1. FROM ANALOG TO DIGITAL DOMAIN Dr.M.A.Kashem Asst. Professor, CSE,DUET

2. M. E. Angoletta - DISP2003 - From analog to digital domain 2 / 30 TOPICS 1.Analog vs. digital: why, what & how 2.Digital system example 3.Sampling & aliasing 4.ADCs: performance & choice 5.Digital data formats

3. M. E. Angoletta - DISP2003 - From analog to digital domain 3 / 30 Analog & digital signals Continuous function continuous Continuous function V of continuous variable t (time, space etc) : V(t). Analog Discrete function discrete Discrete function V k of discrete sampling variable t k, with k = integer: V k = V(t k ). Digital Uniform (periodic) sampling. Sampling frequency f S = 1/ t S

4. M. E. Angoletta - DISP2003 - From analog to digital domain 4 / 30 Digital vs analog proc’ing Digital Signal Processing (DSPing) More flexible. Often easier system upgrade. Data easily stored. Better control over accuracy requirements. Reproducibility. Advantages A/D & signal processors speed: wide-band signals still difficult to treat (real-time systems). Finite word-length effect. Obsolescence (analog electronics has it, too!). Limitations

5. M. E. Angoletta - DISP2003 - From analog to digital domain 5 / 30 DSPing: aim & tools Software Programming languages: Pascal, C / C++... “High level” languages: Matlab, Mathcad, Mathematica… Dedicated tools (ex: filter design s/w packages). Applications Predicting a system’s output. Implementing a certain processing task. Studying a certain signal. General purpose processors (GPP),  -controllers. Digital Signal Processors (DSP). Programmable logic ( PLD, FPGA ). Hardware real-time DSPing FastFaster

6. M. E. Angoletta - DISP2003 - From analog to digital domain 6 / 30 Digital system example ANALOG DOMAIN Filter Antialiasing DIGITAL DOMAIN A/D Digital Processing ANALOG DOMAIN D/A Filter Reconstruction Sometimes steps missing - Filter + A/D (ex: economics); - D/A + filter (ex: digital output wanted). General scheme Topics of this lecture. Digital Processing Filter Antialiasing A/D

7. M. E. Angoletta - DISP2003 - From analog to digital domain 7 / 30 Digital system implementation Sampling rate. Pass / stop bands. KEY DECISION POINTS: Analysis bandwidth, Dynamic range No. of bits. Parameters. 1 23 Digital Processing A/D Antialiasing Filter ANALOG INPUT DIGITAL OUTPUT Digital format. What to use for processing? See slide “DSPing aim & tools”

8. M. E. Angoletta - DISP2003 - From analog to digital domain 8 / 30 Sampling How fast must we sample a continuous signal to preserve its info content? Ex: train wheels in a movie. 25 frames (=samples) per second. Frequency misidentification due to low sampling frequency. Train starts wheels ‘go’ clockwise. Train accelerates wheels ‘go’ counter-clockwise. 1Why? * Sampling: independent variable (ex: time) continuous  discrete. Quantisation: dependent variable (ex: voltage) continuous  discrete. Here we’ll talk about uniform sampling.*

9. M. E. Angoletta - DISP2003 - From analog to digital domain 9 / 30 Sampling - 2 __ s(t) = sin(2  f 0 t) s(t) @ f S f 0 = 1 Hz, f S = 3 Hz __ s 1 (t) = sin(8  f 0 t) __ s 2 (t) = sin(14  f 0 t) s k (t) = sin( 2  (f 0 + k f S ) t ),  k   s(t) @ f S represents exactly all sine-waves s k (t) defined by: 1

10. M. E. Angoletta - DISP2003 - From analog to digital domain 10 / 30 The sampling theorem A signal s(t) with maximum frequency f MAX can be recovered if sampled at frequency f S > 2 f MAX. Condition on f S ? f S > 300 Hz F 1 =25 Hz, F 2 = 150 Hz, F 3 = 50 Hz F1F1 F2F2 F3F3 f MAX Example 1 Theo * * Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel’nikov. Nyquist frequency (rate) f N = 2 f MAX or f MAX or f S,MIN or f S,MIN /2 Naming gets confusing !

