Analog and Telecommunication Electronics

Transcription

1 Politecnico di Torino Electronic Eng. Master Degree Analog and Telecommunication Electronics D1 - A/D/A conversion systems» Sampling, spectrum aliasing» Quantization error» SNRq vs signal type and level» Total error (ENOB) AY /04/ ATLCE - D DDC 2016 DDC 1

2 Contents section D Analog-to-Digital conversion process» Sampling, quantization, errors, SNR Digital to Analog converters» Transfer function, errors, taxonomy, uniform/weighted Analog to Digital converters» Transfer function, errors, taxonomy, SAR, subranging Signal conditioning» Protection, amplifier, filter, mux, Sample/Hold Special Converters» Logarithmic, differential, Sigma-Delta, oversampling 16/04/ ATLCE - D DDC 2016 DDC 2

3 Quick test - section D Describe effects of sampling Aliasing error, SNR A Describe effects of quantization Quantization error, SNR Q Draw block diagrams of various ADC types: Basic: flash, SAR, tracking, ramp /1, 2,..), Differential: Δ, ΣΔ, oversampling, noise shaping Mixed: residue, pipeline, multibit residue/pipeline Sample/Track-Hold Define sampling jitter, evaluate SNR J System level: define and evaluate ENOB 16/04/ ATLCE - D DDC 2016 DDC 3

4 Lesson D1: A/D/A conversion systems Sampling and aliasing Effects of sampling Anti-alias and reconstruction filters Oversampling and signal bandwidth Quantization Quantization error, SNR Q Effect of signal waveform Total system error: ENOB References Elettronica per Telecom.: 4.1 Rappresentaz. numerica. Design with Op Amp : 12.1 Performance Specification 16/04/ ATLCE - D DDC 2016 DDC 4

5 Analog vs Digital Analog signals Continuous in time and amplitude domains Ampl. Digital signals: numbers Discrete in time (sampling) and amplitude (quantization) sequence of numbers time Electronic systems migrate towards digital Which loss of information when A D? Qualitative Quantitative, relevant parameters How to keep under control information loss 16/04/ ATLCE - D DDC 2016 DDC 5

6 Radio systems: where are ADC/DAC? Services V battery, TX power,.. Baseband chain A/D e D/A for voice signals Receiver chain: A/D conversion of I/Q components in the IF channel Transmitter chain D/A conversion of synthesized I/Q components Software Defined Radio architectures Most functions by digital/programmable circuits A/D or D/A conversion very close to antenna 16/04/ ATLCE - D DDC 2016 DDC 6

7 AD and DA in the reference system A/D and D/A converters in the IF chain, with related signal conditioning circuits (filters, amplifiers, ). Voice ADC and DAC (audio CODEC) 16/04/ ATLCE - D DDC 2016 DDC 7

8 ADC in digital radio systems Digital IF filter Digital demodulator O X ADC DEMOD. ADC positions - End of the radio chain: baseband rate. - End of IF (digital demodulator) can use complex algorithms. (same HW for different types of modulation). - Before IF filter: increased computational load, but IF parameters can be mofied in the SW. - RF: speed problems, but this is the trend. 16/04/ ATLCE - D DDC 2016 DDC 8

9 I/Q digital radio Q channel V Q O X ADC V, V Va Input amplifier (LNA) /2 X I channel ADC V I DEMOD. Signal decomposed in I-Q components Image rejection by digital processing. ADC operates at IF 16/04/ ATLCE - D DDC 2016 DDC 9

10 Direct sampling I-Q SDR RF - speed - noise - resolution - linearity ADC /2 NCO DEMOD. Analog front-end: Filter+LNA ADC Universal radio HW: Can change frequency, modulation, (GSM, GPS, UMTS, ) by changing the digital processing SDR (Software Defined Radio) Numeric operation by programmable logic (FPGA) or microprocessors (DSP) 16/04/ ATLCE - D DDC 2016 DDC 10

