DSP Based Corrections of Analog Components in Digital Receivers

Transcription

1 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

2 It s all done with Computer Chips

3 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.

4 We each own a Billion Transistors For Comparison, the Eiffel Tower Contains 18,084 Parts It is Fastened together by 2.5 Million Rivets

5 We each own a Billion Transistors The world manufactures more transistors than it grows grains of rice.

6 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)

7 Digital Translator Sampling Clock Analog IF Variable BW Filter 1-to-8 Filter ADC DAC 2 DC I-Q 2 4-to-1 Line 2 Cancel Balance Filter Cancel Equalizer Analog Synthesizer DDS DDS DC Cancel I-Q Balance Digital Translation Line Cancel Variable BW Filter Equalizer DAC sin(x)/x Compensation (x) sin(x) Comp

8 From Where Did DC Come? Self Mixing in Analog Down Converter ADC Injects DC 2 s Complement Arithmetic Injects DC

9 Self Mixing in Down Converter LNA Parasitic Coupling Frequency Synthesizer Low Pass Filter Low Pass Filter f f f DC Offset: Self Mixing Desired Component: Mixing Spectral Image: I-Q Imbalance

10 DC From A-to-D Converter CLK x q CLK x q X(n) ADC x q (n) X(n) ADC X q (n) x x x q x- x- x x q x Rounding Quantizer Truncating Quantizer

11 Truncating Quantizer Highest Allowable Output Value Not Greater than Measured Value (n+1)q Sampled Signal Values Nq Quantized Sample Values (N-1)q Amplitude Quantization Errors Sampled Signal Values Quantization Errors Quantized Sample Values Sample Time

12 2 s Complement Number Representation (b-2)-bits b-bits Error (b-2)-bits b-bits Error Quantized Number Line -N min Negative numbers: Measure displacement from reference. Reference = -Nmin 0 +N max Positive numbers: Measure displacement from reference. Reference = 0

13 Errors Due to finite Arithmetic x 1 q A q C x q M h h Q x 2 s = x + x +q 1 2 A h Q p = x. h Q +q M Quantize Addition Quantize Coefficient Quantize Multiplication

14 Finite Arithmetic in Radix-2 FFT Algorithm Time Time Freq Freq k 1 + k n n 1 N + N N -Pnt N DFT n 2 N N -Pnt 2 N 2 DFT 2 k 2 -j 2π e N k - + k+ SCL (2, 2 0-1) N 2 N

15 Radix-2 FFT Signal Flow Diagram w 0 w 2 w 0 - w 0 w 1 w 2 w w

16 Algorithm Noise Due to Finite Arithmetic, Coefficient Noise

17 Algorithm Noise due to Finite Arithmetic, Scaling Noise

18 DC Canceller, DC Notch Filter x(z) - f(z) z -1 y(z) α 1-α + H ( Z) Z 1 = Z (1 α)

19 Spectral Response

20 DC Canceller with Embedded Sigma-Delta Converter x(n) y (n) 2 - q_out Q q_loop R1-1 Z - μ -1 Z R2 Suppress Bit Growth with Sigma-Delta Converter in Feedback Path

21 DC Canceller Time Series

22 Spectra Input and Output of Canceller

23 Tunable Notch, Spin the Delay Line x(z) y(z) - f(z) e jφ z -1 α φ 1-α + H ( Z) = Z jφ Z e (1 α) e j φ

24 Spectral Response

25 Tuning With LP-to-BP Transformation x(z) y(z) f(z) - c cz 1 Z c z -1 2α φ 1-α + 1-α + H 2 Z 2cZ + 1 Z) = : c Z 2c(1 α) Z + (1 2α ) ( 2 = cos( φ)

26 Implementing LP-to-BP Transformation x(z) x(z) -1-1 Z Z - - b 1 b 1 x Z y(z) Z -1 Z Y(z) z -1 Y(z) z 1-cZ Z-c y(z) y z z c=cos( θ) - mu