11. M. E. Angoletta - DISP2003 - From analog to digital domain 11 / 30 Frequency domain (hints)  Time & frequency  Time & frequency : two complementary signal descriptions. Signals seen as “projected’ onto time or frequency domains. Warning : formal description makes use of “negative” frequencies ! 1  Bandwidth  Bandwidth : indicates rate of change of a signal. High bandwidth signal changes fast. Ear Ear + brain act as frequency analyser: audio spectrum split into many narrow bands low-power sounds detected out of loud background. Example

12. M. E. Angoletta - DISP2003 - From analog to digital domain 12 / 30 Sampling low-pass signals (a) Band-limited signal: frequencies in [-B, B] (f MAX = B). (a) (b) Time sampling frequency repetition. f S > 2 B no aliasing. (b) 1 (c) aliasing ! (c) f S 2 B aliasing ! Aliasing: signal ambiguity in frequency domain

13. M. E. Angoletta - DISP2003 - From analog to digital domain 13 / 30 Antialiasing filter Filter it before! (a),(b) Out-of-band noise can aliase into band of interest. Filter it before! (a) (b) (c) Passband : depends on bandwidth of interest. Attenuation A MIN : depends on ADC resolution ( number of bits N). A MIN, dB ~ 6.02 N + 1.76 Out-of-band noise magnitude. Other parameters: ripple, stopband frequency... Antialiasing filter (c) Antialiasing filter 1

14. M. E. Angoletta - DISP2003 - From analog to digital domain 14 / 30 Under-sampling (hints) 1 Using spectral replications to reduce sampling frequency f S req’ments. m , selected so that f S > 2B Advantages  Slower ADCs / electronics needed.  Simpler antialiasing filters. f C = 20 MHz, B = 5MHz Without under-sampling f S > 40 MHz. With under-sampling f S = 22.5 MHz (m=1); = 17.5 MHz (m=2); = 11.66 MHz (m=3).Example

15. M. E. Angoletta - DISP2003 - From analog to digital domain 15 / 30 Over-sampling (hints) 1 f OS = over-sampling frequency, w = additional bits required. f OS = 4 w · f S Each additional bit implies over-sampling by a factor of four. It works for: -white noise -white noise with amplitude sufficient to change the input signal randomly from sample to sample by at least LSB. -Input that can take all values between two ADC bits. Caveat Oversampling : sampling at frequencies f S >> 2 f MAX. Over-sampling & averaging may improve ADC resolution ( i.e. SNR, see ) 2

16. M. E. Angoletta - DISP2003 - From analog to digital domain 16 / 30 (Some) ADC parameters 1.Number of bits N (~resolution) 2.Data throughput (~speed) 3.Signal-to-noise ratio (SNR) 4.Signal-to-noise-&-distortion rate (SINAD) 5.Effective Number of Bits (ENOB) 6.Spurious-free dynamic range (SFDR) 7.Integral non-linearity (INL) 8.Differential non-linearity (DNL) 9.… NB: Definitions may be slightly manufacturer-dependent! Different applications have different needs. 2 Static distortion Dynamic distortion Imaging / video Communication Radar systems

17. M. E. Angoletta - DISP2003 - From analog to digital domain 17 / 30 ADC - Number of bits N Continuous input signal digitized into 2 N levels. Uniform, bipolar transfer function (N=3) Quantisation step Quantisation step q = V FSR 2 N Ex: V FSR = 1V, N = 12 q = 244.1  V LSB Voltage ( = q) Scale factor (= 1 / 2 N ) Percentage (= 100 / 2 N ) Quantisation error 2