11 Sampling and quantization The A/D conversion process requires two steps: Sampling: the analog signal A(t) (time continuous) is replaced by a sequence of samples As(ti) which represent the signal at specific values of time (ti). t t Quantization: each sample is translated into a numeric value Di with finite resolution -15, 8, 2, -5, -7, -6, -3, 6, 12, 16/04/ ATLCE - D DDC 2016 DDC 11

12 Sampling in the time domain x(t) t δ x S (t) T S F s = 1/T s is the sampling frequency 16/04/ ATLCE - D DDC 2016 DDC 12

13 Sampling in the frequency domain X(ω) Main spectrum (baseband) X S (ω) 0 2πF Secondary spectra (alias, folding) ω 0 2πF [F s ] 4πF S [2F s ] ω [f] sampling rate Fs = 1/Ts Time-domain) 16/04/ ATLCE - D DDC 2016 DDC 13

14 Numeric example Signal Fa = 5 khz, sampled at Fs = 16 ks/s X(ω) Main spectrum (baseband) Sampling rate F (khz) F S = 16 Band limit (Fs/2) Spectrum of sampled signal? X S (ω)??? F (khz) 16/04/ ATLCE - D DDC 2016 DDC 14

15 Sampled sine signal Continuous sine signal: Single spectral line at Fa Sampled sine signal: Spectral line replicated around K Fs Example: Signal Fa = 5 khz, sampled at Fs = 16 khz» Fa1a = 16 5 = 11 khz, Fa1b = = 21 khz» Fa2a = 32 5 = 27 khz, Fa2b = = 37 khz» Fa3a = 48 5 = 43 khz, Fa3b = = 53 khz» With 6 khz bandwidth, all components outband 16/04/ ATLCE - D DDC 2016 DDC 15

16 Numerical example: D1e-1 X(ω) Main spectrum (baseband) F (khz) F S = 16 Sampling rate X S (ω) Secondary spectra (aliases, folded signals) F (khz) Band limit /04/ ATLCE - D DDC 2016 DDC 16

17 Continuous signal Time domain: continuous sine Frequency domain: one spectral line Sine signal Simulator sampling and quantization in the website 16/04/ ATLCE - D DDC 2016 DDC 17

18 Sampled signal Replication around K Fs Spectral lines, generated by the sampling process (aliased spectra) fundamental 16/04/ ATLCE - D DDC 2016 DDC 18

19 Lesson D1: test D1-1 Draw, in the khz range, the spectrum of: 10 khz sine signal 10 khz sine signal, sampled at 40 ks/s 10 khz sine signal, sampled at 40 ks/s, low-pass filtered, cutoff 15 khz 10 khz sine signal, sampled at 18 ks/s 10 khz sine signal, sampled at 18 ks/s, low-pass filtered, cutoff 15 khz 25 khz sine signal, sampled at 40 ks/s 25 khz sine signal, sampled at 40 ks/s, low-pass filtered, cutoff 30 khz Compare spectra and discuss differences Point out spurious components caused by aliasing 16/04/ ATLCE - D DDC 2016 DDC 19

20 Simulation of the sampling process Virtual experiments Simulator available from the course website: learning material simulators 1. Spectrum of continuous sine signal 2. Spectrum of sampled sine signal 3. Effects of changing the sine signal frequency 4. Effects of changing the sampling frequency 16/04/ ATLCE - D DDC 2016 DDC 20

21 Recovery of x(t) with separate alias Main spectrum (baseband) Secondary spectra (aliases, folded, ) X S (ω) ω ω < 0 not considered 2πF M Useful bandwidth of signal x(t): F M 2πF S 4πF S Sampling rate: F S or 2πF S A lowpass filter can recover X(ω) and rebuild x(t) 16/04/ ATLCE - D DDC 2016 DDC 21

22 Overlapped aliases: no recovery! X S (ω) No lowpass filter can recover X(ω) and rebuild x(t) Overlap area no recovery possible 2πF M 2πF S 4πFS F M > F S /2 Nyquist rule violation: Inband aliasing noise ω 16/04/ ATLCE - D DDC 2016 DDC 22

