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

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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

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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

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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

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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

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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

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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”

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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.*

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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

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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 !

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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

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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

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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

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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

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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

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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

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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

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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

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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 )

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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:

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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.

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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).

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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.

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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

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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

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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

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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

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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.

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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

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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

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

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