18. M. E. Angoletta - DISP2003 - From analog to digital domain 18 / 30 ADC - Quantisation error 2  Quantisation Error e q in [-0.5 q, +0.5 q].  e q limits ability to resolve small signal.  Higher resolution means lower e q. QE for N = 12 V FS = 1

19. M. E. Angoletta - DISP2003 - From analog to digital domain 19 / 30 SNR of ideal ADC 2 (1) Also called SQNR (signal-to-quantisation-noise ratio) e q  Ideal ADC: only quantisation error e q p(e) (p(e) constant, no stuck bits…) e q  e q uncorrelated with signal.  ADC performance constant in time.Assumptions Input(t) = ½ V FSR sin(  t). (sampling frequency f S = 2 f MAX )

20. M. E. Angoletta - DISP2003 - From analog to digital domain 20 / 30 SNR of ideal ADC - 2 (2)Substituting in (1) : One additional bit SNR increased by 6 dB 2 Actually (2) needs correction factor depending on ratio between sampling freq & Nyquist freq. Processing gain due to oversampling. - Real signals have noise. - Forcing input to full scale unwise. - Real ADCs have additional noise (aperture jitter, non-linearities etc). Real SNR lower because:

21. M. E. Angoletta - DISP2003 - From analog to digital domain 21 / 30 Real ADCs: parameters SNR : SNR : ( sine_in RMS )/(ADC out_noise RMS ), with out_noise = output - (DC + first 5 input harmonics) output components. SINAD : SINAD : ( sine_in RMS )/(ADC out_noise_2 RMS ), with out_noise_2 = output - (DC output component). 12-bit ADC chip, 68 dB SINAD in specs ~ 11-bit ideal ADC. Example 2 ENOB : ENOB : N from (2) when setting SNR ideal = SINAD, i.e. ENOB = (SINAD – 1.76 dB) / 6.02. Actual number of bit available to an equivalent ideal ADC SNRSINAD SNR and SINAD often confused in specs.

22. M. E. Angoletta - DISP2003 - From analog to digital domain 22 / 30 ADC selection dilemma Speed & resolution: a tradeoff. a tradeoff. 2 High resolution (bit #) - Higher cost & dissipation. - Tailored onto DSP word width. High speed - Large amount of data to store/analyse. - Lower accuracy & input impedance. * * DIFFICULT * DIFFICULT area moves down & right every year. Rule of thumb: 1 bit improvement every 3 years. may increase SNR. 2 Oversampling & averaging Oversampling & averaging (see ). Dithering Dithering ( = adding small random noise before quantisation).

23. M. E. Angoletta - DISP2003 - From analog to digital domain 23 / 30 Digital data formats 10 (decimal)2 (binary) Important bases: 10 (decimal), 2 (binary), 8 (octal), 16 (hexadecimal). Positional number system with base Positional number system with base b : [.. a 2 a 1 a 0. a -1 a -2.. ] b =.. + a 2 b 2 + a 1 b 1 + a 0 b 0 + a -1 b -1 + a -2 b -2 +.. Integer part Fractional part 3 Early computers (ex: ENIAC) mainly base-10 machines. Mostly turned binary in the ’50s. a) less complex arithmetic h/w; Benefits Benefits b) less storage space needed; c) simpler error analysis.

24. M. E. Angoletta - DISP2003 - From analog to digital domain 24 / 30 Decimal arithmetic BUT Increasing number of applications requires decimal arithmetic. Ex: Banking, Financial Analysis. IEEE 754,1985: binary floating point arithmetic standard specified IEEE 854,1987: standard expanded to include decimal arithmetic. Common decimal fractional numbers only approximated by binary numbers. Ex: 0.1 infinite recurring binary fraction. butNon-integer decimal arithmetic software emulation available but often too slow. 3

