0 0.5 1 1.5 2 2.5 30.95
1
1.05
1.1
1.15
1.2
1.25
1.3Compensating Gain of Transfer Function
Input Magnitude
Com
pens
atin
g G
ain
Transition Diagram Input and Output of Amplifier and input and output of Precompensator
-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Transition Diagram: Input and Compressed Amplifier
-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Transition Diagram: Input and Precompensated
Spectra: Input and Output of Amplifier and Output of Precompensator and Precompensated Amplifier
To Clip or Not to Clip; That is the Question!
s (t)1
s (t)=3 s (t)-s (t)2 1
+LCLIP
-LCLIP
s (t)2
-LCLIP
-LCLIP -LCLIP
LCLIPLCLIP
LCLIP
s2 s3
s1 s1
Band LimitedSubtractive Clipping
L L
-L -L
S (n)1
S (n)3
a1
a1
a2
a2
C(k )1
k1 k1
k1
k2
k2 k2
C(k )2
-
Band-Limited Clipping
LCLIP
s (t)2
s (t)3
s (t)1CLIP-1
-LCLIP
s (t)2
s (t)3
s (t)1
CLIP-2
LCLIP
s (t)2
s (t)3s (t)4
s (t)1
CLIP-2 Filter
Delay
Spectra: Input Signal, Clipping Component and Clipped Signal
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.5
1
1.5
2
2.5Magnitude of Input Signal
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4Magnitude of Input Signal Exceeding Top
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.5
1
1.5
2
2.5Magnitude of Input Signal - Level Exceeding Top
Spectra: Input Signal, Band Limited Clipping Component and Clipped Signal
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.5
1
1.5
2
2.5Magnitude of Input Signal
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4Magnitude of Input Signal - Level Exceeding Top and Filtered Version
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.5
1
1.5
2
2.5Magnitude of Input Signal - Filtered Level Exceeding Top
Spectra: Input, Clip Component, Band Limited Clip, and Band Limited Clip
-4 -2 0 2 4-80
-60
-40
-20
0
Input Spectra to PAR Control
Normalized Spectra
20*lo
g 10(m
ag),
(dB
)
-4 -2 0 2 4-80
-60
-40
-20
0
Spectra; Signal Exceeding Top
Normalized Spectra
20*lo
g 10(m
ag),
(dB
)
-4 -2 0 2 4-80
-60
-40
-20
0
Spectra; Bandlimited Version Exceeding Top
Normalized Spectra
20*lo
g 10(m
ag),
(dB
)
-4 -2 0 2 4-80
-60
-40
-20
0
Spectra; Signal with Bandlimiting Top
Normalized Spectra
20*lo
g 10(m
ag),
(dB
)
Professor harris, may I be excused? My brain is full.
SOFTWAREDEFINED
RADIOMAN
Is Open For Questions