23 Numerical example : D1e-2 Fa = 5 khz (signal useful bandwidth: 6kHz) Fs = 9 ks/s (sampling rate) Fs does not comply the Nyquist rule Draw spectrum of sampled signal 16/04/ ATLCE - D DDC 2016 DDC 23

24 Numerical example : D1e-2 - graph X(ω) Continuous signal F (khz) 0 5 F S = 9 Sampling rate Xs(ω)?????? F (khz) 0 F S = 9 Spectrum of sampled signal 16/04/ ATLCE - D DDC 2016 DDC 24

25 Numeric example : D1e-2 - sol Fa = 5 khz (signal useful bandwidth: 6kHz) Fs = 9 khz (sampling rate) Fs does not comply the Nyquist rule First alias (two sideband): Fa1a = 9 5 = 4 khz, Fa1b = = 14 khz Fa1a within signal bandwidth Cannot be removed by filtering Sampling creates a 4 khz component, not in the original input signal No problem for Fa1b and higher (outband) 16/04/ ATLCE - D DDC 2016 DDC 25

26 Numerical example : D1e-2 graph sol X(ω) Main spectrum (baseband) F (khz) 0 5 F S = 9 Secondary spectra (alias, folding) X S (ω) F (khz) No lowpass filter can recover X(ω) 16/04/ ATLCE - D DDC 2016 DDC 26

27 Anti Aliasing lowpass filter To avoid information loss signals must be sampled at least twice the bandwidth (Nyquist). Input signal must be band-limited, with Fa < Fs/2 Need for a low-pass anti-aliasing filter Real filters have ripple and finite attenuation» Always some energy above Fs/2 1 0 Fs/2 F 16/04/ ATLCE - D DDC 2016 DDC 27

28 Aliasing for real signals Actual signals are not frequency-limited Always some residual HF (over-nyquist) components HF signals are folded to baseband by sampling aliasing NOISE The amount of aliasing noise is related with: Input signal frequency spectrum (modified by antialias filter) Sampling frequency Fs First Main signal alias X(ω) (baseband) Aliasing noise 0 F S F (khz) 16/04/ ATLCE - D DDC 2016 DDC 28

29 Oversampling Sampling only slightly above the Nyquist limit requires steep antialiasing filters expensive! Another choice: sampling at a rate far higher than the Nyquist limit oversampling Example: 3 khz signal, 1 Ms/s alias sent far (2, 3 MHz, ) Relaxed specifications on the anti-alias input filter, but higher bit rate (more samples/s) Higher bit rate brings more heavy digital processing Bit rate can be reduced by digital filters Complexity moved from analog to digital domains More easy & cheap to handle 16/04/ ATLCE - D DDC 2016 DDC 29

30 Which is the actual limit? True Nyquist rule: A signal must be sampled at least twice the signal BANDWIDTH Example: a 1 GHz carrier, 100 khz BW signal can be safely sampled at > 200 ks/s Less stringent specs for RF A/D converters Minimum sampling rate depends on bandwidth, not carrier Tight specs for the sampling circuit Sampling jitter related with carrier, not bandwidth 16/04/ ATLCE - D DDC 2016 DDC 30

31 Undersampling Sampling at a rate lower than the Nyquist limit may create overlapped aliases. Can be done on band-limited signals Undersampling (see lesson B1) Sample rate can be reduced after sampling: decimation (move filtering from analog to digital domain). Baseband alias Sampling rate RF signal S f > 2 B1 F S 2F S 3F S B1 4F S f 16/04/ ATLCE - D DDC 2016 DDC 31

32 Sample/Hold module The A/D converter operates on each sample Conversion requires some time; the input signal (sample) must remain steady during the conversion. Two operations required Sample: read the analog signal value at a specific time Hold:keep that value for some time Sample/Hold (or Track/Hold) unit x S (t) T S = 1/F S t 16/04/ ATLCE - D DDC 2016 DDC 32