27 Spectral Response

28 Self Tuning: Reference Canceling x(n) y(n) * w(n) Z -1 w(n+1) * μ r(n) x(n) y(n) * w(n) Z -1 w(n+1) * μ e jθn e jθn

29 Filters have Same Transfer Function x(n) y(n) * w(n) Z -1 w(n+1) * μ x(z) e jθn e jθn y(z) - f(z) e jφ z -1 α

30 I-Q Imbalance

31 Ideal I-Q Mixing cos( ω 0 t) cos( ω 0 t) I(t) Shape Match ^ I(t) CHANNEL Q(t) Shape n(t) Match ^ Q(t) -sin( ω 0 t) -sin( ω 0 t)

32 Real Sinusoids, Time and Frequency

33 Complex Sinusoids-I

34 Complex Sinusoids-II

35 Imbalance: New Spectral Terms X={ A cos(2 f t) } i Y={ i A sin(2 f t) } A 2 Real A Real 2 _ A cos(2πf A 0 t) π -f 0 2 -f 0 0 _ A Imag Imag 2 f 0 f f 0 f π 0 A -f 0 A 2 Imag Real Real _ A 2 f 0 f _ A 2 _ A α 2 - A _ (1+ ε) -f 0 Imag Real Real _ A α 2 X=F{ } i Y=F{i (1+ ε) A sin(2πf t+ α) } f 0 _ A 2 f (1+ ε) 0 Imag f 0 f _ A α 2 -f 0 -f X+i Y={ A exp(j 2 f t) } 0 π 0 Imag _ A α 2 X+i Y=F{sum} f 0 _ A ε 2 f

36 Arbitrary Signals, Time and Frequency

37 Complex Baseband & Complex Band-Centered

38 Complex Baseband & Real Band-Centered

39 Complex Down Conversion

40 Gain and Phase Imbalance in I-Q Mixers

41 Balanced Mixers

42 Gain Imbalance

43 Phase Imbalance

44 Gain and Phase Imbalance

45 Filter Imbalance

46 Effect of Gain Imbalance

47 Effect of Phase Imbalance

48 Description and Model of Baseband Down Conversion

49 Effect of I-Q Imbalance on 16-QAM Constellation 2.5 I-Q SAMPLES OF 16-QAM SIGNAL WITH I-Q GAIN AND PHASE MISMATCH

50 Input Signal Conditioning and Correction Analog Signals Digital Signals Analog I/Q Down Convert Analog Low Pass Filters A-to-D Converter DC Cancel Phase Balance Gain Balance Fixed Equalizer Synthesizer

51 DC-Cancel and I-Q Imbalance Correction DC I I I 1+ε α Q Q I Q DC Cancel DC Cancel - ^α α estimate 1-ε ^ ε estimate ^ I ^ Q DC Q

52 I-Q Imbalance Correction Algorithms

53 Gain and Phase Parameter Trajectories 0.5 PHASE PARAMETER 0.4 AMPLITUDE GAIN PARAMETER 1 AMPLITUDE TIME (in SAMPLES)

54 Constellation after Correcting Gain and Phase Imbalance 2.5 CONSTELLATION OF I-Q SAMPLES AFTER PHASE AND GAIN ADAPTION

55 That was the Easy part The Easy part is over Consider a channelizer!

56 Analog Block Conversion

57 Digital Second Conversion

58 Description and Model of Block Down Conversion

59 Constellations of channel +k and -k

60 Crosstalk Between Channels k and k Due to gain and Phase Imbalance

61 Constellation after Gradient Descent Correction of Gain and Phase Imbalance

62 Crosstalk Between Channels k and Empty Channel k

63 Constellation after Gradient Descent Correction of Gain and Phase Imbalance

64 Estimating Correcting Gain Terms

65 The Plot Thickens Analog Block Down Conversion With a Frequency Offset Removed by Digital Down Conversion

66 Analog Block Conversion with Frequency Offset

67 Digital Down Conversion and Removal of Frequency Offset

68 Description of Model and Block Down Conversion With frequency Offset

69 Estimating Correcting Gain Terms with Frequency Offset

70 Constellations with I-Q Imbalance with and without Frequency Offset Constellations, With I-Q Imbalance Constellations, With I-Q Imbalance and Frequency Offset