25. M. E. Angoletta - DISP2003 - From analog to digital domain 25 / 30 Fixed-point binary Represent integer or fractional binary numbers. NB: Constant gap between numbers. Binary representation Fractional point (DSPs) 15 14... 0 MSB LSB Sign bit -4 100-3 111-3 0000 000 -3 101-2 110-2 0011 001 -2 110-1 101-1 0102 010 -1 111 0 100 0 0113 011 0 000 1 1004 100 1 001 2 1015 101 2 010 3 1106 110 3 011 4 1117 111 Two’s complement Sign- Magnitude Offset -Binary Unsigned integer Ex: 3-bit formats 3 Decimal equivalent

26. M. E. Angoletta - DISP2003 - From analog to digital domain 26 / 30 Floating-point binary Formats & methods for binary floating-point arithmetic. IEEE 754 standard Definition of IEEE 754 standard between 1977 and 1985. De facto standard before 1985 ! NOT Note: NOT the easiest h/w choice! Wide variety of floating point hardware in ‘60s and ‘70s, different ranges, precision and rounded arithmetic. William Kahan: “Reliable portable software was becoming more expensive to develop than anyone but AT&T and the Pentagon could afford”. 3 PROBLEM

27. M. E. Angoletta - DISP2003 - From analog to digital domain 27 / 30 Floating-point binary - 2 IEEE 754 standard NB: Variable gap between numbers. Large numbers large gaps; small numbers small gaps. 31 30 23 22 0 f e s MSB LSB e = exponent, offset binary, -126 < e < 127 s = sign, 0 = pos, 1 = neg f = fractional part, sign-magnitude + hidden bit Single (32 bits) Double (64 bits) Double-extended ( 80 bits) Precision Coded number x = (-1) s · 2 e · 1. f Single precision range Max = 3.4 · 10 38 Min = 1.175 · 10 -38 3

28. M. E. Angoletta - DISP2003 - From analog to digital domain 28 / 30 Finite word-length effects Dynamic range dB Dynamic range dB = 20 log 10 largest value smallest value Fixed point ~ 180 dB Floating point ~1500 dB High dynamic range wide data set representation with no overflow. NB: Different applications have different needs. Ex: telecomms: 50 dB; HiFi audio: 90 dB. 3 Overflow Overflow : arises when arithmetic operation result has one too many bits to be represented in a certain format.

29. M. E. Angoletta - DISP2003 - From analog to digital domain 29 / 30 Finite word-length effects - 2 Round-off error estimate: Relative error Relative error = (floating - actual value)/actual value (depends on base). The smaller the base, the tighter the error estimate. 3 For integersFor integers within ±16.8 million range: single-precision floating point gives no round-off error. OutsideOutside that range, integers are missing: gaps between consecutive floating point numbers are larger than integers. Round-off Round-off : error caused by rounding math calculation result to nearest quantisation level. Big concern for real numbers real numbers. 0.1 not exactly represented (falls between two floating point numbers). Example

30. M. E. Angoletta - DISP2003 - From analog to digital domain 30 / 30 References 1.On bandwidth, David Slepian, IEEE Proceedings, Vol. 64, No 3, pp 291 - 300. 2.The Shannon sampling theorem - Its various extensions and applications: a tutorial review, A. J. Jerri, IEEE Proceedings, Vol. 65, no 11, pp 1565 – 1598. 3.What every computer scientist should know about floating-point arithmetic, David Goldberg. 4.IEEE Standard for radix-independent floating-point arithmetic, ANSI/IEEE Std 854-1987. Papers 1.Understanding digital signal processing, R. G. Lyons, Addison-Wesley Publishing, 1996. 2.The scientist and engineer’s guide to digital signal processing, S. W. Smith, at http://www.dspguide.com. 3.Discrete-time signal processing, A. V. Oppeheim & R. W. Schafer, Prentice Hall, 1999. Books

31. M. E. Angoletta - DISP2003 - From analog to digital domain 31 / 30 COFFEE BREAK Be back in ~15 minutes Coffee in room #13