33 Effect of hold The Hold operation modifies the spectrum: Pulses become steps, with a width T H The signal spectrum is multiplied by sinf/f: 0s in F H =1/T H» Attenuation of high frequency components Can be corrected with filters» Peaking of HF response T H Va HOLD Vb 0 F S = 1/T H F 16/04/ ATLCE - D DDC 2016 DDC 33

34 Spectrum of signals with Hold As pulses become more narrow, the spectrum envelope becomes more wide. For T H = 0 (delta), the envelope becomes flat For T H = T S (Hold till next sample), the envelope is 0 at F = Fs Th Ts t Example: Squarewave: T H = T S /2, no second harmonic Fs = 1/Ts 1/Th F 16/04/ ATLCE - D DDC 2016 DDC 34

35 Mathematical model Sampling is multiplication with deltas Hold changes deltas into steps Hold operator: 16/04/ ATLCE - D DDC 2016 DDC 35

36 Correction of distortion Hold distorsion Hold distorsion compensation Overall transfer function H 0 (ω) = 1 16/04/ ATLCE - D DDC 2016 DDC 36

37 Reconstruction filter - 1 The signal from DAC is sampled Includes secondary spectra (alias) x(t) t X S (ω) T S = 1/F S ω 0 2πF S 4πF S 16/04/ ATLCE - D DDC 2016 DDC 37

38 Reconstruction filter - 2 The spectral replicas (alias) must be removed to get a continuos signal Need for a low-pass reconstruction filter t X S (ω) ω 0 2πF S 4πF S 16/04/ ATLCE - D DDC 2016 DDC 38

39 Corrected reconstruction filter The reconstruction filter must consider spectral distortion caused by Hold peaking at high band limit H(jω) Effect of hold ω Spectral correction 16/04/ ATLCE - D DDC 2016 DDC 39

40 Lesson D1: A/D and D/A conversion Sampling and aliasing Effects of sampling Anti-alias filter Reconstruction filters Quantization Quantization error Quantization noise, SNRq Signal type and dynamic Total system error: Oversampling and signal bandwidth ENOB 16/04/ ATLCE - D DDC 2016 DDC 40

41 Quantization - 1 Analog signal: any value in the input range (0 S) Digital signal: sequence of numbers Usualy binary (N bits) 2 N possible values (0..2 N -1) D i defines the signal interval, not exact value The difference is the quantization error q S A S 0 D 2 N -1 D i /04/ ATLCE - D DDC 2016 DDC 41

42 Quantization - 2 If the range 0.S is divided in 2 N intervals, the maximum error in the representation of A with D i is A D i+1 A D = S / 2 N A D ε q D i The maximum quantization error ε q is ε q A D /2 = S / 2 N+1 D i-1 16/04/ ATLCE - D DDC 2016 DDC 42

43 With xy representation D (digital) 2 N -1 1 LSB S A D A (analog) 16/04/ ATLCE - D DDC 2016 DDC 43

44 Quantization error ε q Same amplitude for all quantization intervals A D A D = S/2 N = 1 LSB ε q varies within ± A D /2 (1/2 LSB) +A D /2 -A D /2 A D Maximum value of q : 16/04/ ATLCE - D DDC 2016 DDC 44

45 Quantization noise Quantization can be seen as noise ε q added to an ideal A D conversion process x(t) D(t) Sampling Σδ(t-nT s ) ε q (t) Quantization Which are the features of this noise? How to define a signal/(quantization noise) ratio SNR q Which relation with the signal and with N? 16/04/ ATLCE - D DDC 2016 DDC 45

46 Signal amplitude distribution Amplitude distribution probability for the signal to assume a defined value (probability density function: PDF). Triangular wave: flat Squarewave: only extreme values When only statistical parameters are known, signal power can be evaluated from amplitude distribution. Distribution of quantization noise: A d is small Same probability to get any value of input signal within A d, for any input waveform Constant PDF ( q ) 16/04/ ATLCE - D DDC 2016 DDC 46