71 Constellation Trajectories During Adaptive Convergence Constellations, With I-Q Imbalance and Frequency Offset Constellations, Compensated for I-Q Imbalance

72 Processing Task Variable BW Variable Length FIR Filter

73 Narrower Bandwidth Filters Have Narrower Transition Bandwidths Hence Are Longer Filters

74 Spectrum and Group Delay IIR Filter and Equalizer

75 Spectrum and Group Delay 2- BW IIR Filters and Equalizers

76 Pole-Zero Diagrams for IIR Filter and Equalizer

77 Filter Coefficient Variation with Filter Bandwidth H( Z) = Z + bz Z + az 1 + a2

78 Equalizer Coefficient Variation with Filter Bandwidth EZ ( ) = az + az + 1 Z az a

79 Recursive Filter for Variable Bandwidth with Companion All- Pass Equalizer x(n) bw Recursive Filter a(bw) n b(bw) n Recursive Equalizer c(bw) n d(bw) n Coefficient Polynomials y(n)

80 Compare FIR Filter and Equalized IIR Filter

81 Narrower Bandwidth-> Longer Filter, Longer Filter -> More Taps True for Fixed Sample Rate

82 Narrower Bandwidth-> Longer Filter, Longer Filter -> Same Taps at Lower Sample Rate

83 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

84 Spectra at Input and Output of the Three Processing blocks of the Variable Bandwidth Filter

85 Impulse Response of Variable BW FIR Filter

86 Power Amplifier Linearization 1-to-8 Filter Pre-Distort IF Filter PA DAC NL Model LP Filter 2 Gain Correct Table 2 (x) sin(x) Comp DDS Analog Synthesizer IF Filter ADC PA Linearization Peak-to-Average Ratio Control

87 Non Linear Amplifier and Precompensating Gain 3 Nonlinear Transfer Function of Amplifier Output Magnitude dB Compression Point Input Magnitude Compensating Gain of Transfer Function Compensating Gain Input Magnitude

88 Transition Diagram Input and Output of Amplifier and input and output of Precompensator 2 Transition Diagram: Input and Compressed Amplifier 2 Transition Diagram: Input and Precompensated

89 Spectra: Input and Output of Amplifier and Output of Precompensator and Precompensated Amplifier

90 To Clip or Not to Clip; That is the Question! +L CLIP s(t) 1 s(t) 2 L CLIP s 2 s 3 -L CLIP s 1 -L CLIP s 1 L CLIP L CLIP -L CLIP s(t)= 3 s(t)-s(t) 2 1 -L CLIP

91 Band Limited Subtractive Clipping a 1 L S(n) 3 k 2 S(n) 1 k 2 - k 1 k 1 L -L -L C(k ) 1 a 2 a 1 k 1 C(k ) 2 a 2 k 2

92 Band-Limited Clipping s(t) 1 L CLIP CLIP-1 s(t) 2 s(t) 1 s(t) 2 - s(t) 3 CLIP-2 s(t) 3 L CLIP s(t) 1 Delay s(t) 2 L CLIP CLIP-2 Filter s(t) 3 s(t) 4

93 Spectra: Input Signal, Clipping Component and Clipped Signal Magnitude of Input Signal Magnitude of Input Signal Exceeding Top Magnitude of Input Signal - Level Exceeding Top

94 Spectra: Input Signal, Band Limited Clipping Component and Clipped Signal Magnitude of Input Signal Magnitude of Input Signal - Level Exceeding Top and Filtered Version Magnitude of Input Signal - Filtered Level Exceeding Top

95 Spectra: Input, Clip Component, Band Limited Clip, and Band Limited Clip Input Spectra to PAR Control Spectra; Signal Exceeding Top *log 10 (mag), (db) *log 10 (mag), (db) Normalized Spectra 0 Spectra; Bandlimited Version Exceeding Top Normalized Spectra 0 Spectra; Signal with Bandlimiting Top 20*log 10 (mag), (db) *log 10 (mag), (db) Normalized Spectra Normalized Spectra

96 Professor harris, may I be excused? My brain is full.

97 SOFTWARE DEFINED RADIO MAN Is Open For Questions