47 Amplitude distribution example 1 Triangular wave S A time A ρ(a) (PDF) Ps = S 2 /12 16/04/ ATLCE - D DDC 2016 DDC 47

48 Amplitude distribution example 2 Sine wave A A S time ρ(a) (PDF) Ps = S 2 /8 16/04/ ATLCE - D DDC 2016 DDC 48

49 Amplitude distribution example 3 Voice signal A A S time ρ(a) (PDF) Ps = S 2 /36 16/04/ ATLCE - D DDC 2016 DDC 49

50 Quantization noise power From amplitude distribution +A D /2 Small A D : -A D /2 constant amplitude distribution of ε q : ρ(ε q ) = 1/A D A d 2 q D A 2 D 3 2 A d D Quantization error power: P εq 16/04/ ATLCE - D DDC 2016 DDC 50

51 Signal to quantization noise ratio Defined as SNRq= P s /P εq = Signal power Quantization noise ε q power Quantization noise power: related with Full scale amplitude S Bit number N Signal power: related with Waveform Amplitude 16/04/ ATLCE - D DDC 2016 DDC 51

52 SNRq for sinewaves Sine, Vpp = S Veff = S 2 SQR(2) Ps = S 2 /8 Peq = A D2 /12 ; A D = S/2 N Ps/Peq = (2^2N Ad^2 / 8)/(12/Ad^2) = 1,5 2^2N measure in db 10 log 10 : log 10 2 = 0,3 log 10 10(2^2N) = 0,3x2N 6 N db log 10 1,5 = 0,176 db SNRq = (6 N + 1,76) db 16/04/ ATLCE - D DDC 2016 DDC 52

53 SNRq for some signals Signal with PDF close to 0 have lower power With full scale signals Sine, Vpp= S Ps = S 2 /8 Triangle, Vpp = S Ps = S 2 /12 Voice (gaussian pdf, S/2 = 3σ) Ps = S 2 /36 SNRq = (6 N + 1,76) db SNRq = 6 N db SNRq = (6 N - 4,77) db 16/04/ ATLCE - D DDC 2016 DDC 53

54 SNRq and number of bits SNRq = (K + 6N) db: K depends on signal distribution 1 bit adds 6 db to SNRq 6 db 1 bit waveform 16/04/ ATLCE - D DDC 2016 DDC 54

55 SNRq and signal amplitude Previous SNRq applies for full-scale signals If amplitude A < S SNRq decreases with signal amplitude (unity slope, -20dB/decade or -6dB/octave) If amplitude A > S A/D conversion saturates at full scale Overload condition SNRq decreases (very quickly) with increase of signal amplitude 16/04/ ATLCE - D DDC 2016 DDC 55

56 SNRq vs signal amplitude Correct operation Overload 1 bit less: -6dB Low SNRq Quantization Overload 16/04/ ATLCE - D DDC 2016 DDC 56

57 Quantization noise spectrum Flat power distribution from 0 to sampling frequency F S Spectral power density: N(f) = A D2 /12 F S To reduce quantization noise power (and improve SNRq): Increase N (bit number): reduces quantization step amplitude Increase F S (oversampling): spreads quantization noise power over a wider bandwidth Oversampling and filter: If DAC output is filtered more narrow than F S /2, the quantization noise power is reduced 16/04/ ATLCE - D DDC 2016 DDC 57

58 Lesson D1: A/D and D/A conversion Sampling and aliasing Effects of sampling Anti-alias filter Reconstruction filters Quantization Quantization error Quantization noise, SNRq Signal type and dynamic Total system error: Oversampling and signal bandwidth ENOB 16/04/ ATLCE - D DDC 2016 DDC 58

59 A/D conversion system Sampling: Sample/Hold Constraints on signal bandwidth F S > 2 F A Quantization: A/D converter Constraints on signal level use ADC near full scale Signal conditioning to fit these constraints Amplifier» Adapt signal level to ADC full scale Anti-alias filter» Limit signal bandwidth Input protection (avoid damages)» Limit input voltage to avoid damages to the system 16/04/ ATLCE - D DDC 2016 DDC 59

60 A/D conversion system block diagram Conditioning Sampling + quantization A Amplifier: To bring input signal near ADC full scale Antialias filter: To limit signal bandwidth for correct sampling Input protection not shown 16/04/ ATLCE - D DDC 2016 DDC 60

61 Multiple channel system + input protection 16/04/ ATLCE - D DDC 2016 DDC 61

62 Complete A/D D/A chain 16/04/ ATLCE - D DDC 2016 DDC 62

63 Total error Each unit introduces errors and noise Amplifier:» Gain, offset, nonlinearity, band limits Filter:» Outband signal and aliases folded into useful band Sample/Hold:» Sampling jitter A/D converter:» Quantization error Actual accuracy depends from all these elements Not just the bit number N of the A/D 16/04/ ATLCE - D DDC 2016 DDC 63

64 SNRt Key parameter: total Signal/Noise ratio: SNR t Caused by Aliasing, quantization, sampling jitter Other errors (amplifier, mux, ) Errors are nor correlated; evaluation sequence: Find power of single errors (Pni) Add the power: Ptot = Pni SNR 10 1 Pni 10 log ; Evaluate SNR t SNR t P s P P ni s 10 i 16/04/ ATLCE - D DDC 2016 DDC 64

65 Effective Number of Bits: ENOB SNR t expressed as Equivalent Number Of Bits Computed from SNR t (measured or evaluated with full-scale sine input signal) ENOB = (SNR t - 1,76)/6 = SNR t /6-0,3 Includes all noise/error sources (quantization, aliasing, sampling jitter, ) ENOB represents the number of actually useful bits of the A/D conversion system Specific parameters can describe distortion (similar to amplifiers): SNR, SFDR, THD, SINAD, 16/04/ ATLCE - D DDC 2016 DDC 65

66 ADC systems glossary SNR: Signal-to-Noise Ratio. Ps/Pn, no DC; use first five harmonics (sometime the first 9). In dbc (db to carrier, reference is the fundamental), or dbfs (db to full scale, the fundamental is extrapolated to the full-scale). SFDR: Spurious Free Dynamic Range. Ps/Ph (Ph is the highest spur). THD: Total Harmonic Distortion. Ps/Pd (Pd is the power of the first five (or 9) harmonics) SINAD: Signal Noise And Distortion. Ps/(Pn+Pd); no DC, in dbc (db / carrier) or dbfs (db / full scale) ENOB: Effective Number Of Bits. Measure in bit equivalent of converter performance (reversing the theoretical ideal SNR limit based on quantization noise). 16/04/ ATLCE - D DDC 2016 DDC 66

67 Example of harmonic folding Sampling moves all spectral lines into the 0 Fs/2 range Fs/2 (Nyquist) Harmonics folded into baseband Fs/2 (Nyquist) 16/04/ ATLCE - D DDC 2016 DDC 67

68 Sampled spectrum parameters Harm3 is largest spurious; used to evaluate P H (SFDR) 16/04/ ATLCE - D DDC 2016 DDC 68

69 Terminology Sampling rate SFDR Fundamental Largest spur. (H3) Folded harmonic 3 Folded fundamental Spurious F Folded harmonic 6 Harm 2 Harm 3 Harm 6 16/04/ ATLCE - D DDC 2016 DDC 69

70 Lesson D1 test questions How can the aliasing noise be reduced? Define SNRq and how it is related with signal waveforms. Describe the relation between signal amplitude and SNRq. Which is the relation between sampling jitter error and signal frequency? Which parameter best describes the actual precision of an A/D conversion system? Which elements influence the precision of an A/D conversion system? 16/04/ ATLCE - D DDC 2016 DDC 70